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Diodes and rectifiers

Choose a diode model, solve its operating state and trace rectifier energy from source to load. Derive mean and RMS output, finite-resistance capacitor charging, actual switching events and loaded reverse-clamp regulation while preserving startup, device and thermal limits.

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After reading, you can

  • Define diode terminal signs and test ideal conducting or blocked states.
  • Explain depletion and equilibrium without treating the built-in potential as a battery.
  • Differentiate the fixed-temperature junction law and distinguish static and differential resistance.
  • Solve the original monotone load line and close its signed power balance.
  • Derive half-wave and bridge mean, RMS, ripple and source true power factor.
  • Separate DC-component fraction, diode loss and useful-output efficiency.
  • Solve finite-source-resistance reservoir trajectories, events and actual capacitor extrema.
  • Compare periodic ripple with its approximation while keeping startup separate.
  • Verify independent source, resistor, load, storage and charge ledgers.
  • Test loaded reverse-clamp headroom and identify dynamic, material and thermal model limits.
01

Start with the load and a one-way path

A sinusoidal source changes polarity. A load that needs a positive voltage therefore asks a different question from the linear AC networks of the previous lesson: can the connections send current through the load in one direction during both source polarities? A diode makes the circuit depend on its operating state. Its symbol alone does not tell us the resulting voltage, current or energy transfer.

Begin with a resistor and an ideal diode. During the permitted half-cycle the load follows the source; during the other half it receives no current. A bridge can use both half-cycles. The output then has a positive mean, but it still varies strongly. Rectification and smoothing are different operations, and neither follows from calling the output DC.

We will first use simple switch-like models, then return to a junction current law and the load line that fixes an actual operating point. The advanced path adds finite source resistance and capacitor storage, solves the turn-on and turn-off events, and keeps startup distinct from a periodic state. The final regulator example tests whether a nominal voltage clamp has enough current to operate.

02

Choose a first reading and an advanced return

For a first reading, follow the terminal signs, model ladder, load-line argument, half-wave and bridge paths, mean and RMS definitions, and the qualitative capacitor and regulator discussions. Circuit foundations supplies signed port work and Kirchhoff laws; DC networks supplies the node and load-line logic. RC transients supplies the capacitor state and discharge law.

Return to the junction mechanism, differential resistance, clipped-waveform integrals, exact charging events and independent energy and charge ledgers. The AC lesson explains why source power factor uses the actual RMS current and average real product, rather than a single phase angle for a distorted waveform. RLC assumptions also warn us that a source inductance would enlarge the present capacitor model.

Every numerical case is an original stated model. The exponential junction, ideal switching diode, constant forward drop and ideal reverse clamp are separate approximations. None is silently substituted into another model. We do not infer a real device rating, a temperature rise, a certified supply design or a measured material parameter from these synthetic cases.

03

Define anode, cathode and the power signs

The bar on the diode symbol marks its cathode K; the other terminal is the anode A. Define v_D=V_A−V_K and positive i_D from anode to cathode. The absorbed terminal power is p_D=v_D i_D. Forward bias means positive anode-to-cathode voltage; reverse bias means negative voltage. Conventional current and electron motion should not be confused when interpreting that reference.

An ideal diode permits i_D≥0 while its conducting voltage is zero, and permits v_D≤0 while its blocked current is zero. Together these are i_D≥0, v_D≤0 and i_Dv_D=0. They are constraints to solve with the rest of the circuit. A guessed conducting state that needs negative forward current is inconsistent and must be rejected.

For the source we use delivered current i_S and delivered power p_S=v_S i_S. This is deliberately opposite to the source absorbed-power sign used in a complete passive branch ledger. The distinction is explicit wherever a source appears. A reverse-leakage exponential diode has both v_D<0 and i_D<0, so its absorbed power can be positive; a negative current coordinate alone does not imply energy generation.

04

Understand the junction without inventing a battery

In a p–n junction, mobile carriers diffuse from regions of higher concentration. Electrons and holes recombine near the boundary, leaving exposed ionized dopants and a depletion region with an electric field. At thermal equilibrium, drift and diffusion currents balance, giving no net terminal current. The built-in electrostatic potential is part of that equilibrium, not a freely extractable battery at the external terminals.

Applied forward bias lowers the effective barrier and changes carrier injection; reverse bias increases the depletion barrier and suppresses the ordinary majority-carrier contribution. A small reverse current can remain. The quantitative current law depends on the material, doping, geometry, recombination and temperature. A drawing of a barrier explains a mechanism but does not establish a universal turn-on voltage.

The ideal-junction equation used below summarizes a restricted quasistatic regime. It does not resolve the spatial carrier distributions, large-injection effects, series resistance or breakdown. The richer semiconductor transport treatment belongs to later device coverage. Here the junction mechanism explains why the terminal law is nonlinear and why the model must be chosen before the circuit is solved.

05

Choose a model and keep its conditions

The ideal model is useful when conduction direction and topology dominate the question and forward loss is negligible at the scale of interest. The constant-drop model instead takes v_D=V_γ while conducting and i_D=0 below its allowed forward threshold. It captures a specified voltage loss without pretending to describe the detailed curve or its local slope.

The exponential model assigns a continuous current to a fixed-temperature junction. It is useful for operating points and small-signal slopes within its stated regime. The ideal Zener model later supplies a separate reverse-voltage clamp with a nonnegative clamp current; it is not an extension of the forward exponential formula into breakdown.

“Silicon means 0.7 V” is not a constitutive law. Forward voltage varies with current, temperature, structure and the model accuracy required. A particular constant drop is an input approximation, not an observed universal threshold. Similarly, a zero-current ideal switch is not a reason to erase the reverse leakage of an exponential model.

Separate diode models used in the original cases
ModelConducting or current relationBoundary of the claim
Ideal diodei_D≥0, v_D≤0, i_Dv_D=0Direction and topology; no forward loss, leakage or recovery
Specified forward dropv_D=V_γ in forward conductionPiecewise approximation; not the exponential slope
Fixed-temperature junctioni_D=I_s[exp(v_D/(nV_T))−1]Before breakdown and strong series/high-injection effects
Ideal reverse clampOutput at V_Z only when clamp current is nonnegativeSeparate loaded regulation model; no actual rating
06

Derive current and differential resistance

At fixed temperature, write i_D=I_s[exp(v_D/a)−1], where a=nV_T and V_T=k_BT/e. Our original parameters are T=300 K, n=3/2 and I_s=2 nA. The SI constants are k_B=1.380649×10⁻²³ J/K and e=1.602176634×10⁻¹⁹ C. Thus a has voltage units; the exponent v_D/a is dimensionless.

Differentiate while keeping those parameters fixed: g_d=di_D/dv_D=(I_s/a)exp(v_D/a)=(i_D+I_s)/a. Its reciprocal r_d=a/(i_D+I_s) is the local differential resistance. Only when i_D≫I_s may we simplify it to a/i_D. This resistance predicts a sufficiently small perturbation about the chosen operating point; it does not replace the entire curved characteristic by a line through the origin.

The static ratio v_D/i_D answers another question and is undefined at the exact zero-current, zero-voltage point. The differential resistance there is finite, a/I_s. At reverse voltages the model current approaches −I_s while the differential conductance becomes small. Extending that asymptote into avalanche or tunnelling breakdown would misuse the model, rather than predict a device surviving any reverse voltage.

Original fixed-temperature junction; approximations shown to six significant digits
v_D (V)i_D (A)g_d (S)r_d (Ω)
-0.2-1.98849 e-92.96854 e-103.36866 e+9
-0.1-1.84827 e-93.91286 e-92.55568 e+8
005.15756 e-81.93890 e+7
0.30.000004578180.0001181138466.49
0.450.0002191820.00565227176.920
0.550.002889070.074503013.4223
0.60.01048900.2704893.69701
0.650.03808110.9820291.01830
0.70.1382563.565330.280479
07

Linearize locally and keep the temperature boundary

For a small perturbation around (V_D, I_D), Taylor expansion gives δi≈g_d δv. The neglected terms depend on δv/a; a variation small compared with a is a natural requirement for the first-order approximation, in addition to remaining within the physical regime of the junction law. A small-signal model is a local statement about a bias point, not another global turn-on threshold.

The quoted n and I_s are held fixed in the numerical curve. In a real diode, I_s can depend strongly on temperature and n can depend on operating regime. Differentiating the fixed-parameter formula with respect to voltage does not supply a full temperature coefficient. In particular, conclusions from changing V_T alone can fail when the other parameters change.

Terminal capacitance and stored minority-carrier charge also become relevant when the perturbation is fast. A quasistatic differential resistor cannot reproduce those dynamics. We therefore use the slope for a DC operating point and sufficiently slow local changes, then discuss the omitted dynamic and thermal processes explicitly at the end.

08

Let the circuit choose the operating point

Connect the exponential diode in series with R=330 Ω and a signed source V_S. Kirchhoff’s voltage relation is V_S=Ri_D+v_D. It must hold together with the diode current law. The intersection is the operating point; selecting a forward voltage first and then ignoring either equation does not solve the circuit.

Substituting the junction law gives F(v_D)=R I_s[exp(v_D/a)−1]+v_D−V_S. Its derivative is Rg_d+1>0, so there is at most one root. For V_S≥0 the root is bracketed by 0≤v_D≤V_S; for V_S<0 it lies between V_S and zero. Continuity and the endpoint signs give existence within those intervals for this model.

Equivalently, V_S=Ri_D+a ln(1+i_D/I_s) for i_D>−I_s. Its derivative R+a/(i_D+I_s) is positive. This current-domain form makes the reverse-current boundary explicit. The exact V_S=0 case has v_D=i_D=0. In the negative-source cases, leakage remains small and negative; it is not replaced by the ideal blocked-switch value.

Original series load line; every voltage, current and power uses the declared signs, six significant digits
V_S (V)v_D (V)i_D (A)Source delivered (W)Resistor absorbed (W)Diode absorbed (W)
-1-0.999999-2.00000 e-92.00000 e-91.32000 e-152.00000 e-9
-0.25-0.249999-1.99683 e-94.99207 e-101.31582 e-154.99206 e-10
000000
0.250.2495890.000001246253.11562 e-75.12534 e-103.11049 e-7
10.5231350.001445050.001445050.0006890920.000755954
30.5860250.007315080.02194520.01765840.00428682
60.6171230.01631170.09787050.08780410.0100664
09

Close the signed DC power ledger

Multiplying V_S=Ri_D+v_D by i_D gives p_S=Ri_D²+v_Di_D. The resistor term is nonnegative, and the diode term is its absorbed power under our anode-to-cathode reference. In the original reverse cases, the source voltage and delivered current are both negative, so their product is positive. The source still delivers the tiny leakage loss.

The nonlinearity explains why replacing the whole diode by r_d in the DC equation would be wrong: r_d is the tangent slope, while the DC point has a separate voltage and current intercept. It is legitimate to linearize the complete circuit around its solved point and use R+r_d for the appropriate incremental relation; that does not reconstruct the original large-signal voltage.

The monotonic uniqueness result depends on the selected passive resistor and junction law. It is not a theorem about every circuit containing a diode. Feedback, reactive states, negative differential resistance or coupled thermal dynamics can change the dynamical or algebraic problem and require a new existence, stability and operating-domain analysis.

A diode curve and the points selected by its circuit

The current law is fixed at 300 K with the stated empirical parameters. Forward and reverse current scales are separate. Loaded points are isolated operating conditions, not a connected trajectory. The reverse I–V grid covers −0.2 to 0 V; the separate loaded controls extend to −1 V under the same no-breakdown assumption.

Forward junction current and isolated loaded points

The retained fixed-temperature forward I–V samples and nonnegative-source operating points share the same V/A axes. Local gd and rd, and all seven source conditions, are in the complete tables.

  • Retained forward I–V grid
  • Isolated series-load operating points
Forward junction current and isolated loaded pointsThe retained fixed-temperature forward I–V samples and nonnegative-source operating points share the same V/A axes. Local gd and rd, and all seven source conditions, are in the complete tables.Diode current iD (A)00.20.40.600.0250.050.0750.10.1250.15Diode anode-to-cathode voltage vD (V)
The retained fixed-temperature forward I–V samples and nonnegative-source operating points share the same V/A axes. Local gd and rd, and all seven source conditions, are in the complete tables.

Reverse leakage stays nonzero on its own current scale

Current is shown in nA. The retained I–V grid reaches −0.2 V; isolated negative-source load controls extend farther left. There is no claim of breakdown behavior or measured device ratings.

  • Retained I–V grid, −0.2 to 0 V
  • Isolated series-load operating points
Reverse leakage stays nonzero on its own current scaleCurrent is shown in nA. The retained I–V grid reaches −0.2 V; isolated negative-source load controls extend farther left. There is no claim of breakdown behavior or measured device ratings.Diode current iD (nA = 10⁻⁹ A)-1-0.75-0.5-0.250-2-1.5-1-0.50Diode anode-to-cathode voltage vD (V)
Current is shown in nA. The retained I–V grid reaches −0.2 V; isolated negative-source load controls extend farther left. There is no claim of breakdown behavior or measured device ratings.

Short fractions are exact. A leading ≈ marks a value rounded to 12 significant digits. The plots use the labelled units and scales.

Fixed-temperature junction parameters: chosen empirical inputs and exact SI constants are different kinds of information.
Fixed-temperature junction parameters: chosen empirical inputs and exact SI constants are different kinds of information.
QuantityValueSI unit
Boltzmann constant≈ 1.380649e-23J/K
Elementary charge≈ 1.602176634e-19C
Fixed junction temperature300K
Chosen ideality factor3/21
Chosen saturation current1/500000000A
Thermal voltage≈ 0.0258519997864V
n times thermal voltage≈ 0.0387779996797V
Every original fixed-temperature I–V sample, including nonzero reverse leakage and local differential slope.
Every original fixed-temperature I–V sample, including nonzero reverse leakage and local differential slope.
vD (V)iD (A)gd (S)rd (Ω)
-1/5≈ -1.98848858847e-9≈ 2.96854185906e-10≈ 3368657231.32
-19/100≈ -1.98510214816e-9≈ 3.84183092438e-10≈ 2602925583.36
-9/50≈ -1.98071948094e-9≈ 4.97202517340e-10≈ 2011252890.17
-17/100≈ -1.97504751562e-9≈ 6.43470126913e-10≈ 1554073698.49
-4/25≈ -1.96770696501e-9≈ 8.32766910443e-10≈ 1200816203.74
-3/20≈ -1.95820696276e-9≈ 1.07775123992e-9≈ 927857897.968
-7/50≈ -1.94591223890e-9≈ 1.39480534177e-9≈ 716945920.732
-13/100≈ -1.93000063901e-9≈ 1.80513078472e-9≈ 553976481.075
-3/25≈ -1.90940814632e-9≈ 2.33616623937e-9≈ 428051729.859
-11/100≈ -1.88275773040e-9≈ 3.02342231608e-9≈ 330751015.060
-1/10≈ -1.84826726440e-9≈ 3.91285617750e-9≈ 255567788.499
-9/100≈ -1.80363035337e-9≈ 5.06394471733e-9≈ 197474509.660
-2/25≈ -1.74586210441e-9≈ 6.55366181044e-9≈ 152586451.502
-7/100≈ -1.67109952540e-9≈ 8.48162559488e-9≈ 117901926.796
-3/50≈ -1.57434320473e-9≈ 1.09767599874e-8≈ 91101563.7718
-1/20≈ -1.44912299812e-9≈ 1.42059158911e-8≈ 70393208.5523
-1/25≈ -1.28706536683e-9≈ 1.83850286002e-8≈ 54392082.9141
-3/100≈ -1.07733343479e-9≈ 2.37935575024e-8≈ 42028183.4651
-1/50≈ -8.05902321274e-10≈ 3.07931736704e-8≈ 32474729.9744
-1/100≈ -4.54621289958e-10≈ 3.98519449896e-8≈ 25092878.1584
00≈ 5.15756360958e-8≈ 19388999.8398
1/100≈ 5.88362304986e-10≈ 6.67482161630e-8≈ 14981673.7807
1/50≈ 1.34980971094e-9≈ 8.63842833206e-8≈ 11576179.8508
3/100≈ 2.33526059233e-9≈ 1.11796911345e-7≈ 8944790.94250
1/25≈ 3.61061254974e-9≈ 1.44685455570e-7≈ 6911544.74416
1/20≈ 5.26114901581e-9≈ 1.87249189638e-7≈ 5340477.05056
3/50≈ 7.39724220170e-9≈ 2.42334372049e-7≈ 4126529.76770
7/100≈ 1.01617337429e-8≈ 3.13624576908e-7≈ 3188525.62468
2/25≈ 1.37394865916e-8≈ 4.05887016392e-7≈ 2463739.80840
9/100≈ 1.83697468968e-8≈ 5.25291326657e-7≈ 1903705.52349
1/10≈ 2.43621425149e-8≈ 6.79822134527e-7≈ 1470972.99310
11/100≈ 3.21173879821e-8≈ 8.79812993552e-7≈ 1136605.17329
3/25≈ 4.21540804987e-8≈ 0.00000113863739397≈ 878242.718264
13/100≈ 5.51433787871e-8≈ 0.00000147360305480≈ 678608.799527
7/50≈ 7.19538838160e-8≈ 0.00000190710929978≈ 524353.795618
3/20≈ 9.37097225884e-8≈ 0.00000246814491152≈ 405162.596139
4/25≈ 1.21865719084e-7≈ 0.00000319422662611≈ 313064.825089
17/100≈ 1.58304679079e-7≈ 0.00000413390789631≈ 241901.857778
9/50≈ 2.05463294320e-7≈ 0.00000535002568554≈ 186914.990465
19/100≈ 2.66495085343e-7≈ 0.00000692390240758≈ 144427.223426
1/5≈ 3.45481278988e-7≈ 0.00000896078399759≈ 111597.378117
21/100≈ 4.47703722110e-7≈ 0.0000115968777612≈ 86230.1061189
11/50≈ 5.79998081361e-7≈ 0.0000150084606264≈ 66629.0850804
23/100≈ 7.51210947684e-7≈ 0.0000194236668706≈ 51483.5847764
6/25≈ 9.72791412344e-7≈ 0.0000251377435762≈ 39780.8179151
1/4≈ 0.00000125955667347≈ 0.0000325327939525≈ 30738.2145370
13/50≈ 0.00000163068286960≈ 0.0000421033287713≈ 23751.0911651
27/100≈ 0.00000211098739784≈ 0.0000544893345530≈ 18352.2153134
7/25≈ 0.00000273258846574≈ 0.0000705190697903≈ 14180.5614137
29/100≈ 0.00000353705285218≈ 0.0000912644510140≈ 10957.1688526
3/10≈ 0.00000457817549897≈ 0.000118112732395≈ 8466.48773357
31/100≈ 0.00000592557680588≈ 0.000152859272135≈ 6541.96494615
8/25≈ 0.00000766935818212≈ 0.000197827588981≈ 5054.90667481
33/100≈ 0.00000992612717332≈ 0.000256024737102≈ 3905.87257825
17/50≈ 0.0000128467950673≈ 0.000331342389329≈ 3018.02616328
7/20≈ 0.0000166266684083≈ 0.000428817075292≈ 2331.99668954
9/25≈ 0.0000215185092451≈ 0.000554966976710≈ 1801.90901795
37/100≈ 0.0000278494374570≈ 0.000718227801514≈ 1392.31591689
19/50≈ 0.0000360428054267≈ 0.000929516883915≈ 1075.82768781
39/100≈ 0.0000466465078285≈ 0.00120296323209≈ 831.280602205
2/5≈ 0.0000603696196236≈ 0.00155685234211≈ 642.321672359
41/100≈ 0.0000781298122623≈ 0.00201484895837≈ 496.315118731
21/50≈ 0.000101114718840≈ 0.00260757954704≈ 383.497408980
43/100≈ 0.000130861351725≈ 0.00337468030341≈ 296.324365597
11/25≈ 0.000169358883354≈ 0.00436744764436≈ 228.966682930
9/20≈ 0.000219181663206≈ 0.00565226842583≈ 176.920118554
23/50≈ 0.000283661365856≈ 0.00731505926554≈ 136.704292296
47/100≈ 0.000367109781743≈ 0.00946701183083≈ 105.629951443
12/25≈ 0.000475107148790≈ 0.0122520282819≈ 81.6191390513
49/100≈ 0.000614875305741≈ 0.0158563440822≈ 63.0662399109
1/2≈ 0.000795760620186≈ 0.0205209816587≈ 48.7306122404
51/100≈ 0.00102985898490≈ 0.0265578676933≈ 37.6536253418
13/25≈ 0.00133282467635≈ 0.0343706918191≈ 29.0945554795
53/100≈ 0.00172491716407≈ 0.0444819015504≈ 22.4810533081
27/50≈ 0.00223235587171≈ 0.0575676386135≈ 17.3708705808
11/20≈ 0.00288907348319≈ 0.0745029528871≈ 13.4222867853
14/25≈ 0.00373898503847≈ 0.0964203174316≈ 10.3712581190
57/100≈ 0.00483892455461≈ 0.124785357537≈ 8.01376074673
29/50≈ 0.00626244554527≈ 0.161494857832≈ 6.19214762269
59/100≈ 0.00810473958157≈ 0.209003601231≈ 4.78460655277
3/5≈ 0.0104890018007≈ 0.270488521516≈ 3.69701455129
61/100≈ 0.0135746690273≈ 0.350061146511≈ 2.85664378914
31/50≈ 0.0175680813948≈ 0.453042538035≈ 2.20729824695
63/100≈ 0.0227362804150≈ 0.586319114002≈ 1.70555585878
16/25≈ 0.0294248661792≈ 0.758803146688≈ 1.31786485647
13/20≈ 0.0380811078121≈ 0.982028730896≈ 1.01830014595
33/50≈ 0.0492838525848≈ 1.27092307473≈ 0.786829682993
67/100≈ 0.0637822337259≈ 1.64480468959≈ 0.607974920263
17/25≈ 0.0825457653403≈ 2.12867522879≈ 0.469775749007
69/100≈ 0.106829174310≈ 2.75489136088≈ 0.362990720505
7/10≈ 0.138256304517≈ 3.56532847642≈ 0.280479065706
All seven isolated source/load operating points. Different source conditions are not a connected trajectory.
All seven isolated source/load operating points. Different source conditions are not a connected trajectory.
Source VS (V)R (Ω)vD (V)iD (A)gd (S)rd (Ω)Source delivered (W)Resistor absorbed (W)Diode absorbed (W)
-1330≈ -0.999999340000≈ -1.99999999999e-9≈ 3.25795368501e-19≈ 3.06941134431e+18≈ 1.99999999999e-9≈ 1.31999999998e-15≈ 1.99999867999e-9
-1/4330≈ -0.249999341046≈ -1.99682926019e-9≈ 8.17664614068e-11≈ 12229953244.8≈ 4.99207315046e-10≈ 1.31581794113e-15≈ 4.99205999228e-10
033000≈ 5.15756360958e-8≈ 19388999.8398000
1/4330≈ 0.249588738201≈ 0.00000124624787636≈ 0.0000321895891142≈ 31065.9448448≈ 3.11561969090e-7≈ 5.12534143878e-10≈ 3.11049434946e-7
1330≈ 0.523134833044≈ 0.00144504596047≈ 0.0372646338751≈ 26.8350952635≈ 0.00144504596047≈ 0.000689092083199≈ 0.000755953877273
3330≈ 0.586024808038≈ 0.00731507633928≈ 0.188639909219≈ 5.30110518044≈ 0.0219452290178≈ 0.0176584128103≈ 0.00428681620751
6330≈ 0.617122748859≈ 0.0163117492459≈ 0.420644473171≈ 2.37730450245≈ 0.0978704954753≈ 0.0878041439420≈ 0.0100663515333

