Physics · Article
RLC circuits and resonance
Solve two-state RLC motion with full initial conditions, signed power and finite work. Distinguish damping branches, switched memory, observable-specific resonance, exact current-power bandwidth, loaded Q and finite lossless forcing without confusing ideal limits with physical steady states.
Previous article: RC and RL transients
Back to PhysicsAfter reading, you can
- Derive series and parallel RLC equations with explicit associated references and both initial states.
- Convert dimensionless states and frequencies back to their physical SI scales.
- Solve lossless, underdamped, critical and overdamped branches without losing a mode or initial condition.
- Integrate finite source and resistor work and audit physical electric and magnetic storage.
- Carry both states through a compatible finite source switch with correct one-sided derivatives.
- Distinguish current, resistor-power, capacitor-voltage and inductor-voltage maxima from damped natural frequency.
- Derive exact series current-power half-power edges and identify the weak-damping scope of a charge-response linewidth.
- Relate Q to driven storage and equal-phase ringdown while qualifying finite-cycle approximations.
- Solve finite zero-loss forcing and keep small-loss, long-time and cycle-average limits in order.
- Include source/internal/load losses and distinguish useful power from highest loaded Q.
- Use causal kernels and retained unilateral-transform initial terms in their proper domain.
- State where ideal lumped, linear, constant-parameter predictions need a more complete physical model.
Choose the observable before naming resonance
A resonator stores energy in two complementary forms and can exchange that energy repeatedly. In a circuit, a capacitor stores electric energy and an inductor magnetic energy. Resistance removes energy from an isolated passive circuit, while a source can replace it. The response depends on the initial states, the forcing and the quantity being observed. A large capacitor voltage, a large loop current and a slowly decaying ring are related phenomena, but their characteristic frequencies need not be identical.
This lesson follows one original series circuit through natural motion, a DC step, a finite pulse and sinusoidal forcing, then compares a parallel circuit and a loaded source. The preceding transient lesson supplies state continuity and signed energy accounting. Here there are two independent storage states, so one time constant or one initial voltage is generally insufficient. The later AC lesson develops phasors and network power more broadly; we introduce only the sinusoidal algebra needed for the present questions.
On a first reading, follow the state equations, four damping branches, step and switch, energy balance and the distinction between current and capacitor-voltage resonance. Return to the root-coalescence formulas, finite-work integrals, parallel-state mapping, exact bandwidth and lossless limits afterward. The parameters describe ideal teaching circuits, not measured components. Small-looking numerical rounding does not establish the physical validity of a lumped or linear approximation.
Keep two storage states and associated references
For a series loop, orient current i from the positive source terminal through R and L into the positive capacitor terminal. Let v_C be that capacitor voltage. Associated passive references give v_R=Ri, v_L=L di/dt and i=C dv_C/dt. Kirchhoff’s voltage relation is v_s=v_R+v_L+v_C. A negative current is allowed: it means flow opposite the chosen arrow, not a different circuit or a negative stored energy.
With constant positive L and C, the state equations are dv_C/dt=i/C and di/dt=(v_s−Ri−v_C)/L. Specifying v_C(0) and i(0) therefore determines both initial derivatives once the future source is known. A value such as v_C(0)=0 alone cannot determine the response. Conversely, differentiating a voltage equation and solving for current must not discard the initial capacitor charge that was present before differentiation.
The continuity statement has a domain. Integrating i=C dv_C/dt gives CΔv_C=∫i dt; integrating v_L=L di/dt gives LΔi=∫v_L dt. Finite ordinary source steps in these fixed closed circuits have no concentrated charge or voltage-time impulse, so both states are continuous. An imposed impulse or an incompatible ideal connection requires a separate integrated analysis; it is not an ordinary initial derivative.
| Quantity | Definition | SI value or unit |
|---|---|---|
| Capacitance C | Fixed positive storage parameter | 1/200 F |
| Inductance L | Fixed positive storage parameter | 1/2 H |
| State voltage v_C | 4x | V |
| State current i | (2/5)j | A |
| Physical time t | u/20 | s |
| Storage scale | CV*²/2 | 1/25 J |
Derive and scale the series equation
Substitute i=C dv_C/dt into KVL to obtain LC d²v_C/dt²+RC dv_C/dt+v_C=v_s. Dividing by LC gives the natural-frequency scale ω₀=1/√(LC) and decay parameter α=R/(2L). Write ζ=α/ω₀, so the coefficient of the first derivative is 2ζω₀. The parameter ω₀ is the undamped angular frequency; it is not automatically the frequency of every driven maximum or of an actually damped oscillation.
Our series scale is L=1/2 H, C=1/200 F and V*=4 V. Then ω₀=20 rad/s, Z₀=√(L/C)=10 Ω and I*=Cω₀V*=2/5 A. With u=ω₀t, x=v_C/V*, j=i/I* and f=v_s/V*, the system becomes x′=j and j′=f−x−2ζj. Primes here mean derivatives with respect to dimensionless u; they must be multiplied by the appropriate physical scale before being called volts per second or amperes per second.
The same dimensionless curve can represent circuits with different physical clocks and powers, so the scales remain part of the answer. In this circuit R=20ζ Ω, v_C=4x V, i=(2/5)j A, v_R=8ζj V and v_L=4j′ V. Physical derivatives are dv_C/dt=80j V/s and di/dt=8j′ A/s. The frequency ratio r=ω/ω₀ used later is a separate coordinate, not a second clock.
Classify the characteristic roots
For a constant source f, subtract the equilibrium by writing y=x−f. The homogeneous equation y″+2ζy′+y=0 has characteristic roots s=−ζ±√(ζ²−1) in dimensionless time. Physical rates are ω₀s. For 0<ζ<1 the roots are conjugate and the response oscillates under a decaying envelope. At ζ=1 the roots coincide. For ζ>1 both roots are negative real. At ζ=0 the ideal circuit has undamped imaginary roots.
The four selected resistances are 0, 12, 20 and 25 Ω, corresponding to ζ=0, 3/5, 1 and 5/4. They keep L and C fixed while changing damping. The underdamped frequency is ω_d=ω₀√(1−ζ²), defined as an oscillation frequency only for ζ<1. At critical damping a zero imaginary part does not describe an oscillation of infinite period; the solution belongs to the repeated-root branch.
Two roots do not guarantee that both modes appear in a particular experiment. Their coefficients are fixed by the two initial conditions, and one coefficient can vanish. Nor does a negative real root guarantee that a plotted voltage is monotone for every signed preparation. Stability concerns the decay of the homogeneous state; overshoot, zero crossings and a chosen settling threshold are questions about a specified response and observable.
| R | ζ | Physical roots | Natural behavior |
|---|---|---|---|
| 0 Ω | 0 | ±20i s⁻¹ | Undamped |
| 12 Ω | 3/5 | −12±16i s⁻¹ | Underdamped |
| 20 Ω | 1 | −20 s⁻¹, repeated | Critical |
| 25 Ω | 5/4 | −10,−40 s⁻¹ | Overdamped |
Solve arbitrary initial conditions
For 0≤ζ<1 define d=√(1−ζ²), A=x₀−f and B=(j₀+ζA)/d. The solution is x=f+e^(−ζu)[A cos(du)+B sin(du)]. Its derivative is j=e^(−ζu)[j₀ cos(du)−(A+ζj₀)sin(du)/d]. Substitution at u=0 recovers x₀ and j₀. The exponential envelope multiplies a phase-dependent combination; the voltage alone does not generally equal a single positive exponential.
For ζ>1 write s_±=−ζ±√(ζ²−1). Then x=f+a_+e^(s_+u)+a_−e^(s_−u), where a_+=(j₀−s_−A)/(s_+−s_−) and a_−=(s_+A−j₀)/(s_+−s_−). These coefficients solve the two equations a_++a_−=A and s_+a_++s_−a_−=j₀. Their signs can differ, and cancellation can be physically important.
At ζ=1 use the separate expression x=f+e^(−u)[A+(j₀+A)u], with j=e^(−u)[j₀−(j₀+A)u]. The factor u is essential. Merely setting both distinct-root exponents to −u would lose one degree of freedom. The repeated-root form is also the continuous analytic limit of the other branches; expressions with nearly coalescing denominators need care in numerical work, even though the physical state remains well defined.
Read a natural response with reversed current
The natural series preparation has v_C(0)=4 V and i(0)=−1/5 A, or x₀=1 and j₀=−1/2, with the source replaced by a zero-voltage connection completing the loop. The initial current removes positive charge from the capacitor. Initially p_C=v_Ci=−4/5 W, so the capacitor delivers energy. The inductor’s initial energy is positive despite the negative current, because it is Li²/2.
At R=12 Ω the normalized motion is x=e^(−3u/5)[cos(4u/5)+(1/8)sin(4u/5)] and j=e^(−3u/5)[−(1/2)cos(4u/5)−(7/8)sin(4u/5)]. At R=25 Ω the roots are −1/2 and −2, but this preparation excites only the first: x=e^(−u/2), j=−e^(−u/2)/2. The missing fast exponential is an initial-condition cancellation, not the disappearance of a storage element.
At zero resistance, x=cos u−(1/2)sin u and j=−sin u−(1/2)cos u. Voltage and current remain bounded and oscillatory, and their combined energy stays fixed. Positive resistance instead makes both states approach zero. A finite plot ending at u=6 shows a finite portion of either history; it does not turn a small positive-R remainder into exact zero or make an undamped circuit settle.
Balance instantaneous power and storage
Multiply the series KVL equation by i. Since v_Ci=d(Cv_C²/2)/dt and v_Li=d(Li²/2)/dt, source-delivered power obeys v_si=Ri²+dU/dt, where U=Cv_C²/2+Li²/2. This identity fixes the accounting signs without assuming that every element absorbs power at every instant. A capacitor or inductor can deliver stored energy, and an ideal source can absorb returned energy during part of a cycle.
In the chosen series normalization, U_C=x²/25 J and U_L=j²/25 J. Hence U=(x²+j²)/25 J, p_R=(16/5)ζj² W and dU/du=−4ζj²/25 for the natural circuit. For R>0, total free energy cannot increase. It need not follow a pure exponential between arbitrary phases: the fraction in the dissipative current channel changes during the oscillation. The state envelope and the instantaneous total energy are distinct objects.
The initial natural energy is 1/25+1/100=1/20 J. Over complete decay through any of the selected positive series resistances, the resistor receives this energy. At R=0 it receives none and all the energy remains in the LC pair. These are different infinite-time limits, even though the finite-time positive-R motion approaches the lossless motion as R tends to zero. Taking a limit in damping does not automatically commute with waiting forever.
Integrate finite work independently
Over an interval beginning at t₀, define W_R=∫Ri²dt and W_s=∫v_si dt, with U−U₀=W_s−W_R. For constant normalized source f, W_s=(2/25)f(x−x₀) J because i dt=C dv_C. The source work can be negative if it receives charge against the source polarity. By contrast W_R≥0 for these positive resistors, regardless of the sign of current.
The resistor work has an independent closed-form integral. If y=a_+e^(s_+u)+a_−e^(s_−u), then ∫j²du=a_+²s_+²E(2s_+,u)+a_−²s_−²E(2s_−,u)+2a_+a_−s_+s_−E(s_++s_−,u), where E(s,u)=(e^(su)−1)/s and E(0,u)=u. For conjugate roots these terms combine to a real integral of the real current squared; replacing that square by the modulus square of a complex representation would solve a different integral.
At the repeated root, write j=e^(−u)(D+Eu). Then ∫j²du=D²I₀+2DEI₁+E²I₂, with I₀=(1−e^(−2u))/2, I₁=[1−(1+2u)e^(−2u)]/4 and I₂=[1−(1+2u+2u²)e^(−2u)]/4. Multiply the integral by 4ζ/25 J to obtain W_R. Checking the energy identity afterward tests the independently evaluated quantities; it should not be used to define all three and then counted as an additional physical result.
Drive a zero-state DC step
Now prepare x₀=j₀=0 and apply v_s=4 V for t≥0. The equilibrium subtraction has A=−1. In the underdamped case x=1−e^(−ζu)[cos(du)+(ζ/d)sin(du)] and j=e^(−ζu)sin(du)/d. The current can reverse after the capacitor overshoots; a positive source does not force current to remain positive throughout a second-order transient.
For critical damping, x=1−(1+u)e^(−u) and j=ue^(−u). For the overdamped branch use the two real roots and the zero-state coefficients. At the instant of the finite step, both states remain zero, j′(0+)=1, v_L(0+)=4 V and di/dt(0+)=8 A/s. The derivative and inductor voltage jump, while neither stored state does. An infinite initial current is not needed to start the response.
For every selected R>0 the final state is v_C=4 V and i=0. Source work over complete settling is CV_s²=2/25 J; final storage is CV_s²/2=1/25 J; resistor work is the other 1/25 J. This integrated result does not imply a constant half-and-half split at each time. With R=0, x=1−cos u and j=sin u never settle, so the positive-R final ledger must not be assigned to an undamped final state.
State a settling criterion and its limits
A response has not finished merely because a plot looks flat. Settling usually means that a chosen error remains inside a specified band thereafter. An underdamped voltage may enter that band, leave it at a later crest and only then settle. The time depends on the measured state, the error band and the initial preparation. A single first crossing is therefore not a general settling-time definition.
Critical damping separates oscillatory and nonoscillatory homogeneous roots. For the fixed-ω₀ zero-state step family it gives the familiar fastest approach among the nonoscillatory damping branches. That statement is narrower than “critical damping always minimizes settling time”: a prescribed tolerance can favor an underdamped design, and a different nonzero initial current can select or cancel modes. The natural preparation above is an explicit reminder that modal coefficients matter.
Useful engineering decisions may compare overshoot, current stress, dissipated energy and a voltage tolerance simultaneously. The present ideal model gives no component voltage rating, core saturation current or allowable thermal rise. A mathematically attractive damping choice becomes a physical design only after those additional constraints and source/load behavior are specified. They cannot be recovered from Q or a small final numerical remainder alone.
Carry two states through a finite pulse
Keep R=12 Ω and apply the 4 V source only until u_s=3/2, or t_s=3/40 s. At the switch replace it by a zero-voltage connection that leaves the RLC loop closed. The first interval is a zero-state step. Evaluate its terminal x_s and j_s, then use both as the initial conditions of the second, source-free interval. Setting the capacitor voltage to its first-interval value but resetting current to zero would change the physical preparation.
For this pulse, x_s=1−e^(−9/10)[cos(6/5)+(3/4)sin(6/5)] and j_s=(5/4)e^(−9/10)sin(6/5). On the second interval let w=u−3/2. Substitute x₀=x_s, j₀=j_s and f=0 into the natural-response formula, using w as its local coordinate. The horizontal position in the full experiment remains u=3/2+w; neither w nor the frequency ratio r should be mislabeled as seconds.
The switch state should be carried at full mathematical precision. Rounding it for display and using the rounded numbers as a new preparation creates a slightly different trajectory and energy ledger. Resistor work accumulates across both intervals. Source work stops changing after the source is replaced by zero voltage, and the final total resistor work equals the source work supplied during the pulse because initial and final total storage are both zero.
Retain one-sided derivatives and switch power
At u_s, x and j agree on the two sides, but j′=f−x−2ζj changes by −1. Thus di/dt changes by −8 A/s and v_L by −4 V. Resistor and capacitor voltages are unchanged at the instant because they depend on the continuous current and capacitor voltage. These one-sided statements show exactly where the new KVL constraint is met; saying that “everything is continuous” would be wrong.
Source-delivered power changes from 4i_s to zero. Inductor power changes by the same −4i_s, while capacitor power and resistor power retain their one-sided values. There is no instantaneous jump in either stored energy, so the bounded power jump does not represent a finite concentrated work impulse. A single derivative evaluated at the discontinuity cannot replace the two one-sided derivatives.
This is a deliberately closed return circuit. Opening an ideal inductor with nonzero current and pretending its current instantly disappears is a different, incompatible ideal constraint. A real switch may arc or charge parasitic capacitance; a diode return has another constitutive law. Those phenomena require their own models. The finite-pulse calculation neither predicts an arc voltage nor treats a missing return path as a harmless removal of the source.
Two inherited states across a finite pulse
Separate the on and off intervals so that continuous storage states and one-sided algebraic power changes remain visible.
(a) Capacitor state is inherited
- Capacitor voltage — on
- Capacitor voltage — off
(b) Inductor state is inherited
- Series current — on
- Series current — off
(c) One-sided power can jump
- Capacitor absorbed — on
- Capacitor absorbed — off
- Inductor absorbed — on
- Inductor absorbed — off
- Resistor absorbed — on
- Resistor absorbed — off
- Source delivered — on
- Source delivered — off
(d) Finite work from the original zero state
- Total storage — on
- Total storage — off
- Resistor work — on
- Resistor work — off
- Source-delivered work — on
- Source-delivered work — off
- The vertical guide is the actual source switch at u=3/2, or t=3/40 s. Separate paths retain the switch-left and switch-right samples; no line is drawn through a power jump.
- Capacitor voltage and inductor current remain continuous here. Their derivatives and instantaneous powers can change immediately because the ordinary source changes. The off interval keeps a closed RLC loop.
- All work curves use the original zero-state preparation as their reference. Source-delivered work equals total stored energy plus resistor work; off-interval work inherits the on-interval totals.
Finite pulse switch at u_s=3/2: continuous states and one-sided constraints
Two inherited states across a finite pulse — values and conditions
| Quantity and original case | Value | Unit | Meaning and precision |
|---|---|---|---|
| Capacitor state is inherited — Capacitor voltage — on — sample 1 — Physical time | 0 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 1 — Capacitor voltage | 0 | V | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 2 — Capacitor voltage | 1.128066830387213843332339754150807214844631917719568814135946652896398E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — on — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 3 — Capacitor voltage | 4.051803227840372834973059090796931438191428330285466526266431947938548E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — on — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 4 — Capacitor voltage | 8.148752996760205158068944572447046030454887307288953774430208221426214E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — on — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 5 — Capacitor voltage | 1.289476894136240529916233028720884725360252919069867517666955228084591E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — on — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 6 — Capacitor voltage | 1.786668804380303588574073583693021905139020680367170708478033980923781E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — on — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — on — sample 7 — Capacitor voltage | 2.273888882092302411335410377856504135014653321583833417870881509972981E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 1 — Physical time | 3/40 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 1 — Capacitor voltage | 2.273888882092302411335410377856504135014653321583833417870881509972981E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 2 — Physical time | 7/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 2 — Capacitor voltage | 2.614743458200748303458866900079629552339796852637039580833582939810506E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 3 — Physical time | 1/10 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 3 — Capacitor voltage | 2.726801234797980973035080101353471925789472637105687826357763855317618E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 4 — Physical time | 9/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 4 — Capacitor voltage | 2.663342880339351342922480073969792097658720748953843358243458439972615E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 5 — Physical time | 1/8 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 5 — Capacitor voltage | 2.473267705589356267589257261629253150316129542709337202546381838087559E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 6 — Physical time | 11/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 6 — Capacitor voltage | 2.199602557366817956293780909617325021314354540165802069132310850065487E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 7 — Physical time | 3/20 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 7 — Capacitor voltage | 1.878712625010742399080061452440594064957341009123684589221593841281971E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 8 — Physical time | 13/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 8 — Capacitor voltage | 1.540074628882079339187679804840586411836755114925065450370454938523240E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 9 — Physical time | 7/40 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 9 — Capacitor voltage | 1.206478815126277504469152284270088920727900865874815456557428543864425E+0 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 10 — Physical time | 3/16 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 10 — Capacitor voltage | 8.945377675806102569011445559605491944117440611027679030588626305343339E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 11 — Physical time | 1/5 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 11 — Capacitor voltage | 6.153951840831729353425738832585047327292269741481675288247457186773749E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 12 — Physical time | 17/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 12 — Capacitor voltage | 3.755447174409874626805778903573833657415822318004533543593875141447687E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 13 — Physical time | 9/40 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 13 — Capacitor voltage | 1.776863694590002977898151043527540837669440377420938827578263773218028E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 14 — Physical time | 19/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 14 — Capacitor voltage | 2.156470291738589350517617996854667722248936372324580116487123325582650E-2 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 15 — Physical time | 1/4 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 15 — Capacitor voltage | -9.525144142307905751343451309168112526578689631962442217303367184729329E-2 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 16 — Physical time | 21/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 16 — Capacitor voltage | -1.766744452187413612048257427603368431969787700413966397868740928839749E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 17 — Physical time | 11/40 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 17 — Capacitor voltage | -2.275038492715968715047798303256175459905909539967359217744628430138095E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 18 — Physical time | 23/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 18 — Capacitor voltage | -2.529386060117460993984334651800703632882476596538007322090495282651644E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 19 — Physical time | 3/10 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 19 — Capacitor voltage | -2.581942924627850342074061489254258215811411193716784416812514779363962E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 20 — Physical time | 5/16 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 20 — Capacitor voltage | -2.482186416256153329253162472570566198778018056696809698742450282242111E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 21 — Physical time | 13/40 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 21 — Capacitor voltage | -2.274951904481849572151728865005674346015659768153956804095370661654388E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 22 — Physical time | 27/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 22 — Capacitor voltage | -1.999227534091114714436560688878537639234693615293504728765281944703584E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 23 — Physical time | 7/20 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 23 — Capacitor voltage | -1.687575432920825571713826971965768003980376548935599448434726126990609E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 24 — Physical time | 29/80 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 24 — Capacitor voltage | -1.366048111799100579524301058052885735639031218171311391934513846156705E-1 | V | Stored approximation, 70 significant digits |
| Capacitor state is inherited — Capacitor voltage — off — sample 25 — Physical time | 3/8 | s | Exact ratio |
| Capacitor state is inherited — Capacitor voltage — off — sample 25 — Capacitor voltage | -1.054476202546788801038581316658963544100252751425584352903554628201394E-1 | V | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — on — sample 1 — Series current | 0 | A | Exact ratio |
| Inductor state is inherited — Series current — on — sample 2 — Series current | 8.549813884316877983278961192056850471712873699090930452847831891459572E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — on — sample 3 — Series current | 1.442441017249592945821572562194674374924029325078213937994806335954137E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — on — sample 4 — Series current | 1.800159683190113487616615272041921734666479657865729294588735121311389E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — on — sample 5 — Series current | 1.968466849542911162897131740131621846734684181407394965092694689183597E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — on — sample 6 — Series current | 1.987413741626464152964373160877460105128058022349848490076429886002437E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — on — sample 7 — Series current | 1.894694070233171107345978058189728983319924014513190659487463779080022E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 1 — Series current | 1.894694070233171107345978058189728983319924014513190659487463779080022E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 2 — Series current | 8.692489134401000308931318582896219152291121390084803772165520741940177E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 3 — Series current | 6.288790083203947753784685909810179971291440682703297259694166725325603E-3 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 4 — Series current | -5.378571148988771729142223110061013929060859857606929701540508790586361E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 5 — Series current | -9.540084471920636113171656068975056783255618180901764972706898818182146E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 6 — Series current | -1.211055439257661090407513864935945360179136417103127434414346099205647E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 7 — Series current | -1.336427506334560991312123967190139103499118282909572281019109570534153E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 8 — Series current | -1.357517906484423065160051139750169498955768440398229371916281586666683E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 9 — Series current | -1.300221656205803914949321299876923571379329911361849514963517174037306E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 10 — Series current | -1.187932871192866062263124042785394117237200386413009387051932780291598E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 11 — Series current | -1.040936316434710851993024221689549854317924602429876624977710331628632E-1 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 12 — Series current | -8.761236772589312821582408024511335007914638756325733250742841353388340E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 13 — Series current | -7.069656303458696932345078294303356195620670413088048788139064234541576E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 14 — Series current | -5.436749162548275678292153397523108495771374057696630295660565844721129E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 15 — Series current | -3.935021497501577822640503727452815391063032291770114280156739212747317E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 16 — Series current | -2.611140749668983233498503360694213958831949487820177544438054743729154E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 17 — Series current | -1.490125788693075885435862103056002315533896350776973964141875142175740E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 18 — Series current | -5.796139147739855039792996488978378263713948422678690667123506030792141E-3 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 19 — Series current | 1.260444002902821211003306368770132427584909904748824350649013665583254E-3 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 20 — Series current | 6.420383393403498061068398855421760668986801791019804374478146897228513E-3 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 21 — Series current | 9.898083333697454472414937177558668762903726706067106726837288164596201E-3 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 22 — Series current | 1.194274158170449039675733670216125821619179329298398773886159267886967E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 23 — Series current | 1.281594561410755110961108641553268891187842379960119510748466648688900E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 24 — Series current | 1.277441000966475363821255170604381416228786575113236633449008750727885E-2 | A | Stored approximation, 70 significant digits |
| Inductor state is inherited — Series current — off — sample 25 — Series current | 1.205744974071157803570263359425277464426603610880567946370976026749642E-2 | A | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — on — sample 1 — Capacitor absorbed power | 0 | W | Exact ratio |
| One-sided power can jump — Capacitor absorbed — on — sample 2 — Capacitor absorbed power | 9.644761448881933556016829555633158101978078684588103901558146542793567E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — on — sample 3 — Capacitor absorbed power | 5.844487169661251609263747527714613036074153349790343880836725076458701E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — on — sample 4 — Capacitor absorbed power | 1.466905661304233879630478400501643703885905786861672450851548481350402E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — on — sample 5 — Capacitor absorbed power | 2.538292519358743372754785446300208885897951375119563979278700354023870E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — on — sample 6 — Capacitor absorbed power | 3.550850133560740300847700652016624967274220367988982432383182631217708E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — on — sample 7 — Capacitor absorbed power | 4.308323781269419760023073113336533072451225945332971588040884490577826E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 1 — Capacitor absorbed power | 4.308323781269419760023073113336533072451225945332971588040884490577826E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 2 — Capacitor absorbed power | 2.272862909965610075299977951254060705760555254959581595637297577386786E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 3 — Capacitor absorbed power | 1.714828056426582223850005828868858317868662987131964489074500409905207E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 4 — Capacitor absorbed power | -1.432497917605788972716967476813948355267163594931666690081793641787846E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 5 — Capacitor absorbed power | -2.359518283299579719862155854832321079724960463229975181808359843649343E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 6 — Capacitor absorbed power | -2.663840641304146397486738036392198283646229225501274392932523097664645E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 7 — Capacitor absorbed power | -2.510763228562363645825219821622416656726749435956123288468300431680694E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 8 — Capacitor absorbed power | -2.090678886029775137758207926381048098536480096217311311286221287252493E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 9 — Capacitor absorbed power | -1.568689883180704449432834935689390121072080461335379803955162296125302E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 10 — Capacitor absorbed power | -1.062650818632491043191542177183598872981926902268280000271379921605748E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 11 — Capacitor absorbed power | -6.405871960711988376916051412055503243367524865667189162652990088866025E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 12 — Capacitor absorbed power | -3.290236188195642415093573182867687949926570800078347788581090388326313E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 13 — Capacitor absorbed power | -1.256181561884512347939020967369192999685366621405131083918433039931069E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 14 — Capacitor absorbed power | -1.172418805267001138640863690794343787812462384445008911399203915652989E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 15 — Capacitor absorbed power | 3.748164696678283738382216527956300174564883954727677055306414465609067E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 16 — Capacitor absorbed power | 4.613218433358160283726360250656607424024112970053993059657684210668542E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 17 — Capacitor absorbed power | 3.390093528265489459433737037087276778733108796315672304029690499081552E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 18 — Capacitor absorbed power | 1.466067356279429010963818829327993082535885281516461208418938310733058E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 19 — Capacitor absorbed power | -3.254394475184544883774833469520039575204858867964895331380029086941700E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 20 — Capacitor absorbed power | -1.593658844626274947536834088911058975693721477826503543081926375990235E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 21 — Capacitor absorbed power | -2.251766353071507863143795337510359902950261692012519907788746233318017E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 22 — Capacitor absorbed power | -2.387625780267848734144249561849310453039193366735341238133878651997675E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 23 — Capacitor absorbed power | -2.162787496801730630542790756619757952875715411041045895602093562380512E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 24 — Capacitor absorbed power | -1.745045867305006689256477058678759084153535062201373152043161847868096E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Capacitor absorbed — off — sample 25 — Capacitor absorbed power | -1.271429381498430807182097040137098432139542361692298143290634450226887E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — on — sample 1 — Inductor absorbed power | 0 | W | Exact ratio |
