Engineering & Technology · Electrical Circuits & Measurement

Conservation laws, circuit networks and equivalent models

A current reaches a junction and splits between two branches. Knowing the total does not tell us each branch current until the element relationships are included. Circuit analysis combines conservation with models of the connected elements.

Forces, Energy & Systems and Engineering Measurement supply energy, units, model assumptions and measurement records. This chapter defines its own electrical terms and recalls each algebraic relationship before substitution. Exponentials and complex impedance are introduced as supplied representations; readers can follow the physical comparisons first. All numerical networks, curves and specifications are original teaching models. They are not physical build instructions, actual component ratings or approval to work on electrical equipment.

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After reading, you can

Apply charge and voltage balances to a small network and limit an equivalent model to the terminals it represents.

Define nodes and a consistent direction

A node groups points connected by ideal wires and assigned the same potential in a lumped circuit model. Kirchhoff’s current law expresses the node’s charge balance: the signed sum of branch currents is zero. Our ordinary node model does not accumulate an independently specified net charge; capacitive storage is represented through elements and their branch currents.

Kirchhoff’s voltage law gives a signed voltage account around a loop. Follow a consistent traversal and polarity, including source rises and element drops. Lumped models assign voltage differences to elements and neglect propagation delays along ideal connections. When changing magnetic flux, long connections or high frequencies defeat those assumptions, the model needs appropriate induced-source or distributed-field effects.

Combine topology with element laws

Our original network has a 12 V source and 200 Ω series resistor feeding a node. From that node, 300 Ω and 600 Ω branches return to the reference. Parallel elements share terminal voltage; they need not share current. Their combined resistance is 1/(1/300+1/600)=200 Ω. Including the series resistor gives 400 Ω.

The source current is 30 mA. The series drop is 6 V, leaving the junction at 6 V. Branch currents are 6/300=20 mA and 6/600=10 mA. The node balance 30=20+10 mA and either loop’s 12−6−6=0 V give independent consistency checks on the calculation.

Top network: 12 V source,200 Ω series resistor, then 300 Ω and 600 Ω parallel to reference. Source current 30 mA; node 6 V; branch currents 20 and 10 mA. Green circles mark the two terminals of the 600 Ω load. Removing that load from the mathematical network gives 7.2 V open-circuit and 120 Ω source resistance. Bottom: 7.2 V source in series with 120 Ω connected to an open two-terminal port. A 600 Ω load there gives 10 mA and 6 V. Schematic geometry is not physical component placement.

Replace what the external connection can see

A suitable linear two-terminal network can be represented by a Thévenin voltage source in series with a resistance for its external voltage-current relation. Identify the two terminals and which load is excluded before simplifying. An equivalent preserves that terminal behaviour under its assumptions; it need not reproduce every internal voltage, loss or physical hazard.

Remove the 600 Ω branch from the mathematical network and treat its former connections as the port. With no port current, the 200 Ω and 300 Ω divider gives 7.2 V. In the ideal-source model, setting that source voltage to zero makes the two resistors parallel as seen from the port: 120 Ω. This source replacement is an analysis operation, not a procedure for shorting physical supplies. The port model is therefore 7.2 V in series with 120 Ω.

Check the reduced port and the original energy balance

Reconnect a model 600 Ω load to the equivalent. Current is 7.2/(120+600)=0.010 A; its voltage is 6 V. This agrees with the original network. Changing the load would change the port voltage according to the same linear relation.

In the original network, resistor powers are (0.030)²×200=0.18 W, 6²/300=0.12 W and 6²/600=0.06 W. Their sum is 0.36 W, matching source delivery 12×0.030.

The equivalent’s internal 120 Ω resistor is a representation, not a claim about where 0.18 or 0.12 W physically heats the original network. Keep the original detail when internal temperatures or component ratings are the question.

Try the reasoning

Why do the parallel branches have different currents?
Show the answer and reasoning
They share the same 6 V but have different resistance values. Their currents follow the individual element laws.
Does a negative solved branch current invalidate the calculation?
Show the answer and reasoning
No. It can indicate flow opposite to the chosen positive arrow, provided the equations and signs are consistent.
Can a two-terminal equivalent establish every original component’s dissipation?
Show the answer and reasoning
No. It preserves the specified external terminal relationship. Internal behaviour needs the original model or additional information.

What to carry forward

Conservation, element laws and connection definitions work together. State the port and assumptions of every equivalent.

References

Teaching resources and research supporting this lesson’s explanations and examples.