Related complete original table: Separate diode models used in the original cases

Related complete original table: Original fixed-temperature junction; approximations shown to six significant digits

Related complete original table: Original series load line; every voltage, current and power uses the declared signs, six significant digits

10

Trace half-wave and bridge paths

For the unfiltered cases, use v_S=9 sin θ V, θ=ωt, f=60 Hz and R_L=470 Ω. With one ideal half-wave diode and the permitted load polarity, v_o=max(0, v_S) and i_o=v_o/R_L. During the negative half-cycle the diode is blocked and the resistor has no stored energy to sustain its voltage.

A full-wave bridge has two alternative conducting pairs. Each source polarity sends current through the load in the same direction, so the ideal output is |v_S|. The current drawn from the source reverses with its voltage, even though the load current does not. A bridge uses two forward-biased diodes in each conducting path; four is the total bridge count, not the number conducting in series at one time.

Full-wave rectification is a waveform function, not a unique four-diode topology. A suitable centre-tapped source and two diodes is a different architecture with different winding and reverse-voltage conditions. Our numerical full-wave cases refer specifically to the bridge, and no transformer or centre-tap parameters are silently attached to them.

11

Include the stated forward loss

In the constant-drop comparison, V_γ=3/5 V per conducting diode. The half-wave path has total d=3/5 V; the bridge path has d=6/5 V. Let w=v_S for the half-wave circuit and w=|v_S| for the bridge. The output is v_o=max(0, w−d), with i_o=v_o/R_L. Conducting regions require w>d; at the boundary the current is zero.

For a positive source half-cycle, conduction begins at α=asin(d/V_m) and ends at π−α, where V_m=9 V. The bridge repeats the same output segment in the next source half-cycle. Thus a forward drop changes both the instantaneous output and the conduction interval. Simply subtracting d from an already averaged ideal result misses the blocked interval.

The combined diode power while conducting is d i_o, not d²/R_L. Source delivered power is w i_o, load power is v_o i_o, and their instantaneous balance is w i_o=v_o i_o+d i_o. These forward-drop losses belong to this model; the ideal diode has no such loss, and neither model includes leakage, recovery or junction heating dynamics.

12

Derive mean, RMS and ripple separately

For a periodic output, V_DC is the complete-period mean and V_rms² is the complete-period mean square. On a positive conducting segment v_o=V_m sin θ−d. Integrating from α to π−α gives the half-wave mean [2 V_m cos α−d(π−2α)]/(2π). The bridge mean is twice that result because it repeats the segment twice in a full source period.

Squaring before integrating gives the half-wave mean square {V_m²[π−2α+sin(2α)]/2−4 V_m d cos α+d²(π−2α)}/(2π). The bridge mean square is again twice this value; the RMS itself therefore increases by √2 for the same segment, not by two. These formulas assume the present memory-free resistive load and cannot be reused unchanged after adding a capacitor.

Separate v_o=V_DC+v_ac, where the mean of v_ac is zero. Then V_ac, rms²=V_rms²−V_DC² by expanding the square and averaging. The ripple factor is V_ac, rms/V_DC when V_DC>0. It is an RMS ratio, whereas peak-to-peak ripple is the difference between maximum and minimum voltage. Neither is automatically small just because the output never becomes negative.

Original unfiltered rectifiers; all displayed approximations use six significant digits
CaseV_DC (V)V_rms (V)Ripple RMS (V)Ripple factorSource true PF
Half-wave, ideal diodes2.864794.500003.470301.211360.707107
Bridge, ideal diodes5.729586.363962.769830.4834261
Half-wave, specified forward drop2.571164.121863.221621.252990.706498
Bridge, specified forward drops4.580585.305492.677030.5844300.996475
13

Separate a DC component fraction from physical loss

For the ideal half-wave resistor case, V_DC=V_m/π and V_rms=V_m/2. For the ideal bridge, they are 2 V_m/π and V_m/√2. The corresponding fractions V_DC²/V_rms² are 4/π² and 8/π². These compare the mean-output component’s resistor power V_DC²/R_L with the total resistor power V_rms²/R_L.

That component fraction is not the physical source-to-load efficiency. In an ideal diode circuit with only the load resistor absorbing power, every joule delivered by the ideal source is absorbed in that resistor, including heating due to the alternating part of its current. Its physical power-transfer efficiency to the resistor is one. The smaller DC-component fraction describes waveform quality, not missing energy.

With the specified forward drop, the physical ratio becomes P_load/P_source<1 because the diode path absorbs d times the mean load current. Keep that loss ratio separate from V_DC²/V_rms². A real supply has additional losses and a specified useful output function, so its efficiency question also needs the complete system boundary and operating state.

14

Read true power factor at the source

For the half-wave case, source current is i_S=i_o during the positive segment and zero otherwise. For the bridge, i_S has the sign of v_S and magnitude i_o. The source voltage RMS is V_m/√2 in every case. True source PF is ⟨v_S i_S⟩/[(V_m/√2)I_S, rms], using the complete source current rather than the positive load-current mean.

The ideal resistive bridge gives i_S=v_S/R_L, so true PF is one even though the load voltage is rectified. The ideal half-wave case has I_S, rms=V_m/(2 R_L), P_source=V_m²/(4 R_L) and PF=1/√2. Its current includes a DC offset and harmonics, so assigning a single sinusoidal displacement cosine would not supply the required RMS denominator.

For a constant forward drop, source power is the integral of w(w−d)/R_L over the conducting regions, and I_S, rms=V_rms/R_L because changing the source-current sign leaves its square unchanged. The resulting PF changes slightly. Once a reservoir is added, narrow recharge currents change this problem again; a smoother load voltage does not by itself prove a better source power factor.

Rectified output and the current seen by the source

Each curve retains all 49 samples of one complete source period. A bridge keeps load polarity while its source current changes sign. The specified bridge path has two forward drops. Lines join retained values as visual guides; mean, RMS, true PF and power-transfer ratios are separately defined controls below.

Four resistive rectifiers: output voltage and signed source current

Each lane uses its own physical unit. The two bridge source currents reverse during the negative source half-cycle; the load-output voltage stays nonnegative. Nested marker sizes identify cases, not physical magnitude.

  • Half-wave, ideal diode
  • Bridge, ideal diodes
  • Half-wave, specified forward drop
  • Bridge, specified forward drops
Four resistive rectifiers: output voltage and signed source currentEach lane uses its own physical unit. The two bridge source currents reverse during the negative source half-cycle; the load-output voltage stays nonnegative. Nested marker sizes identify cases, not physical magnitude.Output voltage (V)00.511.520246810θ/π = ωt/π (1); complete source period 0 to 2Source delivered current (A)00.511.52-0.02-0.0100.010.02θ/π = ωt/π (1); complete source period 0 to 2
Each lane uses its own physical unit. The two bridge source currents reverse during the negative source half-cycle; the load-output voltage stays nonnegative. Nested marker sizes identify cases, not physical magnitude.

Short fractions are exact. A leading ≈ marks a value rounded to 12 significant digits. The plots use the labelled units and scales.