| One-sided power can jump — Inductor absorbed — on — sample 2 — Inductor absorbed power | 2.446286129760439925534874067512288892030748607522364609855674052660655E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — on — sample 3 — Inductor absorbed power | 2.688552046139398029228611464519251190850034760295758496475225920563710E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — on — sample 4 — Inductor absorbed power | 1.845043209476464358547929545092872698499683213451517910446463112424934E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — on — sample 5 — Inductor absorbed power | 6.857407935136284146039361709448261975069895771109717612749136101190539E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — on — sample 6 — Inductor absorbed power | -3.409712235417261046899274585567507796015222681415376566345176833305116E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — on — sample 7 — Inductor absorbed power | -1.037386244068824205225422755424023341545247186305317536035553791558786E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 1 — Inductor absorbed power | -8.616162525001508634609334988182939274824943244358080173985408907878873E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 2 — Inductor absorbed power | -3.179575318185763494648694684646040778608422017795948354604537787775463E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 3 — Inductor absorbed power | -1.762286713279307403184338762812908721825848386456040619206317582511486E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 4 — Inductor absorbed power | 1.085349586348976219593497074225265580481041371471402918739018946532136E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 5 — Inductor absorbed power | 1.267359742523004810878927853694247049367146265509377165983013167400955E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 6 — Inductor absorbed power | 9.038543089574666563110140091436002074119106439074425213215681719795864E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 7 — Inductor absorbed power | 3.675170529372279755529437508497311186249151254690492008132858087592666E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 8 — Inductor absorbed power | -1.207469536812458298054089482975751915951181112665813301500340755697179E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 9 — Inductor absorbed power | -4.600017431391720508740466378456816383833849690405424139990255797794461E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 10 — Inductor absorbed power | -6.307705891201407702035690881642931843146818541060057957489500575292214E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 11 — Inductor absorbed power | -6.596709017759985404523788270613404593097176474196640720804604526717492E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 12 — Inductor absorbed power | -5.920876186048901382066229988371662883366242554125298473494873204120557E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 13 — Inductor absorbed power | -4.741423267999482107307062737963991909765846737428690040836118505484095E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 14 — Inductor absorbed power | -3.429747094249625115731949555287583245419496422387367680853906728888276E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 15 — Inductor absorbed power | -2.232943771963775578430531803159914479170715838875613726268208008378720E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 16 — Inductor absorbed power | -1.279488565085644000078914402419219989831262080119427474132775112899651E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 17 — Inductor absorbed power | -6.054663367619283192639550232116112260268932403322315007009991784100948E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 18 — Inductor absorbed power | -1.869210104518978928500373768515705414535421815088854362972434502082093E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 19 — Inductor absorbed power | 3.063748185050102423928024593311211129896338538343092006171420277066372E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 20 — Inductor absorbed power | 1.099002969606777944137490412324825562669167766048104758520940665390600E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 21 — Inductor absorbed power | 1.076101708901676096735479674118808220695736636141546114035160097818202E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 22 — Inductor absorbed power | 6.760768624193670572918370988265826312642497940662154499901575990756610E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 23 — Inductor absorbed power | 1.918059529965797264746270810955075393672320474191025145655162508508804E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 24 — Inductor absorbed power | -2.131807458352699244288637593102298151059301479465017137511069234978352E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Inductor absorbed — off — sample 25 — Inductor absorbed power | -4.731557494989975976617174030337557848329616358157159669472434037587751E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — on — sample 1 — Resistor absorbed power | 0 | W | Exact ratio |
| One-sided power can jump — Resistor absorbed — on — sample 2 — Resistor absorbed power | 8.771918094774919322165421137541197156346200852681265322678772384952378E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — on — sample 3 — Resistor absorbed power | 2.496763305892848593131304031487985005238667205038062867420326915606969E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — on — sample 4 — Resistor absorbed power | 3.888689861979755712288053142573170536280329631149726817056928891470218E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — on — sample 5 — Resistor absorbed power | 4.649834085299272864229805343281452303533795773399044119817164792591465E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — on — sample 6 — Resistor absorbed power | 4.739776056486842415699719450049966232839533989551949184557054596122553E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — on — sample 7 — Resistor absorbed power | 4.307838743732088874586261874846406202373717299025108585944524417301046E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 1 — Resistor absorbed power | 4.307838743732088874586261874846406202373717299025108585944524417301046E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 2 — Resistor absorbed power | 9.067124082201534193487167333919800728478667628363667589672402103886776E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 3 — Resistor absorbed power | 4.745865685272517933433293394405040395718539932407613013181717260627865E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 4 — Resistor absorbed power | 3.471483312568127531234704025886827747861222234602637713427746952557098E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 5 — Resistor absorbed power | 1.092158540776574908983228001138074030357814197720598015825346676248388E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 6 — Resistor absorbed power | 1.759986332346679741175724027248598076234318581593831871610954925685059E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 7 — Resistor absorbed power | 2.143246175625135670272276070772685538101834310487074087655014622921427E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 8 — Resistor absorbed power | 2.211425839711020967563616874678623290131598207483892641436255362822211E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 9 — Resistor absorbed power | 2.028691626319876500306881573535071759455465430375922217954187875904748E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 10 — Resistor absorbed power | 1.693421407752631813395111265347892057296608756374285796020329979134969E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 11 — Resistor absorbed power | 1.300258097847197378143983968266890783646470133986382988345759461558352E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 12 — Resistor absorbed power | 9.211112374244543797159803171239350833292813354203646262075963592446870E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 13 — Resistor absorbed power | 5.997604829883994455246083705333184909451213358833821124754551545415164E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 14 — Resistor absorbed power | 3.546988974776325229596035924367017624200742660831868571993827120453575E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 15 — Resistor absorbed power | 1.858127302295947204592310150364284461714227443402846020737566561817813E-2 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 16 — Resistor absorbed power | 8.181667217498279717062783773535592474288507831140281681670066918327964E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 17 — Resistor absorbed power | 2.664569839353793733205813195028835481535823607006642702980301285019396E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 18 — Resistor absorbed power | 4.031427482395499175365549391877123319995365335723931545534961913490355E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 19 — Resistor absorbed power | 1.906462901344424598468088762088284453085203296218033252086088098753283E-5 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 20 — Resistor absorbed power | 4.946558750194970033993436765862334130245537117783987845609857105996353E-4 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 21 — Resistor absorbed power | 1.175664644169831766408315663391551682254525055870973793753586135499815E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 22 — Resistor absorbed power | 1.711548917848481676852412463022727821774943572669125788143721052922014E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 23 — Resistor absorbed power | 1.970981543805150904068163675524250413508483363621943381036577311529631E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 24 — Resistor absorbed power | 1.958226613140276613685340817988988899259465210147874865794268771365932E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Resistor absorbed — off — sample 25 — Resistor absorbed power | 1.744585130997428404843814443170854216972503997508014110237877853985663E-3 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Source delivered — on — sample 1 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — on — sample 2 — Source-delivered power | 3.419925553726751193311584476822740188685149479636372181139132756583829E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Source delivered — on — sample 3 — Source-delivered power | 5.769764068998371783286290248778697499696117300312855751979225343816549E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Source delivered — on — sample 4 — Source-delivered power | 7.200638732760453950466461088167686938665918631462917178354940485245555E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Source delivered — on — sample 5 — Source-delivered power | 7.873867398171644651588526960526487386938736725629579860370778756734389E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Source delivered — on — sample 6 — Source-delivered power | 7.949654966505856611857492643509840420512232089399393960305719544009749E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Source delivered — on — sample 7 — Source-delivered power | 7.578776280932684429383912232758915933279696058052762637949855116320087E-1 | W | Stored approximation, 70 significant digits |
| One-sided power can jump — Source delivered — off — sample 1 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 2 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 3 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 4 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 5 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 6 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 7 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 8 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 9 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 10 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 11 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 12 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 13 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 14 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 15 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 16 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 17 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 18 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 19 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 20 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 21 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 22 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 23 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 24 — Source-delivered power | 0 | W | Exact ratio |
| One-sided power can jump — Source delivered — off — sample 25 — Source-delivered power | 0 | W | Exact ratio |
| Finite work from the original zero state — Total storage — on — sample 1 — Total stored energy | 0 | J | Exact ratio |
| Finite work from the original zero state — Total storage — on — sample 2 — Total stored energy | 1.859296305756937902593879224813515139975419193770864986454949851584406E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — on — sample 3 — Total stored energy | 5.612017955538542842164221358918286477941440824197519823448007096726886E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — on — sample 4 — Total stored energy | 9.761491597513035003984006793717261278597948195196768441222138390915194E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — on — sample 5 — Total stored energy | 1.384403099565159830062498940042542840917638606921768939606655402970820E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — on — sample 6 — Total stored energy | 1.785499699237861386184206647890058531618131394601782190308170310726476E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — on — sample 7 — Total stored energy | 2.190109066969930376666291373268579321046361187053944072792839509233112E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 1 — Total stored energy | 2.190109066969930376666291373268579321046361187053944072792839509233112E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 2 — Total stored energy | 1.898119256430100726627893122663374702081516255746025186036685399115142E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 3 — Total stored energy | 1.859849965541713548052280583951750517751785293533006179143795168268779E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 4 — Total stored energy | 1.845671393575414081758359015833124470684614564490033944523256920316714E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 5 — Total stored energy | 1.756796315206262774058931797835967558642777011244689051991023330815213E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 6 — Total stored energy | 1.576226671832553032337238724937704810484117337142333397103773114059959E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 7 — Total stored energy | 1.328899901765591861277336282766798879557182786483386527628667281475460E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 8 — Total stored energy | 1.053671182238031302853245404524979346890611666947393192168276462160733E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 9 — Total stored energy | 7.865418716537675607944701891772391541805572968867916138832547179788966E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 10 — Total stored energy | 5.528455810221571014777483433020922233691378042601657419509081428566471E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 11 — Total stored energy | 3.655649118663566963728744399763749269828858234196102802303712548680113E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 12 — Total stored energy | 2.271566331628857740092428942432102775643702233199664731642193770672087E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 13 — Total stored energy | 1.328432120954633062623925477407912392365873758297668021037439150186695E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 14 — Total stored energy | 7.401186274415222019987139771414551920973308674729941259082216390289110E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 15 — Total stored energy | 4.097919473779246533296035747476126289702531232707045325951827907991833E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 16 — Total stored energy | 2.484860493479225996721297168224846867731123475913839315663942887967274E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 17 — Total stored energy | 1.849068752366877070113457595742850297113826668344487583945864621721380E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 18 — Total stored energy | 1.683436532829039233212424846927547211416689025366433256790617410157726E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 19 — Total stored energy | 1.670579114220088536036575470992113284548858814160568442484871313278374E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 20 — Total stored energy | 1.643365658557369963533954555205151069727325135483712378526348812326807E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 21 — Total stored energy | 1.538781676128446889530524328253816552943103114380631158252186059108668E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 22 — Total stored energy | 1.355800374485443494606721115327105644746177437150868256536252732599423E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 23 — Total stored energy | 1.122598865408884400550318298742267658650591764970071149937391702908161E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 24 — Total stored energy | 8.744857386750296268398414739376140858289036073611209196745115043824855E-5 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Total storage — off — sample 25 — Total stored energy | 6.414352510588383411915209483585818816777688110185817940198124840818594E-5 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — on — sample 1 — Accumulated resistor work | 0 | J | Exact ratio |
| Finite work from the original zero state — Resistor work — on — sample 2 — Accumulated resistor work | 3.968373550174897840708002834880992897138446416682726418169434542083903E-4 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — on — sample 3 — Accumulated resistor work | 2.491588500142202827781896822675576398441415836373413229084856799150210E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — on — sample 4 — Accumulated resistor work | 6.536014396007375312153882351176830782311826419381139107638278051937234E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — on — sample 5 — Accumulated resistor work | 1.194550688707321229769967117399226609802867231217966095727255053198362E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — on — sample 6 — Accumulated resistor work | 1.787837909522745790963940519495985278659909966132559226647897651121087E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — on — sample 7 — Accumulated resistor work | 2.357668697214674446004529382444428948982945456113722762948923510712850E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 1 — Accumulated resistor work | 2.357668697214674446004529382444428948982945456113722762948923510712850E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 2 — Accumulated resistor work | 2.649658507754504096042927633049633567947790387421641649705077620830820E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 3 — Accumulated resistor work | 2.687927798642891274618540171761257752277521349634660656597967851677183E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 4 — Accumulated resistor work | 2.702106370609190740912461739879883799344692078677632891218506099629248E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 5 — Accumulated resistor work | 2.790981448978342048611888957877040711386529631922977783750739689130749E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 6 — Accumulated resistor work | 2.971551092352051790333582030775303459545189306025333438637989905886003E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 7 — Accumulated resistor work | 3.218877862419012961393484472946209390472123856684280308113095738470502E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 8 — Accumulated resistor work | 3.494106581946573519817575351188028923138694976220273643573486557785229E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 9 — Accumulated resistor work | 3.761235892530837261876350566535769115848749346280875221858508301967065E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 10 — Accumulated resistor work | 3.994932183162447721193072412410916046660168838907501093790854877089315E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 11 — Accumulated resistor work | 4.182212852318248126297946315736633343046420819748056555511391765077951E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 12 — Accumulated resistor work | 4.320621131021719048661577861469797992464936419847700362577543642878753E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 13 — Accumulated resistor work | 4.414934552089141516408428207972217030792719267337900033638019104927293E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 14 — Accumulated resistor work | 4.473765901440452602470949357998862750819573556420367423150940856043071E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 15 — Accumulated resistor work | 4.506798569446812357337860398238247007132281330840596382482244740866044E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 16 — Accumulated resistor work | 4.522929159249812562703607784030759801351995408408528442585123591066289E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 17 — Accumulated resistor work | 4.529287076660936051969686179755579767058168376484221959902304373728748E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 18 — Accumulated resistor work | 4.530943398856314430338696507243732797915139752914002503173856845844385E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 19 — Accumulated resistor work | 4.531071973042403937310455001003087137183818055026061151316914306813178E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 20 — Accumulated resistor work | 4.531344107599031123035481210160956759332033391812829711956499531822694E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 21 — Accumulated resistor work | 4.532389947423320353775515512430470104499875612023860524159241159354875E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 22 — Accumulated resistor work | 4.534219760439750387724753544559737213581844868796158153176400492619968E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 23 — Accumulated resistor work | 4.536551775530515978665317572725585593442800725517966124242389102916880E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 24 — Accumulated resistor work | 4.539032906797854526402422340973632129171017607094055626545017904902137E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Resistor work — off — sample 25 — Accumulated resistor work | 4.541363411674016439258905546229422451212528955057481017801564895105143E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — on — sample 1 — Accumulated source-delivered work | 0 | J | Exact ratio |
| Finite work from the original zero state — Source-delivered work — on — sample 2 — Accumulated source-delivered work | 2.256133660774427686664679508301614429689263835439137628271893305792797E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — on — sample 3 — Accumulated source-delivered work | 8.103606455680745669946118181593862876382856660570933052532863895877095E-3 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — on — sample 4 — Accumulated source-delivered work | 1.629750599352041031613788914489409206090977461457790754886041644285243E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — on — sample 5 — Accumulated source-delivered work | 2.578953788272481059832466057441769450720505838139735035333910456169182E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — on — sample 6 — Accumulated source-delivered work | 3.573337608760607177148147167386043810278041360734341416956067961847563E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — on — sample 7 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 1 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 2 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 3 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 4 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 5 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 6 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 7 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 8 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 9 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 10 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 11 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 12 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 13 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 14 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 15 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 16 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 17 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 18 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 19 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 20 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 21 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 22 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 23 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 24 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Finite work from the original zero state — Source-delivered work — off — sample 25 — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Switch-left limit — Dimensionless time | 3/2 | 1 | Exact ratio |
| Switch-left limit — Local interval coordinate | 3/2 | 1 | Exact ratio |
| Switch-left limit — Physical time | 3/40 | s | Exact ratio |
| Switch-left limit — zeta | 3/5 | 1 | Exact ratio |
| Switch-left limit — R | 12 | ohm | Exact ratio |
| Switch-left limit — f | 1 | 1 | Exact ratio |
| Switch-left limit — x | 5.684722205230756028338525944641260337536633303959583544677203774932453E-1 | 1 | Stored approximation, 70 significant digits |
| Switch-left limit — j | 4.736735175582927768364945145474322458299810036282976648718659447700054E-1 | 1 | Stored approximation, 70 significant digits |
| Switch-left limit — j Prime | -1.368804415930269350376460119210447287496405347499155523139595112172518E-1 | 1 | Stored approximation, 70 significant digits |
| Switch-left limit — v Source | 4 | V | Exact ratio |
| Switch-left limit — Capacitor voltage | 2.273888882092302411335410377856504135014653321583833417870881509972981E+0 | V | Stored approximation, 70 significant digits |
| Switch-left limit — v R | 2.273632884279805328815173669827674779983908817415828791384956534896026E+0 | V | Stored approximation, 70 significant digits |
| Switch-left limit — v L | -5.475217663721077401505840476841789149985621389996622092558380448690070E-1 | V | Stored approximation, 70 significant digits |
| Switch-left limit — Series current | 1.894694070233171107345978058189728983319924014513190659487463779080022E-1 | A | Stored approximation, 70 significant digits |
| Switch-left limit — charge | 1.136944441046151205667705188928252067507326660791916708935440754986491E-2 | C | Stored approximation, 70 significant digits |
| Switch-left limit — dv C dt | 3.789388140466342214691956116379457966639848029026381318974927558160043E+1 | V/s | Stored approximation, 70 significant digits |
| Switch-left limit — di dt | -1.095043532744215480301168095368357829997124277999324418511676089738014E+0 | A/s | Stored approximation, 70 significant digits |
| Switch-left limit — Capacitor energy | 1.292642662025745194460820149342244695551836749757046450721063588962061E-2 | J | Stored approximation, 70 significant digits |
| Switch-left limit — Inductor energy | 8.974664049441851822054712239263346254945244372968976220717759202710513E-3 | J | Stored approximation, 70 significant digits |
| Switch-left limit — Total stored energy | 2.190109066969930376666291373268579321046361187053944072792839509233112E-2 | J | Stored approximation, 70 significant digits |
| Switch-left limit — Capacitor absorbed power | 4.308323781269419760023073113336533072451225945332971588040884490577826E-1 | W | Stored approximation, 70 significant digits |
| Switch-left limit — Inductor absorbed power | -1.037386244068824205225422755424023341545247186305317536035553791558786E-1 | W | Stored approximation, 70 significant digits |
| Switch-left limit — Resistor absorbed power | 4.307838743732088874586261874846406202373717299025108585944524417301046E-1 | W | Stored approximation, 70 significant digits |
| Switch-left limit — Source-delivered power | 7.578776280932684429383912232758915933279696058052762637949855116320087E-1 | W | Stored approximation, 70 significant digits |
| Switch-left limit — resistor Work Interval | 2.357668697214674446004529382444428948982945456113722762948923510712850E-2 | J | Stored approximation, 70 significant digits |
| Switch-left limit — source Work Interval | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Switch-left limit — Accumulated resistor work | 2.357668697214674446004529382444428948982945456113722762948923510712850E-2 | J | Stored approximation, 70 significant digits |
| Switch-left limit — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| Switch-right limit — Dimensionless time | 3/2 | 1 | Exact ratio |
| Switch-right limit — Local interval coordinate | 0 | 1 | Exact ratio |
| Switch-right limit — Physical time | 3/40 | s | Exact ratio |
| Switch-right limit — zeta | 3/5 | 1 | Exact ratio |
| Switch-right limit — R | 12 | ohm | Exact ratio |
| Switch-right limit — f | 0 | 1 | Exact ratio |
| Switch-right limit — x | 5.684722205230756028338525944641260337536633303959583544677203774932453E-1 | 1 | Stored approximation, 70 significant digits |
| Switch-right limit — j | 4.736735175582927768364945145474322458299810036282976648718659447700054E-1 | 1 | Stored approximation, 70 significant digits |
| Switch-right limit — j Prime | -1.136880441593026935037646011921044728749640534749915552313959511217252E+0 | 1 | Stored approximation, 70 significant digits |
| Switch-right limit — v Source | 0 | V | Exact ratio |
| Switch-right limit — Capacitor voltage | 2.273888882092302411335410377856504135014653321583833417870881509972981E+0 | V | Stored approximation, 70 significant digits |
| Switch-right limit — v R | 2.273632884279805328815173669827674779983908817415828791384956534896026E+0 | V | Stored approximation, 70 significant digits |
| Switch-right limit — v L | -4.547521766372107740150584047684178914998562138999662209255838044869007E+0 | V | Stored approximation, 70 significant digits |
| Switch-right limit — Series current | 1.894694070233171107345978058189728983319924014513190659487463779080022E-1 | A | Stored approximation, 70 significant digits |
| Switch-right limit — charge | 1.136944441046151205667705188928252067507326660791916708935440754986491E-2 | C | Stored approximation, 70 significant digits |
| Switch-right limit — dv C dt | 3.789388140466342214691956116379457966639848029026381318974927558160043E+1 | V/s | Stored approximation, 70 significant digits |
| Switch-right limit — di dt | -9.095043532744215480301168095368357829997124277999324418511676089738014E+0 | A/s | Stored approximation, 70 significant digits |
| Switch-right limit — Capacitor energy | 1.292642662025745194460820149342244695551836749757046450721063588962061E-2 | J | Stored approximation, 70 significant digits |
| Switch-right limit — Inductor energy | 8.974664049441851822054712239263346254945244372968976220717759202710513E-3 | J | Stored approximation, 70 significant digits |
| Switch-right limit — Total stored energy | 2.190109066969930376666291373268579321046361187053944072792839509233112E-2 | J | Stored approximation, 70 significant digits |
| Switch-right limit — Capacitor absorbed power | 4.308323781269419760023073113336533072451225945332971588040884490577826E-1 | W | Stored approximation, 70 significant digits |
| Switch-right limit — Inductor absorbed power | -8.616162525001508634609334988182939274824943244358080173985408907878873E-1 | W | Stored approximation, 70 significant digits |
| Switch-right limit — Resistor absorbed power | 4.307838743732088874586261874846406202373717299025108585944524417301046E-1 | W | Stored approximation, 70 significant digits |
| Switch-right limit — Source-delivered power | 0 | W | Exact ratio |
| Switch-right limit — resistor Work Interval | 0 | J | Exact ratio |
| Switch-right limit — source Work Interval | 0 | J | Exact ratio |
| Switch-right limit — Accumulated resistor work | 2.357668697214674446004529382444428948982945456113722762948923510712850E-2 | J | Stored approximation, 70 significant digits |
| Switch-right limit — Accumulated source-delivered work | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits |
| j Prime Jump | -1 | 1 | Exact ratio |
| di Dt Jump | -8 | A/s | Exact ratio |
| v L Jump | -4 | V | Exact ratio |
| Closed pulse after switch-off — long-time limit of normalized capacitor-voltage state x | 0 | 1 | Exact long-time limit as t→∞; x is dimensionless, not a voltage in volts. |
| Closed pulse after switch-off — long-time limit of normalized state derivative j=dx/du | 0 | 1 | Exact long-time limit as t→∞; j is dimensionless, not a current in amperes. |
| Closed pulse after switch-off — long-time total capacitor-plus-inductor stored energy | 0 | J | Exact long-time limit as t→∞; storage energy, distinct from accumulated work. |
| Closed pulse — limiting total resistor work from the original zero-state preparation | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits; accumulated over both on and off intervals, as t→∞. |
| Closed pulse — limiting total source-delivered work from the original zero-state preparation | 4.547777764184604822670820755713008270029306643167666835741763019945962E-2 | J | Stored approximation, 70 significant digits; delivered work over both intervals as t→∞, with the source-delivered sign convention. |
Focus a graph or table region to scroll it locally. Axis ticks are rounded drawing labels; the numerical equivalents retain exact ratios, full stored approximations and qualified limits.
| Quantity | Left | Right | Right minus left |
|---|---|---|---|
| x | x_s | x_s | 0 |
| j | j_s | j_s | 0 |
| v_C | 4x_s V | 4x_s V | 0 V |
| v_R | (24/5)j_s V | (24/5)j_s V | 0 V |
| di/dt | 8[1−x_s−(6/5)j_s] A/s | 8[−x_s−(6/5)j_s] A/s | −8 A/s |
| v_L | 4[1−x_s−(6/5)j_s] V | 4[−x_s−(6/5)j_s] V | −4 V |
Derive the parallel natural equation
For a separate parallel RLC network, let v be the common top-to-return voltage and orient each branch current downward. In the source-free circuit, C dv/dt+v/R+i_L=0 and L di_L/dt=v. Differentiating KCL gives d²v/dt²+[1/(RC)]dv/dt+v/(LC)=0. The natural frequency remains 1/√(LC), but α=1/(2RC), so increasing parallel resistance reduces damping instead of increasing it.
