Equivalent circuit models

June 1, 2006
Designers often use mathematical models to predict how mechanical and electrical systems will behave. The details expressed by these models, like significant

Designers often use mathematical models to predict how mechanical and electrical systems will behave. The details expressed by these models, like significant digits in an equation, go only as deep as the analysis requires. Determining how gears convert torque, for example, is satisfied with a simple model, usually just two parameters (gear ratio and efficiency). Predicting how a gear will behave under stress, on the other hand, calls for a more complex model, typically of finite-element resolution.

In the electrical domain, models follow a similar order. The model of an ac motor drive, for instance, doesn't need to include every last component to predict how the current and voltage output will power its intended load. In fact, all that's needed is a single voltage or current source and an equivalent resistor — a simple combination that can represent many complex, multi-source circuits.


One of the most powerful tools when modeling electrical circuits is superposition. The principle, which applies to any linear system consisting of multiple energy sources, allows the effect of each source to be analyzed independently. Summing the effects of the individual sources working alone produces the net effect of all sources acting together. The condition of linearity means simply that all variables in the system are proportionally related (no exponents, powers, or roots).

Isolating power sources in an electrical circuit is accomplished by “turning off” all independent voltage and current sources except the one of interest. All current sources are replaced with open circuits (representing zero current), while all voltage sources are replaced by short circuits (zero voltage). With all sources “removed,” remaining components in the circuit are more easily simplified to series/parallel impedance combinations.

Thevenin's theorem

Thevenin's theorem, based on superposition, reduces linear circuits to equivalent models consisting of a voltage source in series with a resistor. Thevenin's equivalents are useful when analyzing power systems and other circuits where the load resistance may change. To find a circuit's Thevenin source voltage vT, replace the load resistor with an open circuit. The open-circuit voltage vOC is simply vT because no voltage drops across RT when i = 0. To find the Thevenin equivalent resistance RT, remove all power sources and calculate the total resistance across the load terminals.

Norton's theorem

Norton's theorem, related to Thevenin's, states that a complex linear circuit can reduce to an equivalent current source and parallel resistor. This is the dual of Thevenin's theorem, where instead of voltage, equations focus on current relationships. As such, the first step is finding the source current iN by replacing the load with a short and calculating current through it. Here, iN = iSC because source current is diverted through the short circuit load. To find the equivalent resistance RN, remove all power sources and calculate total resistance at the load.

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