The four resistive rectifier boundaries; half-wave and bridge have different conducting-path drops.
The four resistive rectifier boundaries; half-wave and bridge have different conducting-path drops.
CasePeak source (V)RL (Ω)f (Hz)Total forward drop (V)α (rad)Conduction fraction (1)Mean output (V)RMS output (V)AC ripple RMS (V)Ripple factor (1)
Half-wave, ideal diode94706000≈ 0.500000000000≈ 2.86478897565≈ 4.50000000000≈ 3.47030029320≈ 1.21136332298
Bridge, ideal diodes94706000≈ 1.00000000000≈ 5.72957795131≈ 6.36396103068≈ 2.76982607755≈ 0.483425847609
Half-wave, specified forward drop9470603/5≈ 0.0667161484102≈ 0.478763590393≈ 2.57115753438≈ 4.12185712801≈ 3.22162305633≈ 1.25298547960
Bridge, specified forward drops9470606/5≈ 0.133731589410≈ 0.914863825991≈ 4.58058338996≈ 5.30549082517≈ 2.67702982868≈ 0.584429886058
Physical source-to-load transfer, true source PF and DC-component power fraction have separate meanings.
Physical source-to-load transfer, true source PF and DC-component power fraction have separate meanings.
CaseLoad mean power (W)Diode mean power (W)Source delivered mean (W)Source RMS current (A)True source PF (1)Physical transfer efficiency (1)DC-component power fraction (1)
Half-wave, ideal diode81/1880081/1880≈ 0.00957446808511≈ 0.7071067811871≈ 0.405284734569
Bridge, ideal diodes81/940081/940≈ 0.013540342618511≈ 0.810569469139
Half-wave, specified forward drop≈ 0.0361483110292≈ 0.00328232876729≈ 0.0394306397964≈ 0.00876990878299≈ 0.706498486954≈ 0.916756898081≈ 0.389109205016
Bridge, specified forward drops≈ 0.0598898572256≈ 0.0116951065276≈ 0.0715849637531≈ 0.0112882783514≈ 0.996475414092≈ 0.836626214300≈ 0.745401825752
Half-wave, ideal diode: every full-source-period sample; signed source current is distinct from positive load current.
Half-wave, ideal diode: every full-source-period sample; signed source current is distinct from positive load current.
Sampleθ/π (1)t (s)Source V (V)Output V (V)Load I (A)Source delivered I (A)Source delivered p (W)Load p (W)Diodes p (W)Diode state
0000000000blocked or boundary
11/24≈ 0.000347222222222≈ 1.17473572998≈ 1.17473572998≈ 0.00249943772336≈ 0.00249943772336≈ 0.00293617879850≈ 0.002936178798500conducting
21/12≈ 0.000694444444444≈ 2.32937140592≈ 2.32937140592≈ 0.00495610937430≈ 0.00495610937430≈ 0.0115446194611≈ 0.01154461946110conducting
31/8≈ 0.00104166666667≈ 3.44415089129≈ 3.44415089129≈ 0.00732798061976≈ 0.00732798061976≈ 0.0252386709829≈ 0.02523867098290conducting
41/6≈ 0.00138888888889≈ 4.50000000000≈ 4.50000000000≈ 0.00957446808511≈ 0.00957446808511≈ 0.0430851063830≈ 0.04308510638300conducting
55/24≈ 0.00173611111111≈ 5.47885286108≈ 5.47885286108≈ 0.0116571337470≈ 0.0116571337470≈ 0.0638677205816≈ 0.06386772058160conducting
61/4≈ 0.00208333333333≈ 6.36396103068≈ 6.36396103068≈ 0.0135403426185≈ 0.0135403426185≈ 0.0861702127660≈ 0.08617021276600conducting
77/24≈ 0.00243055555556≈ 7.14018006262≈ 7.14018006262≈ 0.0151918724737≈ 0.0151918724737≈ 0.108472704950≈ 0.1084727049500conducting
81/3≈ 0.00277777777778≈ 7.79422863406≈ 7.79422863406≈ 0.0165834651789≈ 0.0165834651789≈ 0.129255319149≈ 0.1292553191490conducting
93/8≈ 0.00312500000000≈ 8.31491579260≈ 8.31491579260≈ 0.0176913101970≈ 0.0176913101970≈ 0.147101754549≈ 0.1471017545490conducting
105/12≈ 0.00347222222222≈ 8.69333243660≈ 8.69333243660≈ 0.0184964519928≈ 0.0184964519928≈ 0.160795806071≈ 0.1607958060710conducting
1111/24≈ 0.00381944444444≈ 8.92300375236≈ 8.92300375236≈ 0.0189851143667≈ 0.0189851143667≈ 0.169404246733≈ 0.1694042467330conducting
121/2≈ 0.00416666666667≈ 9.00000000000≈ 9.00000000000≈ 0.0191489361702≈ 0.0191489361702≈ 0.172340425532≈ 0.1723404255320conducting
1313/24≈ 0.00451388888889≈ 8.92300375236≈ 8.92300375236≈ 0.0189851143667≈ 0.0189851143667≈ 0.169404246733≈ 0.1694042467330conducting
147/12≈ 0.00486111111111≈ 8.69333243660≈ 8.69333243660≈ 0.0184964519928≈ 0.0184964519928≈ 0.160795806071≈ 0.1607958060710conducting
155/8≈ 0.00520833333333≈ 8.31491579260≈ 8.31491579260≈ 0.0176913101970≈ 0.0176913101970≈ 0.147101754549≈ 0.1471017545490conducting
162/3≈ 0.00555555555556≈ 7.79422863406≈ 7.79422863406≈ 0.0165834651789≈ 0.0165834651789≈ 0.129255319149≈ 0.1292553191490conducting
1717/24≈ 0.00590277777778≈ 7.14018006262≈ 7.14018006262≈ 0.0151918724737≈ 0.0151918724737≈ 0.108472704950≈ 0.1084727049500conducting
183/4≈ 0.00625000000000≈ 6.36396103068≈ 6.36396103068≈ 0.0135403426185≈ 0.0135403426185≈ 0.0861702127660≈ 0.08617021276600conducting
1919/24≈ 0.00659722222222≈ 5.47885286108≈ 5.47885286108≈ 0.0116571337470≈ 0.0116571337470≈ 0.0638677205816≈ 0.06386772058160conducting
205/6≈ 0.00694444444444≈ 4.50000000000≈ 4.50000000000≈ 0.00957446808511≈ 0.00957446808511≈ 0.0430851063830≈ 0.04308510638300conducting
217/8≈ 0.00729166666667≈ 3.44415089129≈ 3.44415089129≈ 0.00732798061976≈ 0.00732798061976≈ 0.0252386709829≈ 0.02523867098290conducting
2211/12≈ 0.00763888888889≈ 2.32937140592≈ 2.32937140592≈ 0.00495610937430≈ 0.00495610937430≈ 0.0115446194611≈ 0.01154461946110conducting
2323/24≈ 0.00798611111111≈ 1.17473572998≈ 1.17473572998≈ 0.00249943772336≈ 0.00249943772336≈ 0.00293617879850≈ 0.002936178798500conducting
241≈ 0.008333333333330000000blocked or boundary
2525/24≈ 0.00868055555556≈ -1.17473572998000000blocked or boundary
2613/12≈ 0.00902777777778≈ -2.32937140592000000blocked or boundary
279/8≈ 0.00937500000000≈ -3.44415089129000000blocked or boundary
287/6≈ 0.00972222222222≈ -4.50000000000000000blocked or boundary
2929/24≈ 0.0100694444444≈ -5.47885286108000000blocked or boundary
305/4≈ 0.0104166666667≈ -6.36396103068000000blocked or boundary
3131/24≈ 0.0107638888889≈ -7.14018006262000000blocked or boundary
324/3≈ 0.0111111111111≈ -7.79422863406000000blocked or boundary
3311/8≈ 0.0114583333333≈ -8.31491579260000000blocked or boundary
3417/12≈ 0.0118055555556≈ -8.69333243660000000blocked or boundary
3535/24≈ 0.0121527777778≈ -8.92300375236000000blocked or boundary
363/2≈ 0.0125000000000≈ -9.00000000000000000blocked or boundary
3737/24≈ 0.0128472222222≈ -8.92300375236000000blocked or boundary
3819/12≈ 0.0131944444444≈ -8.69333243660000000blocked or boundary
3913/8≈ 0.0135416666667≈ -8.31491579260000000blocked or boundary
405/3≈ 0.0138888888889≈ -7.79422863406000000blocked or boundary
4141/24≈ 0.0142361111111≈ -7.14018006262000000blocked or boundary
427/4≈ 0.0145833333333≈ -6.36396103068000000blocked or boundary
4343/24≈ 0.0149305555556≈ -5.47885286108000000blocked or boundary
4411/6≈ 0.0152777777778≈ -4.50000000000000000blocked or boundary
4515/8≈ 0.0156250000000≈ -3.44415089129000000blocked or boundary
4623/12≈ 0.0159722222222≈ -2.32937140592000000blocked or boundary
4747/24≈ 0.0163194444444≈ -1.17473572998000000blocked or boundary
482≈ 0.01666666666670000000blocked or boundary
Bridge, ideal diodes: every full-source-period sample; signed source current is distinct from positive load current.
Bridge, ideal diodes: every full-source-period sample; signed source current is distinct from positive load current.
Sampleθ/π (1)t (s)Source V (V)Output V (V)Load I (A)Source delivered I (A)Source delivered p (W)Load p (W)Diodes p (W)Diode state
0000000000blocked or boundary
11/24≈ 0.000347222222222≈ 1.17473572998≈ 1.17473572998≈ 0.00249943772336≈ 0.00249943772336≈ 0.00293617879850≈ 0.002936178798500conducting
21/12≈ 0.000694444444444≈ 2.32937140592≈ 2.32937140592≈ 0.00495610937430≈ 0.00495610937430≈ 0.0115446194611≈ 0.01154461946110conducting
31/8≈ 0.00104166666667≈ 3.44415089129≈ 3.44415089129≈ 0.00732798061976≈ 0.00732798061976≈ 0.0252386709829≈ 0.02523867098290conducting
41/6≈ 0.00138888888889≈ 4.50000000000≈ 4.50000000000≈ 0.00957446808511≈ 0.00957446808511≈ 0.0430851063830≈ 0.04308510638300conducting
55/24≈ 0.00173611111111≈ 5.47885286108≈ 5.47885286108≈ 0.0116571337470≈ 0.0116571337470≈ 0.0638677205816≈ 0.06386772058160conducting
61/4≈ 0.00208333333333≈ 6.36396103068≈ 6.36396103068≈ 0.0135403426185≈ 0.0135403426185≈ 0.0861702127660≈ 0.08617021276600conducting
77/24≈ 0.00243055555556≈ 7.14018006262≈ 7.14018006262≈ 0.0151918724737≈ 0.0151918724737≈ 0.108472704950≈ 0.1084727049500conducting
81/3≈ 0.00277777777778≈ 7.79422863406≈ 7.79422863406≈ 0.0165834651789≈ 0.0165834651789≈ 0.129255319149≈ 0.1292553191490conducting
93/8≈ 0.00312500000000≈ 8.31491579260≈ 8.31491579260≈ 0.0176913101970≈ 0.0176913101970≈ 0.147101754549≈ 0.1471017545490conducting
105/12≈ 0.00347222222222≈ 8.69333243660≈ 8.69333243660≈ 0.0184964519928≈ 0.0184964519928≈ 0.160795806071≈ 0.1607958060710conducting
1111/24≈ 0.00381944444444≈ 8.92300375236≈ 8.92300375236≈ 0.0189851143667≈ 0.0189851143667≈ 0.169404246733≈ 0.1694042467330conducting
121/2≈ 0.00416666666667≈ 9.00000000000≈ 9.00000000000≈ 0.0191489361702≈ 0.0191489361702≈ 0.172340425532≈ 0.1723404255320conducting
1313/24≈ 0.00451388888889≈ 8.92300375236≈ 8.92300375236≈ 0.0189851143667≈ 0.0189851143667≈ 0.169404246733≈ 0.1694042467330conducting
147/12≈ 0.00486111111111≈ 8.69333243660≈ 8.69333243660≈ 0.0184964519928≈ 0.0184964519928≈ 0.160795806071≈ 0.1607958060710conducting
155/8≈ 0.00520833333333≈ 8.31491579260≈ 8.31491579260≈ 0.0176913101970≈ 0.0176913101970≈ 0.147101754549≈ 0.1471017545490conducting
162/3≈ 0.00555555555556≈ 7.79422863406≈ 7.79422863406≈ 0.0165834651789≈ 0.0165834651789≈ 0.129255319149≈ 0.1292553191490conducting
1717/24≈ 0.00590277777778≈ 7.14018006262≈ 7.14018006262≈ 0.0151918724737≈ 0.0151918724737≈ 0.108472704950≈ 0.1084727049500conducting
183/4≈ 0.00625000000000≈ 6.36396103068≈ 6.36396103068≈ 0.0135403426185≈ 0.0135403426185≈ 0.0861702127660≈ 0.08617021276600conducting
1919/24≈ 0.00659722222222≈ 5.47885286108≈ 5.47885286108≈ 0.0116571337470≈ 0.0116571337470≈ 0.0638677205816≈ 0.06386772058160conducting
205/6≈ 0.00694444444444≈ 4.50000000000≈ 4.50000000000≈ 0.00957446808511≈ 0.00957446808511≈ 0.0430851063830≈ 0.04308510638300conducting
217/8≈ 0.00729166666667≈ 3.44415089129≈ 3.44415089129≈ 0.00732798061976≈ 0.00732798061976≈ 0.0252386709829≈ 0.02523867098290conducting
2211/12≈ 0.00763888888889≈ 2.32937140592≈ 2.32937140592≈ 0.00495610937430≈ 0.00495610937430≈ 0.0115446194611≈ 0.01154461946110conducting
2323/24≈ 0.00798611111111≈ 1.17473572998≈ 1.17473572998≈ 0.00249943772336≈ 0.00249943772336≈ 0.00293617879850≈ 0.002936178798500conducting
241≈ 0.008333333333330000000blocked or boundary
2525/24≈ 0.00868055555556≈ -1.17473572998≈ 1.17473572998≈ 0.00249943772336≈ -0.00249943772336≈ 0.00293617879850≈ 0.002936178798500conducting
2613/12≈ 0.00902777777778≈ -2.32937140592≈ 2.32937140592≈ 0.00495610937430≈ -0.00495610937430≈ 0.0115446194611≈ 0.01154461946110conducting
279/8≈ 0.00937500000000≈ -3.44415089129≈ 3.44415089129≈ 0.00732798061976≈ -0.00732798061976≈ 0.0252386709829≈ 0.02523867098290conducting
287/6≈ 0.00972222222222≈ -4.50000000000≈ 4.50000000000≈ 0.00957446808511≈ -0.00957446808511≈ 0.0430851063830≈ 0.04308510638300conducting
2929/24≈ 0.0100694444444≈ -5.47885286108≈ 5.47885286108≈ 0.0116571337470≈ -0.0116571337470≈ 0.0638677205816≈ 0.06386772058160conducting
305/4≈ 0.0104166666667≈ -6.36396103068≈ 6.36396103068≈ 0.0135403426185≈ -0.0135403426185≈ 0.0861702127660≈ 0.08617021276600conducting
3131/24≈ 0.0107638888889≈ -7.14018006262≈ 7.14018006262≈ 0.0151918724737≈ -0.0151918724737≈ 0.108472704950≈ 0.1084727049500conducting
324/3≈ 0.0111111111111≈ -7.79422863406≈ 7.79422863406≈ 0.0165834651789≈ -0.0165834651789≈ 0.129255319149≈ 0.1292553191490conducting
3311/8≈ 0.0114583333333≈ -8.31491579260≈ 8.31491579260≈ 0.0176913101970≈ -0.0176913101970≈ 0.147101754549≈ 0.1471017545490conducting
3417/12≈ 0.0118055555556≈ -8.69333243660≈ 8.69333243660≈ 0.0184964519928≈ -0.0184964519928≈ 0.160795806071≈ 0.1607958060710conducting
3535/24≈ 0.0121527777778≈ -8.92300375236≈ 8.92300375236≈ 0.0189851143667≈ -0.0189851143667≈ 0.169404246733≈ 0.1694042467330conducting
363/2≈ 0.0125000000000≈ -9.00000000000≈ 9.00000000000≈ 0.0191489361702≈ -0.0191489361702≈ 0.172340425532≈ 0.1723404255320conducting
3737/24≈ 0.0128472222222≈ -8.92300375236≈ 8.92300375236≈ 0.0189851143667≈ -0.0189851143667≈ 0.169404246733≈ 0.1694042467330conducting
3819/12≈ 0.0131944444444≈ -8.69333243660≈ 8.69333243660≈ 0.0184964519928≈ -0.0184964519928≈ 0.160795806071≈ 0.1607958060710conducting
3913/8≈ 0.0135416666667≈ -8.31491579260≈ 8.31491579260≈ 0.0176913101970≈ -0.0176913101970≈ 0.147101754549≈ 0.1471017545490conducting
405/3≈ 0.0138888888889≈ -7.79422863406≈ 7.79422863406≈ 0.0165834651789≈ -0.0165834651789≈ 0.129255319149≈ 0.1292553191490conducting
4141/24≈ 0.0142361111111≈ -7.14018006262≈ 7.14018006262≈ 0.0151918724737≈ -0.0151918724737≈ 0.108472704950≈ 0.1084727049500conducting
427/4≈ 0.0145833333333≈ -6.36396103068≈ 6.36396103068≈ 0.0135403426185≈ -0.0135403426185≈ 0.0861702127660≈ 0.08617021276600conducting
4343/24≈ 0.0149305555556≈ -5.47885286108≈ 5.47885286108≈ 0.0116571337470≈ -0.0116571337470≈ 0.0638677205816≈ 0.06386772058160conducting
4411/6≈ 0.0152777777778≈ -4.50000000000≈ 4.50000000000≈ 0.00957446808511≈ -0.00957446808511≈ 0.0430851063830≈ 0.04308510638300conducting
4515/8≈ 0.0156250000000≈ -3.44415089129≈ 3.44415089129≈ 0.00732798061976≈ -0.00732798061976≈ 0.0252386709829≈ 0.02523867098290conducting
4623/12≈ 0.0159722222222≈ -2.32937140592≈ 2.32937140592≈ 0.00495610937430≈ -0.00495610937430≈ 0.0115446194611≈ 0.01154461946110conducting
4747/24≈ 0.0163194444444≈ -1.17473572998≈ 1.17473572998≈ 0.00249943772336≈ -0.00249943772336≈ 0.00293617879850≈ 0.002936178798500conducting
482≈ 0.01666666666670000000blocked or boundary
Half-wave, specified forward drop: every full-source-period sample; signed source current is distinct from positive load current.
Half-wave, specified forward drop: every full-source-period sample; signed source current is distinct from positive load current.
Sampleθ/π (1)t (s)Source V (V)Output V (V)Load I (A)Source delivered I (A)Source delivered p (W)Load p (W)Diodes p (W)Diode state
0000000000blocked or boundary
11/24≈ 0.000347222222222≈ 1.17473572998≈ 0.574735729980≈ 0.00122284197868≈ 0.00122284197868≈ 0.00143651616448≈ 0.000702810977268≈ 0.000733705187209conducting
21/12≈ 0.000694444444444≈ 2.32937140592≈ 1.72937140592≈ 0.00367951362962≈ 0.00367951362962≈ 0.00857095383655≈ 0.00636324565877≈ 0.00220770817777conducting
31/8≈ 0.00104166666667≈ 3.44415089129≈ 2.84415089129≈ 0.00605138487508≈ 0.00605138487508≈ 0.0208418826110≈ 0.0172110516860≈ 0.00363083092505conducting
41/6≈ 0.00138888888889≈ 4.50000000000≈ 3.90000000000≈ 0.00829787234043≈ 0.00829787234043≈ 0.0373404255319≈ 0.0323617021277≈ 0.00497872340426conducting
55/24≈ 0.00173611111111≈ 5.47885286108≈ 4.87885286108≈ 0.0103805380023≈ 0.0103805380023≈ 0.0568734403334≈ 0.0506451175320≈ 0.00622832280138conducting
61/4≈ 0.00208333333333≈ 6.36396103068≈ 5.76396103068≈ 0.0122637468738≈ 0.0122637468738≈ 0.0780460071949≈ 0.0706877590706≈ 0.00735824812427conducting
77/24≈ 0.00243055555556≈ 7.14018006262≈ 6.54018006262≈ 0.0139152767290≈ 0.0139152767290≈ 0.0993575814661≈ 0.0910084154287≈ 0.00834916603739conducting
81/3≈ 0.00277777777778≈ 7.79422863406≈ 7.19422863406≈ 0.0153068694342≈ 0.0153068694342≈ 0.119305240042≈ 0.110121118381≈ 0.00918412166050conducting
93/8≈ 0.00312500000000≈ 8.31491579260≈ 7.71491579260≈ 0.0164147144523≈ 0.0164147144523≈ 0.136486968431≈ 0.126638139759≈ 0.00984882867141conducting
105/12≈ 0.00347222222222≈ 8.69333243660≈ 8.09333243660≈ 0.0172198562481≈ 0.0172198562481≈ 0.149697934875≈ 0.139366021126≈ 0.0103319137489conducting
1111/24≈ 0.00381944444444≈ 8.92300375236≈ 8.32300375236≈ 0.0177085186221≈ 0.0177085186221≈ 0.158013178113≈ 0.147388066940≈ 0.0106251111732conducting
121/2≈ 0.00416666666667≈ 9.00000000000≈ 8.40000000000≈ 0.0178723404255≈ 0.0178723404255≈ 0.160851063830≈ 0.150127659574≈ 0.0107234042553conducting
1313/24≈ 0.00451388888889≈ 8.92300375236≈ 8.32300375236≈ 0.0177085186221≈ 0.0177085186221≈ 0.158013178113≈ 0.147388066940≈ 0.0106251111732conducting
147/12≈ 0.00486111111111≈ 8.69333243660≈ 8.09333243660≈ 0.0172198562481≈ 0.0172198562481≈ 0.149697934875≈ 0.139366021126≈ 0.0103319137489conducting
155/8≈ 0.00520833333333≈ 8.31491579260≈ 7.71491579260≈ 0.0164147144523≈ 0.0164147144523≈ 0.136486968431≈ 0.126638139759≈ 0.00984882867141conducting
162/3≈ 0.00555555555556≈ 7.79422863406≈ 7.19422863406≈ 0.0153068694342≈ 0.0153068694342≈ 0.119305240042≈ 0.110121118381≈ 0.00918412166050conducting
1717/24≈ 0.00590277777778≈ 7.14018006262≈ 6.54018006262≈ 0.0139152767290≈ 0.0139152767290≈ 0.0993575814661≈ 0.0910084154287≈ 0.00834916603739conducting
183/4≈ 0.00625000000000≈ 6.36396103068≈ 5.76396103068≈ 0.0122637468738≈ 0.0122637468738≈ 0.0780460071949≈ 0.0706877590706≈ 0.00735824812427conducting
1919/24≈ 0.00659722222222≈ 5.47885286108≈ 4.87885286108≈ 0.0103805380023≈ 0.0103805380023≈ 0.0568734403334≈ 0.0506451175320≈ 0.00622832280138conducting
205/6≈ 0.00694444444444≈ 4.50000000000≈ 3.90000000000≈ 0.00829787234043≈ 0.00829787234043≈ 0.0373404255319≈ 0.0323617021277≈ 0.00497872340426conducting
217/8≈ 0.00729166666667≈ 3.44415089129≈ 2.84415089129≈ 0.00605138487508≈ 0.00605138487508≈ 0.0208418826110≈ 0.0172110516860≈ 0.00363083092505conducting
2211/12≈ 0.00763888888889≈ 2.32937140592≈ 1.72937140592≈ 0.00367951362962≈ 0.00367951362962≈ 0.00857095383655≈ 0.00636324565877≈ 0.00220770817777conducting
2323/24≈ 0.00798611111111≈ 1.17473572998≈ 0.574735729980≈ 0.00122284197868≈ 0.00122284197868≈ 0.00143651616448≈ 0.000702810977268≈ 0.000733705187209conducting
241≈ 0.008333333333330000000blocked or boundary
2525/24≈ 0.00868055555556≈ -1.17473572998000000blocked or boundary
2613/12≈ 0.00902777777778≈ -2.32937140592000000blocked or boundary
279/8≈ 0.00937500000000≈ -3.44415089129000000blocked or boundary
287/6≈ 0.00972222222222≈ -4.50000000000000000blocked or boundary
2929/24≈ 0.0100694444444≈ -5.47885286108000000blocked or boundary
305/4≈ 0.0104166666667≈ -6.36396103068000000blocked or boundary
3131/24≈ 0.0107638888889≈ -7.14018006262000000blocked or boundary
324/3≈ 0.0111111111111≈ -7.79422863406000000blocked or boundary
3311/8≈ 0.0114583333333≈ -8.31491579260000000blocked or boundary
3417/12≈ 0.0118055555556≈ -8.69333243660000000blocked or boundary
3535/24≈ 0.0121527777778≈ -8.92300375236000000blocked or boundary
363/2≈ 0.0125000000000≈ -9.00000000000000000blocked or boundary
3737/24≈ 0.0128472222222≈ -8.92300375236000000blocked or boundary
3819/12≈ 0.0131944444444≈ -8.69333243660000000blocked or boundary
3913/8≈ 0.0135416666667≈ -8.31491579260000000blocked or boundary
405/3≈ 0.0138888888889≈ -7.79422863406000000blocked or boundary
4141/24≈ 0.0142361111111≈ -7.14018006262000000blocked or boundary
427/4≈ 0.0145833333333≈ -6.36396103068000000blocked or boundary
4343/24≈ 0.0149305555556≈ -5.47885286108000000blocked or boundary
4411/6≈ 0.0152777777778≈ -4.50000000000000000blocked or boundary
4515/8≈ 0.0156250000000≈ -3.44415089129000000blocked or boundary
4623/12≈ 0.0159722222222≈ -2.32937140592000000blocked or boundary
4747/24≈ 0.0163194444444≈ -1.17473572998000000blocked or boundary
482≈ 0.01666666666670000000blocked or boundary
Bridge, specified forward drops: every full-source-period sample; signed source current is distinct from positive load current.
Bridge, specified forward drops: every full-source-period sample; signed source current is distinct from positive load current.
Sampleθ/π (1)t (s)Source V (V)Output V (V)Load I (A)Source delivered I (A)Source delivered p (W)Load p (W)Diodes p (W)Diode state
0000000000blocked or boundary
11/24≈ 0.000347222222222≈ 1.17473572998000000blocked or boundary
21/12≈ 0.000694444444444≈ 2.32937140592≈ 1.12937140592≈ 0.00240291788494≈ 0.00240291788494≈ 0.00559728821196≈ 0.00271378675003≈ 0.00288350146193conducting
31/8≈ 0.00104166666667≈ 3.44415089129≈ 2.24415089129≈ 0.00477478913040≈ 0.00477478913040≈ 0.0164450942392≈ 0.0107153472827≈ 0.00572974695647conducting
41/6≈ 0.00138888888889≈ 4.50000000000≈ 3.30000000000≈ 0.00702127659574≈ 0.00702127659574≈ 0.0315957446809≈ 0.0231702127660≈ 0.00842553191489conducting
55/24≈ 0.00173611111111≈ 5.47885286108≈ 4.27885286108≈ 0.00910394225761≈ 0.00910394225761≈ 0.0498791600852≈ 0.0389544293761≈ 0.0109247307091conducting
61/4≈ 0.00208333333333≈ 6.36396103068≈ 5.16396103068≈ 0.0109871511291≈ 0.0109871511291≈ 0.0699218016238≈ 0.0567372202689≈ 0.0131845813549conducting
77/24≈ 0.00243055555556≈ 7.14018006262≈ 5.94018006262≈ 0.0126386809843≈ 0.0126386809843≈ 0.0902424579819≈ 0.0750760408008≈ 0.0151664171812conducting
81/3≈ 0.00277777777778≈ 7.79422863406≈ 6.59422863406≈ 0.0140302736895≈ 0.0140302736895≈ 0.109355160934≈ 0.0925188325069≈ 0.0168363284274conducting
93/8≈ 0.00312500000000≈ 8.31491579260≈ 7.11491579260≈ 0.0151381187077≈ 0.0151381187077≈ 0.125872182313≈ 0.107706439863≈ 0.0181657424492conducting
105/12≈ 0.00347222222222≈ 8.69333243660≈ 7.49333243660≈ 0.0159432605034≈ 0.0159432605034≈ 0.138600063679≈ 0.119468151075≈ 0.0191319126041conducting
1111/24≈ 0.00381944444444≈ 8.92300375236≈ 7.72300375236≈ 0.0164319228774≈ 0.0164319228774≈ 0.146622109493≈ 0.126903802040≈ 0.0197183074528conducting
121/2≈ 0.00416666666667≈ 9.00000000000≈ 7.80000000000≈ 0.0165957446809≈ 0.0165957446809≈ 0.149361702128≈ 0.129446808511≈ 0.0199148936170conducting
1313/24≈ 0.00451388888889≈ 8.92300375236≈ 7.72300375236≈ 0.0164319228774≈ 0.0164319228774≈ 0.146622109493≈ 0.126903802040≈ 0.0197183074528conducting
147/12≈ 0.00486111111111≈ 8.69333243660≈ 7.49333243660≈ 0.0159432605034≈ 0.0159432605034≈ 0.138600063679≈ 0.119468151075≈ 0.0191319126041conducting
155/8≈ 0.00520833333333≈ 8.31491579260≈ 7.11491579260≈ 0.0151381187077≈ 0.0151381187077≈ 0.125872182313≈ 0.107706439863≈ 0.0181657424492conducting
162/3≈ 0.00555555555556≈ 7.79422863406≈ 6.59422863406≈ 0.0140302736895≈ 0.0140302736895≈ 0.109355160934≈ 0.0925188325069≈ 0.0168363284274conducting
1717/24≈ 0.00590277777778≈ 7.14018006262≈ 5.94018006262≈ 0.0126386809843≈ 0.0126386809843≈ 0.0902424579819≈ 0.0750760408008≈ 0.0151664171812conducting
183/4≈ 0.00625000000000≈ 6.36396103068≈ 5.16396103068≈ 0.0109871511291≈ 0.0109871511291≈ 0.0699218016238≈ 0.0567372202689≈ 0.0131845813549conducting
1919/24≈ 0.00659722222222≈ 5.47885286108≈ 4.27885286108≈ 0.00910394225761≈ 0.00910394225761≈ 0.0498791600852≈ 0.0389544293761≈ 0.0109247307091conducting
205/6≈ 0.00694444444444≈ 4.50000000000≈ 3.30000000000≈ 0.00702127659574≈ 0.00702127659574≈ 0.0315957446809≈ 0.0231702127660≈ 0.00842553191489conducting
217/8≈ 0.00729166666667≈ 3.44415089129≈ 2.24415089129≈ 0.00477478913040≈ 0.00477478913040≈ 0.0164450942392≈ 0.0107153472827≈ 0.00572974695647conducting
2211/12≈ 0.00763888888889≈ 2.32937140592≈ 1.12937140592≈ 0.00240291788494≈ 0.00240291788494≈ 0.00559728821196≈ 0.00271378675003≈ 0.00288350146193conducting
2323/24≈ 0.00798611111111≈ 1.17473572998000000blocked or boundary
241≈ 0.008333333333330000000blocked or boundary
2525/24≈ 0.00868055555556≈ -1.17473572998000000blocked or boundary
2613/12≈ 0.00902777777778≈ -2.32937140592≈ 1.12937140592≈ 0.00240291788494≈ -0.00240291788494≈ 0.00559728821196≈ 0.00271378675003≈ 0.00288350146193conducting
279/8≈ 0.00937500000000≈ -3.44415089129≈ 2.24415089129≈ 0.00477478913040≈ -0.00477478913040≈ 0.0164450942392≈ 0.0107153472827≈ 0.00572974695647conducting
287/6≈ 0.00972222222222≈ -4.50000000000≈ 3.30000000000≈ 0.00702127659574≈ -0.00702127659574≈ 0.0315957446809≈ 0.0231702127660≈ 0.00842553191489conducting
2929/24≈ 0.0100694444444≈ -5.47885286108≈ 4.27885286108≈ 0.00910394225761≈ -0.00910394225761≈ 0.0498791600852≈ 0.0389544293761≈ 0.0109247307091conducting
305/4≈ 0.0104166666667≈ -6.36396103068≈ 5.16396103068≈ 0.0109871511291≈ -0.0109871511291≈ 0.0699218016238≈ 0.0567372202689≈ 0.0131845813549conducting
3131/24≈ 0.0107638888889≈ -7.14018006262≈ 5.94018006262≈ 0.0126386809843≈ -0.0126386809843≈ 0.0902424579819≈ 0.0750760408008≈ 0.0151664171812conducting
324/3≈ 0.0111111111111≈ -7.79422863406≈ 6.59422863406≈ 0.0140302736895≈ -0.0140302736895≈ 0.109355160934≈ 0.0925188325069≈ 0.0168363284274conducting
3311/8≈ 0.0114583333333≈ -8.31491579260≈ 7.11491579260≈ 0.0151381187077≈ -0.0151381187077≈ 0.125872182313≈ 0.107706439863≈ 0.0181657424492conducting
3417/12≈ 0.0118055555556≈ -8.69333243660≈ 7.49333243660≈ 0.0159432605034≈ -0.0159432605034≈ 0.138600063679≈ 0.119468151075≈ 0.0191319126041conducting
3535/24≈ 0.0121527777778≈ -8.92300375236≈ 7.72300375236≈ 0.0164319228774≈ -0.0164319228774≈ 0.146622109493≈ 0.126903802040≈ 0.0197183074528conducting
363/2≈ 0.0125000000000≈ -9.00000000000≈ 7.80000000000≈ 0.0165957446809≈ -0.0165957446809≈ 0.149361702128≈ 0.129446808511≈ 0.0199148936170conducting
3737/24≈ 0.0128472222222≈ -8.92300375236≈ 7.72300375236≈ 0.0164319228774≈ -0.0164319228774≈ 0.146622109493≈ 0.126903802040≈ 0.0197183074528conducting
3819/12≈ 0.0131944444444≈ -8.69333243660≈ 7.49333243660≈ 0.0159432605034≈ -0.0159432605034≈ 0.138600063679≈ 0.119468151075≈ 0.0191319126041conducting
3913/8≈ 0.0135416666667≈ -8.31491579260≈ 7.11491579260≈ 0.0151381187077≈ -0.0151381187077≈ 0.125872182313≈ 0.107706439863≈ 0.0181657424492conducting
405/3≈ 0.0138888888889≈ -7.79422863406≈ 6.59422863406≈ 0.0140302736895≈ -0.0140302736895≈ 0.109355160934≈ 0.0925188325069≈ 0.0168363284274conducting
4141/24≈ 0.0142361111111≈ -7.14018006262≈ 5.94018006262≈ 0.0126386809843≈ -0.0126386809843≈ 0.0902424579819≈ 0.0750760408008≈ 0.0151664171812conducting
427/4≈ 0.0145833333333≈ -6.36396103068≈ 5.16396103068≈ 0.0109871511291≈ -0.0109871511291≈ 0.0699218016238≈ 0.0567372202689≈ 0.0131845813549conducting
4343/24≈ 0.0149305555556≈ -5.47885286108≈ 4.27885286108≈ 0.00910394225761≈ -0.00910394225761≈ 0.0498791600852≈ 0.0389544293761≈ 0.0109247307091conducting
4411/6≈ 0.0152777777778≈ -4.50000000000≈ 3.30000000000≈ 0.00702127659574≈ -0.00702127659574≈ 0.0315957446809≈ 0.0231702127660≈ 0.00842553191489conducting
4515/8≈ 0.0156250000000≈ -3.44415089129≈ 2.24415089129≈ 0.00477478913040≈ -0.00477478913040≈ 0.0164450942392≈ 0.0107153472827≈ 0.00572974695647conducting
4623/12≈ 0.0159722222222≈ -2.32937140592≈ 1.12937140592≈ 0.00240291788494≈ -0.00240291788494≈ 0.00559728821196≈ 0.00271378675003≈ 0.00288350146193conducting
4747/24≈ 0.0163194444444≈ -1.17473572998000000blocked or boundary
482≈ 0.01666666666670000000blocked or boundary