Choose the same L=1/2 H and C=1/200 F, but R=25/3 Ω. Then α=12 s⁻¹ and ζ=3/5. Set v(0)=4 V and dv/dt(0)=−40 V/s. KCL determines i_L(0)=−7/25 A; it is not an extra freely chosen third initial condition. Using x=v/4 and j=dx/du produces the same normalized voltage equation and initial x,j as the selected underdamped series natural response.
At the initial instant i_C=−1/5 A, i_R=12/25 A and i_L=−7/25 A, whose sum is zero. A negative inductor branch current is fully consistent with a positive common voltage and a positive inductor current derivative. The differentiation used to obtain the voltage equation must be paired with this original KCL constraint so that no unsupported integration constant is introduced.
Keep the parallel energy ledger physical
The shared voltage shape does not make j the inductor current. In the parallel circuit i_C=(2/5)j, i_R=(12/25)x and i_L=−(2/5)j−(12/25)x, all in amperes. Therefore U_C=x²/25 J but U_L=i_L²/4 J. Copying the series expression (x²+j²)/25 J would attach energy to the wrong coordinate and fail the physical initial preparation.
The parallel initial energy is 1/25+49/2500=149/2500 J. Initial powers are p_C=−4/5 W, p_R=48/25 W and p_L=−28/25 W. Their sum vanishes: both storage elements initially supply the resistor. At later times the partition can change. Since the source is absent and R>0, the combined storage decreases and the complete resistor work tends to 149/2500 J.
Here W_R=∫v²/R dt=(12/125)∫x²du J. Its integral can be evaluated from the same exponential modes with the coefficients of x, rather than those of the series current j. The equality U−U₀+W_R=0 then checks the physical branch mapping. Differential-equation similarity is useful, but it does not identify energy, current direction or the meaning of an observed variable without the circuit relations.
States, damping and physical energy
Compare the same original state preparation under four damping branches, then distinguish electric, magnetic and dissipated energy.
(a) Natural response from reversed initial current
- Lossless ζ=0
- Underdamped ζ=3/5
- Critical ζ=1
- Overdamped ζ=5/4
(b) Zero-state unit-source step
- Lossless ζ=0
- Underdamped ζ=3/5
- Critical ζ=1
- Overdamped ζ=5/4
(c) Critical natural response: storage and loss
- Capacitor energy
- Inductor energy
- Total storage
- Resistor work
(d) Parallel response: its physical energy ledger
- Capacitor energy
- Inductor energy
- Resistor work
- Each branch uses the same L=1/2 H and C=1/200 F. Physical time is on every axis; ζ changes the resistance. A source-free natural response and a zero-state step use different initial/source conditions.
- The series natural preparation has capacitor voltage 4 V and series current −1/5 A. The parallel preparation retains its own KCL mapping and initial inductor current −7/25 A; normalized derivative j is not that inductor current.
- Electric and magnetic storage can exchange while total storage decreases through a positive resistor. Accumulated resistor work is distinct from instantaneous resistor power. Lossless oscillations do not settle.
Original series state and SI scales
Four original series damping branches with fixed L and C
States, damping and physical energy — values and conditions
| Quantity and original case | Value | Unit | Meaning and precision |
|---|---|---|---|
| Natural response from reversed initial current — Lossless ζ=0 — sample 1 — Physical time | 0 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 1 — Capacitor voltage | 4 | V | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 2 — Capacitor voltage | 3.380841768333533277384684388277978407429754800375839703239422836166974E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 3 — Capacitor voltage | 2.551479170353084863918550459984175831802974052557774861613210247012733E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 4 — Capacitor voltage | 1.563477955448615211780871106440550304685400314884630111603482356580800E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 5 — Capacitor voltage | 4.782672538567658562987417865113084156841155608749467793348856015405808E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 6 — Capacitor voltage | -6.366797891300977669068274843199464458440892371167572102111175025470947E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 7 — Capacitor voltage | -1.712041166537297221530687336545899809072938741590844166209367979354999E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 8 — Capacitor voltage | -2.680956116345842158994033649706496622955872112954524772837151184985926E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 9 — Capacitor voltage | -3.483182199839932938782312649826538444468513027197960099778545918189794E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 10 — Capacitor voltage | -4.068840884666798838475489580761127868764182610589375987442715986480075E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 11 — Capacitor voltage | -4.401518750395647847437720566241731201121465738225727686903646031925585E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 12 — Capacitor voltage | -4.460531498634517573539760702862414515008034606826317616116345608082568E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 13 — Capacitor voltage | -4.242210002521516273287780784541265769268582914966884483560109014178161E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 14 — Capacitor voltage | -3.760128435261968123420455215564748046232333402355817360211321193242437E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 15 — Capacitor voltage | -3.044260293783945654553892906599770681909867591246670035652443555237261E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 16 — Capacitor voltage | -2.139114791873555345464138462462257193600151658238210537189029964549282E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 17 — Capacitor voltage | -1.100969492838591155811394543367343337424708611911927845057207588608330E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 18 — Capacitor voltage | 5.628756801995852275194774697204092021945599011486639134331867712390486E-3 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 19 — Capacitor voltage | 1.111877037607075286856342729821937983685615072887250142363003685661172E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 20 — Capacitor voltage | 2.148994189502662108330526760687494126226849820226558570892876292852697E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 21 — Capacitor voltage | 3.052497291179181995652865498366217180042613456938067335984781174814811E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 22 — Capacitor voltage | 3.766210895820546793268716235521808528654477253602004159364783181847117E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 23 — Capacitor voltage | 4.245759748305823812573523076347515097645614134683370352540369119395670E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 24 — Capacitor voltage | 4.461327823644600043740616870242636826092875618072150303694768016808745E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Lossless ζ=0 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Natural response from reversed initial current — Lossless ζ=0 — sample 25 — Capacitor voltage | 4.399512142999315827805720084915487141063740445480459429483848443520806E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 1 — Physical time | 0 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 1 — Capacitor voltage | 4 | V | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 2 — Capacitor voltage | 3.459702622745434716502817964982076754929893123273496595944013740137382E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 3 — Capacitor voltage | 2.873599168591166243591907809822969668718842504432346378375953637229077E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 4 — Capacitor voltage | 2.285044858728922740384797906734334529621271440338239975262611617201684E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 5 — Capacitor voltage | 1.726289681092303888635201101213304351272404990226434999786697427323610E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 6 — Capacitor voltage | 1.219624324806464334943739835868248042296950308457905046483751076075000E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 7 — Capacitor voltage | 7.787640827911120349916005930486313733253846711595712523853866004870081E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 8 — Capacitor voltage | 4.103347078246363975973851357576362449755402011322168265021547632298669E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 9 — Capacitor voltage | 1.153539833771655317879042789204468430729126639131420657197189482848309E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 10 — Capacitor voltage | -1.093694641609900160805710117324068715844063157352568979038213832416126E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 11 — Capacitor voltage | -2.699738009010205732954733569671959598809436434378139541243394698548420E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 12 — Capacitor voltage | -3.744505129315230761462841412811042989278360231563333054413867243876629E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 13 — Capacitor voltage | -4.317347890523498684323988757968931398823971965093271963266524555278866E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 14 — Capacitor voltage | -4.509809678152314090102690992078654173833306935937339379101593819600313E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 15 — Capacitor voltage | -4.410140036462100152065873554111802919653232462807486333559510085609411E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 16 — Capacitor voltage | -4.099407961451672418501535463002590073775501836315201074367176236301474E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 17 — Capacitor voltage | -3.649008267668380826415351293809257654512083163710431704602300227179009E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 18 — Capacitor voltage | -3.119333917430448977477416304129209142736863171404999972636321708630145E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 19 — Capacitor voltage | -2.559383433384153196049600654344794138683077135970186396775256211226106E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 20 — Capacitor voltage | -2.007082126052035186008376093002566698172513544321482242920023229798014E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 21 — Capacitor voltage | -1.490114873480520665611276559468583971463847353726184971750803760762820E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 22 — Capacitor voltage | -1.027093134429767189704026355052114662389739391991145904279407935597148E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 23 — Capacitor voltage | -6.289068806058741686496219390250810368392254483545855047867342259530443E-2 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 24 — Capacitor voltage | -3.001408999259952315034318278795865966686891537330581171896511244388939E-2 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Natural response from reversed initial current — Underdamped ζ=3/5 — sample 25 — Capacitor voltage | -4.046270890144391452570448352904254425091143749084319749519211514130739E-3 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 1 — Physical time | 0 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 1 — Capacitor voltage | 4 | V | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 2 — Capacitor voltage | 3.504603523821321907103266201402442912835475306917636634085928148207105E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 3 — Capacitor voltage | 3.032653298563167118018997674955902267209590677435934778414460793675283E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 4 — Capacitor voltage | 2.598016040075580889259256030187973521336119685250621753176118692776605E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 5 — Capacitor voltage | 2.207276647028653929573142620968765204674866786190607007047020810184769E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 6 — Capacitor voltage | 1.862281179591235652111755273210944418155480150133631806074484372878901E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 7 — Capacitor voltage | 1.561911121039008802532963295348087649395201405527555301206847231322276E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 8 — Capacitor voltage | 1.303304575878338450105379439997782620110406400049287782629860404404271E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 9 — Capacitor voltage | 1.082682265892901535151995959779875227261052367276607051745270981232587E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 10 — Capacitor voltage | 8.958934087758468626573503585459338267821741237068618707651747769840063E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 11 — Capacitor voltage | 7.387649876150891565257580702044382705402370891389298396118256946370228E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 12 — Capacitor voltage | 6.073146814637219406730852428005416184320239032483002580728065550883982E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 13 — Capacitor voltage | 4.978706836786394297934241565006177663169959218842321556762772760606067E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 14 — Capacitor voltage | 4.071291822330811038124482703097595042315126503282115298573065404274812E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 15 — Capacitor voltage | 3.321712176455035081376492159998182957882658547242270733847424539354920E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 16 — Capacitor voltage | 2.704540773441047447157386286549788039057780356403326453878021534499978E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 17 — Capacitor voltage | 2.197876666648101635246162552788949065429448106417071372351991292710052E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 18 — Capacitor voltage | 1.783029238624906909160868232000192999993739594280015834604567352617270E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 19 — Capacitor voltage | 1.444169549971499844498607457300968610300104775750272929433909457338161E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 20 — Capacitor voltage | 1.167978852421285613904653034228802764502454711690585743670389146090623E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 21 — Capacitor voltage | 9.433125798719653935290467792407793948389419038297119602427744201756932E-2 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 22 — Capacitor voltage | 7.608901678813007200915617544999704869178371363953585695089138008838622E-2 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 23 — Capacitor voltage | 6.130157157696100490197054027081152612585984976407299131020571613610775E-2 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 24 — Capacitor voltage | 4.933310234589983955378947907411869586206578246638796404116964364463750E-2 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Critical ζ=1 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Natural response from reversed initial current — Critical ζ=1 — sample 25 — Capacitor voltage | 3.966003482666173476872267889306668626410367336854312081405798410041911E-2 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 1 — Physical time | 0 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 1 — Capacitor voltage | 4 | V | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 2 — Capacitor voltage | 3.529987610338381611459568572916202944888019299962602967081237276832437E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 3 — Capacitor voltage | 3.115203132285619472980681067913282589187089161704565896965269465072982E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 4 — Capacitor voltage | 2.749157115163888794180809356586054361738608095090084276730627558982497E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 5 — Capacitor voltage | 2.426122638850533694415198139964721813767672541948747822731568634940226E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 6 — Capacitor voltage | 2.141045714075960967826490032088185621734176500144087564995601666022770E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 7 — Capacitor voltage | 1.889466210964058828552186203773071651880814316545906729582631776564804E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 8 — Capacitor voltage | 1.667448078714033610341188729345569378474903319529405276177004312017699E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 9 — Capacitor voltage | 1.471517764685769286382095080645843469783244524127071338031347206789846E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 10 — Capacitor voltage | 1.298609869433398919188272549889887967600648425618868224501325999234673E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 11 — Capacitor voltage | 1.146019187440760401299541706591350411172603169313004188353528844848555E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 12 — Capacitor voltage | 1.011358383218985911243949999277301111874135554529120353864342767653650E+0 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 13 — Capacitor voltage | 8.925206405937193157331218830560500853686865174443173149753412750413007E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 14 — Capacitor voltage | 7.876467008167762002451568723608070057810901440202032420337998451848016E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 15 — Capacitor voltage | 6.950957738017805067228690346654840640588834133596201507359255490156114E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 16 — Capacitor voltage | 6.134198673797138531822579128403170171852248151122500325430845771544202E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 17 — Capacitor voltage | 5.413411329464507675759979798899376136305261836383035258726354906162935E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 18 — Capacitor voltage | 4.777318730668784719835077904475256432685968732576878029159081738786887E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 19 — Capacitor voltage | 4.215968982474573471328707569627923890739642935091114685953763656395324E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 20 — Capacitor voltage | 3.720579568426539467994582936150167494029973094466453178058155474849908E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 21 — Capacitor voltage | 3.283399944955951806781146978686392313512164840617465953830336420608990E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 22 — Capacitor voltage | 2.897590281370058509533754134946352563732474179077650547217475928027959E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 23 — Capacitor voltage | 2.557114448268302908097201022318069972345363803150737928727606547740624E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 24 — Capacitor voltage | 2.256645580151093932250415555128525984067953014324770533022179695738122E-1 | V | Stored approximation, 70 significant digits |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Natural response from reversed initial current — Overdamped ζ=5/4 — sample 25 — Capacitor voltage | 1.991482734714557719173696626002471065267983687536928622705109104242427E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 1 — Physical time | 0 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 1 — Capacitor voltage | 0 | V | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 2 — Capacitor voltage | 1.243503131574208634216182020232432007834632388502286754074875028442298E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 3 — Capacitor voltage | 4.896697524385091355348736695846813920334192115610237880095565267361969E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 4 — Capacitor voltage | 1.073244524504716454752644987999661824637834895796910069969267119117000E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 5 — Capacitor voltage | 1.838790776527441130396253570228093585070758317528311089319610978475598E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 6 — Capacitor voltage | 2.738710550418925338209845790247847945088057168068972769940003438186842E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 7 — Capacitor voltage | 3.717051193329188359647240594262925163659635889746612522309418331236308E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 8 — Capacitor voltage | 4.712984222597968361530707775770525435879683405166193091544250273370341E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 9 — Capacitor voltage | 5.664587346188569547990272918003048759064003084302179563020599895127860E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 10 — Capacitor voltage | 6.512694490890956355653556229585613227155261124768933152537660382394790E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 11 — Capacitor voltage | 7.204574462187734859334011161869406657714271395071280540298391966648096E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 12 — Capacitor voltage | 7.697209514529854176386637958106853912843539089583877177056638062821747E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 13 — Capacitor voltage | 7.959969986401781829086291178925045209574716386462353315256343731713317E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 14 — Capacitor voltage | 7.976518704322184877492116900686398048270147280760438129384671178485080E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 15 — Capacitor voltage | 7.745826749163185350794630506687041852079831063127837006483953853378560E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 16 — Capacitor voltage | 7.282237429358242890332449609162844189437181452865667753402532541283563E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 17 — Capacitor voltage | 6.614574483454447658556672732391001525696534386584872988028041135410951E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 18 — Capacitor voltage | 5.784349959655171196656310882179348319740173924819364250571120350972790E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 19 — Capacitor voltage | 4.843183197723118823921927299175321572043148003360779280579378056717823E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 20 — Capacitor voltage | 3.849591388448093781138014750506664400268674869587078447317045467171210E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 21 — Capacitor voltage | 2.865351258147094942133443313945770766662309630991136220278563733655394E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 22 — Capacitor voltage | 1.951658091032637267392735143323607773054126027290626926490550072316331E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 23 — Capacitor voltage | 1.165320902834959999890315274696625060048966429292983118071635006311479E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 24 — Capacitor voltage | 5.552303313539166162229419482842297647292372563035276329533744238609480E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Lossless ζ=0 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Zero-state unit-source step — Lossless ζ=0 — sample 25 — Capacitor voltage | 1.593188533985359178173908083083023781922492831559492074828542957858166E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 1 — Physical time | 0 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 1 — Capacitor voltage | 0 | V | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 2 — Capacitor voltage | 1.128066830387213843332339754150807214844631917719568814135946652896398E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 3 — Capacitor voltage | 4.051803227840372834973059090796931438191428330285466526266431947938548E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 4 — Capacitor voltage | 8.148752996760205158068944572447046030454887307288953774430208221426214E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 5 — Capacitor voltage | 1.289476894136240529916233028720884725360252919069867517666955228084591E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 6 — Capacitor voltage | 1.786668804380303588574073583693021905139020680367170708478033980923781E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 7 — Capacitor voltage | 2.273888882092302411335410377856504135014653321583833417870881509972981E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 8 — Capacitor voltage | 2.727550141239469687792100875494710273824260044408996462247177605100146E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 9 — Capacitor voltage | 3.131981557582018256532386010433165069608615470134234478984407050111473E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 10 — Capacitor voltage | 3.478218180015371858729374531214496700704209479682738735686479262115236E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 11 — Capacitor voltage | 3.762744599725596797505490290350137875676382461779204720213337066172151E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 12 — Capacitor voltage | 3.986271361747121544867854493310346926453375220532972777610344830989268E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 13 — Capacitor voltage | 4.152601507103044810415471830297098199971994330707518007092475351254952E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 14 — Capacitor voltage | 4.267624770121549026979780680335296685661015159334061912617632543623385E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 15 — Capacitor voltage | 4.338460372708295761001538294703253990336516336009049935541835593975898E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 16 — Capacitor voltage | 4.372755947595982115630519087175045895115953540785506638745341892649570E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 17 — Capacitor voltage | 4.378139783808769732848064173608642608405609435927372249038082784849526E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 18 — Capacitor voltage | 4.361816079188109007548432383667730292194957452333426131969732345134037E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 19 — Capacitor voltage | 4.330287876562045108205286934649852283738938368449611889850301728576755E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 20 — Capacitor voltage | 4.289189473038934920484956860303843362883504523057307713782503776879212E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 21 — Capacitor voltage | 4.243208931285216703488103781611572865070729439689425513368801922128604E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 22 — Capacitor voltage | 4.196081502377240754425693344414709051918974770744109998958467799765595E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 23 — Capacitor voltage | 4.150635934537172861343284343283025062415018481930636327263619941835716E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 24 — Capacitor voltage | 4.108877473176362908149998918487659928906709792679625399760682816868872E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Zero-state unit-source step — Underdamped ζ=3/5 — sample 25 — Capacitor voltage | 4.072093584099260073997880785724426462157797249077933448169050250640359E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 1 — Physical time | 0 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 1 — Capacitor voltage | 0 | V | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 2 — Capacitor voltage | 1.059960846429756587741486651083967635161385478692926287934131686587719E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 3 — Capacitor voltage | 3.608160417241994583772027900529172793484911870768782659026470475896609E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 4 — Capacitor voltage | 6.934341308128970500336741433971246092085749460446632232303943910115934E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 5 — Capacitor voltage | 1.056964470628461427235809838708313060433510951745857323937305586420308E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 6 — Capacitor voltage | 1.421456828258289097076031160169461574861642869045740576204560099090752E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 7 — Capacitor voltage | 1.768698398515701710667195292359874786578283706389206712561646812396748E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 8 — Capacitor voltage | 2.088486622045103606512110154669918823838070613261044585476204740207069E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 9 — Capacitor voltage | 2.375976601160647697272006060330187159108421449085089422382093528151120E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 10 — Capacitor voltage | 2.629810080695763621818170039870924735509616046095387727065026811671520E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 11 — Capacitor voltage | 2.850810019265416867626598557459762690270742305783886916159382252786853E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 12 — Capacitor voltage | 3.041082081899386409463549616630723760370488573818473276727147544597266E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 13 — Capacitor voltage | 3.203406906114176912330521349599011573892806524985228550917956358303029E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 14 — Capacitor voltage | 3.340838466860725831922702800450865564577550947087657523088170363117411E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 15 — Capacitor voltage | 3.456447098398266986683846737454842788710110419542173879915875984469195E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 16 — Capacitor voltage | 3.553162828735826943513127483091774150068714549811624325011457311691308E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 17 — Capacitor voltage | 3.633687222225316394125639574535175155761758648930488104608001451214991E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 18 — Capacitor voltage | 3.700451087911015639260974137023967576001051748160957339786432684760299E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 19 — Capacitor voltage | 3.755602076158669257084851045687528389026136114873030735018876861065850E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 20 — Capacitor voltage | 3.801011010328225413927355408983241010492174382452715021448748515851227E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 21 — Capacitor voltage | 3.838289272021948789680734837844437818027609959343477949672667242255595E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 22 — Capacitor voltage | 3.868812040020465393087661766465522329841752217862869212153635551571748E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 23 — Capacitor voltage | 3.893743942599934258169917730197260021381842927075606815062310092030747E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 24 — Capacitor voltage | 3.914064918494238989164366713870890013659627346671453223928285136877083E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Critical ζ=1 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Zero-state unit-source step — Critical ζ=1 — sample 25 — Capacitor voltage | 3.930594939053341964154735311937133299037818571605049538575398527824267E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 1 — Physical time | 0 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 1 — Capacitor voltage | 0 | V | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 2 — Capacitor voltage | 1.020573991656690828589746160999700114051984473661119848022065092034925E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 3 — Capacitor voltage | 3.369017451810971314864569363309043710116292924362692500567564488326387E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 4 — Capacitor voltage | 6.319640599793880463366281522372775461380847123613267360176103463704374E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 5 — Capacitor voltage | 9.456168591814386633050684733436834528532786719477707449821203169517964E-1 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 6 — Capacitor voltage | 1.254719046063917103124051523171965581471503494495132111800208992656606E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 7 — Capacitor voltage | 1.547094476538406819236208282502653499667847034190021981313327934721676E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 8 — Capacitor voltage | 1.816999072944379853864796750690733955462342950291002307325781214756460E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 9 — Capacitor voltage | 2.062397165603953191882163920836530363238223391235205675651003627532539E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 10 — Capacitor voltage | 2.283332169473124516410494112529390080227854455849204140966784918531596E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 11 — Capacitor voltage | 2.480958346077766754396125789109064017434995220952467862769035582298839E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 12 — Capacitor voltage | 2.656971184292637540999352937876559542045673469495757127408450151229454E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 13 — Capacitor voltage | 2.813277482110596056919897712499688776697093282854955506767383131953301E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 14 — Capacitor voltage | 2.951808984501601829602968041295147530731880491917466297417008272969552E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 15 — Capacitor voltage | 3.074421477551698679313512135225233624756787081557087870806520779208371E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 16 — Capacitor voltage | 3.182844289320578640534458782997617786683448695293586886279738867184566E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 17 — Capacitor voltage | 3.278659106241935659017097878981124329851712955327222391317212113508824E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 18 — Capacitor voltage | 3.363295460402842896254481523977281897491600584358730500147276887897549E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 19 — Capacitor voltage | 3.438035348742172443222169172970583526336143812430903293942877488540817E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 20 — Capacitor voltage | 3.504022526649645005056017194272450407684154236603033357662836140263523E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 21 — Capacitor voltage | 3.562273873912223072231227858195895092345361912036159959448300572326336E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 22 — Capacitor voltage | 3.613691344416458528272642663469956225517138895938632234676763894838653E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 23 — Capacitor voltage | 3.659073675831946606466123377714071444859388743642513992335623293284068E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 24 — Capacitor voltage | 3.699127429437985649980916514457195317454004940455170181015925899797959E+0 | V | Stored approximation, 70 significant digits |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Zero-state unit-source step — Overdamped ζ=5/4 — sample 25 — Capacitor voltage | 3.734477160987863408389852026277242265340683806980829367976619544108845E+0 | V | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 1 — Capacitor energy | 1/25 | J | Exact ratio |
| Critical natural response: storage and loss — Capacitor energy — sample 2 — Capacitor energy | 3.070561464795206706994235145892851045549710560903883963144641553596224E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 3 — Capacitor energy | 2.299246507321514509972023563509130421536319568948548965673980010609134E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 4 — Capacitor energy | 1.687421836122500581307933560152844692650172947043162423625254598124959E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 5 — Capacitor energy | 1.218017549129514227045995454752359630668683913186182933213429853886660E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 6 — Capacitor energy | 8.670227979649310239781466240593754702868060282255496034333232110670615E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 7 — Capacitor energy | 6.098915875063333014969445917132567637383200043081843907034396631742432E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 8 — Capacitor energy | 4.246507043763539166532447363634040713202262347326766563157218871334415E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 9 — Capacitor energy | 2.930502222197468846994883403718598753905930808556095163135988390280069E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 10 — Capacitor energy | 2.006562499720016610865853630576826578734280193229946522146153044090041E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 11 — Capacitor energy | 1.364434267314807087068799805687555910392040968039404799636870143468413E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 12 — Capacitor energy | 9.220778058034551154004735432401233721431419068679312442910109802139541E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 13 — Capacitor energy | 6.196880441665896057612918577041669728766198963834862627196560015690486E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 14 — Capacitor energy | 4.143854275644434058099127309287095868938913841322547547382612569854771E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 15 — Capacitor energy | 2.758442945802411529209486655338079945083016695494062868468803601215677E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 16 — Capacitor energy | 1.828635198801274784130987792703493485263731575692902167363355438530628E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 17 — Capacitor energy | 1.207665460449042619757000852811099669519240481393149697960472692481906E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 18 — Capacitor energy | 7.947983164478288061286462128401030753801231513112422904915617259776828E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 19 — Capacitor energy | 5.214064222662210966275150183343929151548457164207674969581866407383698E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 20 — Capacitor energy | 3.410436499258358198906063363091919756022965582057811412199612852479468E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 21 — Capacitor energy | 2.224596558361757725243984262466979901657986354461683493694493976898972E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 22 — Capacitor energy | 1.447384618946084984880170791446240927277317107668315209909513539493141E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 23 — Capacitor energy | 9.394706694513183363357478515326744481340151414599007518945164626357224E-6 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 24 — Capacitor energy | 6.084387467677570631570401796795077440215943546472619162538149882689760E-6 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Capacitor energy — sample 25 — Capacitor energy | 3.932295906130054245556677234435540679183353161528481904363843760959925E-6 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 1 — Inductor energy | 1/100 | J | Exact ratio |
| Critical natural response: storage and loss — Inductor energy — sample 2 — Inductor energy | 9.477041558009897243809367734237194585029970866987296182545189980235258E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 3 — Inductor energy | 8.277287426357452235899284828632869517530750448214776276426328038192884E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 4 — Inductor energy | 6.833361154545663511081714417147883466104006149183054442779956637034958E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 5 — Inductor energy | 5.413411329464507675759979798899376136305261836383035258726354906162935E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 6 — Inductor energy | 4.155553055334876505457389144899965271788833626406480347816519532333253E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 7 — Inductor energy | 3.111691772991496436208900978128861039481224511776450972976732975378792E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 8 — Inductor energy | 2.283677121312836618446338359998750783544327751229061129520104370806507E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 9 — Inductor energy | 1.648407499986076226434621914591711799072086079812803529263993469532539E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 10 — Inductor energy | 1.173387759351843623655118559057036995868835130297096755165051434087256E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 11 — Inductor energy | 8.253985073879697193379159318356819704840741658509979652124276176537315E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 12 — Inductor energy | 5.747022335340094209559738150388580574299360915381842935331785887760102E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 13 — Inductor energy | 3.966003482666173476872267889306668626410367336854312081405798410041911E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 14 — Inductor energy | 2.715587042315740233085369143727824730438426530934730705654365153487594E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 15 — Inductor energy | 1.846560980247895321206350570928797318609292168058009027652670179326197E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 16 — Inductor energy | 1.247896610146049521873887699746618427561828164130695051830191518543585E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 17 — Inductor energy | 8.386565697562795970534728144521525482772503343007984013614393697791013E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 18 — Inductor energy | 5.608096920855880056043727677799767299882148955652125601708459538498530E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 19 — Inductor energy | 3.733146573622056372303509894583523238682623176740406339168081865641583E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 20 — Inductor energy | 2.474788625662100805516196871159980179610629345553610750416454319563290E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 21 — Inductor energy | 1.634397471449454655281294560179821968565051199196338893326567003435979E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 22 — Inductor energy | 1.075642552724498353805120980563496527405853974188700364082575460384320E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 23 — Inductor energy | 7.056468583878791059588506084845421410428824840298810092007701430463871E-6 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 24 — Inductor energy | 4.615523895876117575873905213177535331859961337542704858990958651905135E-6 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Inductor energy — sample 25 — Inductor energy | 3.010664053130822781754331007614710832499754764295243958028567879484943E-6 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 1 — Total stored energy | 1/20 | J | Exact ratio |
| Critical natural response: storage and loss — Total storage — sample 2 — Total stored energy | 4.018265620596196431375171919316570504052707647602613581399160551619749E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 3 — Total stored energy | 3.126975249957259733561952046372417373289394613770026593316612814428423E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 4 — Total stored energy | 2.370757951577066932416105001867633039260573561961467867903250261828455E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 5 — Total stored energy | 1.759358682075964994621993434642297244299210096824486459086065344502954E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 6 — Total stored energy | 1.282578103498418674523885538549371997465689390866197638214975164300387E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 7 — Total stored energy | 9.210607648054829451178346895261428676864424554858294880011129607121224E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 8 — Total stored energy | 6.530184165076375784978785723632791496746590098555827692677323242140922E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 9 — Total stored energy | 4.578909722183545073429505318310310552978016888368898692399981859812608E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 10 — Total stored energy | 3.179950259071860234520972189633863574603115323527043277311204478177297E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 11 — Total stored energy | 2.189832774702776806406715737523237880876115133890402764849297761122145E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 12 — Total stored energy | 1.496780039337464536356447358278981429573077998406115537824189568989964E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 13 — Total stored energy | 1.016288392433206953448518646634833835517656630068917470860235842573240E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 14 — Total stored energy | 6.859441317960174291184496453014920599377340372257278253036977723342364E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 15 — Total stored energy | 4.605003926050306850415837226266877263692308863552071896121473780541873E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 16 — Total stored energy | 3.076531808947324306004875492450111912825559739823597219193546957074213E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 17 — Total stored energy | 2.046322030205322216810473667263252217796490815693948099321912062261007E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 18 — Total stored energy | 1.355608008533416811733018980620079805368338046876454850662407679827536E-4 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 19 — Total stored energy | 8.947210796284267338578660077927452390231080340948081308749948273025281E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 20 — Total stored energy | 5.885225124920459004422260234251899935633594927611422162616067172042759E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 21 — Total stored energy | 3.858994029811212380525278822646801870223037553658022387021060980334951E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 22 — Total stored energy | 2.523027171670583338685291772009737454683171081857015573992088999877462E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 23 — Total stored energy | 1.645117527839197442294598460017216589176897625489781761095286605682109E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 24 — Total stored energy | 1.069991136355368820744430700997261277207590488401532402152910853459489E-5 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Total storage — sample 25 — Total stored energy | 6.942959959260877027311008242050251511683107925823725862392411640444868E-6 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 1 — Accumulated resistor work | 0 | J | Exact ratio |
| Critical natural response: storage and loss — Resistor work — sample 2 — Accumulated resistor work | 9.817343794038035686248280806834294959472923523973864186008394483802506E-3 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 3 — Accumulated resistor work | 1.873024750042740266438047953627582626710605386229973406683387185571577E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 4 — Accumulated resistor work | 2.629242048422933067583894998132366960739426438038532132096749738171545E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 5 — Accumulated resistor work | 3.240641317924035005378006565357702755700789903175513540913934655497046E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 6 — Accumulated resistor work | 3.717421896501581325476114461450628002534310609133802361785024835699613E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 7 — Accumulated resistor work | 4.078939235194517054882165310473857132313557544514170511998887039287878E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 8 — Accumulated resistor work | 4.346981583492362421502121427636720850325340990144417230732267675785908E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 9 — Accumulated resistor work | 4.542109027781645492657049468168968944702198311163110130760001814018739E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 10 — Accumulated resistor work | 4.682004974092813976547902781036613642539688467647295672268879552182270E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 11 — Accumulated resistor work | 4.781016722529722319359328426247676211912388486610959723515070223887786E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 12 — Accumulated resistor work | 4.850321996066253546364355264172101857042692200159388446217581043101004E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 13 — Accumulated resistor work | 4.898371160756679304655148135336516616448234336993108252913976415742676E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 14 — Accumulated resistor work | 4.931405586820398257088155035469850794006226596277427217469630222766576E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 15 — Accumulated resistor work | 4.953949960739496931495841627737331227363076911364479281038785262194581E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 16 — Accumulated resistor work | 4.969234681910526756939951245075498880871744402601764027808064530429258E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 17 — Accumulated resistor work | 4.979536779697946777831895263327367477822035091843060519006780879377390E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 18 — Accumulated resistor work | 4.986443919914665831882669810193799201946316619531235451493375923201725E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 19 — Accumulated resistor work | 4.991052789203715732661421339922072547609768919659051918691250051726975E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 20 — Accumulated resistor work | 4.994114774875079540995577739765748100064366405072388577837383932827957E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 21 — Accumulated resistor work | 4.996141005970188787619474721177353198129776962446341977612978939019665E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 22 — Accumulated resistor work | 4.997476972828329416661314708227990262545316828918142984426007911000123E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 23 — Accumulated resistor work | 4.998354882472160802557705401539982783410823102374510218238904713394318E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 24 — Accumulated resistor work | 4.998930008863644631179255569299002738722792409511598467597847089146541E-2 | J | Stored approximation, 70 significant digits |
| Critical natural response: storage and loss — Resistor work — sample 25 — Accumulated resistor work | 4.999305704004073912297268899175794974848831689207417627413760758835956E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 1 — Physical time | 0 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 1 — Capacitor energy | 1/25 | J | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 2 — Capacitor energy | 2.992385559457909942746231221060506713544363974312405179620562013275307E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 3 — Capacitor energy | 2.064393045431960468955065188810635654065141174242006013083200030478721E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 4 — Capacitor energy | 1.305357501600870621030607789438748216780012642044430092195179054540267E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 5 — Capacitor energy | 7.450190157614420654856904191564768425097808415609292902782933472815918E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 6 — Capacitor energy | 3.718708734149060038305303185604036341494099176473458312844283400367538E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 7 — Capacitor energy | 1.516183741613705000011574197552243666890436877336562580200018926482968E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 8 — Capacitor energy | 4.209364311138242971483556388423061112279301455181641381308530433485385E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 9 — Capacitor energy | 3.326635370244845456587394490321400017417725944482363839697928349913683E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 10 — Capacitor energy | 2.990419922715519890521162502961153723142781725427576910938214730061850E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 11 — Capacitor energy | 1.822146329323597422833825086502105328908730065934827746120910457882672E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 12 — Capacitor energy | 3.505329665867018261832935214835389096291081606609460900947500078855339E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 13 — Capacitor energy | 4.659873201951925995160487188422680138373719151365172619838386536488714E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 14 — Capacitor energy | 5.084595833289069699826239820470820632691736058636955099109919434415044E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 15 — Capacitor energy | 4.862333785301483514687082672785863103737212868886819631089852207451209E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 16 — Capacitor energy | 4.201286408603339134385661077287075904733883931885122345749343102687339E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 17 — Capacitor energy | 3.328815334378049402908637655689480540685139990987500297086678522797356E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 18 — Capacitor energy | 2.432561022107997769787958470052352926827546805893063724862798631944514E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 19 — Capacitor energy | 1.637610889770314035138567900006793523116223063028035315779758930045527E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 20 — Capacitor energy | 1.007094665179389414880925386421314220791132867692181116680027720899588E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 21 — Capacitor energy | 5.551105840419670276133364464100790418346360281215518940573572182995639E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 22 — Capacitor energy | 2.637300766981909539279154400510607351308595914141729881609333894847050E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 23 — Capacitor energy | 9.888096611835281663078465048512461032904718725008981203942707267679111E-6 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 24 — Capacitor energy | 2.252113995209657109094336739426976862466130059040338247162674521371375E-6 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Capacitor energy — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Parallel response: its physical energy ledger — Capacitor energy — sample 25 — Capacitor energy | 4.093077029107471490655522645196316827764769961903635517596737039277050E-8 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 1 — Inductor energy | 49/2500 | J | Exact ratio |
| Parallel response: its physical energy ledger — Inductor energy — sample 2 — Inductor energy | 8.705393206278503681655427222522153419073209541413211848611887754293452E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 3 — Inductor energy | 2.883328915328609066329176854319624946519333266861594540748082589651426E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 4 — Inductor energy | 4.610201742131062934247147050761952897350103314954700341183586881048287E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 5 — Inductor energy | 1.263826511944891645323400218835793082785984948339492441991342196964110E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 6 — Inductor energy | 4.797893880479086024233713999215071706254050049129654502882928231147414E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 7 — Inductor energy | 1.177918300042013301062772350068264727105996500737667659434934800791487E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 8 — Inductor energy | 1.736869808174475661436004698997196255428935818077524018137752914285442E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 9 — Inductor energy | 2.014735079294900692917117407218158674860389860899738422802612534758216E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 10 — Inductor energy | 2.011770502588104405153482468605349663782536115757669408037025392858648E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 11 — Inductor energy | 1.799369958235623477780616297528090016448971966276780957567867032853438E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 12 — Inductor energy | 1.469757218632625196947332269749034562870086166887902466278997788209333E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 13 — Inductor energy | 1.105853851937881707086482075522931801772962306749374361324119088419615E-3 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 14 — Inductor energy | 7.674277706724783342341773062208942551012115496671335882413831297710139E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 15 — Inductor energy | 4.884752475042996584664735315308865033471387771664587184418293465659291E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 16 — Inductor energy | 2.810633144899291717652703210282444780325291593982077246602066793068386E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 17 — Inductor energy | 1.418718430809590617314297566542188092970048427298520023811004157507963E-4 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 18 — Inductor energy | 5.891013840209007800823988076752840855702994145669685876703291565417659E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 19 — Inductor energy | 1.701663446276726036682474235409575815012467493089165853537270639317770E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 20 — Inductor energy | 1.621816291614858480877861535117281208988612440672385503671256760359419E-6 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 21 — Inductor energy | 8.234324693534032136622045421991751294010848416239460075298447542816932E-7 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 22 — Inductor energy | 6.122488637166470349398756394810184066977093184264582287169608114079170E-6 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 23 — Inductor energy | 1.226587371101424372328674910670775808900582399832778341513783115220407E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 24 — Inductor energy | 1.661073346403423153544651103316887463816137977833920146489503812249550E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Inductor energy — sample 25 — Inductor energy | 1.833086443044381901061071846237413601929575463232882417049582437476908E-5 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 1 — Accumulated resistor work | 0 | J | Exact ratio |
| Parallel response: its physical energy ledger — Resistor work — sample 2 — Accumulated resistor work | 2.097075119914239689088226056687277944548315071546273635518249211295348E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 3 — Accumulated resistor work | 3.607274063035178624412017125757401851282925499071834532841991710556137E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 4 — Accumulated resistor work | 4.608540480977818749626920740053632254246486324806022904392985076649250E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 5 — Accumulated resistor work | 5.213717157726613042868986180624687364407433173490731217279715310521444E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 6 — Accumulated resistor work | 5.540150187780303135927132541447445648788049581861357623686742377651772E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 7 — Accumulated resistor work | 5.690589795834428169892565345237949160600356662192576976036504627272555E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 8 — Accumulated resistor work | 5.744219376071170004141563966216049763334313403640431184373139404236602E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 9 — Accumulated resistor work | 5.755199856700265085251700864787862732496543287965543793880040818174265E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 10 — Accumulated resistor work | 5.755832529818474039594130590636503879898603606698805482285359245984073E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 11 — Accumulated resistor work | 5.761841540883201677993600119382169945066015502712973626782004192135829E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 12 — Accumulated resistor work | 5.777970981478067297686937420876742652750080567245115144362625220390513E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 13 — Accumulated resistor work | 5.802815882786692569339746920563480018438966577811410837669204225793151E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 14 — Accumulated resistor work | 5.832411264599861469578319871173202368162961484446917090184762492678748E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 15 — Accumulated resistor work | 5.862529137396555199006481820119052718627913993594485931844918543268895E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 16 — Accumulated resistor work | 5.889880804464973691479616357124304793149408244741328004076485901042443E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 17 — Accumulated resistor work | 5.912524662348123599797770647777683313663448115817139796791023173196947E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 18 — Accumulated resistor work | 5.929783375938711014501296427222723629876021537795399676874668722115137E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 19 — Accumulated resistor work | 5.941922227656020133611931846764522488953825301876630480988665140060227E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 20 — Accumulated resistor work | 5.949766871719044620003102959982275129671189810079010950282832597114968E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 21 — Accumulated resistor work | 5.954366550912644989402500415081679292068713531234622086458673443341576E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 22 — Accumulated resistor work | 5.956750450369301443425780969960008374241993694767431811889673705293745E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 23 — Accumulated resistor work | 5.957784602967715047461363478584477978087808945727666323538091946158012E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 24 — Accumulated resistor work | 5.958113715254075611135545915222740414849937249016262046028794228735613E-2 | J | Stored approximation, 70 significant digits |
| Parallel response: its physical energy ledger — Resistor work — sample 25 — Accumulated resistor work | 5.958162820479926510627448272631117390081242659766805213947432820825484E-2 | J | Stored approximation, 70 significant digits |
| Original physical scales — L | 1/2 | H | Exact ratio |
| Original physical scales — C | 1/200 | F | Exact ratio |
| Original physical scales — voltage Scale | 4 | V | Exact ratio |
| Original physical scales — omega0 | 20 | rad/s | Exact ratio |
| Original physical scales — impedance Scale | 10 | ohm | Exact ratio |
| Original physical scales — current Scale | 2/5 | A | Exact ratio |
| Original physical scales — time Scale | 1/20 | s | Exact ratio |
| Original physical scales — storage Scale | 1/25 | J | Exact ratio |
Focus a graph or table region to scroll it locally. Axis ticks are rounded drawing labels; the numerical equivalents retain exact ratios, full stored approximations and qualified limits.
Retain initial terms in the causal response
The normalized linear equation also admits a causal input-response form. For ζ<1 the zero-state impulse-response kernel is h(u)=e^(−ζu)sin(du)/d for u≥0. It satisfies h(0)=0 and h′(0+)=1. Thus x(u)=x_zero-input(u)+∫₀ᵘh(u−s)f(s)ds, with the zero-input term chosen from the actual initial states. This is a mathematical kernel; it does not claim that every real source can deliver a physical impulse.
At critical damping h(u)=ue^(−u); above critical damping h(u)=[e^(s_+u)−e^(s_−u)]/(s_+−s_−). These expressions keep the second-order memory even when the input is not a single step. They also explain why the two-state pulse can be solved either by interval propagation or by superposed delayed zero-state steps. Superposition applies to voltages and currents, while their quadratic energies do not superpose in the same way.
With p the Laplace variable conjugate to dimensionless u, the unilateral transform gives (p²+2ζp+1)X(p)=𝓕(p)+(p+2ζ)x₀+j₀. The initial terms cannot be deleted unless the states are actually zero. Setting p to an imaginary frequency later extracts a particular sinusoidal response only where its denominator and physical domain permit it; it does not determine an undamped initial-value response or erase a natural component that never decays.
Find the domain of a sinusoidal response
Let the source be v_s=4 cos(ωt) V with ω>0 and R>0. A unique attracting sinusoidal response remains after the homogeneous part decays. Write r=ω/ω₀ and D=(1−r²)²+(2ζr)². Substitution of x=a cos(ru)+b sin(ru) gives a=(1−r²)/D and b=2ζr/D. These coefficients retain the sign of the in-phase component as the driving frequency crosses ω₀.
The normalized current is j=rb cos(ru)−ra sin(ru). Consequently the capacitor-voltage peak amplitude is 4/√D V, the current peak amplitude is (2/5)r/√D A and the inductor-voltage peak amplitude is 4r²/√D V. Peak amplitudes are positive magnitudes; the signed cosine and sine coefficients still determine the actual time-dependent voltages and current. In particular, adding positive amplitudes is not KVL.
For a lossless circuit the same nonresonant algebra gives a particular sinusoid, but its free component does not decay, so it is not an attracting response selected by waiting. At lossless resonance D=0 and this steady-sinusoid ansatz fails altogether. We solve that case at finite time later. A denominator or a formal complex substitution is useful only with its initial-value and existence conditions stated.
Locate current and resistor-power resonance
The source peak amplitude is fixed while frequency changes. In the ideal series circuit, I_peak=V_peak/√[R²+(ωL−1/(ωC))²]. The denominator is minimized when the inductive and capacitive reactances cancel: ω=ω₀. Thus current amplitude and mean resistor power both peak at ω₀ for every positive constant R, even when the natural roots are critically damped or overdamped.
At that frequency I_peak=4/R A and the current is in phase with the source. Capacitor and inductor voltages oppose one another and cancel in their sum, while their separate amplitudes can exceed the source amplitude. This is reactive energy exchange, not extra net average energy delivered without a source. The resistor consumes the source’s mean delivery once the periodic state is established.
The mean resistor power is R I_peak²/2; the factor one-half uses peak, not RMS, amplitudes. Relative to P_max=4²/(2R), its frequency dependence is P/P_max=(2ζr)²/D. The normalized current amplitude is 2ζr/√D. These two curves are different quantities even though their maxima share a location: half power corresponds to current amplitude 1/√2 of its resonant value, not one-half.
Locate capacitor-voltage resonance
Capacitor-voltage amplitude is proportional to 1/√D rather than r/√D. Differentiating D with respect to r gives 4r(r²+2ζ²−1). There is a positive-frequency interior maximum only when ζ<1/√2, at r_C=√(1−2ζ²). The damped free frequency r_d=√(1−ζ²) is different. Neither should be relabeled as the current-resonance frequency r=1.
For ζ=3/5, the capacitor-voltage maximum occurs at r_C=√7/5, while the free oscillation has r_d=4/5. Its peak voltage amplitude is 2/[ζ√(1−ζ²)]=25/6 V. At r=1 the capacitor amplitude is instead 4Q=10/3 V, where Q=5/6. An observable-specific maximum can therefore remain meaningful even when the quality factor is not large.
For ζ≥1/√2, the capacitor amplitude decreases from its DC boundary value of 4 V; there is no positive interior capacitor peak. Writing an imaginary value of √(1−2ζ²) into a frequency table would not locate a physical resonance. The r=0 boundary is a statement about a static voltage response, and cycle-averaged energies need a separate limiting argument because a nonzero-frequency cycle ceases to exist at exactly zero frequency.
Separate inductor peaks from a boundary supremum
Inductor-voltage amplitude is proportional to r²/√D. Differentiating its squared magnitude shows a finite interior maximum at r_L=1/√(1−2ζ²), again only when ζ<1/√2. In that domain r_Cr_L=1, and the two peak voltage magnitudes are equal even though they occur on opposite sides of ω₀. This reciprocal placement belongs to the stated ideal series model.
For ζ=3/5 the inductor peak is at r_L=5/√7, and its amplitude is 25/6 V. For ζ≥1/√2 the inductor amplitude instead rises toward the source amplitude of 4 V as frequency tends to infinity; it has no finite positive-frequency interior maximum. An asymptotic supremum is not an ordinary point at the right edge of a finite plot, and infinity is not a sampled frequency.
The high-frequency limit itself has a physical boundary. A real inductor has winding capacitance and frequency-dependent loss, while leads and propagation may cease to be negligible. The constant-L, constant-C, constant-R prediction describes this circuit idealization only. It cannot justify taking a real device to arbitrarily high frequency or interpreting every later parasitic resonance as the same single lumped mode.
| Observable | Normalized location | Domain |
|---|---|---|
| Current amplitude | 1 | R>0 |
| Mean resistor power | 1 | R>0, fixed source amplitude |
| Capacitor-voltage amplitude | √(1−2ζ²) | Interior only if ζ<1/√2; otherwise DC boundary |
| Inductor-voltage amplitude | 1/√(1−2ζ²) | Interior only if ζ<1/√2; otherwise high-frequency supremum |
| Free ringing | √(1−ζ²) | Oscillatory branch only if ζ<1 |
Derive the exact current-power bandwidth
Half maximum mean resistor power means (2ζr)²/D=1/2, so (1−r²)²=(2ζr)². The two positive solutions are r_−=√(1+ζ²)−ζ and r_+=√(1+ζ²)+ζ. Their product is exactly one and their difference is exactly 2ζ. They are symmetric on a logarithmic frequency axis, not generally at equal additive distances from r=1.