Related complete original table: Original unfiltered rectifiers; all displayed approximations use six significant digits

15

Add a reservoir and retain source resistance

Now use ideal diodes, a source amplitude of 9 V at 60 Hz, R_s=20 Ω, R_L=1000 Ω and an ideal capacitor across the load. These are new cases, not the previous 470 Ω forward-drop cases with a component added informally. The capacitor voltage v_C is a continuous state, and its current is C dv_C/dt. It can supply the load while the diode path is blocked.

When the path conducts, i_D=(w−v_C)/R_s; when blocked, i_D=0. Combining the branches gives i_D=max[0, (w−v_C)/R_s] and C dv_C/dt=i_D−v_C/R_L. Here w is the positive half-cycle source for a half-wave charging interval or the rectified source for a bridge. The ideal diode prohibits the capacitor discharging back through the source.

Finite R_s makes the charging current finite. For nonnegative v_C and w≤V_m, 0≤i_D≤V_m/R_s=9/20 A in these cases. Removing R_s while retaining an ideal source and ideal capacitor changes the mathematical limit; an instantaneous connection of unequal voltages cannot be described by this finite-current differential equation with R_s set to zero.

16

Derive the off and on trajectories

Write θ=ωt with ω=2πf. While blocked, dv_C/dθ=−λv_C, where λ=1/(ωR_LC). Thus a blocked interval starting at θ₀ has v_C(θ)=v_C(θ₀)exp[−λ(θ−θ₀)]. This is the RC discharge law with the actual angular clock, not a linear voltage ramp unless the interval is small compared with the discharge time.

During a positive conducting interval, dv_C/dθ+a v_C=b sin θ, where a=(1/R_s+1/R_L)/(ωC) and b=V_m/(ωR_sC). A particular solution is x_p(θ)=b[a sin θ−cos θ]/(a²+1). Direct differentiation verifies x_p′+a x_p=b sin θ. The general on-trajectory adds the homogeneous term K exp(−aθ).

If conduction begins at α, continuity and zero initial diode current give v_C(α)=V_m sin α. Therefore v_C(θ)=x_p(θ)+[V_m sin α−x_p(α)]exp[−a(θ−α)] until turn-off. This expression solves the on equation, but it is physically valid only while its implied i_D is nonnegative. Switching events must be checked rather than inferred from a sketch.

17

Solve the periodic conduction events

Let P be the rectified waveform period in θ: P=2π for half-wave and P=π for the bridge. Turn-off at β satisfies v_C, on(β)=V_m sin β. Before the next turn-on, the capacitor discharges from that value. Periodicity therefore adds V_m sin α=V_m sin β exp[−λ(P+α−β)]. These two equations determine α and β together for the selected periodic state.

The continuous waveform can be followed from θ=0: its initial periodic voltage is V_m sin β exp[−λ(P−β)], then the off trajectory reaches α, the on trajectory reaches β, and the final off trajectory returns to the same voltage at P. The first source zero is not a zero capacitor voltage in a periodic state. For an initial uncharged capacitor, α=0 instead and the periodic matching equation must be removed.

At either switching event i_D=0 and v_C is continuous. Immediately after turn-on the source exceeds the capacitor enough for positive current; immediately after turn-off the blocked state is consistent. The capacitor derivative also matches at a zero-current event under this ideal finite-resistance model. A regular sample grid alone can miss these events, so their separate labelled records and one-sided branch conditions are retained.

18

Find capacitor extrema rather than source extrema

At the start of periodic conduction, i_D is zero while the load still draws v_C/R_L. The capacitor therefore continues falling briefly after turn-on. Its minimum occurs when i_D first equals the load current. During the on interval, dv_C/dt=0 gives v_C=w/(1+R_s/R_L), not v_C=w. This interior event is different from the turn-on crossing.

At the source peak θ=π/2 the capacitor can still be charging. Its maximum occurs later, when the falling charging current again equals the load current. The maximum and minimum solve the same derivative-zero condition on different branches of the on trajectory. Turn-off follows the maximum, when the current has fallen all the way to zero.

Our periodic cases have α<θ_min<π/2<θ_max<β. The uncharged startup case instead begins at the interval minimum v_C=0, then rises to its first maximum. Calling that whole first-cycle variation a steady ripple would discard its initial condition. Peak-to-peak periodic ripple uses the actual capacitor extrema, rather than the source amplitude or a pair of nearby grid points.

Turn-on, turn-off and capacitor extrema occur at different times

This detailed view is the periodic 100 µF bridge, with 20 Ω source resistance and 1000 Ω load. The two lanes share the same angular clock. All regular samples and four separate events are retained; the diode-current lane is not a new load-current projection. The marked capacitor minimum follows turn-on, and its maximum follows the source peak.

One bridge period: actual switching and capacitor extrema

Source and capacitor voltage share the upper V axis; diode current has a separate A axis. Dotted event lines and the event list distinguish turn-on, turn-off, capacitor minimum and maximum.

  • Source voltage
  • Capacitor voltage
  • Diode current iD

Diamond: a separately retained event record. Larger outer and smaller inner marks preserve coincident voltage controls.

One bridge period: actual switching and capacitor extremaSource and capacitor voltage share the upper V axis; diode current has a separate A axis. Dotted event lines and the event list distinguish turn-on, turn-off, capacitor minimum and maximum.Source / capacitor voltage (V)00.250.50.7510246810θ/π (1); one bridge output period 0 to 1Turn-on αCapacitor minimumCapacitor maximumTurn-off βDiode current iD (A)00.250.50.75100.010.020.030.04θ/π (1); same bridge output periodTurn-on αCapacitor minimumCapacitor maximumTurn-off β
Source and capacitor voltage share the upper V axis; diode current has a separate A axis. Dotted event lines and the event list distinguish turn-on, turn-off, capacitor minimum and maximum.

Short fractions are exact. A leading ≈ marks a value rounded to 12 significant digits. The plots use the labelled units and scales.

Finite-source-resistance reservoir boundaries and actual interval endpoints.
Finite-source-resistance reservoir boundaries and actual interval endpoints.
CaseC (F)Rs (Ω)RL (Ω)Peak source (V)f (Hz)Interval (s)Initial vC (V)End vC (V)
Half-wave, 100 µF, periodic1/100002010009601/60≈ 7.25333693406≈ 7.25333693406
Bridge, 100 µF, periodic1/100002010009601/120≈ 8.08598424755≈ 8.08598424755
Bridge, 470 µF, periodic47/1000002010009601/120≈ 8.12971571050≈ 8.12971571050
Half-wave, 100 µF, first cycle from zero1/100002010009601/600≈ 6.56002533234
Turn-on, turn-off and the actual capacitor extrema are separate events. Startup has its interval minimum at the initial zero.
Turn-on, turn-off and the actual capacitor extrema are separate events. Startup has its interval minimum at the initial zero.
CaseTurn-on α (rad)Turn-off β (rad)Conduction fraction (1)Minimum θ (rad)Maximum θ (rad)Minimum vC (V)Maximum vC (V)
Half-wave, 100 µF, periodic≈ 0.905602490373≈ 2.01602487247≈ 0.176729211030≈ 0.931079719224≈ 1.97066430144≈ 7.07880852653≈ 8.12746234474
Bridge, 100 µF, periodic≈ 1.06230356212≈ 1.95478855641≈ 0.284086796951≈ 1.09850734794≈ 1.89968445960≈ 7.85761152634≈ 8.35060633673
Bridge, 470 µF, periodic≈ 1.11438158427≈ 2.00038181635≈ 0.282022633031≈ 1.15670841066≈ 1.95408313929≈ 8.07779720146≈ 8.18329831321
Half-wave, 100 µF, first cycle from zero0≈ 2.19330891694≈ 0.3490759558590≈ 2.164306055990≈ 7.31455490065
Bridge, 100 µF, periodic: all regular retained samples. Adjacent samples are not exact extrema.
Bridge, 100 µF, periodic: all regular retained samples. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
Sample 00000≈ 8.0859842475500≈ -80.859842475500≈ 0.0653831412517≈ -0.0653831412517≈ 0.00326915706258blocked
Sample 11/48≈ 0.0654498469498≈ 0.000173611111111≈ 0.588628163071≈ 8.0719582593200≈ -80.719582593200≈ 0.0651565101401≈ -0.0651565101401≈ 0.00325782550701blocked
Sample 21/24≈ 0.130899693900≈ 0.000347222222222≈ 1.17473572998≈ 8.0579566006300≈ -80.579566006300≈ 0.0649306645776≈ -0.0649306645776≈ 0.00324653322888blocked
Sample 31/16≈ 0.196349540849≈ 0.000520833333333≈ 1.75581289815≈ 8.0439792292900≈ -80.439792292900≈ 0.0647056018412≈ -0.0647056018412≈ 0.00323528009206blocked
Sample 41/12≈ 0.261799387799≈ 0.000694444444444≈ 2.32937140592≈ 8.0300261031600≈ -80.300261031600≈ 0.0644813192175≈ -0.0644813192175≈ 0.00322406596087blocked
Sample 55/48≈ 0.327249234749≈ 0.000868055555556≈ 2.89295518773≈ 8.0160971802000≈ -80.160971802000≈ 0.0642578140024≈ -0.0642578140024≈ 0.00321289070012blocked
Sample 61/8≈ 0.392699081699≈ 0.00104166666667≈ 3.44415089129≈ 8.0021924184100≈ -80.021924184100≈ 0.0640350835013≈ -0.0640350835013≈ 0.00320175417507blocked
Sample 77/48≈ 0.458148928649≈ 0.00121527777778≈ 3.98059821197≈ 7.9883117759000≈ -79.883117759000≈ 0.0638131250290≈ -0.0638131250290≈ 0.00319065625145blocked
Sample 81/6≈ 0.523598775598≈ 0.00138888888889≈ 4.50000000000≈ 7.9744552108100≈ -79.744552108100≈ 0.0635919359093≈ -0.0635919359093≈ 0.00317959679546blocked
Sample 93/16≈ 0.589048622548≈ 0.00156250000000≈ 5.00013209718≈ 7.9606226814000≈ -79.606226814000≈ 0.0633715134756≈ -0.0633715134756≈ 0.00316857567378blocked
Sample 105/24≈ 0.654498469498≈ 0.00173611111111≈ 5.47885286108≈ 7.9468141459500≈ -79.468141459500≈ 0.0631518550703≈ -0.0631518550703≈ 0.00315759275352blocked
Sample 1111/48≈ 0.719948316448≈ 0.00190972222222≈ 5.93411233590≈ 7.9330295628600≈ -79.330295628600≈ 0.0629329580452≈ -0.0629329580452≈ 0.00314664790226blocked
Sample 121/4≈ 0.785398163397≈ 0.00208333333333≈ 6.36396103068≈ 7.9192688905800≈ -79.192688905800≈ 0.0627148197613≈ -0.0627148197613≈ 0.00313574098806blocked
Sample 1313/48≈ 0.850848010347≈ 0.00225694444444≈ 6.76655826731≈ 7.9055320876200≈ -79.055320876200≈ 0.0624974375884≈ -0.0624974375884≈ 0.00312487187942blocked
Sample 147/24≈ 0.916297857297≈ 0.00243055555556≈ 7.14018006262≈ 7.8918191125900≈ -78.918191125900≈ 0.0622808089059≈ -0.0622808089059≈ 0.00311404044530blocked
Sample 155/16≈ 0.981747704247≈ 0.00260416666667≈ 7.48322651072≈ 7.8781299241600≈ -78.781299241600≈ 0.0620649311020≈ -0.0620649311020≈ 0.00310324655510blocked
Sample 161/3≈ 1.04719755120≈ 0.00277777777778≈ 7.79422863406≈ 7.8644644810600≈ -78.644644810600≈ 0.0618498015739≈ -0.0618498015739≈ 0.00309249007869blocked
Sample 1717/48≈ 1.11264739815≈ 0.00295138888889≈ 8.07185467379≈ 7.85814595510≈ 0.0106854359348≈ 0.0106854359348≈ 28.2728997969≈ 0.0862512859917≈ 0.00228357082233≈ 0.0617504578516≈ 0.0222172573178≈ 0.00308752289258conducting
Sample 183/8≈ 1.17809724510≈ 0.00312500000000≈ 8.31491579260≈ 7.87333141594≈ 0.0220792188333≈ 0.0220792188333≈ 142.058874173≈ 0.183586845365≈ 0.00974983808575≈ 0.0619893475852≈ 0.111847659694≈ 0.00309946737926conducting
Sample 1919/48≈ 1.24354709205≈ 0.00329861111111≈ 8.52237116546≈ 7.90592799820≈ 0.0308221583626≈ 0.0308221583626≈ 229.162303644≈ 0.262677873686≈ 0.0190001089225≈ 0.0625036975128≈ 0.181174067251≈ 0.00312518487564conducting
Sample 205/12≈ 1.30899693900≈ 0.00347222222222≈ 8.69333243660≈ 7.95146396087≈ 0.0370934237864≈ 0.0370934237864≈ 291.419598256≈ 0.322465464187≈ 0.0275184417640≈ 0.0632257791211≈ 0.231721243302≈ 0.00316128895605conducting
Sample 217/16≈ 1.37444678595≈ 0.00364583333333≈ 8.82706752363≈ 8.00577923997≈ 0.0410644141828≈ 0.0410644141828≈ 330.586349428≈ 0.362478356810≈ 0.0337257222436≈ 0.0640925012392≈ 0.264660133327≈ 0.00320462506196conducting
Sample 2211/24≈ 1.43989663290≈ 0.00381944444444≈ 8.92300375236≈ 8.06501215763≈ 0.0428995797366≈ 0.0428995797366≈ 348.345675789≈ 0.382793110964≈ 0.0368074788315≈ 0.0650444211028≈ 0.280941211030≈ 0.00325222105514conducting
Sample 2323/48≈ 1.50534647985≈ 0.00399305555556≈ 8.98073030915≈ 8.12558748906≈ 0.0427571410042≈ 0.0427571410042≈ 346.315535152≈ 0.383990352149≈ 0.0365634621371≈ 0.0660251720424≈ 0.281401717970≈ 0.00330125860212conducting
Sample 241/2≈ 1.57079632679≈ 0.00416666666667≈ 9.00000000000≈ 8.18420571525≈ 0.0407897142377≈ 0.0407897142377≈ 326.055085224≈ 0.367107428139≈ 0.0332760157518≈ 0.0669812231895≈ 0.266850189198≈ 0.00334906115947conducting
Sample 2525/48≈ 1.63624617374≈ 0.00434027777778≈ 8.98073030915≈ 8.23783330296≈ 0.0371448503093≈ 0.0371448503093≈ 289.070170063≈ 0.333587883001≈ 0.0275947980899≈ 0.0678618975274≈ 0.238131187384≈ 0.00339309487637conducting
Sample 2613/24≈ 1.70169602069≈ 0.00451388888889≈ 8.92300375236≈ 8.28369386666≈ 0.0319654942854≈ 0.0319654942854≈ 236.818004188≈ 0.285228225455≈ 0.0204358564982≈ 0.0686195840765≈ 0.196172784880≈ 0.00343097920382conducting
Sample 279/16≈ 1.76714586764≈ 0.00468750000000≈ 8.82706752363≈ 8.31926007878≈ 0.0253903722424≈ 0.0253903722424≈ 170.711121637≈ 0.224122530234≈ 0.0128934200522≈ 0.0692100882584≈ 0.142019021924≈ 0.00346050441292conducting
Sample 287/12≈ 1.83259571459≈ 0.00486111111111≈ 8.69333243660≈ 8.34224620652≈ 0.0175543115042≈ 0.0175543115042≈ 92.1206529769≈ 0.152605465602≈ 0.00616307704773≈ 0.0695930717702≈ 0.0768493167838≈ 0.00347965358851conducting
Sample 2929/48≈ 1.89804556154≈ 0.00503472222222≈ 8.52237116546≈ 8.35060116339≈ 0.00858850010337≈ 0.00858850010337≈ 2.37898939985≈ 0.0731943856355≈ 0.00147524668051≈ 0.0697325397900≈ 0.00198659916501≈ 0.00348662698950conducting
Sample 305/8≈ 1.96349540849≈ 0.00520833333333≈ 8.31491579260≈ 8.3426609405500≈ -83.426609405500≈ 0.0695999915690≈ -0.0695999915690≈ 0.00347999957845blocked
Sample 3131/48≈ 2.02894525544≈ 0.00538194444444≈ 8.07185467379≈ 8.3281897196600≈ -83.281897196600≈ 0.0693587440066≈ -0.0693587440066≈ 0.00346793720033blocked
Sample 322/3≈ 2.09439510239≈ 0.00555555555556≈ 7.79422863406≈ 8.3137436006100≈ -83.137436006100≈ 0.0691183326567≈ -0.0691183326567≈ 0.00345591663284blocked
Sample 3311/16≈ 2.15984494934≈ 0.00572916666667≈ 7.48322651072≈ 8.2993225398800≈ -82.993225398800≈ 0.0688787546209≈ -0.0688787546209≈ 0.00344393773104blocked
Sample 3417/24≈ 2.22529479629≈ 0.00590277777778≈ 7.14018006262≈ 8.2849264939800≈ -82.849264939800≈ 0.0686400070107≈ -0.0686400070107≈ 0.00343200035053blocked
Sample 3535/48≈ 2.29074464324≈ 0.00607638888889≈ 6.76655826731≈ 8.2705554195400≈ -82.705554195400≈ 0.0684020869477≈ -0.0684020869477≈ 0.00342010434739blocked
Sample 363/4≈ 2.35619449019≈ 0.00625000000000≈ 6.36396103068≈ 8.2562092732400≈ -82.562092732400≈ 0.0681649915635≈ -0.0681649915635≈ 0.00340824957818blocked
Sample 3737/48≈ 2.42164433714≈ 0.00642361111111≈ 5.93411233590≈ 8.2418880118300≈ -82.418880118300≈ 0.0679287179996≈ -0.0679287179996≈ 0.00339643589998blocked
Sample 3819/24≈ 2.48709418409≈ 0.00659722222222≈ 5.47885286108≈ 8.2275915921500≈ -82.275915921500≈ 0.0676932634073≈ -0.0676932634073≈ 0.00338466317036blocked
Sample 3913/16≈ 2.55254403104≈ 0.00677083333333≈ 5.00013209718≈ 8.2133199711200≈ -82.133199711200≈ 0.0674586249480≈ -0.0674586249480≈ 0.00337293124740blocked
Sample 405/6≈ 2.61799387799≈ 0.00694444444444≈ 4.50000000000≈ 8.1990731057100≈ -81.990731057100≈ 0.0672247997927≈ -0.0672247997927≈ 0.00336123998964blocked
Sample 4141/48≈ 2.68344372494≈ 0.00711805555556≈ 3.98059821197≈ 8.1848509529800≈ -81.848509529800≈ 0.0669917851225≈ -0.0669917851225≈ 0.00334958925612blocked
Sample 427/8≈ 2.74889357189≈ 0.00729166666667≈ 3.44415089129≈ 8.1706534700700≈ -81.706534700700≈ 0.0667595781279≈ -0.0667595781279≈ 0.00333797890639blocked
Sample 4343/48≈ 2.81434341884≈ 0.00746527777778≈ 2.89295518773≈ 8.1564806141800≈ -81.564806141800≈ 0.0665281760095≈ -0.0665281760095≈ 0.00332640880047blocked
Sample 4411/12≈ 2.87979326579≈ 0.00763888888889≈ 2.32937140592≈ 8.1423323426000≈ -81.423323426000≈ 0.0662975759773≈ -0.0662975759773≈ 0.00331487879886blocked
Sample 4515/16≈ 2.94524311274≈ 0.00781250000000≈ 1.75581289815≈ 8.1282086126700≈ -81.282086126700≈ 0.0660677752512≈ -0.0660677752512≈ 0.00330338876256blocked
Sample 4623/24≈ 3.01069295969≈ 0.00798611111111≈ 1.17473572998≈ 8.1141093818500≈ -81.141093818500≈ 0.0658387710606≈ -0.0658387710606≈ 0.00329193855303blocked
Sample 4747/48≈ 3.07614280664≈ 0.00815972222222≈ 0.588628163071≈ 8.1000346076100≈ -81.000346076100≈ 0.0656105606445≈ -0.0656105606445≈ 0.00328052803223blocked
Sample 481≈ 3.14159265359≈ 0.008333333333330≈ 8.0859842475500≈ -80.859842475500≈ 0.0653831412517≈ -0.0653831412517≈ 0.00326915706258blocked
Bridge, 100 µF, periodic: actual separate event records. Adjacent samples are not exact extrema.
Bridge, 100 µF, periodic: actual separate event records. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
alpha≈ 0.338141725952≈ 1.06230356212≈ 0.00281784771627≈ 7.86131382626≈ 7.8613138262600≈ -78.613138262600≈ 0.0618002550749≈ -0.0618002550749≈ 0.00309001275374turn on
beta≈ 0.622228522903≈ 1.95478855641≈ 0.00518523769086≈ 8.34458795422≈ 8.3445879542200≈ -83.445879542200≈ 0.0696321481256≈ -0.0696321481256≈ 0.00348160740628turn off
minimum≈ 0.349665748895≈ 1.09850734794≈ 0.00291388124079≈ 8.01476375686≈ 7.85761152634≈ 0.00785761152634≈ 0.007857611526340≈ 0.0629769000768≈ 0.00123484117798≈ 0.06174205889880≈ 0.00308710294494conducting
maximum≈ 0.604688344120≈ 1.89968445960≈ 0.00503906953433≈ 8.51761846346≈ 8.35060633673≈ 0.00835060633673≈ 0.008350606336730≈ 0.0711272787148≈ 0.00139465252382≈ 0.06973262619100≈ 0.00348663130955conducting
One-sided branch conditions for the selected periodic bridge. The state at the exact switching event is distinct from its neighboring conducting or blocked intervals.
One-sided branch conditions for the selected periodic bridge. The state at the exact switching event is distinct from its neighboring conducting or blocked intervals.
Time sideCondition
Immediately before turn-onblocked, iD=0
At turn-oniD=0, continuous vC
Immediately after turn-onconducting, iD>0
Immediately before turn-offconducting, iD>0
At turn-offiD=0, continuous vC
Immediately after turn-offblocked, iD=0
19