Therefore Δω=ω₀(r_+−r_−)=2ζω₀=R/L and Q=ω₀/Δω=ω₀L/R=1/(2ζ). These bandwidth identities are exact for current squared or resistor power in an ideal series RLC circuit driven by a frequency-independent voltage amplitude with constant R. No large-Q assumption was used in solving the half-power equation.
At ζ=3/5 the edges are (√34−3)/5 and (√34+3)/5 in normalized frequency, giving Δω=24 rad/s and Q=5/6. They must not be called the half-height edges of the capacitor-voltage squared response. If the source amplitude, load, resistance or effective reactances vary with frequency, the measured power curve changes and this exact elementary bandwidth relation need not survive unchanged.
Ask which observable is resonant
Use the same positive-frequency cases to compare capacitor voltage, inductor voltage, current and resistor power, with exact analytic peak and half-power controls.
(a) Capacitor-voltage peak depends on damping
- ζ=1/10
- ζ=3/5
- ζ=1
Additional hollow markers show the analytic controls described in the notes and numerical tables.
(b) Finite inductor peaks and a boundary supremum
- ζ=1/10
- ζ=3/5
- ζ=1
Additional hollow markers show the analytic controls described in the notes and numerical tables.
(c) Current peaks at r=1 for each positive R
- ζ=1/10
- ζ=3/5
- ζ=1
(d) Exact current-power half-power controls
- ζ=1/10
- ζ=3/5
- ζ=1
Additional hollow markers show the analytic controls described in the notes and numerical tables.
- Connected sample points illustrate the frequency response. Distinct analytic markers locate extrema or half-power crossings; a sampled maximum is not substituted for an analytic one.
- For ζ=1 the capacitor-voltage maximum is at the DC boundary r=0. Its inductor-voltage response has no finite interior maximum: 4 V is approached only as r tends to infinity. No infinity is plotted as a finite point.
- Current and resistor-power maxima are at r=1 for these R>0 cases. The exact current-power half-power width is R/L in angular-frequency units; the displacement/charge Lorentzian approximation has narrower conditions.
Observable-specific frequencies in the ideal series circuit
Energy and Q statements with their actual conditions
Ask which observable is resonant — values and conditions
| Quantity and original case | Value | Unit | Meaning and precision |
|---|---|---|---|
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 1 — Frequency ratio ω/ω₀ | 1/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 1 — Peak capacitor voltage | 4.260611433292060370397412677353799680956748166263272727273143503300623E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 2 — Frequency ratio ω/ω₀ | 1/2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 2 — Peak capacitor voltage | 5.286548803640718227839951807630066449510873531476670806039092748640283E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 3 — Frequency ratio ω/ω₀ | 3/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 3 — Peak capacitor voltage | 320/37 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 4 — Frequency ratio ω/ω₀ | 4/5 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 4 — Peak capacitor voltage | 1.015346165133619043478990867517280874401663534787913832136407683488293E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 5 — Frequency ratio ω/ω₀ | 33/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 5 — Peak capacitor voltage | 1.112720436572671623894604124584495751291204534243693416718817196809016E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 6 — Frequency ratio ω/ω₀ | 17/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 6 — Peak capacitor voltage | 1.229133897784931149323610672714770160953971283556430211260599971379853E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 7 — Frequency ratio ω/ω₀ | 7/8 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 7 — Peak capacitor voltage | 1.367518245679275421846359060369369018689033003634881561081501002605510E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 8 — Frequency ratio ω/ω₀ | 9/10 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 8 — Peak capacitor voltage | 1.528321439801740212954591343234422127659732521559369232478768618084268E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 9 — Frequency ratio ω/ω₀ | 37/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 9 — Peak capacitor voltage | 1.704534680134064365361430614251610552078706130736968718760309167238310E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 10 — Frequency ratio ω/ω₀ | 19/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 10 — Peak capacitor voltage | 1.873043266591944321297617514765419962251997150217136512521891577621553E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 11 — Frequency ratio ω/ω₀ | 39/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 11 — Peak capacitor voltage | 1.988527213500938735337914613560599076975555784927640140059654808121628E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 12 — Frequency ratio ω/ω₀ | 1 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 12 — Peak capacitor voltage | 20 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 13 — Frequency ratio ω/ω₀ | 41/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 13 — Peak capacitor voltage | 1.894312075651197506178182019507065778246044952091760794046494287824866E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 14 — Frequency ratio ω/ω₀ | 21/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 14 — Peak capacitor voltage | 1.711744002151045984214635639254422483743747715831937485475173840100256E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 15 — Frequency ratio ω/ω₀ | 43/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 15 — Peak capacitor voltage | 1.507084327861209036290729547096789598145878871797033997951791643018561E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 16 — Frequency ratio ω/ω₀ | 11/10 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 16 — Peak capacitor voltage | 1.315191898442858311135066863827473959369723263737602418685222280321369E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 17 — Frequency ratio ω/ω₀ | 9/8 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 17 — Peak capacitor voltage | 1.149057012112268624529900811662546924403524406389956861084669088074090E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 18 — Frequency ratio ω/ω₀ | 23/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 18 — Peak capacitor voltage | 1.009810471232692748731750813235457057616355722273681888975205444720029E+1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 19 — Frequency ratio ω/ω₀ | 47/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 19 — Peak capacitor voltage | 8.942027952117047926084516935318904795156374552553577018136857791548436E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 20 — Frequency ratio ω/ω₀ | 6/5 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 20 — Peak capacitor voltage | 7.980868844676221521334627271180169338230341626949509340758810755152099E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 21 — Frequency ratio ω/ω₀ | 5/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 21 — Peak capacitor voltage | 6.498215456855161878265541552110597596170646622642648525673009174325074E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 22 — Frequency ratio ω/ω₀ | 3/2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 22 — Peak capacitor voltage | 3.111639366337655905988102893239983419633954138473246221165076373521696E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 23 — Frequency ratio ω/ω₀ | 2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 23 — Peak capacitor voltage | 1.321637200910179556959987951907516612377718382869167701509773187160071E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 24 — Frequency ratio ω/ω₀ | 3 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 24 — Peak capacitor voltage | 4.985996549442281937290031568987664464133367594524356343066307981073762E-1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 25 — Frequency ratio ω/ω₀ | 4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1/10 — sample 25 — Peak capacitor voltage | 2.662882145807537731498382923346124800597967603914545454545714689562889E-1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 1 — Frequency ratio ω/ω₀ | 1/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 1 — Peak capacitor voltage | 4.063676361383783187440455853235104758848957474362548646827933679306640E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 2 — Frequency ratio ω/ω₀ | 1/2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 2 — Peak capacitor voltage | 4.164633650362828413976076536339390741151492769184402527173510651491501E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 3 — Frequency ratio ω/ω₀ | 3/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 3 — Peak capacitor voltage | 3.997190462836722745417749698778987793244311917594914079575340694303200E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 4 — Frequency ratio ω/ω₀ | 4/5 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 4 — Peak capacitor voltage | 3.901371573204352131448241244858313440454089191567708786259658106745045E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 5 — Frequency ratio ω/ω₀ | 33/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 5 — Peak capacitor voltage | 3.845264463852253592973351644083360049971427280488044951422907758700426E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 6 — Frequency ratio ω/ω₀ | 17/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 6 — Peak capacitor voltage | 3.784028888096306136487344785983621620858239468634460618515883354082426E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 7 — Frequency ratio ω/ω₀ | 7/8 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 7 — Peak capacitor voltage | 3.718025089487700474071237440488642809576408168072582329274403271380198E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 8 — Frequency ratio ω/ω₀ | 9/10 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 8 — Peak capacitor voltage | 3.647686020700425582793858078358014744311350363118877786657909233228796E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 9 — Frequency ratio ω/ω₀ | 37/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 9 — Peak capacitor voltage | 3.573502881082972081028807229050777870500008314304638285597877122502181E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 10 — Frequency ratio ω/ω₀ | 19/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 10 — Peak capacitor voltage | 3.496009009532727546218471525998157519997625204344352249622582785878075E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 11 — Frequency ratio ω/ω₀ | 39/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 11 — Peak capacitor voltage | 3.415763184741494480543568768875708991961755345458451090855561067076928E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 12 — Frequency ratio ω/ω₀ | 1 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 12 — Peak capacitor voltage | 10/3 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 13 — Frequency ratio ω/ω₀ | 41/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 13 — Peak capacitor voltage | 3.249281502639270173405417184671241761332903896167125844039993061063292E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 14 — Frequency ratio ω/ω₀ | 21/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 14 — Peak capacitor voltage | 3.164150749101912570309302311154496569122335417482803539425294848797359E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 15 — Frequency ratio ω/ω₀ | 43/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 15 — Peak capacitor voltage | 3.078454359800865189913508449399025900618115745560240063481384079105618E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 16 — Frequency ratio ω/ω₀ | 11/10 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 16 — Peak capacitor voltage | 2.992667594954890248460690867967203744075761323599760335972041559921190E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 17 — Frequency ratio ω/ω₀ | 9/8 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 17 — Peak capacitor voltage | 2.907221937886392944331974867986946785460365339379700520057954450278482E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 18 — Frequency ratio ω/ω₀ | 23/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 18 — Peak capacitor voltage | 2.822501680873983806504078615882352189448381792966390019237152226048563E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 19 — Frequency ratio ω/ω₀ | 47/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 19 — Peak capacitor voltage | 2.738842566397270202921281888165583523731032007591692059139309236749387E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 20 — Frequency ratio ω/ω₀ | 6/5 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 20 — Peak capacitor voltage | 2.656532141809274770543010597039809437372616074958414260383898029343336E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 21 — Frequency ratio ω/ω₀ | 5/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 21 — Peak capacitor voltage | 2.496877806850785364126874396709320601890617082603333623206181188316829E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 22 — Frequency ratio ω/ω₀ | 3/2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 22 — Peak capacitor voltage | 1.825266590385101564347716433032582859482720064975806259517506935341422E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 23 — Frequency ratio ω/ω₀ | 2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 23 — Peak capacitor voltage | 1.041158412590707103494019134084847685287873192296100631793377662872875E+0 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 24 — Frequency ratio ω/ω₀ | 3 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 24 — Peak capacitor voltage | 4.559607525875531978492322597947518430389187953898597233322386541535995E-1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 25 — Frequency ratio ω/ω₀ | 4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=3/5 — sample 25 — Peak capacitor voltage | 2.539797725864864492150284908271940474280598421476592904267458549566650E-1 | V | Stored approximation, 70 significant digits |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 1 — Frequency ratio ω/ω₀ | 1/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 1 — Peak capacitor voltage | 64/17 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 2 — Frequency ratio ω/ω₀ | 1/2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 2 — Peak capacitor voltage | 16/5 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 3 — Frequency ratio ω/ω₀ | 3/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 3 — Peak capacitor voltage | 64/25 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 4 — Frequency ratio ω/ω₀ | 4/5 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 4 — Peak capacitor voltage | 100/41 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 5 — Frequency ratio ω/ω₀ | 33/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 5 — Peak capacitor voltage | 6400/2689 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 6 — Frequency ratio ω/ω₀ | 17/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 6 — Peak capacitor voltage | 1600/689 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 7 — Frequency ratio ω/ω₀ | 7/8 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 7 — Peak capacitor voltage | 256/113 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 8 — Frequency ratio ω/ω₀ | 9/10 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 8 — Peak capacitor voltage | 400/181 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 9 — Frequency ratio ω/ω₀ | 37/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 9 — Peak capacitor voltage | 6400/2969 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 10 — Frequency ratio ω/ω₀ | 19/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 10 — Peak capacitor voltage | 1600/761 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 11 — Frequency ratio ω/ω₀ | 39/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 11 — Peak capacitor voltage | 6400/3121 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 12 — Frequency ratio ω/ω₀ | 1 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 12 — Peak capacitor voltage | 2 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 13 — Frequency ratio ω/ω₀ | 41/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 13 — Peak capacitor voltage | 6400/3281 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 14 — Frequency ratio ω/ω₀ | 21/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 14 — Peak capacitor voltage | 1600/841 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 15 — Frequency ratio ω/ω₀ | 43/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 15 — Peak capacitor voltage | 6400/3449 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 16 — Frequency ratio ω/ω₀ | 11/10 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 16 — Peak capacitor voltage | 400/221 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 17 — Frequency ratio ω/ω₀ | 9/8 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 17 — Peak capacitor voltage | 256/145 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 18 — Frequency ratio ω/ω₀ | 23/20 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 18 — Peak capacitor voltage | 1600/929 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 19 — Frequency ratio ω/ω₀ | 47/40 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 19 — Peak capacitor voltage | 6400/3809 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 20 — Frequency ratio ω/ω₀ | 6/5 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 20 — Peak capacitor voltage | 100/61 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 21 — Frequency ratio ω/ω₀ | 5/4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 21 — Peak capacitor voltage | 64/41 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 22 — Frequency ratio ω/ω₀ | 3/2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 22 — Peak capacitor voltage | 16/13 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 23 — Frequency ratio ω/ω₀ | 2 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 23 — Peak capacitor voltage | 4/5 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 24 — Frequency ratio ω/ω₀ | 3 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 24 — Peak capacitor voltage | 2/5 | V | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 25 — Frequency ratio ω/ω₀ | 4 | 1 | Exact ratio |
| Capacitor-voltage peak depends on damping — ζ=1 — sample 25 — Peak capacitor voltage | 4/17 | V | Exact ratio |
| Capacitor maximum ζ=1/10 — x coordinate | 9.899494936611665341611821069467886549987703127638636512236758165935127E-1 | 1 | Stored approximation, 70 significant digits |
| Capacitor maximum ζ=1/10 — y coordinate | 2.010075630518424150978747113133749505410356694296577937611322512797869E+1 | V | Stored approximation, 70 significant digits |
| Capacitor maximum ζ=3/5 — x coordinate | 5.291502622129181181003231507278520851420518366164900360736668918402138E-1 | 1 | Stored approximation, 70 significant digits |
| Capacitor maximum ζ=3/5 — y coordinate | 25/6 | V | Exact ratio |
| Capacitor maximum ζ=1 — x coordinate | 0 | 1 | Exact ratio |
| Capacitor maximum ζ=1 — y coordinate | 4 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 1 — Peak inductor voltage | 2.662882145807537731498382923346124800597967603914545454545714689562889E-1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 2 — Peak inductor voltage | 1.321637200910179556959987951907516612377718382869167701509773187160071E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 3 — Peak inductor voltage | 180/37 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 4 — Peak inductor voltage | 6.498215456855161878265541552110597596170646622642648525673009174325074E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 5 — Peak inductor voltage | 7.573453471422746240132649322953224207225760861196138317542449545781362E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 6 — Peak inductor voltage | 8.880492411496127553863087110364214412892442523695208276357834793219436E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 7 — Peak inductor voltage | 1.047006156848195244851118655595298154933790893407956195203024205119843E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 8 — Peak inductor voltage | 1.237940366239409572493218988019881923404383342463089078307802580648257E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 9 — Peak inductor voltage | 1.458442485689708822612374069319034278622342933111818859989289531218279E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 10 — Peak inductor voltage | 1.690421548099229749971099807075791515932427428070965702551007148803452E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 11 — Peak inductor voltage | 1.890343682334329885280605079516044497549887718046837908144209351970622E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 12 — Peak inductor voltage | 20 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 13 — Peak inductor voltage | 1.990211624481039379928452484244610983269750977791406184245098061146000E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 14 — Peak inductor voltage | 1.887197762371528197596635792278000788327481856704711077736379158710532E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 15 — Peak inductor voltage | 1.741624326384609692563474332863727479357331271220447413883039217463324E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 16 — Peak inductor voltage | 1.591382197115858556473430905231243490837365149122498926609118959188857E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 17 — Peak inductor voltage | 1.454275280954589977920655714760410951198210576837289152310284314593770E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 18 — Peak inductor voltage | 1.335474348205236160197740450503891958697630442706944298169709200642239E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 19 — Peak inductor voltage | 1.234558734139159929295043619382466293281276961661928227066519928845656E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 20 — Peak inductor voltage | 1.149245113633375899072186327049944384705169194280729345069268748741902E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 21 — Peak inductor voltage | 1.015346165133619043478990867517280874401663534787913832136407683488293E+1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 22 — Peak inductor voltage | 7.001188574259725788473231509789962694176396811564803997621421840423816E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 23 — Peak inductor voltage | 5.286548803640718227839951807630066449510873531476670806039092748640283E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 24 — Peak inductor voltage | 4.487396894498053743561028412088898017720030835071920708759677182966386E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1/10 — sample 25 — Peak inductor voltage | 4.260611433292060370397412677353799680956748166263272727273143503300623E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 1 — Peak inductor voltage | 2.539797725864864492150284908271940474280598421476592904267458549566650E-1 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 2 — Peak inductor voltage | 1.041158412590707103494019134084847685287873192296100631793377662872875E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 3 — Peak inductor voltage | 2.248419635345656544297484205563180633699925453647139169761129140545550E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 4 — Peak inductor voltage | 2.496877806850785364126874396709320601890617082603333623206181188316829E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 5 — Peak inductor voltage | 2.617183125709440101717487462754236934011802692782175595062216593265477E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 6 — Peak inductor voltage | 2.733960871649581183612106607873166621070078016088397796877725723324553E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 7 — Peak inductor voltage | 2.846612959139020675460791165374117151081937503680570845850715004650464E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 8 — Peak inductor voltage | 2.954625676767344722063025043469991942892193794126291007192906478915325E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 9 — Peak inductor voltage | 3.057578402626617986830273185356571815446569613926906133114683612940928E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 10 — Peak inductor voltage | 3.155148131103286610462170552213337161797856746920777905284380964254962E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 11 — Peak inductor voltage | 3.247109877494883190566730060912470860483643675276440068244567739390005E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 12 — Peak inductor voltage | 10/3 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 13 — Peak inductor voltage | 3.413776378710383225934066429645223375500382155910586589894517709779621E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 14 — Peak inductor voltage | 3.488476200884858608766005798047832467457374797774790902216387570799089E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 15 — Peak inductor voltage | 3.557538819544874835093798201836749306401810008463052423360674476416429E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 16 — Peak inductor voltage | 3.621127789895417200637435950240316530331671201555710006526170287504640E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 17 — Peak inductor voltage | 3.679452765137466070170155692295979525348274882652433470698348601133703E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 18 — Peak inductor voltage | 3.732758472955843584101643969504410770545484921198050800441133818949224E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 19 — Peak inductor voltage | 3.781314518232231173908194806848608752451156065481279849149208814987123E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 20 — Peak inductor voltage | 3.825406284205355669581935259737325589816567147940116534952813162254404E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 21 — Peak inductor voltage | 3.901371573204352131448241244858313440454089191567708786259658106745045E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 22 — Peak inductor voltage | 4.106849828366478519782361974323311433836120146195564083914390604518198E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 23 — Peak inductor voltage | 4.164633650362828413976076536339390741151492769184402527173510651491501E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 24 — Peak inductor voltage | 4.103646773287978780643090338152766587350269158508737509990147887382396E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=3/5 — sample 25 — Peak inductor voltage | 4.063676361383783187440455853235104758848957474362548646827933679306640E+0 | V | Stored approximation, 70 significant digits |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 1 — Peak inductor voltage | 4/17 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 2 — Peak inductor voltage | 4/5 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 3 — Peak inductor voltage | 36/25 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 4 — Peak inductor voltage | 64/41 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 5 — Peak inductor voltage | 4356/2689 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 6 — Peak inductor voltage | 1156/689 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 7 — Peak inductor voltage | 196/113 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 8 — Peak inductor voltage | 324/181 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 9 — Peak inductor voltage | 5476/2969 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 10 — Peak inductor voltage | 1444/761 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 11 — Peak inductor voltage | 6084/3121 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 12 — Peak inductor voltage | 2 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 13 — Peak inductor voltage | 6724/3281 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 14 — Peak inductor voltage | 1764/841 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 15 — Peak inductor voltage | 7396/3449 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 16 — Peak inductor voltage | 484/221 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 17 — Peak inductor voltage | 324/145 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 18 — Peak inductor voltage | 2116/929 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 19 — Peak inductor voltage | 8836/3809 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 20 — Peak inductor voltage | 144/61 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 21 — Peak inductor voltage | 100/41 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 22 — Peak inductor voltage | 36/13 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 23 — Peak inductor voltage | 16/5 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 24 — Peak inductor voltage | 18/5 | V | Exact ratio |
| Finite inductor peaks and a boundary supremum — ζ=1 — sample 25 — Peak inductor voltage | 64/17 | V | Exact ratio |
| Interior inductor maximum ζ=1/10 — x coordinate | 1.010152544552210749144063374435498627549765625269248623697628384279095E+0 | 1 | Stored approximation, 70 significant digits |
| Interior inductor maximum ζ=1/10 — y coordinate | 2.010075630518424150978747113133749505410356694296577937611322512797869E+1 | V | Stored approximation, 70 significant digits |
| Interior inductor maximum ζ=3/5 — x coordinate | 1.889822365046136136072582681170900304078756559344607271691667470857906E+0 | 1 | Stored approximation, 70 significant digits |
| Interior inductor maximum ζ=3/5 — y coordinate | 25/6 | V | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 1 — Current / resonant current | 5.325764291615075462996765846692249601195935207829090909091429379125779E-2 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 2 — Current / resonant current | 1.321637200910179556959987951907516612377718382869167701509773187160071E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 3 — Current / resonant current | 12/37 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 4 — Current / resonant current | 4.061384660534476173915963470069123497606654139151655328545630733953171E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 5 — Current / resonant current | 4.589971800862270448565242013911044974076218703755235343965120936837189E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 6 — Current / resonant current | 5.223819065585957384625345359037773184054377955114828397857549878364374E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 7 — Current / resonant current | 5.982892324846829970577820889115989456764519390902606829731566886399105E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 8 — Current / resonant current | 6.877446479107830958295661044554899574468796347017161546154458781379207E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 9 — Current / resonant current | 7.883472895620047689796616590913698803364015854658480324266429898477182E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 10 — Current / resonant current | 8.896955516311735526163683195135744820696986463531398434478984993702378E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 11 — Current / resonant current | 9.694070165817076334772333741107920500255834451522245682790817189592936E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 12 — Current / resonant current | 1 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 13 — Current / resonant current | 9.708349387712387219163182849973712113510980379470274069488283225102439E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 14 — Current / resonant current | 8.986656011292991417126837106085718039654675508117671798744662660526345E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 15 — Current / resonant current | 8.100578262253998570062671315645244090034098935909057738990880081224763E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 16 — Current / resonant current | 7.233555441435720711242867751051106776533477950556813302768722541767531E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 17 — Current / resonant current | 6.463445693131511012980692065601826449769824785943507343601263620416757E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 18 — Current / resonant current | 5.806410209587983305207567176103878081294045403073670861607431307140169E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 19 — Current / resonant current | 5.253441421868765656574653699499856567154370049625226498155403952534706E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 20 — Current / resonant current | 4.788521306805732912800776362708101602938204976169705604455286453091260E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 21 — Current / resonant current | 4.061384660534476173915963470069123497606654139151655328545630733953171E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 22 — Current / resonant current | 2.333729524753241929491077169929987564725465603854934665873807280141272E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 23 — Current / resonant current | 1.321637200910179556959987951907516612377718382869167701509773187160071E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 24 — Current / resonant current | 7.478994824163422905935047353481496696200051391786534514599461971610643E-2 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1/10 — sample 25 — Current / resonant current | 5.325764291615075462996765846692249601195935207829090909091429379125779E-2 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 1 — Current / resonant current | 3.047757271037837390580341889926328569136718105771911485120950259479980E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 2 — Current / resonant current | 6.246950475544242620964114804509086111727239153776603790760265977237251E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 3 — Current / resonant current | 8.993678541382626177189936822252722534799701814588556679044516562182200E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 4 — Current / resonant current | 9.363291775690445115475778987659952257089814059762501087023179456188108E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 5 — Current / resonant current | 9.517029548034327642609045319106316123679282519207911254771696702783554E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 6 — Current / resonant current | 9.649273664645580648042729204258235133188510645017874577215502552910185E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 7 — Current / resonant current | 9.759815859905213744436998281282687375138071441190528614345308587373021E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 8 — Current / resonant current | 9.848752255891149073543416811566639809640645980420970023976354929717750E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 9 — Current / resonant current | 9.916470495005247524854940060615908590637523072195371242534109014943552E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 10 — Current / resonant current | 9.963625677168273506722643849094748931993231832381403911424360939752513E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 11 — Current / resonant current | 9.991107315368871355589938648961448801488134385465969440752516121200014E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 12 — Current / resonant current | 1 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 13 — Current / resonant current | 9.991540620615755783221657842864068416098679480713911970422978662769622E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 14 — Current / resonant current | 9.967074859671024596474302280136664192735356565070831149189678773711682E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 15 — Current / resonant current | 9.928015310357790237471064749311858529493423279431774204727463655115617E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 16 — Current / resonant current | 9.875803063351137819920279864291772355450012367879209108707737147739926E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 17 — Current / resonant current | 9.811874040366576187120415179455945400928733020406489255195596269689875E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 18 — Current / resonant current | 9.737630799015244132439071224794115053596917185734045566368175179867542E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 19 — Current / resonant current | 9.654420046550377465297518655783681921151887826760714508466065059541590E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 20 — Current / resonant current | 9.563515710513389173954838149343313974541417869850291337382032905636010E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 21 — Current / resonant current | 9.363291775690445115475778987659952257089814059762501087023179456188108E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 22 — Current / resonant current | 8.213699656732957039564723948646622867672240292391128167828781209036397E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 23 — Current / resonant current | 6.246950475544242620964114804509086111727239153776603790760265977237251E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 24 — Current / resonant current | 4.103646773287978780643090338152766587350269158508737509990147887382396E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=3/5 — sample 25 — Current / resonant current | 3.047757271037837390580341889926328569136718105771911485120950259479980E-1 | 1 | Stored approximation, 70 significant digits |
| Current peaks at r=1 for each positive R — ζ=1 — sample 1 — Current / resonant current | 8/17 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 2 — Current / resonant current | 4/5 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 3 — Current / resonant current | 24/25 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 4 — Current / resonant current | 40/41 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 5 — Current / resonant current | 2640/2689 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 6 — Current / resonant current | 680/689 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 7 — Current / resonant current | 112/113 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 8 — Current / resonant current | 180/181 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 9 — Current / resonant current | 2960/2969 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 10 — Current / resonant current | 760/761 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 11 — Current / resonant current | 3120/3121 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 12 — Current / resonant current | 1 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 13 — Current / resonant current | 3280/3281 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 14 — Current / resonant current | 840/841 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 15 — Current / resonant current | 3440/3449 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 16 — Current / resonant current | 220/221 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 17 — Current / resonant current | 144/145 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 18 — Current / resonant current | 920/929 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 19 — Current / resonant current | 3760/3809 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 20 — Current / resonant current | 60/61 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 21 — Current / resonant current | 40/41 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 22 — Current / resonant current | 12/13 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 23 — Current / resonant current | 4/5 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 24 — Current / resonant current | 3/5 | 1 | Exact ratio |
| Current peaks at r=1 for each positive R — ζ=1 — sample 25 — Current / resonant current | 8/17 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 1 — Mean power / peak power | 16/5641 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 2 — Mean power / peak power | 4/229 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 3 — Mean power / peak power | 144/1369 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 4 — Mean power / peak power | 16/97 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 5 — Mean power / peak power | 69696/330817 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 6 — Mean power / peak power | 4624/16945 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 7 — Mean power / peak power | 3136/8761 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 8 — Mean power / peak power | 324/685 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 9 — Mean power / peak power | 87616/140977 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 10 — Mean power / peak power | 5776/7297 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 11 — Mean power / peak power | 97344/103585 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 12 — Mean power / peak power | 1 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 13 — Mean power / peak power | 107584/114145 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 14 — Mean power / peak power | 7056/8737 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 15 — Mean power / peak power | 118336/180337 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 16 — Mean power / peak power | 484/925 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 17 — Mean power / peak power | 5184/12409 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 18 — Mean power / peak power | 8464/25105 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 19 — Mean power / peak power | 141376/512257 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 20 — Mean power / peak power | 36/157 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 21 — Mean power / peak power | 16/97 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 22 — Mean power / peak power | 36/661 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 23 — Mean power / peak power | 4/229 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 24 — Mean power / peak power | 9/1609 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1/10 — sample 25 — Mean power / peak power | 16/5641 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 1 — Mean power / peak power | 64/689 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 2 — Mean power / peak power | 16/41 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 3 — Mean power / peak power | 5184/6409 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 4 — Mean power / peak power | 64/73 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 5 — Mean power / peak power | 2509056/2770177 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 6 — Mean power / peak power | 18496/19865 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 7 — Mean power / peak power | 12544/13169 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 8 — Mean power / peak power | 11664/12025 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 9 — Mean power / peak power | 350464/356393 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 10 — Mean power / peak power | 23104/23273 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 11 — Mean power / peak power | 3504384/3510625 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 12 — Mean power / peak power | 1 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 13 — Mean power / peak power | 430336/431065 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 14 — Mean power / peak power | 254016/255697 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 15 — Mean power / peak power | 473344/480233 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 16 — Mean power / peak power | 1936/1985 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 17 — Mean power / peak power | 186624/193849 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 18 — Mean power / peak power | 33856/35705 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 19 — Mean power / peak power | 565504/606713 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 20 — Mean power / peak power | 1296/1417 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 21 — Mean power / peak power | 64/73 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 22 — Mean power / peak power | 1296/1921 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 23 — Mean power / peak power | 16/41 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 24 — Mean power / peak power | 81/481 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=3/5 — sample 25 — Mean power / peak power | 64/689 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 1 — Mean power / peak power | 64/289 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 2 — Mean power / peak power | 16/25 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 3 — Mean power / peak power | 576/625 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 4 — Mean power / peak power | 1600/1681 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 5 — Mean power / peak power | 6969600/7230721 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 6 — Mean power / peak power | 462400/474721 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 7 — Mean power / peak power | 12544/12769 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 8 — Mean power / peak power | 32400/32761 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 9 — Mean power / peak power | 8761600/8814961 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 10 — Mean power / peak power | 577600/579121 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 11 — Mean power / peak power | 9734400/9740641 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 12 — Mean power / peak power | 1 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 13 — Mean power / peak power | 10758400/10764961 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 14 — Mean power / peak power | 705600/707281 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 15 — Mean power / peak power | 11833600/11895601 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 16 — Mean power / peak power | 48400/48841 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 17 — Mean power / peak power | 20736/21025 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 18 — Mean power / peak power | 846400/863041 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 19 — Mean power / peak power | 14137600/14508481 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 20 — Mean power / peak power | 3600/3721 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 21 — Mean power / peak power | 1600/1681 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 22 — Mean power / peak power | 144/169 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 23 — Mean power / peak power | 16/25 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 24 — Mean power / peak power | 9/25 | 1 | Exact ratio |
| Exact current-power half-power controls — ζ=1 — sample 25 — Mean power / peak power | 64/289 | 1 | Exact ratio |
| Lower half-power ζ=1/10 — x coordinate | 9.049875621120890270219264912759576186945023470026377290572828297328491E-1 | 1 | Stored approximation, 70 significant digits |
| Lower half-power ζ=1/10 — y coordinate | 1/2 | 1 | Exact ratio |
| Upper half-power ζ=1/10 — x coordinate | 1.104987562112089027021926491275957618694502347002637729057282829732849E+0 | 1 | Stored approximation, 70 significant digits |
| Upper half-power ζ=1/10 — y coordinate | 1/2 | 1 | Exact ratio |
| Lower half-power ζ=3/5 — x coordinate | 5.661903789690600941748305755091166153042796669771943908900013489735620E-1 | 1 | Stored approximation, 70 significant digits |
| Lower half-power ζ=3/5 — y coordinate | 1/2 | 1 | Exact ratio |
| Upper half-power ζ=3/5 — x coordinate | 1.766190378969060094174830575509116615304279666977194390890001348973562E+0 | 1 | Stored approximation, 70 significant digits |
| Upper half-power ζ=3/5 — y coordinate | 1/2 | 1 | Exact ratio |
| Lower half-power ζ=1 — x coordinate | 4.142135623730950488016887242096980785696718753769480731766797379907325E-1 | 1 | Stored approximation, 70 significant digits |
| Lower half-power ζ=1 — y coordinate | 1/2 | 1 | Exact ratio |
| Upper half-power ζ=1 — x coordinate | 2.414213562373095048801688724209698078569671875376948073176679737990732E+0 | 1 | Stored approximation, 70 significant digits |
| Upper half-power ζ=1 — y coordinate | 1/2 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — zeta | 1/10 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — Q | 5 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — alpha | 2 | 1/s | Exact ratio |
| Analytic frequency controls — case 1 — omega0 | 20 | rad/s | Exact ratio |
| Analytic frequency controls — case 1 — half Power Width | 1/5 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — half Power Product | 1 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — current Power Peak Location | 1 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — driven Resonance Stored Energy | 1 | J | Exact ratio |
| Analytic frequency controls — case 1 — driven Resonance Mean Power | 4 | W | Exact ratio |
| Analytic frequency controls — case 1 — omega0 Energy Over Mean Power | 5 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — half Power Lower Omega | 1.809975124224178054043852982551915237389004694005275458114565659465698E+1 | rad/s | Stored approximation, 70 significant digits |
| Analytic frequency controls — case 1 — half Power Upper Omega | 2.209975124224178054043852982551915237389004694005275458114565659465698E+1 | rad/s | Stored approximation, 70 significant digits |
| Analytic frequency controls — case 1 — bandwidth Omega | 4 | rad/s | Exact ratio |
| Analytic frequency controls — case 1 — Q From Bandwidth | 5 | 1 | Exact ratio |
| Analytic frequency controls — case 1 — damped Omega Or Null | 1.989974874213239909468959642002412010356253127353612158235209287669891E+1 | rad/s | Stored approximation, 70 significant digits |
| Analytic frequency controls — case 1 — peak Status | both have finite positive-frequency interior peaks | Declared model condition | |
| Analytic frequency controls — case 2 — zeta | 3/5 | 1 | Exact ratio |
| Analytic frequency controls — case 2 — Q | 5/6 | 1 | Exact ratio |
| Analytic frequency controls — case 2 — alpha | 12 | 1/s | Exact ratio |
| Analytic frequency controls — case 2 — omega0 | 20 | rad/s | Exact ratio |
| Analytic frequency controls — case 2 — half Power Width | 6/5 | 1 | Exact ratio |
| Analytic frequency controls — case 2 — half Power Product | 1 | 1 | Exact ratio |
| Analytic frequency controls — case 2 — current Power Peak Location | 1 | 1 | Exact ratio |
| Analytic frequency controls — case 2 — driven Resonance Stored Energy | 1/36 | J | Exact ratio |
| Analytic frequency controls — case 2 — driven Resonance Mean Power | 2/3 | W | Exact ratio |
| Analytic frequency controls — case 2 — omega0 Energy Over Mean Power | 5/6 | 1 | Exact ratio |
| Analytic frequency controls — case 2 — half Power Lower Omega | 1.132380757938120188349661151018233230608559333954388781780002697947124E+1 | rad/s | Stored approximation, 70 significant digits |
| Analytic frequency controls — case 2 — half Power Upper Omega | 3.532380757938120188349661151018233230608559333954388781780002697947124E+1 | rad/s | Stored approximation, 70 significant digits |
| Analytic frequency controls — case 2 — bandwidth Omega | 24 | rad/s | Exact ratio |
| Analytic frequency controls — case 2 — Q From Bandwidth | 5/6 | 1 | Exact ratio |
| Analytic frequency controls — case 2 — damped Omega Or Null | 16 | rad/s | Exact ratio |
| Analytic frequency controls — case 2 — peak Status | both have finite positive-frequency interior peaks | Declared model condition | |
| Analytic frequency controls — case 3 — zeta | 1 | 1 | Exact ratio |
| Analytic frequency controls — case 3 — Q | 1/2 | 1 | Exact ratio |
| Analytic frequency controls — case 3 — alpha | 20 | 1/s | Exact ratio |
| Analytic frequency controls — case 3 — omega0 | 20 | rad/s | Exact ratio |
| Analytic frequency controls — case 3 — half Power Width | 2 | 1 | Exact ratio |
| Analytic frequency controls — case 3 — half Power Product | 1 | 1 | Exact ratio |
| Analytic frequency controls — case 3 — current Power Peak Location | 1 | 1 | Exact ratio |
| Analytic frequency controls — case 3 — driven Resonance Stored Energy | 1/100 | J | Exact ratio |
| Analytic frequency controls — case 3 — driven Resonance Mean Power | 2/5 | W | Exact ratio |
| Analytic frequency controls — case 3 — omega0 Energy Over Mean Power | 1/2 | 1 | Exact ratio |
| Analytic frequency controls — case 3 — half Power Lower Omega | 8.284271247461900976033774484193961571393437507538961463533594759814650E+0 | rad/s | Stored approximation, 70 significant digits |
| Analytic frequency controls — case 3 — half Power Upper Omega | 4.828427124746190097603377448419396157139343750753896146353359475981465E+1 | rad/s | Stored approximation, 70 significant digits |
| Analytic frequency controls — case 3 — bandwidth Omega | 40 | rad/s | Exact ratio |
| Analytic frequency controls — case 3 — Q From Bandwidth | 1/2 | 1 | Exact ratio |
| Analytic frequency controls — case 3 — damped Omega Or Null | No finite value | rad/s | Critical response has no oscillatory damped frequency. |
| Analytic frequency controls — case 3 — inductor Peak Location Or Supremum | No finite value | 1 | No finite interior maximum; high-frequency supremum at r->infinity. |
| Analytic frequency controls — case 3 — inductor Peak Amplitude Or Supremum | 4 | V | Exact ratio |
| Analytic frequency controls — case 3 — peak Status | capacitor maximum at DC boundary; inductor supremum only at infinity | Declared model condition | |
| DC state — Capacitor voltage | 4 | V | Exact ratio |
| DC state — Series current | 0 | A | Exact ratio |
| DC state — Capacitor energy | 1/25 | J | Exact ratio |
| Positive-frequency cycle-average limit — mean U C | 1/50 | J | Exact ratio |
| Positive-frequency cycle-average limit — mean U L | 0 | J | Exact ratio |
Focus a graph or table region to scroll it locally. Axis ticks are rounded drawing labels; the numerical equivalents retain exact ratios, full stored approximations and qualified limits.
Identify the narrowband approximation
The charge or displacement response has a denominator of the form (ω₀²−ω²)²+γ²ω², with γ=R/L=2α for the circuit. When γ≪ω₀ and ω is close to ω₀, replace ω₀²−ω² by 2ω₀(ω₀−ω) and γω by γω₀. The squared-response peak then has the approximate form constant/[(ω−ω₀)²+(γ/2)²], a narrow Lorentzian.
Its full width at half maximum is approximately γ. The approximations also put its center near ω₀. Away from weak damping, the capacitor-voltage peak shifts and may cease to be an interior maximum, so a narrowband charge-response formula is not an exact global statement. The exact current-power width derived above happens to equal γ for a different algebraic reason: its numerator contains ω².
This distinction is useful beyond circuits. Spectroscopy, mechanics and filters often report a linewidth or quality factor for a particular observable and a particular fitting regime. Before comparing them, ask which amplitude or power was measured, how the background and source were held fixed and whether the single-mode weak-damping model applies. A shared symbol Q does not erase those experimental and mathematical conditions.
Relate Q to ringdown and driven storage
In an underdamped free response, compare states one damped period T_d=2π/ω_d apart. Both physical states are multiplied by e^(−αT_d), because the phase returns to the same value. Their combined quadratic energy is therefore multiplied by e^(−2αT_d). The logarithmic amplitude decrement is αT_d=2πζ/√(1−ζ²); this is exact for the stated free mode, not an assertion that energy is a pure exponential at arbitrary intermediate phases.
If the cycle loss is measured as the finite difference U(t)−U(t+T_d), the quotient 2πU(t)/[U(t)−U(t+T_d)] equals 2π/[1−e^(−4πζ/√(1−ζ²))]. It approaches Q=1/(2ζ) when damping is weak, but is not identically Q at finite damping. Using leading-cycle energy, mid-cycle energy or a continuous loss rate produces distinct exact expressions before the small-loss approximation.
For the driven ideal series circuit exactly at resonance, x=Q sin u and j=Q cos u, so U=Q²/25 J is constant throughout the cycle. Mean source delivery equals mean resistor loss, and ω₀U/P_mean=Q exactly. This driven rate definition and the free finite-cycle quotient illuminate the same weak-loss physics without being interchangeable exact measurements for every damping and preparation.
| Question | Result | Condition |
|---|---|---|
| Natural series storage | U=(x²+j²)/25 J | Series state mapping |
| Natural parallel storage | U=x²/25+i_L²/4 J | Parallel branch mapping |
| Free equal-phase energy ratio | e^(−2αT_d) | Underdamped free motion |
| Free finite-cycle quotient | 2π/(1−e^(−2αT_d)) | Not exactly Q at finite damping |
| Driven resonance energy rate | ω₀U/P_mean=Q | Ideal series, R>0, r=1 |
| Current-power bandwidth | Q=ω₀/Δω | Constant series R and fixed voltage amplitude |
Keep DC separate from a cycle-average limit
For every positive driving frequency, a periodic average over a full cycle gives ⟨U_C⟩=1/(50D) J and ⟨U_L⟩=r²/(50D) J. Capacitor and inductor average energies are equal at r=1, but not at an arbitrary frequency. The factor one-half comes from averaging the square of a sinusoid; it is not a universal factor in the instantaneous energy formula.
As r tends to zero through positive values, the cycle-averaged capacitor energy tends to 1/50 J. At exactly zero frequency the same expression 4 cos(ωt) becomes a constant 4 V source, and a settled capacitor stores 1/25 J. There is no contradiction: a nonzero-frequency cycle samples all source phases, while a fixed DC source has one value. The cycle duration becomes unbounded as frequency tends to zero.
The order of operations must therefore remain visible. First establish a periodic state and average one increasingly long cycle, or first set ω=0 and solve the static final-state problem; the answers describe different questions. A row labeled “DC” must not contain a nonzero-frequency cycle-average limit merely because both have a small frequency ratio on an axis.
Solve finite lossless off-resonance forcing
For R=0, zero initial states and f=cos(ru), the finite-time solution at r≠1 is x=[cos(ru)−cos u]/(1−r²) and j=[sin u−r sin(ru)]/(1−r²). The second cosine is the homogeneous term required by the initial conditions. It persists forever in the ideal circuit. Thus even off resonance, merely waiting does not select the particular forced sinusoid independently of preparation.
The selected off-resonance case r=1/2 displays two fixed frequencies and bounded beating. There is no resistor loss; source work equals the change in total LC energy, but it can be evaluated independently by integrating cos(ru)j. An antiderivative from zero is {[(1−cos((1+r)u))/(1+r)+(1−cos((1−r)u))/(1−r)]/2−[1−cos(2ru)]/4}/(1−r²), multiplied by 2/25 J for source work.
This ideal beat is not evidence for a finite linewidth or a decaying envelope. Detuning and damping change different parts of the problem. Nearby-frequency oscillations can create a slowly varying amplitude without any irreversible energy loss, while a dissipative ringdown has an actual net resistor-work ledger. Observing a varying peak alone does not identify its cause.
Resolve resonant forcing at finite time
At r=1 the off-resonance expression has a removable finite-time limit. Taking that limit, or solving with a resonant trial function, gives x=(u/2)sin u and j=[sin u+u cos u]/2. Both initial states are zero. The derivative j′=cos u−(u/2)sin u satisfies j′=f−x, so the apparent singularity of the steady-sinusoid denominator has not produced an infinite value at a finite time.
The independent source-power integral is W_s=(2/25){u²/8+u sin(2u)/8+[1−cos(2u)]/16} J. It equals (x²+j²)/25 J and resistor work is exactly zero. The envelope grows secularly, but instantaneous energy exchange still follows the signed source power; the energy is not asserted to increase monotonically at every phase.
At u=2π and 4π the capacitor voltage is exactly zero, while j=π and 2π respectively. The stored energies are π²/25 and 4π²/25 J. These are finite-time phase controls, at physical times π/10 and π/5 s. They show quadratic growth across equal-phase observations without assigning a fictitious infinite sample to the plot or claiming that a real source and inductor can sustain unlimited amplitude.
Keep small-loss and long-time limits in order
For each positive R, resonant sinusoidal forcing eventually reaches a bounded periodic state with current peak 4/R. That amplitude grows without bound as R tends to zero. At exactly R=0, however, the zero-state response grows with elapsed time and has no bounded attracting periodic state. “First wait for steady state, then remove loss” and “remove loss, then solve a finite-time initial-value problem” are different operations.
At any fixed finite time, the positive-R initial-value solution approaches the lossless one as R tends to zero. The time required to forget a generic initial state grows as damping decreases. A finite experiment may therefore be dominated by its transient even when a steady-state formula exists mathematically. Near a sharp resonance, source stability, turn-on history and observation duration become part of the interpretation.
The same caution applies to complete free decay: every positive R eventually dissipates the initial energy, while R=0 dissipates none. These limit distinctions do not require new circuit elements or a paradoxical source of energy. They follow from changing a stability condition before taking an infinite-time limit. No result here licenses unlimited real voltages, currents or stored energies.
Include the source and load in the experiment
A source labeled 4 V need not maintain 4 V across the RLC load. Model a fixed ideal generator with peak emf 4 V and a series source resistance R_s=1 Ω. Add internal resonator resistance R_i=1 Ω and a series useful load R_L. At the ideal reactance-cancellation frequency ω₀=20 rad/s, current peak is 4/(R_s+R_i+R_L) A, not 4/R_L.
The generator’s mean delivery is 4I_peak/2, while each resistor absorbs R_k I_peak²/2. The terminal voltage after the source resistor has peak amplitude 4−R_s I_peak at this resonant, in-phase condition. Distinguishing ideal emf from terminal voltage prevents accidentally changing the source while comparing loads. Off resonance those terminal amplitudes require the full frequency-dependent relation, not this in-phase subtraction of magnitudes.
The original loads 0, 2 and 8 Ω have total resistances 2, 4 and 10 Ω. Their current peaks are 2, 1 and 2/5 A. All source-resistor, internal-resistor and useful-load powers remain in the ledger. A terminal equivalent can predict current without making internal loss disappear; useful delivered power and total generator delivery are different measures.
Trade loaded Q against useful load power
The loaded quality factor of this series model is Q_loaded=ω₀L/(R_s+R_i+R_L)=10/(2+R_L). If intrinsic Q refers only to R_i, it is 10. Inverse contributions add: 1/Q_loaded=R_i/10+R_s/10+R_L/10. A zero load resistance contributes zero inverse Q; its isolated loss Q is not a finite zero, and no division by zero is needed to state the additive loss rule.
The useful load power is P_L=8R_L/(2+R_L)² W. It is maximized at R_L=R_s+R_i=2 Ω for the fixed generator. The selected load powers are 0, 1 and 16/25 W, whereas loaded Q falls from 5 to 5/2 to 1. Highest loaded Q therefore does not mean highest useful load power. The conclusion uses a fixed source and this particular series loading, not a universal prescription for every resonator coupling geometry.
At the matching load, the useful load receives one-half of the ideal generator’s mean delivery in this simple all-resistive resonant ledger; the other half is split between source and internal resistance. That statement concerns the explicitly modeled generator and losses. Additional source conversion losses, current compliance and frequency-dependent coupling would change the practical efficiency question. They are not hidden inside the symbol Q.
Finite ideal responses and a real source boundary
Retain initial data in lossless forcing and compare useful load power with loaded Q at the same explicitly defined source.
(a) Lossless forcing at finite time
- Off resonance r=1/2
- Resonance r=1
(b) Initial-data response remains part of the solution
- Off resonance r=1/2
- Resonance r=1
(c) A source/load tradeoff at ω₀
- Three original load cases
(d) Loaded Q is a different objective
- Three original load cases
- Every plotted lossless value is at a finite time. Resonant growth does not create an attracting sinusoidal steady state; detuned forcing retains its homogeneous initial-data response. Small loss and long time cannot be interchanged without a domain statement.
- The load comparison uses ideal source peak 4 V, source resistance 1 Ω, internal resistance 1 Ω and ω₀=20 rad/s. Only the three original load cases are plotted as isolated markers; no intervening measurements are implied.
- Maximum useful load power and maximum loaded Q are different objectives. Source and internal losses remain in the circuit and in the worked example. Zero load resistance has zero inverse load-Q contribution; its isolated loss-Q is not a finite quotient.