Compare like boundaries and test the ripple approximation

The periodic comparison uses 100 µF in both half-wave and bridge circuits, then 470 µF in the same bridge with all other stated parameters fixed. More frequent bridge recharge reduces the discharge interval, and the larger bridge capacitance reduces the voltage variation further in these cases. Finite source resistance and the actual load voltage remain in the calculation.

The familiar estimate ΔV≈I_DC/(f_ripple C) treats the load current as nearly constant, the blocked discharge as nearly linear, and recharge as short compared with its repetition interval. Here f_ripple=f for half-wave and 2 f for the bridge, with I_DC=V_DC/R_L. It is an approximation under these conditions, not the differential equation or an identity for arbitrary load and capacitance.

The table compares that estimate with the actual event-based ripple. It overestimates the selected periodic cases because the finite recharge interval is appreciable. Capacitance alone does not fix recharge duration, source current peaks, losses or startup behaviour. We make no general monotonic peak-current claim from these three operating states.

Original periodic reservoir comparison, ideal diodes and 20 Ω source resistance; six significant digits
Caseα (rad)β (rad)V_DC (V)Actual minimum (V)Actual maximum (V)Actual peak-to-peak (V)Small-ripple estimate (V)
Half-wave, 100 µF, periodic0.9056022.016027.597637.078818.127461.048651.26627
Bridge, 100 µF, periodic1.062301.954798.105167.857618.350610.4929950.675430
Bridge, 470 µF, periodic1.114382.000388.130598.077808.183300.1055010.144159
20

Integrate source, heat and storage independently

The instantaneous ledger is p_S=w i_D, p_Rs=R_s i_D², p_L=v_C²/R_L and p_C=v_C(i_D−v_C/R_L). Multiplying the capacitor equation by v_C gives p_C=d(Cv_C²/2)/dt, so p_S=p_Rs+p_L+dE_C/dt. The source term uses delivered power; the other terms use absorbed power.

Integrate each term from its own expression, using the on and off trajectories. On an interval, v_C is a linear combination of sin θ, cos θ and an exponential, so v_C, v_C², wv_C and w² have elementary integrals. Off intervals are exponential integrals. Calculating the source energy, resistor heat and load energy separately permits a balance check; defining one as the remainder of the others would not be an independent check.

Likewise Q_D=∫i_Ddt, Q_L=∫v_C/R_L dt and C[v_C(end)−v_C(start)] satisfy Q_D−Q_L=ΔQ_C. In a periodic state both storage increments vanish. The capacitor’s mean current is therefore zero, while its mean voltage is positive. Zero mean capacitor current is not zero mean capacitor voltage, and zero net cycle storage does not mean that no energy moved through the capacitor.

Separately integrated reservoir energy and charge over the stated interval; six significant digits
CaseSource delivered (J)Source resistance heat (J)Load energy (J)Storage increment (J)Diode charge (C)Load charge (C)
Half-wave, 100 µF, periodic0.001095080.0001313530.00096373000.0001266270.000126627
Bridge, 100 µF, periodic0.0005940620.00004642130.00054764100.00006754300.0000675430
Bridge, 470 µF, periodic0.0005978700.00004697300.00055089700.00006775500.0000677550
Half-wave, 100 µF, first cycle from zero0.005173310.002373190.0006484300.002151700.0007538900.0000978873
21

Keep the first cycle and periodic state separate

For the original half-wave 100 µF startup case, v_C(0)=0 at a rising source zero. The on trajectory begins at α=0, not at the periodic α. Its first turn-off and subsequent discharge follow from that initial state. At one source period the capacitor is charged, so its stored energy and charge have increased.

The first-cycle source energy must supply resistor heat, load energy and that storage increment. Imposing a periodic zero-storage balance would incorrectly assign the charging energy to a loss or omit it. The interval mean voltage is useful as an explicitly stated first-cycle average; it is not evidence that the waveform has reached a repeating operating condition.

Finite R_s provides the same current upper bound, but it does not make startup equivalent to periodic recharge. Different initial capacitor voltages, source phase, source impedance or load state change the initial-value problem. An ideal model omitting inductance, equivalent series resistance and device dynamics cannot establish every switching stress of a real supply.

A first cycle carries a different state and storage ledger

Both half-wave traces use the same 100 µF capacitor, source, resistors and 1/60 s interval. One starts from zero; the other starts in its periodic state. Energy has its own joule axis. Tables retain every other reservoir case, independent charge integrals, actual ripple and the two startup nonapplicability masks. Coincident event or endpoint records are the same physical point, not extra observations.

Same half-wave interval, different initial capacitor state

All 49 regular and four event records are retained for each trace. Startup begins at zero; the periodic state begins charged. Joined samples are visual guides to the stated trajectories.

  • Periodic half-wave, 100 µF
  • First cycle from zero, 100 µF

Diamond: a separately retained event record. Larger outer and smaller inner marks preserve coincident voltage controls.

Same half-wave interval, different initial capacitor stateAll 49 regular and four event records are retained for each trace. Startup begins at zero; the periodic state begins charged. Joined samples are visual guides to the stated trajectories.Capacitor voltage vC (V)00.0050.010.01502468Physical time t (s); both intervals 0 to 1/60 s
All 49 regular and four event records are retained for each trace. Startup begins at zero; the periodic state begins charged. Joined samples are visual guides to the stated trajectories.

Source work supplies heat, load work and changed capacitor storage

Both intervals are 1/60 s in the same half-wave 100 µF circuit. Each bar shows a separately calculated energy in joules. A zero storage increment is a visible dot, not an omitted term.

  • Source delivered
  • Rs heat
  • Load energy
  • Stored-energy increment
Source work supplies heat, load work and changed capacitor storageBoth intervals are 1/60 s in the same half-wave 100 µF circuit. Each bar shows a separately calculated energy in joules. A zero storage increment is a visible dot, not an omitted term.Energy over 1/60 s (J)00.0010.0020.0030.0040.0050.00109508Source0.000131353Rs heat0.00096373Load0ΔEC0.00517331Source0.00237319Rs heat0.00064843Load0.0021517ΔECPeriodic half-wave, 100 µFFirst cycle from zero, 100 µFSource work = Rs heat + load energy + stored-energy increment
Both intervals are 1/60 s in the same half-wave 100 µF circuit. Each bar shows a separately calculated energy in joules. A zero storage increment is a visible dot, not an omitted term.

Short fractions are exact. A leading ≈ marks a value rounded to 12 significant digits. The plots use the labelled units and scales.

Periodic ripple uses actual event-based extrema. No periodic ripple approximation is assigned to the first-cycle state.
Periodic ripple uses actual event-based extrema. No periodic ripple approximation is assigned to the first-cycle state.
CaseInterval mean vC (V)Actual variation (V)Periodic small-ripple estimate (V)Estimate / actual (1)Diode-current upper bound (A)
Half-wave, 100 µF, periodic≈ 7.59763428919≈ 1.04865381821≈ 1.26627238153≈ 1.207521833759/20
Bridge, 100 µF, periodic≈ 8.10516247648≈ 0.492994810390≈ 0.675430206373≈ 1.370055408579/20
Bridge, 470 µF, periodic≈ 8.13059484865≈ 0.105501111749≈ 0.144159483132≈ 1.366426199149/20
Half-wave, 100 µF, first cycle from zero≈ 5.87323606867≈ 7.31455490065Not assigned: startup finite interval is not a periodic small-ripple stateNot assigned: periodic ripple approximation not assigned to startup9/20
Separate integrals over each labelled interval; capacitor work equals its endpoint storage change.
Separate integrals over each labelled interval; capacitor work equals its endpoint storage change.
CaseSource delivered (J)Source resistor heat (J)Load energy (J)Capacitor work (J)Stored-energy increment (J)
Half-wave, 100 µF, periodic≈ 0.00109508343277≈ 0.000131353418924≈ 0.00096373001385100
Bridge, 100 µF, periodic≈ 0.000594062023075≈ 0.0000464213166021≈ 0.00054764070647300
Bridge, 470 µF, periodic≈ 0.000597869945461≈ 0.0000469729897260≈ 0.00055089695573500
Half-wave, 100 µF, first cycle from zero≈ 0.00517331390539≈ 0.00237318690841≈ 0.000648430378930≈ 0.00215169661805≈ 0.00215169661805
Diode charge supplies load charge and the capacitor charge increment; periodic zero change does not mean zero mean capacitor voltage.
Diode charge supplies load charge and the capacitor charge increment; periodic zero change does not mean zero mean capacitor voltage.
CaseDiode charge (C)Load charge (C)Capacitor increment (C)
Half-wave, 100 µF, periodic≈ 0.000126627238153≈ 0.0001266272381530
Bridge, 100 µF, periodic≈ 0.0000675430206373≈ 0.00006754302063730
Bridge, 470 µF, periodic≈ 0.0000677549570721≈ 0.00006775495707210
Half-wave, 100 µF, first cycle from zero≈ 0.000753889801046≈ 0.0000978872678112≈ 0.000656002533234
Half-wave, 100 µF, periodic: all regular retained samples. Adjacent samples are not exact extrema.
Half-wave, 100 µF, periodic: all regular retained samples. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
Sample 00000≈ 7.2533369340600≈ -72.533369340600≈ 0.0526108966790≈ -0.0526108966790≈ 0.00263054483395blocked
Sample 11/24≈ 0.130899693900≈ 0.000347222222222≈ 1.17473572998≈ 7.2281954101100≈ -72.281954101100≈ 0.0522468088868≈ -0.0522468088868≈ 0.00261234044434blocked
Sample 21/12≈ 0.261799387799≈ 0.000694444444444≈ 2.32937140592≈ 7.2031410317400≈ -72.031410317400≈ 0.0518852407231≈ -0.0518852407231≈ 0.00259426203616blocked
Sample 31/8≈ 0.392699081699≈ 0.00104166666667≈ 3.44415089129≈ 7.1781734968800≈ -71.781734968800≈ 0.0515261747513≈ -0.0515261747513≈ 0.00257630873757blocked
Sample 41/6≈ 0.523598775598≈ 0.00138888888889≈ 4.50000000000≈ 7.1532925045100≈ -71.532925045100≈ 0.0511695936551≈ -0.0511695936551≈ 0.00255847968276blocked
Sample 55/24≈ 0.654498469498≈ 0.00173611111111≈ 5.47885286108≈ 7.1284977546700≈ -71.284977546700≈ 0.0508154802383≈ -0.0508154802383≈ 0.00254077401192blocked
Sample 61/4≈ 0.785398163397≈ 0.00208333333333≈ 6.36396103068≈ 7.1037889484100≈ -71.037889484100≈ 0.0504638174236≈ -0.0504638174236≈ 0.00252319087118blocked
Sample 77/24≈ 0.916297857297≈ 0.00243055555556≈ 7.14018006262≈ 7.07959741247≈ 0.00302913250747≈ 0.00302913250747≈ -40.5046490500≈ 0.0216285515369≈ 0.000183512874957≈ 0.0501206995227≈ -0.0286756608607≈ 0.00250603497613conducting
Sample 81/3≈ 1.04719755120≈ 0.00277777777778≈ 7.79422863406≈ 7.12197413276≈ 0.0336127250651≈ 0.0336127250651≈ 264.907509324≈ 0.261985264171≈ 0.0225963057261≈ 0.0507225155477≈ 0.188666442898≈ 0.00253612577738conducting
Sample 93/8≈ 1.17809724510≈ 0.00312500000000≈ 8.31491579260≈ 7.25074893554≈ 0.0532083428531≈ 0.0532083428531≈ 459.575939176≈ 0.442422890288≈ 0.0566225549836≈ 0.0525733601262≈ 0.333226975178≈ 0.00262866800631conducting
Sample 105/12≈ 1.30899693900≈ 0.00347222222222≈ 8.69333243660≈ 7.42992040837≈ 0.0631706014115≈ 0.0631706014115≈ 557.406810031≈ 0.549163038290≈ 0.0798104976538≈ 0.0552037172747≈ 0.414148823362≈ 0.00276018586374conducting
Sample 1111/24≈ 1.43989663290≈ 0.00381944444444≈ 8.92300375236≈ 7.62810992264≈ 0.0647446914864≈ 0.0647446914864≈ 571.165815638≈ 0.577717125079≈ 0.0838375015135≈ 0.0581880609918≈ 0.435691562574≈ 0.00290940304959conducting
Sample 121/2≈ 1.57079632679≈ 0.00416666666667≈ 9.00000000000≈ 7.81820835600≈ 0.0590895821998≈ 0.0590895821998≈ 512.713738438≈ 0.531806239798≈ 0.0698315744910≈ 0.0611243818979≈ 0.400850283410≈ 0.00305621909489conducting
Sample 1313/24≈ 1.70169602069≈ 0.00451388888889≈ 8.92300375236≈ 7.97709422893≈ 0.0472954761718≈ 0.0472954761718≈ 393.183819429≈ 0.422017711351≈ 0.0447372413264≈ 0.0636340323372≈ 0.313646437688≈ 0.00318170161686conducting
Sample 147/12≈ 1.83259571459≈ 0.00486111111111≈ 8.69333243660≈ 8.08540463244≈ 0.0303963902079≈ 0.0303963902079≈ 223.109855755≈ 0.264245924950≈ 0.0184788107534≈ 0.0653737680703≈ 0.180393346126≈ 0.00326868840352conducting
Sample 155/8≈ 1.96349540849≈ 0.00520833333333≈ 8.31491579260≈ 8.12734322628≈ 0.00937862831612≈ 0.00937862831612≈ 12.5128508984≈ 0.0779825046986≈ 0.00175917338184≈ 0.0660537079177≈ 0.0101696233990≈ 0.00330268539589conducting
Sample 162/3≈ 2.09439510239≈ 0.00555555555556≈ 7.79422863406≈ 8.1057423358200≈ -81.057423358200≈ 0.0657030588148≈ -0.0657030588148≈ 0.00328515294074blocked
Sample 1717/24≈ 2.22529479629≈ 0.00590277777778≈ 7.14018006262≈ 8.0776462033900≈ -80.776462033900≈ 0.0652483681872≈ -0.0652483681872≈ 0.00326241840936blocked
Sample 183/4≈ 2.35619449019≈ 0.00625000000000≈ 6.36396103068≈ 8.0496474578100≈ -80.496474578100≈ 0.0647968241950≈ -0.0647968241950≈ 0.00323984120975blocked
Sample 1919/24≈ 2.48709418409≈ 0.00659722222222≈ 5.47885286108≈ 8.0217457615000≈ -80.217457615000≈ 0.0643484050621≈ -0.0643484050621≈ 0.00321742025311blocked
Sample 205/6≈ 2.61799387799≈ 0.00694444444444≈ 4.50000000000≈ 7.9939407780800≈ -79.939407780800≈ 0.0639030891635≈ -0.0639030891635≈ 0.00319515445817blocked
Sample 217/8≈ 2.74889357189≈ 0.00729166666667≈ 3.44415089129≈ 7.9662321723300≈ -79.662321723300≈ 0.0634608550234≈ -0.0634608550234≈ 0.00317304275117blocked
Sample 2211/12≈ 2.87979326579≈ 0.00763888888889≈ 2.32937140592≈ 7.9386196101700≈ -79.386196101700≈ 0.0630216813150≈ -0.0630216813150≈ 0.00315108406575blocked
Sample 2323/24≈ 3.01069295969≈ 0.00798611111111≈ 1.17473572998≈ 7.9111027587000≈ -79.111027587000≈ 0.0625855468587≈ -0.0625855468587≈ 0.00312927734294blocked
Sample 241≈ 3.14159265359≈ 0.008333333333330≈ 7.8836812861700≈ -78.836812861700≈ 0.0621524306220≈ -0.0621524306220≈ 0.00310762153110blocked
Sample 2525/24≈ 3.27249234749≈ 0.00868055555556≈ -1.17473572998≈ 7.8563548619800≈ -78.563548619800≈ 0.0617223117174≈ -0.0617223117174≈ 0.00308611558587blocked
Sample 2613/12≈ 3.40339204139≈ 0.00902777777778≈ -2.32937140592≈ 7.8291231566700≈ -78.291231566700≈ 0.0612951694023≈ -0.0612951694023≈ 0.00306475847012blocked
Sample 279/8≈ 3.53429173529≈ 0.00937500000000≈ -3.44415089129≈ 7.8019858419200≈ -78.019858419200≈ 0.0608709830776≈ -0.0608709830776≈ 0.00304354915388blocked
Sample 287/6≈ 3.66519142919≈ 0.00972222222222≈ -4.50000000000≈ 7.7749425905700≈ -77.749425905700≈ 0.0604497322866≈ -0.0604497322866≈ 0.00302248661433blocked
Sample 2929/24≈ 3.79609112309≈ 0.0100694444444≈ -5.47885286108≈ 7.7479930765500≈ -77.479930765500≈ 0.0600313967143≈ -0.0600313967143≈ 0.00300156983572blocked
Sample 305/4≈ 3.92699081699≈ 0.0104166666667≈ -6.36396103068≈ 7.7211369749700≈ -77.211369749700≈ 0.0596159561863≈ -0.0596159561863≈ 0.00298079780932blocked
Sample 3131/24≈ 4.05789051089≈ 0.0107638888889≈ -7.14018006262≈ 7.6943739620400≈ -76.943739620400≈ 0.0592033906677≈ -0.0592033906677≈ 0.00296016953339blocked
Sample 324/3≈ 4.18879020479≈ 0.0111111111111≈ -7.79422863406≈ 7.6677037150900≈ -76.677037150900≈ 0.0587936802624≈ -0.0587936802624≈ 0.00293968401312blocked
Sample 3311/8≈ 4.31968989869≈ 0.0114583333333≈ -8.31491579260≈ 7.6411259125800≈ -76.411259125800≈ 0.0583868052119≈ -0.0583868052119≈ 0.00291934026059blocked
Sample 3417/12≈ 4.45058959259≈ 0.0118055555556≈ -8.69333243660≈ 7.6146402340700≈ -76.146402340700≈ 0.0579827458944≈ -0.0579827458944≈ 0.00289913729472blocked
Sample 3535/24≈ 4.58148928649≈ 0.0121527777778≈ -8.92300375236≈ 7.5882463602500≈ -75.882463602500≈ 0.0575814828239≈ -0.0575814828239≈ 0.00287907414120blocked
Sample 363/2≈ 4.71238898038≈ 0.0125000000000≈ -9.00000000000≈ 7.5619439729100≈ -75.619439729100≈ 0.0571829966494≈ -0.0571829966494≈ 0.00285914983247blocked
Sample 3737/24≈ 4.84328867428≈ 0.0128472222222≈ -8.92300375236≈ 7.5357327549200≈ -75.357327549200≈ 0.0567872681536≈ -0.0567872681536≈ 0.00283936340768blocked
Sample 3819/12≈ 4.97418836818≈ 0.0131944444444≈ -8.69333243660≈ 7.5096123902900≈ -75.096123902900≈ 0.0563942782524≈ -0.0563942782524≈ 0.00281971391262blocked
Sample 3913/8≈ 5.10508806208≈ 0.0135416666667≈ -8.31491579260≈ 7.4835825640900≈ -74.835825640900≈ 0.0560040079935≈ -0.0560040079935≈ 0.00280020039968blocked
Sample 405/3≈ 5.23598775598≈ 0.0138888888889≈ -7.79422863406≈ 7.4576429625000≈ -74.576429625000≈ 0.0556164385561≈ -0.0556164385561≈ 0.00278082192781blocked
Sample 4141/24≈ 5.36688744988≈ 0.0142361111111≈ -7.14018006262≈ 7.4317932727800≈ -74.317932727800≈ 0.0552315512494≈ -0.0552315512494≈ 0.00276157756247blocked
Sample 427/4≈ 5.49778714378≈ 0.0145833333333≈ -6.36396103068≈ 7.4060331832900≈ -74.060331832900≈ 0.0548493275120≈ -0.0548493275120≈ 0.00274246637560blocked
Sample 4343/24≈ 5.62868683768≈ 0.0149305555556≈ -5.47885286108≈ 7.3803623834400≈ -73.803623834400≈ 0.0544697489110≈ -0.0544697489110≈ 0.00272348744555blocked
Sample 4411/6≈ 5.75958653158≈ 0.0152777777778≈ -4.50000000000≈ 7.3547805637500≈ -73.547805637500≈ 0.0540927971409≈ -0.0540927971409≈ 0.00270463985705blocked
Sample 4515/8≈ 5.89048622548≈ 0.0156250000000≈ -3.44415089129≈ 7.3292874157900≈ -73.292874157900≈ 0.0537184540232≈ -0.0537184540232≈ 0.00268592270116blocked
Sample 4623/12≈ 6.02138591938≈ 0.0159722222222≈ -2.32937140592≈ 7.3038826322000≈ -73.038826322000≈ 0.0533467015050≈ -0.0533467015050≈ 0.00266733507525blocked
Sample 4747/24≈ 6.15228561328≈ 0.0163194444444≈ -1.17473572998≈ 7.2785659067000≈ -72.785659067000≈ 0.0529775216582≈ -0.0529775216582≈ 0.00264887608291blocked
Sample 482≈ 6.28318530718≈ 0.01666666666670≈ 7.2533369340600≈ -72.533369340600≈ 0.0526108966790≈ -0.0526108966790≈ 0.00263054483395blocked
Half-wave, 100 µF, periodic: actual separate event records. Adjacent samples are not exact extrema.
Half-wave, 100 µF, periodic: actual separate event records. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
alpha≈ 0.288262225638≈ 0.905602490373≈ 0.00240218521365≈ 7.08117445630≈ 7.0811744563000≈ -70.811744563000≈ 0.0501430316806≈ -0.0501430316806≈ 0.00250715158403turn on
beta≈ 0.641720647699≈ 2.01602487247≈ 0.00534767206416≈ 8.12261036284≈ 8.1226103628400≈ -81.226103628400≈ 0.0659767991066≈ -0.0659767991066≈ 0.00329883995533turn off
minimum≈ 0.296371879454≈ 0.931079719224≈ 0.00246976566212≈ 7.22038469706≈ 7.07880852653≈ 0.00707880852653≈ 0.007078808526530≈ 0.0511117207584≈ 0.00100219060311≈ 0.05010953015530≈ 0.00250547650777conducting
maximum≈ 0.627281929498≈ 1.97066430144≈ 0.00522734941248≈ 8.29001159163≈ 8.12746234474≈ 0.00812746234474≈ 0.008127462344740≈ 0.0673767570485≈ 0.00132111288330≈ 0.06605564416520≈ 0.00330278220826conducting
Bridge, 470 µF, periodic: all regular retained samples. Adjacent samples are not exact extrema.
Bridge, 470 µF, periodic: all regular retained samples. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
Sample 00000≈ 8.1297157105000≈ -17.297267469200≈ 0.0660922775336≈ -0.0660922775336≈ 0.0155316852204blocked
Sample 11/48≈ 0.0654498469498≈ 0.000173611111111≈ 0.588628163071≈ 8.1267132672400≈ -17.290879292000≈ 0.0660434685280≈ -0.0660434685280≈ 0.0155202151041blocked
Sample 21/24≈ 0.130899693900≈ 0.000347222222222≈ 1.17473572998≈ 8.1237119328400≈ -17.284493474100≈ 0.0659946955677≈ -0.0659946955677≈ 0.0155087534584blocked
Sample 31/16≈ 0.196349540849≈ 0.000520833333333≈ 1.75581289815≈ 8.1207117068700≈ -17.278110014600≈ 0.0659459586261≈ -0.0659459586261≈ 0.0154973002771blocked
Sample 41/12≈ 0.261799387799≈ 0.000694444444444≈ 2.32937140592≈ 8.1177125889400≈ -17.271728912600≈ 0.0658972576767≈ -0.0658972576767≈ 0.0154858555540blocked
Sample 55/48≈ 0.327249234749≈ 0.000868055555556≈ 2.89295518773≈ 8.1147145786400≈ -17.265350167300≈ 0.0658485926928≈ -0.0658485926928≈ 0.0154744192828blocked
Sample 61/8≈ 0.392699081699≈ 0.00104166666667≈ 3.44415089129≈ 8.1117176755500≈ -17.258973777800≈ 0.0657999636479≈ -0.0657999636479≈ 0.0154629914573blocked
Sample 77/48≈ 0.458148928649≈ 0.00121527777778≈ 3.98059821197≈ 8.1087218792800≈ -17.252599743100≈ 0.0657513705155≈ -0.0657513705155≈ 0.0154515720711blocked
Sample 81/6≈ 0.523598775598≈ 0.00138888888889≈ 4.50000000000≈ 8.1057271894000≈ -17.246228062500≈ 0.0657028132689≈ -0.0657028132689≈ 0.0154401611182blocked
Sample 93/16≈ 0.589048622548≈ 0.00156250000000≈ 5.00013209718≈ 8.1027336055100≈ -17.239858735100≈ 0.0656542918818≈ -0.0656542918818≈ 0.0154287585922blocked
Sample 105/24≈ 0.654498469498≈ 0.00173611111111≈ 5.47885286108≈ 8.0997411272000≈ -17.233491760000≈ 0.0656058063277≈ -0.0656058063277≈ 0.0154173644870blocked
Sample 1111/48≈ 0.719948316448≈ 0.00190972222222≈ 5.93411233590≈ 8.0967497540700≈ -17.227127136300≈ 0.0655573565800≈ -0.0655573565800≈ 0.0154059787963blocked
Sample 121/4≈ 0.785398163397≈ 0.00208333333333≈ 6.36396103068≈ 8.0937594857000≈ -17.220764863200≈ 0.0655089426123≈ -0.0655089426123≈ 0.0153946015139blocked
Sample 1313/48≈ 0.850848010347≈ 0.00225694444444≈ 6.76655826731≈ 8.0907703216900≈ -17.214404939800≈ 0.0654605643983≈ -0.0654605643983≈ 0.0153832326336blocked
Sample 147/24≈ 0.916297857297≈ 0.00243055555556≈ 7.14018006262≈ 8.0877822616200≈ -17.208047365200≈ 0.0654122219114≈ -0.0654122219114≈ 0.0153718721492blocked
Sample 155/16≈ 0.981747704247≈ 0.00260416666667≈ 7.48322651072≈ 8.0847953051000≈ -17.201692138500≈ 0.0653639151254≈ -0.0653639151254≈ 0.0153605200545blocked
Sample 161/3≈ 1.04719755120≈ 0.00277777777778≈ 7.79422863406≈ 8.0818094517200≈ -17.195339259000≈ 0.0653156440139≈ -0.0653156440139≈ 0.0153491763433blocked
Sample 1717/48≈ 1.11264739815≈ 0.00295138888889≈ 8.07185467379≈ 8.0788247010600≈ -17.188988725700≈ 0.0652674085504≈ -0.0652674085504≈ 0.0153378410094blocked
Sample 183/8≈ 1.17809724510≈ 0.00312500000000≈ 8.31491579260≈ 8.07802667004≈ 0.0118444561279≈ 0.0118444561279≈ 8.01367969763≈ 0.0984856553129≈ 0.00280582281933≈ 0.0652545148819≈ 0.0304253176116≈ 0.0153348109973conducting
Sample 1919/48≈ 1.24354709205≈ 0.00329861111111≈ 8.52237116546≈ 8.08136372018≈ 0.0220503722639≈ 0.0220503722639≈ 29.7212947738≈ 0.187921456769≈ 0.00972437833952≈ 0.0653084395778≈ 0.112888638852≈ 0.0153474833008conducting
Sample 205/12≈ 1.30899693900≈ 0.00347222222222≈ 8.69333243660≈ 8.08810050981≈ 0.0302615963396≈ 0.0302615963396≈ 47.1776507016≈ 0.263074117042≈ 0.0183152842604≈ 0.0654173698568≈ 0.179341462925≈ 0.0153730819163conducting
Sample 217/16≈ 1.37444678595≈ 0.00364583333333≈ 8.82706752363≈ 8.09749892264≈ 0.0364784300495≈ 0.0364784300495≈ 60.3849598445≈ 0.321997565203≈ 0.0266135171776≈ 0.0655694888021≈ 0.229814559223≈ 0.0154088298685conducting
Sample 2211/24≈ 1.43989663290≈ 0.00381944444444≈ 8.92300375236≈ 8.10882268147≈ 0.0407090535446≈ 0.0407090535446≈ 69.3621933259≈ 0.363247037534≈ 0.0331445408100≈ 0.0657530052795≈ 0.264349491444≈ 0.0154519562407conducting
Sample 2323/48≈ 1.50534647985≈ 0.00399305555556≈ 8.98073030915≈ 8.12134025405≈ 0.0429695027547≈ 0.0429695027547≈ 74.1450265970≈ 0.385897515758≈ 0.0369275633397≈ 0.0659561675221≈ 0.283013784896≈ 0.0154996993677conducting
Sample 241/2≈ 1.57079632679≈ 0.00416666666667≈ 9.00000000000≈ 8.13432774319≈ 0.0432836128407≈ 0.0432836128407≈ 74.7857129735≈ 0.389552515567≈ 0.0374694228109≈ 0.0661672878336≈ 0.285915804922≈ 0.0155493126409conducting
Sample 2525/48≈ 1.63624617374≈ 0.00434027777778≈ 8.98073030915≈ 8.14707174866≈ 0.0416829280242≈ 0.0416829280242≈ 71.3528856925≈ 0.374343135081≈ 0.0347493297733≈ 0.0663747780779≈ 0.273219027229≈ 0.0155980728483conducting
Sample 2613/24≈ 1.70169602069≈ 0.00451388888889≈ 8.92300375236≈ 8.15887218873≈ 0.0382065781816≈ 0.0382065781816≈ 63.9312893466≈ 0.340917440480≈ 0.0291948523270≈ 0.0665671953921≈ 0.245155392761≈ 0.0156432909171conducting
Sample 279/16≈ 1.76714586764≈ 0.00468750000000≈ 8.82706752363≈ 8.16904506883≈ 0.0329011227401≈ 0.0329011227401≈ 52.6214418537≈ 0.290420432030≈ 0.0216496775511≈ 0.0667332973365≈ 0.202037457142≈ 0.0156823248741conducting
Sample 287/12≈ 1.83259571459≈ 0.00486111111111≈ 8.69333243660≈ 8.17692518575≈ 0.0258203625427≈ 0.0258203625427≈ 37.5392284191≈ 0.224464995218≈ 0.0133338224368≈ 0.0668621054933≈ 0.144269067287≈ 0.0157125947909conducting
Sample 2929/48≈ 1.89804556154≈ 0.00503472222222≈ 8.52237116546≈ 8.18186875551≈ 0.0170251204972≈ 0.0170251204972≈ 18.8154292376≈ 0.145094396014≈ 0.00579709455887≈ 0.0669429763324≈ 0.0723543251223≈ 0.0157315994381conducting
Sample 305/8≈ 1.96349540849≈ 0.00520833333333≈ 8.31491579260≈ 8.18325595365≈ 0.00658299194738≈ 0.00658299194738≈ -3.40481703462≈ 0.0547370237059≈ 0.000866715659586≈ 0.0669656780030≈ -0.0130953699567≈ 0.0157369343307conducting
Sample 3131/48≈ 2.02894525544≈ 0.00538194444444≈ 8.07185467379≈ 8.1809272981200≈ -17.406228293900≈ 0.0669275714571≈ -0.0669275714571≈ 0.0157279792924blocked
Sample 322/3≈ 2.09439510239≈ 0.00555555555556≈ 7.79422863406≈ 8.1779059415400≈ -17.399799875600≈ 0.0668781455887≈ -0.0668781455887≈ 0.0157163642133blocked
Sample 3311/16≈ 2.15984494934≈ 0.00572916666667≈ 7.48322651072≈ 8.1748857008000≈ -17.393373831500≈ 0.0668287562212≈ -0.0668287562212≈ 0.0157047577120blocked
Sample 3417/24≈ 2.22529479629≈ 0.00590277777778≈ 7.14018006262≈ 8.1718665754900≈ -17.386950160600≈ 0.0667794033276≈ -0.0667794033276≈ 0.0156931597820blocked
Sample 3535/48≈ 2.29074464324≈ 0.00607638888889≈ 6.76655826731≈ 8.1688485651900≈ -17.380528862100≈ 0.0667300868811≈ -0.0667300868811≈ 0.0156815704171blocked
Sample 363/4≈ 2.35619449019≈ 0.00625000000000≈ 6.36396103068≈ 8.1658316695000≈ -17.374109935100≈ 0.0666808068546≈ -0.0666808068546≈ 0.0156699896108blocked
Sample 3737/48≈ 2.42164433714≈ 0.00642361111111≈ 5.93411233590≈ 8.1628158880000≈ -17.367693378700≈ 0.0666315632213≈ -0.0666315632213≈ 0.0156584173570blocked
Sample 3819/24≈ 2.48709418409≈ 0.00659722222222≈ 5.47885286108≈ 8.1598012202700≈ -17.361279192100≈ 0.0665823559544≈ -0.0665823559544≈ 0.0156468536493blocked
Sample 3913/16≈ 2.55254403104≈ 0.00677083333333≈ 5.00013209718≈ 8.1567876659200≈ -17.354867374300≈ 0.0665331850269≈ -0.0665331850269≈ 0.0156352984813blocked
Sample 405/6≈ 2.61799387799≈ 0.00694444444444≈ 4.50000000000≈ 8.1537752245200≈ -17.348457924500≈ 0.0664840504120≈ -0.0664840504120≈ 0.0156237518468blocked
Sample 4141/48≈ 2.68344372494≈ 0.00711805555556≈ 3.98059821197≈ 8.1507638956700≈ -17.342050841800≈ 0.0664349520829≈ -0.0664349520829≈ 0.0156122137395blocked
Sample 427/8≈ 2.74889357189≈ 0.00729166666667≈ 3.44415089129≈ 8.1477536789500≈ -17.335646125400≈ 0.0663858900129≈ -0.0663858900129≈ 0.0156006841530blocked
Sample 4343/48≈ 2.81434341884≈ 0.00746527777778≈ 2.89295518773≈ 8.1447445739600≈ -17.329243774400≈ 0.0663368641750≈ -0.0663368641750≈ 0.0155891630811blocked
Sample 4411/12≈ 2.87979326579≈ 0.00763888888889≈ 2.32937140592≈ 8.1417365802800≈ -17.322843787800≈ 0.0662878745427≈ -0.0662878745427≈ 0.0155776505175blocked
Sample 4515/16≈ 2.94524311274≈ 0.00781250000000≈ 1.75581289815≈ 8.1387296975100≈ -17.316446164900≈ 0.0662389210891≈ -0.0662389210891≈ 0.0155661464559blocked
Sample 4623/24≈ 3.01069295969≈ 0.00798611111111≈ 1.17473572998≈ 8.1357239252300≈ -17.310050904700≈ 0.0661900037875≈ -0.0661900037875≈ 0.0155546508901blocked
Sample 4747/48≈ 3.07614280664≈ 0.00815972222222≈ 0.588628163071≈ 8.1327192630300≈ -17.303658006400≈ 0.0661411226112≈ -0.0661411226112≈ 0.0155431638136blocked
Sample 481≈ 3.14159265359≈ 0.008333333333330≈ 8.1297157105000≈ -17.297267469200≈ 0.0660922775336≈ -0.0660922775336≈ 0.0155316852204blocked
Bridge, 470 µF, periodic: actual separate event records. Adjacent samples are not exact extrema.
Bridge, 470 µF, periodic: actual separate event records. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
alpha≈ 0.354718675254≈ 1.11438158427≈ 0.00295598896045≈ 8.07874563087≈ 8.0787456308700≈ -17.188820491200≈ 0.0652661309683≈ -0.0652661309683≈ 0.0153375407775turn on
beta≈ 0.636741308286≈ 2.00038181635≈ 0.00530617756905≈ 8.18224621996≈ 8.1822462199600≈ -17.409034510500≈ 0.0669491532040≈ -0.0669491532040≈ 0.0157330510029turn off
minimum≈ 0.368191722545≈ 1.15670841066≈ 0.00306826435454≈ 8.23935314549≈ 8.07779720146≈ 0.00807779720146≈ 0.008077797201460≈ 0.0665558237806≈ 0.00130501615256≈ 0.06525080762800≈ 0.0153339397926conducting
maximum≈ 0.622003981661≈ 1.95408313929≈ 0.00518336651384≈ 8.34696427948≈ 8.18329831321≈ 0.00818329831321≈ 0.008183298313210≈ 0.0683056987087≈ 0.00133932742566≈ 0.06696637128300≈ 0.0157370972515conducting
Half-wave, 100 µF, first cycle from zero: all regular retained samples. Adjacent samples are not exact extrema.
Half-wave, 100 µF, first cycle from zero: all regular retained samples. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
Sample 00000000000000conducting
Sample 11/24≈ 0.130899693900≈ 0.000347222222222≈ 1.17473572998≈ 0.0963460158524≈ 0.0539194857064≈ 0.0539194857064≈ 538.231396905≈ 0.0633411464015≈ 0.0581462187769≈ 0.00000928255477064≈ 0.00518564506985≈ 4.64127738532e-7conducting
Sample 21/12≈ 0.261799387799≈ 0.000694444444444≈ 2.32937140592≈ 0.362584432805≈ 0.0983393486559≈ 0.0983393486559≈ 979.767642231≈ 0.229068866836≈ 0.193412549881≈ 0.000131467470913≈ 0.0355248494839≈ 0.00000657337354564conducting
Sample 31/8≈ 0.392699081699≈ 0.00104166666667≈ 3.44415089129≈ 0.766320459021≈ 0.133891521613≈ 0.133891521613≈ 1331.25201154≈ 0.461142603500≈ 0.358538791198≈ 0.000587247045914≈ 0.102016565256≈ 0.0000293623522957conducting
Sample 41/6≈ 0.523598775598≈ 0.00138888888889≈ 4.50000000000≈ 1.27732473398≈ 0.161133763301≈ 0.161133763301≈ 1598.56438567≈ 0.725101934854≈ 0.519281793510≈ 0.00163155847604≈ 0.204188582868≈ 0.0000815779238022conducting
Sample 55/24≈ 0.654498469498≈ 0.00173611111111≈ 5.47885286108≈ 1.86731732620≈ 0.180576776744≈ 0.180576776744≈ 1787.09459418≈ 0.989353589907≈ 0.652159445984≈ 0.00348687399673≈ 0.333707269927≈ 0.000174343699837conducting
Sample 61/4≈ 0.785398163397≈ 0.00208333333333≈ 6.36396103068≈ 2.50983737239≈ 0.192706182915≈ 0.192706182915≈ 1901.96345542≈ 1.22637463844≈ 0.742713458670≈ 0.00629928363584≈ 0.477361896133≈ 0.000314964181792conducting
Sample 77/24≈ 0.916297857297≈ 0.00243055555556≈ 7.14018006262≈ 3.18018127661≈ 0.197999939301≈ 0.197999939301≈ 1948.19758024≈ 1.41375521900≈ 0.784079519262≈ 0.0101135529521≈ 0.619562146781≈ 0.000505677647604conducting
Sample 81/3≈ 1.04719755120≈ 0.00277777777778≈ 7.79422863406≈ 3.85539434901≈ 0.196941714252≈ 0.196941714252≈ 1930.86319903≈ 1.53500874847≈ 0.775720776254≈ 0.0148640655864≈ 0.744423906627≈ 0.000743203279319conducting
Sample 93/8≈ 1.17809724510≈ 0.00312500000000≈ 8.31491579260≈ 4.51430247489≈ 0.190030665886≈ 0.190030665886≈ 1855.16363411≈ 1.58008898485≈ 0.722233079538≈ 0.0203789268348≈ 0.837476978477≈ 0.00101894634174conducting
Sample 105/12≈ 1.30899693900≈ 0.00347222222222≈ 8.69333243660≈ 5.13757191921≈ 0.177788025869≈ 0.177788025869≈ 1726.50453950≈ 1.54557041213≈ 0.632171642851≈ 0.0263946452251≈ 0.887004124054≈ 0.00131973226125conducting
Sample 1111/24≈ 1.43989663290≈ 0.00381944444444≈ 8.92300375236≈ 5.70778673294≈ 0.160760850971≈ 0.160760850971≈ 1550.53064238≈ 1.43446967645≈ 0.516881024101≈ 0.0325788293887≈ 0.885009822961≈ 0.00162894146943conducting
Sample 121/2≈ 1.57079632679≈ 0.00416666666667≈ 9.00000000000≈ 6.20953446967≈ 0.139523276517≈ 0.139523276517≈ 1333.13742047≈ 1.25570948865≈ 0.389334893799≈ 0.0385583183300≈ 0.827816276521≈ 0.00192791591650conducting
Sample 1313/24≈ 1.70169602069≈ 0.00451388888889≈ 8.92300375236≈ 6.62949207192≈ 0.114675584022≈ 0.114675584022≈ 1080.46091950≈ 1.02325066653≈ 0.263009791416≈ 0.0439501651317≈ 0.716290709986≈ 0.00219750825658conducting
Sample 147/12≈ 1.83259571459≈ 0.00486111111111≈ 8.69333243660≈ 6.95650486663≈ 0.0868413784988≈ 0.0868413784988≈ 798.848736322≈ 0.754940972543≈ 0.150828500391≈ 0.0483929599594≈ 0.555719512192≈ 0.00241964799797conducting
Sample 155/8≈ 1.96349540849≈ 0.00520833333333≈ 8.31491579260≈ 7.18165263456≈ 0.0566631579022≈ 0.0566631579022≈ 494.815052676≈ 0.471149386500≈ 0.0642142692690≈ 0.0515761345634≈ 0.355358982667≈ 0.00257880672817conducting
Sample 162/3≈ 2.09439510239≈ 0.00555555555556≈ 7.79422863406≈ 7.29829769990≈ 0.0247965467081≈ 0.0247965467081≈ 174.982490082≈ 0.193269954378≈ 0.0122973745729≈ 0.0532651493163≈ 0.127707430489≈ 0.00266325746582conducting
Sample 1717/24≈ 2.22529479629≈ 0.00590277777778≈ 7.14018006262≈ 7.3055428421000≈ -73.055428421000≈ 0.0533709562178≈ -0.0533709562178≈ 0.00266854781089blocked
Sample 183/4≈ 2.35619449019≈ 0.00625000000000≈ 6.36396103068≈ 7.2802203619800≈ -72.802203619800≈ 0.0530016085190≈ -0.0530016085190≈ 0.00265008042595blocked
Sample 1919/24≈ 2.48709418409≈ 0.00659722222222≈ 5.47885286108≈ 7.2549856546600≈ -72.549856546600≈ 0.0526348168494≈ -0.0526348168494≈ 0.00263174084247blocked
Sample 205/6≈ 2.61799387799≈ 0.00694444444444≈ 4.50000000000≈ 7.2298384159200≈ -72.298384159200≈ 0.0522705635203≈ -0.0522705635203≈ 0.00261352817601blocked
Sample 217/8≈ 2.74889357189≈ 0.00729166666667≈ 3.44415089129≈ 7.2047783425600≈ -72.047783425600≈ 0.0519088309654≈ -0.0519088309654≈ 0.00259544154827blocked
Sample 2211/12≈ 2.87979326579≈ 0.00763888888889≈ 2.32937140592≈ 7.1798051324500≈ -71.798051324500≈ 0.0515496017399≈ -0.0515496017399≈ 0.00257748008700blocked
Sample 2323/24≈ 3.01069295969≈ 0.00798611111111≈ 1.17473572998≈ 7.1549184845000≈ -71.549184845000≈ 0.0511928585199≈ -0.0511928585199≈ 0.00255964292600blocked
Sample 241≈ 3.14159265359≈ 0.008333333333330≈ 7.1301180986900≈ -71.301180986900≈ 0.0508385841012≈ -0.0508385841012≈ 0.00254192920506blocked
Sample 2525/24≈ 3.27249234749≈ 0.00868055555556≈ -1.17473572998≈ 7.1054036759900≈ -71.054036759900≈ 0.0504867613988≈ -0.0504867613988≈ 0.00252433806994blocked
Sample 2613/12≈ 3.40339204139≈ 0.00902777777778≈ -2.32937140592≈ 7.0807749184500≈ -70.807749184500≈ 0.0501373734458≈ -0.0501373734458≈ 0.00250686867229blocked
Sample 279/8≈ 3.53429173529≈ 0.00937500000000≈ -3.44415089129≈ 7.0562315291400≈ -70.562315291400≈ 0.0497904033928≈ -0.0497904033928≈ 0.00248952016964blocked
Sample 287/6≈ 3.66519142919≈ 0.00972222222222≈ -4.50000000000≈ 7.0317732121500≈ -70.317732121500≈ 0.0494458345071≈ -0.0494458345071≈ 0.00247229172536blocked
Sample 2929/24≈ 3.79609112309≈ 0.0100694444444≈ -5.47885286108≈ 7.0073996726000≈ -70.073996726000≈ 0.0491036501716≈ -0.0491036501716≈ 0.00245518250858blocked
Sample 305/4≈ 3.92699081699≈ 0.0104166666667≈ -6.36396103068≈ 6.9831106166400≈ -69.831106166400≈ 0.0487638338842≈ -0.0487638338842≈ 0.00243819169421blocked
Sample 3131/24≈ 4.05789051089≈ 0.0107638888889≈ -7.14018006262≈ 6.9589057514300≈ -69.589057514300≈ 0.0484263692573≈ -0.0484263692573≈ 0.00242131846287blocked
Sample 324/3≈ 4.18879020479≈ 0.0111111111111≈ -7.79422863406≈ 6.9347847851500≈ -69.347847851500≈ 0.0480912400164≈ -0.0480912400164≈ 0.00240456200082blocked
Sample 3311/8≈ 4.31968989869≈ 0.0114583333333≈ -8.31491579260≈ 6.9107474269900≈ -69.107474269900≈ 0.0477584299996≈ -0.0477584299996≈ 0.00238792149998blocked
Sample 3417/12≈ 4.45058959259≈ 0.0118055555556≈ -8.69333243660≈ 6.8867933871400≈ -68.867933871400≈ 0.0474279231572≈ -0.0474279231572≈ 0.00237139615786blocked
Sample 3535/24≈ 4.58148928649≈ 0.0121527777778≈ -8.92300375236≈ 6.8629223768100≈ -68.629223768100≈ 0.0470997035501≈ -0.0470997035501≈ 0.00235498517751blocked
Sample 363/2≈ 4.71238898038≈ 0.0125000000000≈ -9.00000000000≈ 6.8391341082000≈ -68.391341082000≈ 0.0467737553500≈ -0.0467737553500≈ 0.00233868776750blocked
Sample 3737/24≈ 4.84328867428≈ 0.0128472222222≈ -8.92300375236≈ 6.8154282945200≈ -68.154282945200≈ 0.0464500628377≈ -0.0464500628377≈ 0.00232250314188blocked
Sample 3819/12≈ 4.97418836818≈ 0.0131944444444≈ -8.69333243660≈ 6.7918046499400≈ -67.918046499400≈ 0.0461286104030≈ -0.0461286104030≈ 0.00230643052015blocked
Sample 3913/8≈ 5.10508806208≈ 0.0135416666667≈ -8.31491579260≈ 6.7682628896700≈ -67.682628896700≈ 0.0458093825437≈ -0.0458093825437≈ 0.00229046912719blocked
Sample 405/3≈ 5.23598775598≈ 0.0138888888889≈ -7.79422863406≈ 6.7448027298800≈ -67.448027298800≈ 0.0454923638649≈ -0.0454923638649≈ 0.00227461819325blocked
Sample 4141/24≈ 5.36688744988≈ 0.0142361111111≈ -7.14018006262≈ 6.7214238877100≈ -67.214238877100≈ 0.0451775390783≈ -0.0451775390783≈ 0.00225887695391blocked
Sample 427/4≈ 5.49778714378≈ 0.0145833333333≈ -6.36396103068≈ 6.6981260813100≈ -66.981260813100≈ 0.0448648930011≈ -0.0448648930011≈ 0.00224324465006blocked
Sample 4343/24≈ 5.62868683768≈ 0.0149305555556≈ -5.47885286108≈ 6.6749090297900≈ -66.749090297900≈ 0.0445544105560≈ -0.0445544105560≈ 0.00222772052780blocked
Sample 4411/6≈ 5.75958653158≈ 0.0152777777778≈ -4.50000000000≈ 6.6517724532400≈ -66.517724532400≈ 0.0442460767697≈ -0.0442460767697≈ 0.00221230383848blocked
Sample 4515/8≈ 5.89048622548≈ 0.0156250000000≈ -3.44415089129≈ 6.6287160727100≈ -66.287160727100≈ 0.0439398767726≈ -0.0439398767726≈ 0.00219699383863blocked
Sample 4623/12≈ 6.02138591938≈ 0.0159722222222≈ -2.32937140592≈ 6.6057396102400≈ -66.057396102400≈ 0.0436357957982≈ -0.0436357957982≈ 0.00218178978991blocked
Sample 4747/24≈ 6.15228561328≈ 0.0163194444444≈ -1.17473572998≈ 6.5828427888000≈ -65.828427888000≈ 0.0433338191820≈ -0.0433338191820≈ 0.00216669095910blocked
Sample 482≈ 6.28318530718≈ 0.01666666666670≈ 6.5600253323400≈ -65.600253323400≈ 0.0430339323610≈ -0.0430339323610≈ 0.00215169661805blocked
Half-wave, 100 µF, first cycle from zero: actual separate event records. Adjacent samples are not exact extrema.
Half-wave, 100 µF, first cycle from zero: actual separate event records. Adjacent samples are not exact extrema.
Recordθ/π (1)θ (rad)t (s)Source V (V)Capacitor vC (V)Diode iD (A)Source delivered I (A)dvC/dt (V/s)Source delivered p (W)Rs p (W)Load p (W)Capacitor p (W)Stored energy (J)Branch
alpha0000000000000turn on
beta≈ 0.698151911718≈ 2.19330891694≈ 0.00581793259765≈ 7.31174387335≈ 7.3117438733500≈ -73.117438733500≈ 0.0534615984695≈ -0.0534615984695≈ 0.00267307992347turn off
Initial minimum0000000000000conducting
maximum≈ 0.688920014350≈ 2.16430605599≈ 0.00574100011959≈ 7.46084599867≈ 7.31455490065≈ 0.00731455490065≈ 0.007314554900650≈ 0.0545727676626≈ 0.00107005426789≈ 0.05350271339470≈ 0.00267513566973conducting