Original source/load comparison at ω₀=20 rad/s; source peak 4 V and R_s=R_i=1 Ω
Finite ideal responses and a real source boundary — values and conditions
| Quantity and original case | Value | Unit | Meaning and precision |
|---|---|---|---|
| Lossless forcing at finite time — Off resonance r=1/2 — sample 1 — Physical time | 0 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 1 — Capacitor voltage | 0 | V | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 2 — Capacitor voltage | 1.241879760996494346906744442349955719423632770621783057108350554934480E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 3 — Capacitor voltage | 4.870925857081176961510072900819175883332746302810601501360920318559563E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 4 — Capacitor voltage | 1.060366682871964825800736209223845142691465204377266820169183053123346E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 5 — Capacitor voltage | 1.798828032118575993148506534191216257383118807956383068413405935652535E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 6 — Capacitor voltage | 2.643417371253062595957313341349561182251682367541934775537922243290395E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 7 — Capacitor voltage | 3.525075558432629206526127475017684452029067991932936603120201616159078E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 8 — Capacitor voltage | 4.369295540335025177275912355939549284762491975922126228240294968044038E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 9 — Capacitor voltage | 5.101062092881504556792025797033273565324326355698491298267985222203015E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 10 — Capacitor voltage | 5.649867410780828082481909868363163379357403367562670715582553684451871E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 11 — Capacitor voltage | 5.954485215691746028165553828828744950168285636003077027144518037948339E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 12 — Capacitor voltage | 5.967200461981070552239358398307935340498063117480916544283630992716676E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 13 — Capacitor voltage | 5.657225057430124625918734112882826727886773995620987723929233867302678E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 14 — Capacitor voltage | 5.013080218952586128873474874070002191990983052559053878365265139514210E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 15 — Capacitor voltage | 4.043790035420289319018563641222021888266863543948858553252938106677626E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 16 — Capacitor voltage | 2.778804539466595204414699589880346578011377871447996329122203877908770E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 17 — Capacitor voltage | 1.266649516354504147421866419183937022176708403043591233343254987044121E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 18 — Capacitor voltage | -4.276205055493985219459148490542792339120562625028235434208800595115850E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 19 — Capacitor voltage | -2.226015057557116708975505240547055540149484161877538495944376434235956E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 20 — Capacitor voltage | -4.042029996504897496800999980332205009282470819377644064506141909298243E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 21 — Capacitor voltage | -5.785630938720853222934090463898181188069282352106859093359770977323603E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 22 — Capacitor voltage | -7.368494179775067160104850121638978658553559885580365584575382559289136E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 23 — Capacitor voltage | -8.709184815593192235328430244546971803726096880387858745313337408680358E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 24 — Capacitor voltage | -9.737955005241798256199560093731393332113902452530836692768943898017671E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 25 — Capacitor voltage | -1.040086817733766121502520049415565710850995613774187214369798591457000E+1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 26 — Physical time | 5/16 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 26 — Capacitor voltage | -1.066299607096632529939959527490283334202253919756919196259095386169112E+1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 27 — Physical time | 13/40 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 27 — Capacitor voltage | -1.051049227631237183604982655528356735112512090701948031794346765131790E+1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 28 — Physical time | 27/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 28 — Capacitor voltage | -9.951414883565456242723872522146194760210125645993383758103160216674465E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 29 — Physical time | 7/20 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 29 — Capacitor voltage | -9.015247688715205204479227458085026534604112838624459152877879168845419E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 30 — Physical time | 29/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 30 — Capacitor voltage | -7.751149378622481561373500861065545897639006235493416883590008839962752E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 31 — Physical time | 3/8 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 31 — Capacitor voltage | -6.225038267597794845625235938467235337232258688114545396596312758023549E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 32 — Physical time | 31/80 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 32 — Capacitor voltage | -4.515691639292009040747848151197119521814829443543005095006393562658799E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 33 — Physical time | 2/5 | s | Exact ratio |
| Lossless forcing at finite time — Off resonance r=1/2 — sample 33 — Capacitor voltage | -2.710099130959991406775076273421630393679007525870384115498910960837639E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 1 — Physical time | 0 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 1 — Capacitor voltage | 0 | V | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 2 — Physical time | 1/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 2 — Capacitor voltage | 1.237019796272614647984243524246945979466954901934829053382724152471991E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 3 — Physical time | 1/40 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 3 — Capacitor voltage | 4.794255386042030002732879352155713880818033679406006751886166131255350E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 4 — Physical time | 3/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 4 — Capacitor voltage | 1.022458140035001250099862929169840903007573591988844863820437893226651E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 5 — Physical time | 1/20 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 5 — Capacitor voltage | 1.682941969615793013305004643260597999245126121596742131345503419983821E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 6 — Physical time | 1/16 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 6 — Capacitor voltage | 2.372461548388965535871227117590123125945040086309730550338892580450316E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 7 — Physical time | 3/40 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 7 — Capacitor voltage | 2.992484959812163292825170113424461968119954277766347465849924472178036E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 8 — Physical time | 7/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 8 — Capacitor voltage | 3.443950814058779145560820279387949577383330238629580442262576595327274E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 9 — Physical time | 1/10 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 9 — Capacitor voltage | 3.637189707302726781584079463646979370809019885791561073515892046123869E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 10 — Physical time | 9/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 10 — Capacitor voltage | 3.501329385995645586349350040144907943620073343095996378536375109191890E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 11 — Physical time | 1/8 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 11 — Capacitor voltage | 2.992360720519782470259273510930811358517985857886117866513135163193721E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 12 — Physical time | 11/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 12 — Capacitor voltage | 2.099135456287824342171087548077791655952362672416711207414195749467258E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 13 — Physical time | 3/20 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 13 — Capacitor voltage | 8.467200483592033326044688168486616790815995855135935049112958473945321E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 14 — Physical time | 13/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 14 — Capacitor voltage | -7.032683744457044507329004766453625066228951048150174998133874520385892E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 15 — Physical time | 7/40 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 15 — Capacitor voltage | -2.455482593827338936842581600305449095594872151584084397910286043494546E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 16 — Physical time | 3/16 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 16 — Capacitor voltage | -4.286709890567578293256166800127201234388861729852964560800634662753555E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 17 — Physical time | 1/5 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 17 — Capacitor voltage | -6.054419962463426010981112756094632753087303098691780571883334187210484E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 18 — Physical time | 17/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 18 — Capacitor voltage | -7.607409544942959957958899041725347749989007976281116281248171929412019E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 19 — Physical time | 9/40 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 19 — Capacitor voltage | -8.797771058985873498502215130487668000779433843116132403240717840705479E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 20 — Physical time | 19/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 20 — Capacitor voltage | -9.493281495266090474975572178172253000853742276614775836497128360113560E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 21 — Physical time | 1/4 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 21 — Capacitor voltage | -9.589242746631384688931544061559939733524615439646017781316724542351026E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 22 — Physical time | 21/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 22 — Capacitor voltage | -9.018812180979216318472619738938435583970167224686313200740499584358101E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 23 — Physical time | 11/40 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 23 — Capacitor voltage | -7.760943581274310968551110930742770867320193101869944088366022691389318E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 24 — Physical time | 23/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 24 — Capacitor voltage | -5.845209391241470794790463206529482897227149027660148135726819033850735E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 25 — Physical time | 3/10 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 25 — Capacitor voltage | -3.352985978387110473738665359342737115535938371818451821800216435839734E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 26 — Physical time | 5/16 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 26 — Capacitor voltage | -4.147402068444602109680708848661585917270813847270830056386665917263231E-1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 27 — Physical time | 13/40 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 27 — Capacitor voltage | 2.796559845141601815860294737971512614608022117288200597675392303633724E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 28 — Physical time | 27/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 28 — Capacitor voltage | 6.075594996038337767035890736032171994268001923051749721557987949948726E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 29 — Physical time | 7/20 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 29 — Capacitor voltage | 9.197812382063047265557987282310892491116180146964861045209960678650100E+0 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 30 — Physical time | 29/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 30 — Capacitor voltage | 1.193467274566682914711423215947481947786762832416785090123298472739249E+1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 31 — Physical time | 3/8 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 31 — Capacitor voltage | 1.406999965162108286922695697223570854647747093262045551326111264567584E+1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 32 — Physical time | 31/80 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 32 — Capacitor voltage | 1.541628107622322986959466742541676213574228000820918912316948294099722E+1 | V | Stored approximation, 70 significant digits |
| Lossless forcing at finite time — Resonance r=1 — sample 33 — Physical time | 2/5 | s | Exact ratio |
| Lossless forcing at finite time — Resonance r=1 — sample 33 — Capacitor voltage | 1.582973194597410844492997757192461875386270529417218020508344351774205E+1 | V | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 1 — Series current | 0 | A | Exact ratio |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 2 — Series current | 9.870218269968484512966792026311197978638588136860063025801998569656680E-2 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 3 — Series current | 1.897192314543688189199272441551342880720575347984628105868502388684458E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 4 — Series current | 2.658679975894988725582865476916063236000938066456271774823251209054655E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 5 — Series current | 3.209377149364240034751244554786737629768860676416377216418364818288762E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 6 — Series current | 3.500991908721894063710886346458714826844185941471304601936668419261465E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 7 — Series current | 3.502269901826065853733879438674881893533121219160004625830198362580240E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 8 — Series current | 3.201142377364924687543334418350483673021720674620611216445765464292019E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 9 — Series current | 2.605663650249245024372099760515175162085191685593091922893851501520065E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 10 — Series current | 1.743676799137992853076045926721828108933153245611658721375743480831351E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 11 — Series current | 6.612257836062047300139161528967341147444754896814798039191920882596328E-2 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 12 — Series current | -5.801895277826461304962017908039983291298797619250953793432608306312002E-2 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 13 — Series current | -1.907346587958186631307290041400711368034093059780225743056558777585337E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 14 — Series current | -3.239790958933462234194672441912108909253753550346021459758723124425233E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 15 — Series current | -4.494806406008470919926401432147351369888154202067554163941228677197577E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 16 — Series current | -5.592555784251683627167328423981273009367336320150012539981590744467910E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 17 — Series current | -6.461073113177435195043461479827741415930881989655561096932817488889569E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 18 — Series current | -7.040796016734984152466206337170630546716278474761298816501520606196632E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 19 — Series current | -7.288355819248307605837964545560045004088596851088668907719758785124367E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 20 — Series current | -7.179388293077407180497667480901601266027791382378018757229298299568174E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 21 — Series current | -6.710188515813955818235102705328400582422720692017139012175925176290532E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 22 — Series current | -5.898104761235395294575486012267965556526029745482444581264927311152969E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 23 — Series current | -4.780644381848308029441065929125121223404875526926863173711621062233492E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 24 — Series current | -3.413344395631833331993168610966895341678448067182455281390337689823631E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 25 — Series current | -1.866536011887250580596948522751732797607794647703131256316227681437452E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 26 — Series current | -2.212008675318974348661230505807940984483663809390250892797532686458191E-2 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 27 — Series current | 1.435826961881971801679259575227435947171658347529525373328730156173257E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 28 — Series current | 3.017018559902018801291829312474979180150793849945021076538236424847248E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 29 — Series current | 4.439350467005861410438311955282415842556591351828169692617236846531294E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 30 — Series current | 5.629228122812319703685605235027335625509481701271265774256111542963064E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 31 — Series current | 6.526830059444857302216221785729034588752473835546104915221954598552675E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 32 — Series current | 7.089606358560235284077919722912327836007912040097528691952165800627092E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Off resonance r=1/2 — sample 33 — Series current | 7.294717302812511485303696776006417168983336130954653592322259234984177E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 1 — Series current | 0 | A | Exact ratio |
| Initial-data response remains part of the solution — Resonance r=1 — sample 2 — Series current | 9.792641293643682512659951344458729916888490559176530369271537231332676E-2 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 3 — Series current | 1.836433639098778716662857453034972428155251932990945403374844094567021E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 4 — Series current | 2.460810823357399662934242035059914686437576703394618542188775354633326E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 5 — Series current | 2.763546581352072448106877858146551206709746962832586586685697930746022E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 6 — Series current | 2.686275144699344092315828075167193535075996339004676459058611915493476E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 7 — Series current | 2.207201578211217592148016296585780772668575934534272252167885899691459E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 8 — Series current | 1.344110698974651481123956570136761430681465728267912726191681922416537E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 9 — Series current | 1.540075074627938428017668138204409263405068585936009737373461279340749E-2 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 10 — Series current | -1.270634908476483417288317407108300238940747279544606961699812326108854E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 11 — Series current | -2.808773789526755586063804547964433778735644900684653528767735893032632E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 12 — Series current | -4.320341098373886095378504447641363527995370730935390679393169791119081E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 13 — Series current | -5.657714963482938299427947162771347254668208051188998804580750315105131E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 14 — Series current | -6.678233163583767179996351648737046830476803209640332729423440650280898E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 15 — Series current | -7.256763266414814060131340986789594411309667832354881732178429541553779E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 16 — Series current | -7.297317817531392964241654163880919851031745018750584253843251091640962E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 17 — Series current | -6.742753957524751819858623653805661239664894547842691119026915817624522E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 18 — Series current | -5.581722380724405841826166281507667591209989114071999922169082964502361E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 19 — Series current | -3.852222430207211464602606452141733092825846083809782804245982369994097E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 20 — Series current | -1.641365125514978619671326543647486477133627505082948330587904744555532E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 21 — Series current | 9.187733053059857068800829028235851366393028345929558930402457573913095E-2 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 22 — Series current | 3.659028524186143112432618869930113294024315897469473232106972806006184E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 23 — Series current | 6.384286866063076187837794643540140927170761755467942954690999606936285E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 24 — Series current | 8.887154642358973068395483080155972835581330013751680118610906090730081E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 25 — Series current | 1.096321244340654050092471668185130334616726242189574374058473437333593E+0 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 26 — Series current | 1.242675929471112898198498743144695171194141216504616032675393971913756E+0 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 27 — Series current | 1.312587911063993654711596379942236524065996848950982098489861988595452E+0 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 28 — Series current | 1.295567380086377026094189999980054548512566050329639048195719901047338E+0 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 29 — Series current | 1.186860475824384311477076348725581852693012750143675590292630299977285E+0 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 30 — Series current | 9.881062270709086450988969415987451799425908901616066310483851953202877E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 31 — Series current | 7.075529721074864880471217638876528148439959903091965587769114288217982E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 32 — Series current | 3.598010095120773292167279652843839285182656276001676219217635660745245E-1 | A | Stored approximation, 70 significant digits |
| Initial-data response remains part of the solution — Resonance r=1 — sample 33 — Series current | -3.492840476910528582852149128085375775162768069568170799800012194131290E-2 | A | Stored approximation, 70 significant digits |
| A source/load tradeoff at ω₀ — Three original load cases — sample 1 — Load resistance | 0 | ohm | Exact ratio |
| A source/load tradeoff at ω₀ — Three original load cases — sample 1 — Mean useful load power | 0 | W | Exact ratio |
| A source/load tradeoff at ω₀ — Three original load cases — sample 2 — Load resistance | 2 | ohm | Exact ratio |
| A source/load tradeoff at ω₀ — Three original load cases — sample 2 — Mean useful load power | 1 | W | Exact ratio |
| A source/load tradeoff at ω₀ — Three original load cases — sample 3 — Load resistance | 8 | ohm | Exact ratio |
| A source/load tradeoff at ω₀ — Three original load cases — sample 3 — Mean useful load power | 16/25 | W | Exact ratio |
| Loaded Q is a different objective — Three original load cases — sample 1 — Loaded Q | 5 | 1 | Exact ratio |
| Loaded Q is a different objective — Three original load cases — sample 2 — Loaded Q | 5/2 | 1 | Exact ratio |
| Loaded Q is a different objective — Three original load cases — sample 3 — Loaded Q | 1 | 1 | Exact ratio |
| Exact lossless phase controls — case 1 — Dimensionless time | 6.283185307179586476925286766559005768394338798750211641949889184615633E+0 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Local interval coordinate | 6.283185307179586476925286766559005768394338798750211641949889184615633E+0 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Physical time | 3.141592653589793238462643383279502884197169399375105820974944592307816E-1 | s | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — zeta | 0 | 1 | Exact ratio |
| Exact lossless phase controls — case 1 — R | 0 | ohm | Exact ratio |
| Exact lossless phase controls — case 1 — f | 1 | 1 | Exact ratio |
| Exact lossless phase controls — case 1 — x | 0 | 1 | Exact ratio |
| Exact lossless phase controls — case 1 — j | 3.141592653589793238462643383279502884197169399375105820974944592307816E+0 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — j Prime | 1.000000000000000000000000000000000000000000000000000000000000000000000E+0 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — v Source | 4 | V | Exact ratio |
| Exact lossless phase controls — case 1 — Capacitor voltage | 0 | V | Exact ratio |
| Exact lossless phase controls — case 1 — v R | 0 | V | Exact ratio |
| Exact lossless phase controls — case 1 — v L | 4.000000000000000000000000000000000000000000000000000000000000000000000E+0 | V | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Series current | 1.256637061435917295385057353311801153678867759750042328389977836923127E+0 | A | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — charge | 0 | C | Exact ratio |
| Exact lossless phase controls — case 1 — dv C dt | 2.513274122871834590770114706623602307357735519500084656779955673846253E+2 | V/s | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — di dt | 8.000000000000000000000000000000000000000000000000000000000000000000000E+0 | A/s | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Capacitor energy | 0 | J | Exact ratio |
| Exact lossless phase controls — case 1 — Inductor energy | 3.947841760435743447533796399950460454125479762896316250565339750488018E-1 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Total stored energy | 3.947841760435743447533796399950460454125479762896316250565339750488018E-1 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Capacitor absorbed power | 0 | W | Exact ratio |
| Exact lossless phase controls — case 1 — Inductor absorbed power | 5.026548245743669181540229413247204614715471039000169313559911347692506E+0 | W | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Resistor absorbed power | 0 | W | Exact ratio |
| Exact lossless phase controls — case 1 — Source-delivered power | 5.026548245743669181540229413247204614715471039000169313559911347692506E+0 | W | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — resistor Work Interval | 0 | J | Exact ratio |
| Exact lossless phase controls — case 1 — source Work Interval | 3.947841760435743447533796399950460454125479762896316250565339750488018E-1 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — Accumulated resistor work | 0 | J | Exact ratio |
| Exact lossless phase controls — case 1 — Accumulated source-delivered work | 3.947841760435743447533796399950460454125479762896316250565339750488018E-1 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 1 — u Expression | 2*pi | Declared model condition | |
| Exact lossless phase controls — case 1 — Frequency ratio ω/ω₀ | 1 | 1 | Exact ratio |
| Exact lossless phase controls — case 2 — Dimensionless time | 1.256637061435917295385057353311801153678867759750042328389977836923127E+1 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Local interval coordinate | 1.256637061435917295385057353311801153678867759750042328389977836923127E+1 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Physical time | 6.283185307179586476925286766559005768394338798750211641949889184615633E-1 | s | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — zeta | 0 | 1 | Exact ratio |
| Exact lossless phase controls — case 2 — R | 0 | ohm | Exact ratio |
| Exact lossless phase controls — case 2 — f | 1 | 1 | Exact ratio |
| Exact lossless phase controls — case 2 — x | 0 | 1 | Exact ratio |
| Exact lossless phase controls — case 2 — j | 6.283185307179586476925286766559005768394338798750211641949889184615633E+0 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — j Prime | 1.000000000000000000000000000000000000000000000000000000000000000000000E+0 | 1 | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — v Source | 4 | V | Exact ratio |
| Exact lossless phase controls — case 2 — Capacitor voltage | 0 | V | Exact ratio |
| Exact lossless phase controls — case 2 — v R | 0 | V | Exact ratio |
| Exact lossless phase controls — case 2 — v L | 4.000000000000000000000000000000000000000000000000000000000000000000000E+0 | V | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Series current | 2.513274122871834590770114706623602307357735519500084656779955673846253E+0 | A | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — charge | 0 | C | Exact ratio |
| Exact lossless phase controls — case 2 — dv C dt | 5.026548245743669181540229413247204614715471039000169313559911347692506E+2 | V/s | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — di dt | 8.000000000000000000000000000000000000000000000000000000000000000000000E+0 | A/s | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Capacitor energy | 0 | J | Exact ratio |
| Exact lossless phase controls — case 2 — Inductor energy | 1.579136704174297379013518559980184181650191905158526500226135900195207E+0 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Total stored energy | 1.579136704174297379013518559980184181650191905158526500226135900195207E+0 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Capacitor absorbed power | 0 | W | Exact ratio |
| Exact lossless phase controls — case 2 — Inductor absorbed power | 1.005309649148733836308045882649440922943094207800033862711982269538501E+1 | W | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Resistor absorbed power | 0 | W | Exact ratio |
| Exact lossless phase controls — case 2 — Source-delivered power | 1.005309649148733836308045882649440922943094207800033862711982269538501E+1 | W | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — resistor Work Interval | 0 | J | Exact ratio |
| Exact lossless phase controls — case 2 — source Work Interval | 1.579136704174297379013518559980184181650191905158526500226135900195207E+0 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — Accumulated resistor work | 0 | J | Exact ratio |
| Exact lossless phase controls — case 2 — Accumulated source-delivered work | 1.579136704174297379013518559980184181650191905158526500226135900195207E+0 | J | Stored approximation, 70 significant digits |
| Exact lossless phase controls — case 2 — u Expression | 4*pi | Declared model condition | |
| Exact lossless phase controls — case 2 — Frequency ratio ω/ω₀ | 1 | 1 | Exact ratio |
| attracting Steady State | No finite value | 1 | No damping: no attracting sinusoidal steady response. |
| resonant Large Time Amplitude | No finite value | V | Resonant envelope grows without bound as time increases; every retained finite-time record is finite. |
| Source/load comparison — case 1 — Rs | 1 | ohm | Exact ratio |
| Source/load comparison — case 1 — Ri | 1 | ohm | Exact ratio |
| Source/load comparison — case 1 — total Resistance | 2 | ohm | Exact ratio |
| Source/load comparison — case 1 — omega | 20 | rad/s | Exact ratio |
| Source/load comparison — case 1 — ideal Source Peak | 4 | V | Exact ratio |
| Source/load comparison — case 1 — terminal Peak | 2 | V | Exact ratio |
| Source/load comparison — case 1 — current Peak | 2 | A | Exact ratio |
| Source/load comparison — case 1 — mean Source Delivered | 4 | W | Exact ratio |
| Source/load comparison — case 1 — mean Source Resistor Loss | 2 | W | Exact ratio |
| Source/load comparison — case 1 — mean Internal Loss | 2 | W | Exact ratio |
| Source/load comparison — case 1 — Qintrinsic | 10 | 1 | Exact ratio |
| Source/load comparison — case 1 — inverse Q Source | 1/10 | 1 | Exact ratio |
| Source/load comparison — case 1 — inverse Q Internal | 1/10 | 1 | Exact ratio |
| Source/load comparison — case 1 — inverse Q Load | 0 | 1 | Exact ratio |
| Source/load comparison — case 1 — load Q Or Null | No finite value | 1 | Zero load resistance contributes zero inverse Q; its isolated Q is not finite. |
| Source/load comparison — case 2 — Rs | 1 | ohm | Exact ratio |
| Source/load comparison — case 2 — Ri | 1 | ohm | Exact ratio |
| Source/load comparison — case 2 — total Resistance | 4 | ohm | Exact ratio |
| Source/load comparison — case 2 — omega | 20 | rad/s | Exact ratio |
| Source/load comparison — case 2 — ideal Source Peak | 4 | V | Exact ratio |
| Source/load comparison — case 2 — terminal Peak | 3 | V | Exact ratio |
| Source/load comparison — case 2 — current Peak | 1 | A | Exact ratio |
| Source/load comparison — case 2 — mean Source Delivered | 2 | W | Exact ratio |
| Source/load comparison — case 2 — mean Source Resistor Loss | 1/2 | W | Exact ratio |
| Source/load comparison — case 2 — mean Internal Loss | 1/2 | W | Exact ratio |
| Source/load comparison — case 2 — Qintrinsic | 10 | 1 | Exact ratio |
| Source/load comparison — case 2 — inverse Q Source | 1/10 | 1 | Exact ratio |
| Source/load comparison — case 2 — inverse Q Internal | 1/10 | 1 | Exact ratio |
| Source/load comparison — case 2 — inverse Q Load | 1/5 | 1 | Exact ratio |
| Source/load comparison — case 2 — load Q Or Null | 5 | 1 | Exact ratio |
| Source/load comparison — case 3 — Rs | 1 | ohm | Exact ratio |
| Source/load comparison — case 3 — Ri | 1 | ohm | Exact ratio |
| Source/load comparison — case 3 — total Resistance | 10 | ohm | Exact ratio |
| Source/load comparison — case 3 — omega | 20 | rad/s | Exact ratio |
| Source/load comparison — case 3 — ideal Source Peak | 4 | V | Exact ratio |
| Source/load comparison — case 3 — terminal Peak | 18/5 | V | Exact ratio |
| Source/load comparison — case 3 — current Peak | 2/5 | A | Exact ratio |
| Source/load comparison — case 3 — mean Source Delivered | 4/5 | W | Exact ratio |
| Source/load comparison — case 3 — mean Source Resistor Loss | 2/25 | W | Exact ratio |
| Source/load comparison — case 3 — mean Internal Loss | 2/25 | W | Exact ratio |
| Source/load comparison — case 3 — Qintrinsic | 10 | 1 | Exact ratio |
| Source/load comparison — case 3 — inverse Q Source | 1/10 | 1 | Exact ratio |
| Source/load comparison — case 3 — inverse Q Internal | 1/10 | 1 | Exact ratio |
| Source/load comparison — case 3 — inverse Q Load | 4/5 | 1 | Exact ratio |
| Source/load comparison — case 3 — load Q Or Null | 5/4 | 1 | Exact ratio |
| Equal-phase ringdown controls — case 1 — zeta | 1/10 | 1 | Exact ratio |
| Equal-phase ringdown controls — case 1 — Q | 5 | 1 | Exact ratio |
| Equal-phase ringdown controls — case 1 — d | 9.949874371066199547344798210012060051781265636768060791176046438349454E-1 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 1 — damped Period | 3.157419416998276454765447939389913555399960242205858353336933605589089E-1 | s | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 1 — state Scale One Cycle | 5.318020829442596511890967731396322993169541862029128742297039810783817E-1 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 1 — energy Ratio One Cycle | 2.828134554238532217935371681740309740382751562971804418881339589816018E-1 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 1 — log State Decrement | 6.314838833996552909530895878779827110799920484411716706673867211178178E-1 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 1 — leading Cycle Energy Quotient | 8.760880073249162422196228121928132870069447531988214947029704373912550E+0 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 2 — zeta | 1/200 | 1 | Exact ratio |
| Equal-phase ringdown controls — case 2 — Q | 100 | 1 | Exact ratio |
| Equal-phase ringdown controls — case 2 — d | 9.999874999218740234222409439036845177392055795635565837115252399761970E-1 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 2 — damped Period | 3.141631924234289229206676660583473759209614397723417198976641934740531E-1 | s | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 2 — state Scale One Cycle | 9.690720457438979208633440236791743862798679065329999741135707017978508E-1 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 2 — energy Ratio One Cycle | 9.391006298422633844379620337954179405118031187918936602877884003411466E-1 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 2 — log State Decrement | 3.141631924234289229206676660583473759209614397723417198976641934740531E-2 | 1 | Stored approximation, 70 significant digits |
| Equal-phase ringdown controls — case 2 — leading Cycle Energy Quotient | 1.031732395738312046574181674738414939678181019595484636879074045855090E+2 | 1 | Stored approximation, 70 significant digits |
| Qualified limits and branch controls — asymptotic — case 1 — family | naturalSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 1 — zeta | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 1 — state Limit | No finite value | 1 | Undamped periodic state does not settle. |
| Qualified limits and branch controls — asymptotic — case 2 — family | stepSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 2 — zeta | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 2 — state Limit | No finite value | 1 | Undamped periodic state does not settle. |
| Qualified limits and branch controls — asymptotic — case 3 — family | naturalSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 3 — zeta | 3/5 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 3 — limit — x | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 3 — limit — j | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 3 — limit — Total stored energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 3 — limit — Accumulated resistor work | 1/20 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 3 — limit — source Work | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 4 — family | stepSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 4 — zeta | 3/5 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 4 — limit — x | 1 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 4 — limit — j | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 4 — limit — Capacitor energy | 1/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 4 — limit — Inductor energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 4 — limit — Accumulated resistor work | 1/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 4 — limit — source Work | 2/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 5 — family | naturalSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 5 — zeta | 1 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 5 — limit — x | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 5 — limit — j | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 5 — limit — Total stored energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 5 — limit — Accumulated resistor work | 1/20 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 5 — limit — source Work | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 6 — family | stepSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 6 — zeta | 1 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 6 — limit — x | 1 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 6 — limit — j | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 6 — limit — Capacitor energy | 1/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 6 — limit — Inductor energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 6 — limit — Accumulated resistor work | 1/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 6 — limit — source Work | 2/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 7 — family | naturalSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 7 — zeta | 5/4 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 7 — limit — x | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 7 — limit — j | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 7 — limit — Total stored energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 7 — limit — Accumulated resistor work | 1/20 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 7 — limit — source Work | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 8 — family | stepSeries | Declared model condition | |
| Qualified limits and branch controls — asymptotic — case 8 — zeta | 5/4 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 8 — limit — x | 1 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 8 — limit — j | 0 | 1 | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 8 — limit — Capacitor energy | 1/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 8 — limit — Inductor energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 8 — limit — Accumulated resistor work | 1/25 | J | Exact ratio |
| Qualified limits and branch controls — asymptotic — case 8 — limit — source Work | 2/25 | J | Exact ratio |
| Qualified limits and branch controls — parallel Final — Capacitor energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — parallel Final — Inductor energy | 0 | J | Exact ratio |
| Qualified limits and branch controls — parallel Final — Accumulated resistor work | 149/2500 | J | Exact ratio |
| Qualified limits and branch controls — critical Repeated Root — zeta | 1 | 1 | Exact ratio |
| Qualified limits and branch controls — critical Repeated Root — root | -1 | 1 | Exact ratio |
| Qualified limits and branch controls — critical Repeated Root — root Multiplicity | 2 | 1 | Exact ratio |
| Qualified limits and branch controls — overdamped Roots — zeta | 5/4 | 1 | Exact ratio |
| Qualified limits and branch controls — overdamped Roots — root Plus | -1/2 | 1 | Exact ratio |
| Qualified limits and branch controls — overdamped Roots — root Minus | -2 | 1 | Exact ratio |
Focus a graph or table region to scroll it locally. Axis ticks are rounded drawing labels; the numerical equivalents retain exact ratios, full stored approximations and qualified limits.