Related complete original table: Original periodic reservoir comparison, ideal diodes and 20 Ω source resistance; six significant digits

Related complete original table: Separately integrated reservoir energy and charge over the stated interval; six significant digits

22

Treat reverse breakdown as a separate regime

A reverse-biased junction can conduct strongly through tunnelling in a sufficiently strong field or through avalanche multiplication. Which mechanism dominates depends on the device and operating condition; practical “Zener” references can involve both. Breakdown is a change of regime that the prebreakdown exponential approximation does not describe.

A reverse-voltage reference requires current limiting and adequate operating headroom. Orient the ideal clamp with cathode at the positive output and anode at ground. If I_Z denotes positive reverse current from output to ground, the original diode coordinates are v_D=−v_o and i_D=−I_Z, giving absorbed power v_o I_Z≥0. Reverse conduction here dissipates power; it does not provide an additional energy source.

Our loaded example has V_Z=5.1 V, R_s=330 Ω and R_L=1000 Ω. First solve with the clamp off: v_free=V_S R_L/(R_s+R_L). If v_free<V_Z, it stays off. Otherwise the clamped candidate has v_o=V_Z and I_Z=(V_S−V_Z)/R_s−V_Z/R_L. It is valid only if I_Z≥0; merely having V_S>V_Z is insufficient.

23

Check loaded regulation and the complete power balance

The threshold source voltage is V_Z(1+R_s/R_L)=6.783 V. Below that value, the source and resistor cannot supply the stated load current at 5.1 V while leaving a nonnegative clamp current. Above it, the ideal model holds the output at V_Z and the excess resistor current enters the clamp.

For every state, I_R=(V_S−v_o)/R_s and I_L=v_o/R_L obey I_R=I_L+I_Z. Source delivered power V_SI_R equals R_sI_R²+v_oI_L+v_oI_Z. A larger source voltage can increase both resistor and clamp power while the ideal output stays fixed. The table uses exact fractions so that the sign and current boundary are not hidden by rounding.

A real regulator has finite dynamic resistance, a specified current range, temperature coefficients, leakage, noise and power limits. The exact 5.1 V plateau is the selected ideal model, not a guarantee about a physical part. A power calculation becomes a rating assessment only after actual device, thermal and environmental specifications are supplied.

The load sets the reverse-clamp headroom

Only the four stated source conditions are marked. The 6.783 V threshold and 5.1 V guide are analytic controls, not additional solved cases or a line interpolated through the points. A source above the nominal clamp voltage can still leave it off. The original anode-to-cathode voltage and current signs remain in the exact ledger.

Four loaded clamp states and their exact headroom threshold

Isolated output-voltage and clamp-current controls use separate V and mA lanes. The vertical reference is the loaded regulation threshold; zero clamp current remains visibly marked.

  • Clamp off
  • Clamped
Four loaded clamp states and their exact headroom thresholdIsolated output-voltage and clamp-current controls use separate V and mA lanes. The vertical reference is the loaded regulation threshold; zero clamp current remains visibly marked.Output voltage (V)24681012012345Source voltage VS (V); four isolated operating conditionsThreshold 6.783 VReverse-clamp current IZ (mA = 10⁻³ A)24681012051015Source voltage VS (V); four isolated operating conditionsThreshold 6.783 V
Isolated output-voltage and clamp-current controls use separate V and mA lanes. The vertical reference is the loaded regulation threshold; zero clamp current remains visibly marked.

Short fractions are exact. A leading ≈ marks a value rounded to 12 significant digits. The plots use the labelled units and scales.

The reverse clamp has a loaded threshold, not just a nominal voltage.
The reverse clamp has a loaded threshold, not just a nominal voltage.
QuantityValueSI unit
Vz51/10V
Rs330ohm
RL1000ohm
regulationThreshold6783/1000V
All four loaded reverse-clamp states and exact absorbed-power signs.
All four loaded reverse-clamp states and exact absorbed-power signs.
Source VS (V)Unclamped divider (V)Output (V)Resistor I (A)Load I (A)Clamp IZ (A)Diode vD (V)Diode iD (A)Source delivered (W)Rs heat (W)Load power (W)Clamp absorbed (W)Clamp state
3300/133300/1333/13303/13300-300/13309/1330297/17689090/176890Off
6600/133600/1333/6653/6650-600/133018/665594/88445360/176890Off
9900/13351/1013/110051/10000739/110000-51/10-739/110000117/1100507/110002601/10000037689/1100000Clamped
121200/13351/1023/110051/100001739/110000-51/10-1739/11000069/2751587/110002601/10000088689/1100000Clamped

Related complete original table: Original loaded ideal 5.1 V reverse clamp; exact SI fractions

Original loaded ideal 5.1 V reverse clamp; exact SI fractions
V_S (V)Statev_o (V)I_R (A)I_L (A)I_Z (A)Clamp absorbed power (W)
3off300/1333/13303/133000
6off600/1333/6653/66500
9clamped51/1013/110051/10000739/11000037689/1100000
12clamped51/1023/110051/100001739/11000088689/1100000
24

Read the omitted dynamic and thermal limits

A real diode has series resistance and reverse leakage; its forward voltage and current do not follow one ideal exponential across all regimes. Depletion capacitance depends on bias, and forward-injected minority carriers can store charge. On switching from forward to reverse bias, removing that charge can produce reverse-recovery current before blocking is re-established. The waveform depends on the previous current and the external circuit.

A Schottky diode uses a metal–semiconductor barrier and majority-carrier conduction. It can avoid the minority-carrier storage mechanism of an ordinary p–n diode, but that does not imply zero reverse transient current in every circuit: voltage-dependent capacitance still draws displacement current. Structure, leakage, forward drop and reverse-voltage capability have device-specific tradeoffs.

Electrical dissipation changes junction temperature through a thermal path with finite impedance and time dependence. Under a maintained voltage, a temperature-dependent current rise can reinforce heating; a current-limiting network changes that feedback. A fixed-300 K curve therefore does not predict a self-consistent hot operating point. Peak/repetitive current, reverse voltage, surge duration, recovery conditions and thermal limits must be interpreted with their stated test conditions.

25

Keep topology and reverse-stress claims bounded

For a half-wave reservoir, a blocked diode can see the charged capacitor opposed to a negative source. With the ideal nonnegative capacitor voltage bounded by V_m, its reverse terminal magnitude can approach twice the source peak. An actual peak value requires the stated trajectory and source phase; it is not automatically the same as the unfiltered resistor case.

In a bridge, the blocked devices’ individual terminal potentials depend on the bridge connections and any common-mode or parasitic constraints. We do not assign each floating ideal diode a reverse rating from the rectified differential output alone. The present reservoir equations determine the load voltage and conducting-path current, rather than every unspecified internal device stress.

Adding source inductance, capacitor ESR, leakage, a transformer, recovery charge or an active load changes the equations and sometimes the switching states. The existing RLC and induction lessons explain why those additions can introduce new stored energy and ringing. The correct next step is to extend the physical model for the question, preserving its initial conditions and energy boundary.

26

Connect nonlinear devices to circuit and energy reasoning

The same reference directions, KCL and real work p=vi used for linear circuits continue to apply. What changes is the constitutive relation and the validity of a guessed state. A load line combines a device law with the surrounding network; a capacitor adds memory; a periodic solution adds a boundary condition that cannot be imposed on arbitrary startup.

Rectification connects Fourier and RMS ideas to real energy transfer. A positive mean voltage, a small ripple, a high source PF and a high useful-output efficiency are distinct properties. Each needs its own definition and system boundary. The ideal bridge and the capacitor examples show why improving one waveform feature does not answer all the others.

For a new diode circuit, name the terminals and signs, choose the model, solve a candidate state, test its inequalities, and close the relevant current, charge and energy balances. Retain units and averaging intervals. Then compare the required accuracy, dynamics and stresses with the assumptions. This discipline prepares the later transistor and amplifier lessons without treating a nonlinear device as an unexplained switch.

Worked examples

A tangent resistance is not a DC ratio

Use the original fixed-300 K junction at v_D=0.600 V. Compare its current and differential resistance with the static ratio.

  1. Calculate V_T=k_BT/e and a=(3/2)V_T, retaining volts so that 0.600/a is dimensionless. The model parameters, including I_s=2 nA, remain fixed for this voltage derivative.
  2. The exponential law gives i_D≈0.0104890 A and g_d≈0.270489 S. Its reciprocal is r_d≈3.69701 Ω. These are local quantities at the stated bias.
  3. The static ratio is 0.600/i_D. It is a line from the origin to the operating point, whereas r_d is the tangent slope dv_D/di_D. They need not agree for a nonlinear characteristic.
  4. At v_D=0 the same model has i_D=0 exactly, so v_D/i_D is undefined, but r_d=a/I_s remains finite. The distinction cannot be fixed by inventing a zero-current phase or threshold.

Use the differential slope for sufficiently small voltage changes about this bias, and the original nonlinear law for the DC operating point.

The resistor and junction select one point

Solve the original 330 Ω series circuit at V_S=3 V and interpret the negative-source control.

  1. F(v)=330 I_s[exp(v/a)−1]+v−3 has derivative 330 g_d+1>0 and opposite endpoint signs at 0 and 3 V. It has one root within that bracket.
  2. The resulting v_D≈0.586025 V and i_D≈0.00731508 A satisfy both the junction law and the resistor voltage drop. Assigning a universal 0.7 V would give another current and would fail the selected law.
  3. Source power≈0.0219452 W splits into resistor heat≈0.0176584 W and diode absorption≈0.00428682 W. The displayed values are rounded to six significant digits.
  4. At V_S=−1 V the current is approximately−2 nA rather than exactlyzero. Both source voltage and delivered current are negative, and the diode voltage and current are negative, so delivered source power and absorbed diode power are positive. No breakdown is part of this case.

There is one operating point for each listed source in this model; signs and the power ledger remain valid in reverse bias.

Integrate before comparing half and full wave

Find the ideal half-wave and bridge mean and RMS values for V_m=9 V, then include the stated diode drops.

  1. For the ideal half-wave, integrate 9 sinθ over 0..π and divide by 2π to obtain 9/πV. Integrating 81 sin²θ gives mean square 81/4 V² and RMS 9/2 V.
  2. The ideal bridge repeats the positive output segment twice. Its mean is 18/πV, mean square 81/2 V² and RMS 9/√2 V. Doubling the mean square multiplies RMS by√2, not two.
  3. For the half-wave drop use d=0.6 V; for the bridge use d=1.2 V because two diodes conduct in series. Set α=asin(d/9), integrate(9 sinθ−d) and its square overα..π−α, and only then apply the half/full multiplicity.
  4. The resulting half-drop mean/RMS are approximately 2.57116 V/4.12186 V; the bridge-drop values are 4.58058 V/5.30549 V. Both conduction intervals and forward losses differ from their ideal cases.

Mean, mean square, RMS and conduction duration are separately derived; a constant drop is not subtracted from every ideal summary statistic.

An ideal bridge can have unit physical transfer efficiency

Compare source true PF and DC-component power fraction for the ideal resistor rectifiers.

  1. The ideal half-wave delivers P=81/(4 R_L), with I_S, rms=9/(2 R_L) and V_S, rms=9/√2. Their ratio gives true PF=1/√2.
  2. The ideal bridge draws i_S=v_S/R_L, giving P=81/(2 R_L), I_S, rms=9/(√2 R_L) and PF=1. The load current stays positive while the source current changes sign.
  3. For the half-wave, V_DC²/V_rms²=4/π²≈0.405285; for the bridge it is 8/π²≈0.810569. Those are fractions of the resistor heating associated with the DC component, not fractions of delivered energy that escaped the load.
  4. With ideal diodes and no other dissipative element, P_source=P_load in both circuits. The physical transfer efficiency to the resistor is one. A real or specified-drop diode introduces a separate positive absorption term.

PF, DC-component fraction and physical efficiency answer different questions; none can be substituted for another.

Find events and actual ripple with finite source resistance

Compare the original periodic 100 µF half-wave and bridge, then the 470 µF bridge.

  1. Use R_s=20 Ω, R_L=1000 Ω, V_m=9 V, f=60 Hz. Solve the on-trajectory turn-off crossing together with the periodic off-decay matching equation. Use P=2π for half-wave andπ for the bridge.
  2. For the 100 µF bridge, α≈1.06230 rad andβ≈1.95479 rad. The capacitor minimum occurs later than turn-on, atθ≈1.09851 rad; its maximum is later than the source peak, atθ≈1.89968 rad.
  3. Actual peak-to-peak ripple is approximately 1.04865 V for the 100 µF half-wave, 0.492995 V for the 100 µF bridge and 0.105501 V for the 470 µF bridge. The respective positive mean voltages are 7.59763 V, 8.10516 V and 8.13059 V.
  4. For the 100 µF bridge, I_DC=V_DC/1000 andf_ripple=120 Hz give the small-ripple estimate 0.675430 V, about 1.37006 times the actual ripple. The finite recharge interval makes the instantaneous-recharge approximation inaccurate here.

Use event-based capacitor extrema and the stated source boundary; the compact ripple estimate is an approximation with testable conditions.

The first cycle must charge the capacitor

Start the original half-wave 100 µF capacitor at zero voltage and follow one source period.

  1. Set v_C(0)=0 andα=0, retaining the finite 20 Ω source resistance. The first turn-off isβ≈2.19331 rad, rather than the periodic event. The first maximum occurs atθ≈2.16431 rad.
  2. At the end of 1/60 s, v_C≈6.56003 V. Thus ΔE_C=Cv_C²/2≈0.00215170 J andΔQ_C=Cv_C≈0.000656003 C. Neither increment may be discarded as though this were a periodic state.
  3. Separately integrated source energy is approximately 0.00517331 J, source-resistance heat 0.00237319 J and load energy 0.000648430 J. Their balance includes the positive storage increment. The source current remains bounded by 9/20 A in this model.
  4. Diode charge≈0.000753890 C equals load charge≈0.0000978873 C plus stored charge. The first-cycle voltage variation and interval mean are not assigned the periodic small-ripple formula.

Startup has a different state and storage ledger even when its source, resistor, load and capacitor values match a periodic example.

A source above 5.1 V may still leave the clamp off

Use the original ideal 5.1 V clamp, 330 Ω series resistor and 1000 Ω load at 6 V and 9 V.

  1. With the clamp off, the divider gives v_free=V_S·1000/1330. At 6 V this is 600/133 V, below 5.1 V, so the clamp remains off even though 6 V exceeds its nominal voltage.
  2. The onset condition is V_S=V_Z(1+R_s/R_L)=6.783 V. At 9 V the valid clamped output is 51/10 V and the source-resistor current is(9−5.1)/330=13/1100 A.
  3. The load current is 51/10000 A, leaving I_Z=739/110000 A>0. Clamp absorption is V_ZI_Z=37689/1100000 W; its original diode voltage and current are both negative under anode-to-cathode references.
  4. Check source delivered power against series heat, load power and clamp absorption. The ideal result does not establish a real part’s power rating, temperature or regulation tolerance.

Loaded headroom and a nonnegative clamp current determine the state; nominal source voltage alone does not.

Try it yourself

  1. What does the symbol bar identify, and what signs give absorbed diode power?

    Show the answer and reasoning

    The cathode. Use v_D=V_A−V_K, i_D from anode to cathode, and p_D=v_Di_D.

    Reference directions precede state assumptions. A negative current under a stated reference is not automatically negative absorption; voltage and current signs must be multiplied.

  2. Why is the built-in junction potential not a usable battery?

    Show the answer and reasoning

    It belongs to equilibrium carrier and electrostatic conditions with zero net terminal current.

    Diffusion and drift balance, and the complete terminal/contact equilibrium does not provide extractable work around a closed cycle. Applied bias changes the nonequilibrium boundary.

  3. Can the 0.6 V constant-drop case supply the exponential small-signal resistance?

    Show the answer and reasoning

    No; they are separate constitutive models.

    The exponential slope follows from differentiating its fixed-parameter current law. A piecewise constant conducting voltage is selected for another approximation and cannot supply that derivative.

  4. Differentiate the fixed-temperature current law and state when r_d≈a/i_D is valid.

    Show the answer and reasoning

    g_d=(i_D+I_s)/a and r_d=a/(i_D+I_s); simplify only for i_D≫I_s.

    At zero bias the differential resistance is finite a/I_s while the static ratio v_D/i_D is undefined. The local slope does not replace the global current law.

  5. Why does the original 330 Ω series junction circuit have one DC solution for each listed source?

    Show the answer and reasoning

    Its residual F(v) is continuous, bracketed, and has derivative 330 g_d+1>0.

    Strict monotonicity gives at most one root, while opposite endpoint signs give existence. This argument belongs to the selected passive resistor and monotone junction law.

  6. Why is the original bridge total forward drop 1.2 V rather than 0.6 V?

    Show the answer and reasoning

    Two forward-biased diodes lie in the conducting source-to-load path.

    The alternate pair conducts on the opposite polarity while load current keeps its direction. All four devices do not conduct in series at once; other full-wave architectures need separate topology analysis.

  7. For an ideal 9 V-peak half-wave output, find the mean, RMS and ripple RMS.

    Show the answer and reasoning

    V_DC=9/πV, V_rms=9/2 V, and ripple RMS=9√(1/4−1/π²)V.

    Average the waveform and its square separately over the full 2π source period. The alternating remainder has zero mean, so its mean square subtracts V_DC² from the total.

  8. Does the ideal bridge DC-component fraction 8/π² mean that the remaining energy was lost in its diodes?

    Show the answer and reasoning

    No. The ideal diodes absorb zero power and all delivered energy reaches the resistor.

    The resistor also heats from the output’s alternating component. The DC-component fraction classifies that waveform contribution, while a physical efficiency uses source and useful-load power boundaries.

  9. Why can a rectified load voltage coexist with source PF=1 in the ideal resistive bridge?

    Show the answer and reasoning

    The source current is v_S/R_L, proportional to the complete signed source voltage.

    Source current reverses when source voltage reverses, while load current is rerouted to keep its direction. PF uses the source real product and RMS magnitudes at that port.

  10. Write the capacitor voltage during a blocked interval of the reservoir model.

    Show the answer and reasoning

    v_C(t)=v_C(t₀)exp[−(t−t₀)/(R_LC)].

    The diode supplies no current, leaving Cdv_C/dt=−v_C/R_L. A linear ramp is only a short-interval approximation to this exponential.

  11. Why is reservoir capacitor voltage still falling just after periodic turn-on?

    Show the answer and reasoning

    At turn-on i_D=0 while the load current is positive, so Cdv_C/dt<0.

    The minimum occurs later when i_D=v_C/R_L. The maximum obeys the same derivative-zero condition later in the pulse, after the source peak; neither is simply a source crossing.

  12. State the conditions behind ΔV≈I_DC/(f_ripple C) and the bridge ripple frequency.

    Show the answer and reasoning

    Small variation, nearly constant load current and a short recharge interval; f_ripple=2 f for this bridge.

    Finite source resistance can make recharge occupy an appreciable interval. The exact capacitor trajectory then tests the approximation; startup is a separate initial-value problem.

  13. What averages or increments vanish in a periodic reservoir, and what need not vanish?

    Show the answer and reasoning

    ΔQ_C and ΔE_C per period vanish, so mean capacitor current is zero. Mean capacitor voltage need not be zero.

    Integrating Cdv_C/dt gives zero net charge change when endpoints match. Positive voltage can persist while charge and energy are alternately supplied and removed within the period.

  14. Find the source threshold for the original 5.1 V, 330 Ω, 1000 Ω ideal clamp.

    Show the answer and reasoning

    V_S, threshold=5.1(1+330/1000)=6.783 V.

    The series resistor must supply the load current at 5.1 V before any nonnegative reverse-clamp current is available. A 6 V source therefore leaves this loaded clamp off.

  15. Does majority-carrier Schottky conduction imply zero reverse transient current?

    Show the answer and reasoning

    No. It avoids the ordinary minority-carrier storage mechanism, but junction capacitance can still draw current.

    Capacitive displacement current and reverse recovery from stored minority charge are different physical contributions. The observed transient depends on device structure, bias history and the surrounding circuit.

  16. Why does a fixed 300 K curve not predict the final temperature or a safe device rating?

    Show the answer and reasoning

    It holds temperature and empirical junction parameters fixed and omits the thermal path and actual device specifications.

    Electrical losses can change temperature, which changes the terminal law and feedback. Ratings also depend on waveform duration, repetition, reverse voltage and stated thermal/test conditions.

What to carry forward

  • The cathode bar, voltage polarity and current reference must be defined before a conducting state is tested.
  • Ideal, constant-drop, exponential and reverse-clamp laws are separate models with separate domains.
  • Differential resistance is a local slope; the surrounding network fixes the nonlinear operating point.
  • A bridge uses alternate two-diode paths to keep load current in one direction while source current changes sign.
  • Mean voltage, RMS voltage, AC ripple and peak-to-peak variation are different waveform quantities.
  • DC-component power fraction, source true PF and physical transfer efficiency answer different questions.
  • A reservoir capacitor adds state; finite source resistance sets its charging equation and finite current boundary.
  • Turn-on, turn-off and capacitor extrema are different events; actual minima and maxima require their own conditions.
  • Periodic charge and storage increments vanish, while startup must retain the energy needed to charge the capacitor.
  • A loaded clamp needs headroom and nonnegative clamp current; real recovery, capacitance, thermal and rating limits require a larger device model.

References

Sources for the concepts and teaching approaches in this article.

  1. Solar cell operating principles — p–n junction lecture slidesTU Delft OpenCourseWareCourse notes · PDFocw.tudelft.nlRead source
  2. Massachusetts Institute of Technology2 sources
    1. MIT 6.334 Power Electronics, Spring 2007Massachusetts Institute of Technology / MIT OpenCourseWareCourse notes · PDFChapter 1 — Introduction and Analysis Methods (6.334, Spring 2007)Chapter 2 — Introduction to Rectifiers (6.334, Spring 2007)Chapter 3 — Power Factor and Measures of Distortion (6.334, Spring 2007)ocw.mit.edu
    2. Lecture 3: Digital circuits, power supplies and regulation (IAP 2020)MIT 6.117 teaching materialsCourse notes · PDFweb.mit.eduRead source
  3. Diode Application Handbook: Fundamentals, Characteristics, Applications — Design Engineer’s Guide (May 2022)Nexperia UK LtdReference guide · PDFnexperia.comRead source
  4. Do Schottky barrier diodes (SBDs) have reverse recovery time (t_rr)?Toshiba Electronic Devices & Storage CorporationReference guidetoshiba.semicon-storage.comRead source
  5. Special Publication 330, Section 2 — The International System of UnitsNational Institute of Standards and TechnologyReference guidenist.govRead source