| R_L | I_peak | Q_loaded | P_load | P_generator |
|---|---|---|---|---|
| 0 Ω | 2 A | 5 | 0 W | 4 W |
| 2 Ω | 1 A | 5/2 | 1 W | 2 W |
| 8 Ω | 2/5 A | 1 | 16/25 W | 4/5 W |
State where the lumped model stops
An ideal capacitor has fixed C and no leakage or dielectric memory; an ideal inductor has fixed L and no winding resistance, parasitic capacitance or magnetic saturation. A resistor has a frequency-independent ohmic law. The model is useful when these approximations are appropriate over the relevant frequencies, amplitudes and temperatures. A nominal part label alone does not establish that domain.
At higher frequencies, lead lengths and propagation can prevent one voltage or current from describing an entire element. Radiation or coupling to nearby structures can remove energy from the selected circuit. At larger amplitudes, source clipping, heating and nonlinear magnetic or dielectric behavior can shift peaks and generate harmonics. A single second-order transfer function cannot represent all of those effects, and a real high-Q peak need not remain symmetric or amplitude-independent.
The present finite steps and ideal switching are mathematical limits of a specified topology. They preserve state because the relevant integrated impulses vanish. An actual opening switch, incompatible ideal state constraint or imposed impulse changes the problem and may require parasitic fields or a distributional treatment. Endpoint energy and integrated jump laws are meaningful, but an undefined product of an impulse with a discontinuous state is not repaired by casually calling it instantaneous power.
Connect resonance with other systems and audit the answer
The series charge equation Lq̈+Rq̇+q/C=v_s has the same linear form as a mass-spring-damper equation. The analogy identifies inertia, restoring response and damping, but the measured quantity still matters: charge corresponds to displacement and current to velocity. The distinction explains why a charge maximum and a current maximum can occur at different frequencies without describing different underlying resonators.
Filters, sensors and mechanical modes use the same separation between natural dynamics and a driven observable. A narrow line can indicate weak loss only after source, measurement and mode assumptions are stated. Coupled resonators and distributed cavities have additional modes and boundary conditions; they are connections to broader physics, not silently included in the single-mode numerical cases here. The next AC lesson develops phase conventions, RMS power and general frequency-domain networks in their own scope.
For any RLC answer, identify the topology and orientation, both initial states, the future source, the physical time and frequency scales, the observable and the applicable limit. Then check KVL or KCL, state continuity or a justified jump law, signed instantaneous power and independently integrated work. Finally ask whether the claimed peak, bandwidth, settling time or quality factor belongs to that observable and domain. Those questions turn a resonance formula into an interpretable physical prediction.
Worked examples
A reversed natural current changes the storage-power direction
For L=1/2 H,C=1/200 F,R=12 Ω,v_C(0)=4 V and i(0)=−1/5 A, find the natural state and initial power ledger.
- The scales give u=20t,ζ=3/5,x₀=1,j₀=−1/2 and d=4/5. Thus B=(−1/2+3/5)/(4/5)=1/8.
- The state is x=e^(−3u/5)[cos(4u/5)+(1/8)sin(4u/5)] and j=e^(−3u/5)[−(1/2)cos(4u/5)−(7/8)sin(4u/5)]. Multiply by 4 V and 2/5 A respectively to recover physical values.
- At the initial instant v_R=−12/5 V and v_L=−8/5 V, whose sum with v_C=4 V is zero. Therefore di/dt=−16/5 A/s. The initially negative current is becoming more negative.
- Initial powers are p_C=−4/5 W,p_R=12/25 W and p_L=8/25 W. Their sum is zero. Initial storage is 1/20 J, which the positive resistor ultimately receives; the capacitor initially supplies both resistor heating and increasing inductor storage.
The response satisfies both signed initial states. Negative current does not make energy negative, and the capacitor and inductor initially have different power directions.
Two real roots need not produce two visible exponentials
Keep the natural preparation but set R=25 Ω. Determine which modes are present.
- Now ζ=5/4 and √(ζ²−1)=3/4, so the dimensionless roots are s_+=−1/2 and s_−=−2, corresponding to physical rates−10 and−40 s⁻¹.
- The coefficient equations are a_++a_−=1 and−a_+/2−2a_−=−1/2. Solving gives a_+=1,a_−=0.
- Hence v_C=4e^(−10t) V and i=−(1/5)e^(−10t) A. At t=0, v_R=−5 V and v_L=1 V, giving KVL−5+1+4=0.
- The state contains only the slow mode because of the specific initial current. Another initial current would generally excite both roots. Stored energy decreases as e^(−20t) for this single-mode preparation; that special result does not make every damped RLC energy a pure exponential.
The fast mode is absent by initial-condition cancellation. Root multiplicity, mode excitation and a visible trace are separate questions.
A critical step retains its repeated-root factor
For R=20 Ω and zero initial states, apply 4 V. Find the state at u=1 and the full-settling energy ledger.
- Critical damping means ζ=1. The zero-state solution is x=1−(1+u)e^(−u),j=ue^(−u), not a single exponential approach.
- At u=1, or t=1/20 s, v_C=4(1−2/e) V and i=2/(5e) A. The energies are U_C=(1−2/e)²/25 J and U_L=1/(25e²) J.
- Source work to this finite time is W_s=(2/25)(1−2/e) J. The independently integrated resistor result is W_R=[1−5/e²]/25 J; subtracting the two energies from source work gives the same expression.
- At complete settling U_C→1/25 J,U_L→0,W_R→1/25 J and W_s→2/25 J. These are limits, not exact values at u=1 or any other finite time.
The repeated-root factor determines the state, while finite work and final work remain distinct. The integrated half-energy result requires positive resistance and settling.
Carry a pulse state into a closed return loop
With R=12 Ω and zero initial states, remove the 4 V source at u_s=3/2 by replacing it with a closed zero-voltage connection. What is inherited and what jumps?
- The first-interval state is x_s=1−e^(−9/10)[cos(6/5)+(3/4)sin(6/5)] and j_s=(5/4)e^(−9/10)sin(6/5). The physical switch time is 3/40 s.
- For the off interval use w=u−3/2, f=0 and initial conditions x_s,j_s. Both states are continuous; keeping only x_s would discard magnetic memory.
- Before the switch j′=1−x_s−(6/5)j_s; afterward j′=−x_s−(6/5)j_s. Thus Δj′=−1,Δ(di/dt)=−8 A/s and Δv_L=−4 V.
- Source work supplied before removal is 2x_s/25 J and then stays fixed. As both states eventually decay to zero, total resistor work tends 2x_s/25 J. There is no concentrated switching-energy loss in this compatible finite-source step.
The new interval inherits two unrounded states; the derivatives and source/inductor powers have distinct one-sided values.
The same voltage equation has a different inductor energy
For the parallel network R=25/3 Ω,L=1/2 H,C=1/200 F, let v(0)=4 V and dv/dt(0)=−40 V/s. Find the branch currents and complete-decay energy.
- The normalized derivative is j₀=(dv/dt)/(80 V/s)=−1/2. Thus i_C=C dv/dt=−1/5 A and i_R=v/R=12/25 A.
- KCL fixes i_L=−i_C−i_R=−7/25 A. Its derivative is v/L=8 A/s. The initial inductor current is not an independent third state after v and its derivative have been specified.
- Initial energy is C v²/2+L i_L²/2=1/25+49/2500=149/2500 J. It is not the series normalized energy 1/20 J.
- Initial powers−4/5,48/25 and−28/25 W sum to zero. As the source-free positive-R network decays, resistor work tends 149/2500 J.
KCL and the physical branch mapping determine the parallel energy ledger; a shared voltage shape alone does not.
One circuit has different ringing and voltage-peak frequencies
For R=12 Ω, identify current resonance, damped ringing, capacitor and inductor voltage peaks, and the resistor-power half-width controls.
- The damping ratio is 3/5 and Q=5/6. Current and resistor power peak at ω=20 rad/s. Free ringing instead has ω_d=20(4/5)=16 rad/s.
- The capacitor peak has r_C=√7/5 and therefore ω_C=4√7 rad/s. The inductor peak has r_L=5/√7 and therefore ω_L=100/√7 rad/s. Their common peak voltage magnitude is 25/6 V.
- Half-power edges are r_±=(√34±3)/5, so ω_±=4(√34±3) rad/s. Their difference is 24 rad/s and their product in normalized coordinates is 1.
- At current resonance the peak current is 1/3 A, P_max=2/3 W and each reactive voltage peak is 10/3 V. These are not the separate capacitor/inductor maximum voltages. Q=20/24=5/6 verifies the exact current-power bandwidth relation.
The distinct frequencies answer distinct questions. The exact bandwidth belongs to resistor power under the fixed-amplitude, constant-R series experiment.
A singular steady formula has a finite resonant initial-value solution
Start the lossless circuit from zero states with v_s=4 cos(20t) V. Find the states at u=2π and 4π and compare energies.
- At resonance x=(u/2)sin u and j=(sin u+u cos u)/2. These satisfy x(0)=j(0)=0 and the equation x″+x=cos u.
- At u=2π, t=π/10 s, x=0 and j=π. Hence v_C=0 and i=2π/5 A. Storage is entirely magnetic, U=π²/25 J.
- At u=4π, t=π/5 s, x=0 and j=2π, so i=4π/5 A and U=4π²/25 J. Doubling the time between these equal-phase observations quadruples the stored energy.
- The independent source-work integral has the same phase values, while resistor work is zero. This is finite-time secular growth, not an infinite finite-time voltage or a bounded attracting sinusoid. Its unrestricted late-time growth is an ideal-model limitation.
The steady-sinusoid ansatz fails at zero-loss resonance; the initial-value solution remains finite at every finite time.
The largest Q does not give the largest useful load power
At resonance, compare R_L=0,2,8 Ω for an ideal 4 V peak source with R_s=R_i=1 Ω.
- Total resistances are 2,4,10 Ω, so peak currents are 2,1,2/5 A. Generator mean deliveries are 4,2,4/5 W.
- Mean useful load powers R_L I_peak²/2 are 0,1,16/25 W. Each 1 Ω source/internal resistor consumes 2,1/2,2/25 W respectively, completing each power ledger.
- Loaded Q values are 5,5/2,1. Intrinsic Q based only on R_i is 10; source and load coupling add loss even though intrinsic resistance is unchanged.
- The function P_L=8R_L/(2+R_L)² increases up to R_L=2 Ω and then decreases. At this matched load P_L=1 W is half the ideal generator delivery 2 W. The unloaded highest-Q case delivers no useful load power.
Source resistance, internal loss and useful load must remain separate. Resonant current enhancement and maximum delivered power are different design objectives.
A slow cycle and a DC state store different average energies
Why does the positive-frequency limit of average capacitor energy give 1/50 J while a settled 4 V DC preparation gives 1/25 J?
- For every positive frequency, the squared capacitor voltage averages to half its squared peak. Thus ⟨U_C⟩=C V_C,peak²/4=1/(50D) J.
- As r→0+ the peak tends 4 V and D→1, so the cycle-averaged energy tends 1/50 J. The averaging interval grows without bound.
- At exactly ω=0, the source is 4 V at all times. After positive-R settling, U_C=C(4 V)²/2=1/25 J; there is no phase sweep and no sinusoidal factor one-half.
- Taking the positive-frequency cycle average before the limit and setting frequency to zero before solving the static state therefore ask different questions. Neither value is a rounding error or a change of capacitance.
The operations differ. A DC row must contain the static energy, while the low-frequency cycle-average limit needs its own label.
Try it yourself
Recover ω₀,Z₀,I* and the physical time corresponding to u=3/2 from the original L,C and voltage scale.
Show the answer and reasoning
ω₀=20 rad/s,Z₀=10 Ω,I*=2/5 A and t=3/40 s.
Use ω₀=1/√(LC),Z₀=√(L/C),I*=Cω₀V* and t=u/ω₀. The normalized coordinate has no time unit. In particular, treating u=3/2 as 1.5 s would stretch the physical history by a factor of 20 and change all physical derivatives.
For the R=12 Ω natural preparation, determine initial di/dt,p_C,p_L and p_R. Is the inductor initially delivering energy?
Show the answer and reasoning
di/dt=−16/5 A/s,p_C=−4/5 W,p_L=8/25 W,p_R=12/25 W. The inductor initially absorbs energy.
KVL gives v_L=−v_C−Ri=−8/5 V. Dividing by L gives the derivative, and multiplying v_L by the negative current gives positive p_L. Power sign depends on both voltage and current; negative current alone does not imply delivery. The three powers sum to zero.
Why is x=1−e^(−u) not the zero-state critical step response even though it approaches the correct final voltage?
Show the answer and reasoning
Its derivative at zero is 1, contradicting j₀=0. The correct response is x=1−(1+u)e^(−u).
The final value does not impose the second initial condition. A repeated characteristic root supplies two independent functions e^(−u) and ue^(−u). Solving both initial conditions gives j=ue^(−u), which begins at zero and has j′(0+)=1.
In the R=25 Ω natural case, verify the absence of the root −2 mode. Does the circuit become first order?
Show the answer and reasoning
a_+=1,a_−=0 for x₀=1,j₀=−1/2. The system remains second order.
Solve a_++a_−=1 and−a_+/2−2a_−=−1/2. The preparation lies along one eigenmode, so only that exponential appears. Changing the initial current would generally restore the other coefficient; neither L nor C has been removed.
Compare complete resistor work in the natural series case for R=12 Ω and R=0. Why can the limits differ?
Show the answer and reasoning
It is 1/20 J for R=12 Ω and exactly 0 for R=0.
The initial energy is Cv₀²/2+Li₀²/2=1/20 J. Every selected positive R eventually dissipates it, but a lossless closed loop retains it. At fixed finite time the response varies continuously with damping, while the infinite-time energy destination need not commute with the zero-damping limit.
Find source work, resistor work and final storage after the zero-state 4 V step has settled for positive R. Can these values be assigned to the R=0 final state?
Show the answer and reasoning
W_s=2/25 J,W_R=1/25 J,U_final=1/25 J. The lossless step has no settled final state.
Source work is V_s times delivered charge, CV_s²; final capacitor storage is CV_s²/2. Their difference is resistor work, consistent with the independent integral. At R=0 the capacitor voltage oscillates between 0 and 8 V, so importing the settled ledger would assert a limit that does not exist.
When the pulse source drops from 4 V to 0 through a closed return path, which states and derivatives jump?
Show the answer and reasoning
v_C and i are continuous; Δ(di/dt)=−8 A/s and Δv_L=−4 V.
The source has a finite step, not a voltage-time impulse, so the integrated state laws forbid a storage-state jump. Subtract the two KVL equations at the same inherited x_s,j_s. Only the source and inductor voltage change instantaneously; their bounded one-sided power change is not finite concentrated energy.
Why would using U=(x²+j²)/25 J in the parallel case give the wrong initial energy?
Show the answer and reasoning
There j is the normalized voltage derivative, not normalized inductor current. The correct initial energy is 149/2500 J.
KCL gives i_L=−(2/5)j−(12/25)x. At x=1,j=−1/2 this is−7/25 A, so U_C=1/25 J and U_L=49/2500 J. The same voltage differential equation does not identify the physical energy coordinates.
At ζ=1, should the capacitor-voltage peak be assigned the value √(1−2ζ²) for its frequency? What does the response actually do?
Show the answer and reasoning
No positive interior peak exists. The capacitor amplitude is largest at the DC boundary r=0 and equals 4 V there.
A stationary positive frequency requires r²=1−2ζ²>0. At ζ=1 the right side is negative; this is a failed existence condition, not an imaginary physical driving frequency. The damped-root and driven-peak domains must be classified before evaluating a square root.
For ζ=1, is the high-frequency inductor amplitude of 4 V a finite-frequency maximum?
Show the answer and reasoning
It is an asymptotic supremum, not a finite interior maximum.
The squared inductor response is r⁴/D and tends 1 as r→∞. At this damping it increases toward that limit without attaining an interior peak. A finite last point is neither infinity nor the exact supremum. The physical prediction also remains subject to the lumped model’s high-frequency limits.
For ζ=3/5, find the two positive half-power ratios and explain their symmetry.
Show the answer and reasoning
r_±=(√34±3)/5, r_+r_−=1 and r_+−r_−=6/5.
Solve |1−r²|=2ζr on the two sides of 1. Their reciprocal product makes them symmetric on a logarithmic axis; their arithmetic midpoint is√34/5, not 1. The width gives Δω=20(6/5)=24 rad/s. Half power means a current amplitude factor 1/√2, not 1/2.
For ζ=1/10, write the exact equal-phase free-cycle energy ratio and the leading-cycle energy quotient. Must that quotient equal Q=5?
Show the answer and reasoning
The ratio is exp(−4π/√99); the quotient is 2π/[1−exp(−4π/√99)], which is not exactly 5.
The damped period is π/√99 s and α=2 s⁻¹. State amplitude scales by exp(−αT_d), so quadratic energy scales by exp(−2αT_d). Replacing a finite cycle difference by a continuous small-loss rate is an approximation. The separate driven-resonance rate ratio ω₀U/P_mean does equal Q exactly in the stated ideal circuit.
Why does the r=1/2 lossless solution retain a cosine at the natural frequency even after a long wait?
Show the answer and reasoning
It is the homogeneous component required to satisfy the zero initial conditions, and no resistance damps it.
The solution x=[cos(u/2)−cos u]/(3/4) contains both forcing and natural frequencies. Omitting cos u would change the initial voltage and generally the current. A bounded particular solution can exist without being an attracting solution; beating is not by itself evidence of dissipative ringdown.
In the zero-state lossless resonant case, compare the energy at u=2π and 4π. Does this imply infinite energy at either time?
Show the answer and reasoning
The energies are π²/25 and 4π²/25 J; both are finite.
At these phases x=0 and j=u/2, so U=j²/25=u²/100 J at those observations. The factor of four is secular growth across equal-phase points. It does not state that energy is a pure u² function at every intervening phase, nor does it establish a bounded steady sinusoid or an unlimited real source.
Which selected load gives maximum useful power, and which gives largest loaded Q? Why are they different?
Show the answer and reasoning
R_L=2 Ω gives 1 W maximum useful power; R_L=0 gives the largest loaded Q,5, but zero useful load power.
With fixed ideal source and R_s+R_i=2 Ω, P_L=8R_L/(2+R_L)² peaks at matching resistance 2 Ω. Q_loaded=10/(2+R_L) instead decreases monotonically with load resistance. The source/internal losses remain in both calculations; comparing only Q omits the objective of useful power delivery.
State the two initial terms in the unilateral normalized Laplace equation, then explain why a low-frequency cycle average is not DC storage.
Show the answer and reasoning
The initial contribution is(p+2ζ)x₀+j₀. For r→0+ the average capacitor energy tends 1/50 J, whereas a settled constant 4 V source gives 1/25 J.
Transforming x″ and x′ retains both inherited states, so the numerator cannot be reduced to the input unless both are zero. Separately, positive-frequency averaging samples the square of a sinusoid and gives a factor one-half. Setting frequency to zero first removes the phase cycle. Both examples show why a formal operation must retain the initial and limiting conditions of the question.
What to carry forward
- RLC motion generally has two independent storage states: capacitor voltage and inductor current.
- For the ideal series circuit, ω₀=1/√(LC),α=R/(2L),ζ=α/ω₀ and Q=1/(2ζ); a parallel resistor changes the damping relation.
- The critical branch needs the repeated-root factor, and a particular preparation can cancel one of two distinct modes.
- State continuity follows from integrated element laws and the absence of the relevant impulse, not from a universal rule about switches.
- Signed storage power, positive resistor loss and source-delivered work belong to one complete finite-window ledger.
- A shared normalized voltage equation does not make series and parallel energy coordinates identical.
- Current/power resonance, reactive-voltage peaks and damped ringing are observable-specific frequencies.
- Current-power bandwidth Δω=R/L is exact for the stated fixed-voltage, constant-series-R model; the charge-response Lorentzian width is narrowband.
- Q from a driven energy-loss rate and a free finite-cycle energy difference require their respective definitions and regimes.
- DC storage and the low-frequency limit of a cycle average are different questions.
- Lossless resonance has a finite secular initial-value solution at each finite time, without a bounded attracting steady sinusoid.
- Source and load resistances change current, delivered power and loaded Q; the largest Q need not maximize useful power.
- Constant L,C,R and ideal switching are model conditions, not universal component behavior or empirical performance guarantees.
References
Sources for the concepts and teaching approaches in this article.
Massachusetts Institute of Technology4 sources
- MIT 6.002: Second-Order SystemsAnant Agarwal and Jeffrey Lang, Massachusetts Institute of Technology / MIT OpenCourseWareCourse notes · PDF
- MIT 6.002: Damped Second-Order SystemsMassachusetts Institute of Technology / MIT OpenCourseWareCourse notes · PDF
- MIT 6.200: Driven RLC Networks, Lecture 17Massachusetts Institute of Technology, 6.200 course materialsCourse notes · PDF
- MIT 6.200: Rederiving Relationships — RLC Circuits and QMassachusetts Institute of Technology, 6.200 Spring2026 course materialsCourse notes
California Institute of Technology2 sources
- The Feynman Lectures on Physics, Volume I, Chapter 23: ResonanceRichard P. Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on Physics, Caltech online editionTextbook chapter
- The Feynman Lectures on Physics, Volume II, Chapter 22: AC CircuitsRichard P. Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on Physics, Caltech online editionTextbook chapter