Circuit theorems

Nodal and mesh analysis can solve any linear circuit, but they treat every circuit as a fresh system of equations. Often an engineer needs something more insightful: how will the output change if the load changes? What does this complicated network look like from the two terminals where something will be connected? Which source is responsible for which part of the response? The circuit theorems answer these questions. They follow from one property, linearity, and they let engineers replace complicated networks with simple equivalents.

This chapter covers linearity and proportionality, superposition, source transformation, Thévenin's and Norton's theorems, and maximum power transfer. These ideas run through all of electrical engineering. A power system engineer models the entire grid behind a substation as a Thévenin source to calculate fault currents; an electronics designer models an amplifier's output as a Thévenin source to see how a load will pull it down; and a sensor designer uses them to understand how much signal reaches a measuring circuit.

Linearity

A circuit is linear if it contains only linear elements (resistors, capacitors, inductors and linear dependent sources) and independent sources. In a linear circuit, every voltage and current is a linear function of the independent sources. Two properties follow:

• Homogeneity (proportionality): if every independent source is multiplied by a constant \(k\), every response is multiplied by \(k\).
• Additivity: the response to several sources acting together is the sum of the responses to each one acting alone.

Power is not linear: \(p = i^2R\), so doubling the current quadruples the power. Linearity applies to voltages and currents; powers must be computed from the total voltages and currents, never added source by source.

Superposition

Additivity gives the superposition theorem: in a linear circuit with several independent sources, any voltage or current is the algebraic sum of the contributions from each independent source acting alone. To find each contribution, turn off all the other independent sources:

• A turned-off voltage source becomes a short circuit (zero volts).
• A turned-off current source becomes an open circuit (zero amperes).
• Dependent sources are never turned off; they stay in the circuit because they respond to circuit variables.

Superposition usually takes more work than one nodal analysis, so it is rarely the fastest way to solve a whole circuit. Its value is insight: it shows how much each source contributes, which matters when, for example, a signal and a bias supply act on the same circuit, or when an engineer needs to know how much of a measurement error comes from one source of interference.

Source transformation

A voltage source \(v_s\) in series with a resistor \(R\) is equivalent, at its terminals, to a current source \(i_s = v_s/R\) in parallel with the same resistor \(R\). Both have the same open-circuit voltage (\(v_s\)) and the same short-circuit current (\(v_s/R\)), and since both are linear, the same terminal behaviour everywhere in between. The current source arrow points toward the terminal that was the voltage source's + side.

Repeated source transformations, combined with series and parallel reduction, can simplify many circuits without writing a single simultaneous equation. The transformation does not apply to an ideal source with no series or parallel resistance, and it preserves only the terminal behaviour: the power dissipated inside the two equivalents is generally different.

Thévenin's theorem

In 1883 the French telegraph engineer Léon Charles Thévenin showed that any linear two-terminal network, however complicated, behaves at its terminals exactly like a single voltage source in series with a single resistor. The voltage is the Thévenin voltage \(V_{\text{Th}}\) and the resistor is the Thévenin resistance \(R_{\text{Th}}\).

The two values are found as follows:

• \(V_{\text{Th}}\) is the open-circuit voltage \(v_{\text{oc}}\) across the terminals with the load removed.
• \(R_{\text{Th}}\) is the resistance seen looking into the terminals with all independent sources turned off (voltage sources shorted, current sources opened). If the network contains dependent sources, they stay in, and \(R_{\text{Th}}\) is found with a test source: apply a test voltage \(v_t\) (or current \(i_t\)) to the terminals, calculate the resulting current \(i_t\) (or voltage), and \(R_{\text{Th}} = v_t/i_t\).
• Alternatively, \(R_{\text{Th}} = v_{\text{oc}}/i_{\text{sc}}\), where \(i_{\text{sc}}\) is the short-circuit current with the terminals shorted. This works with or without dependent sources.

The great value of the Thévenin equivalent is that once it is found, the effect of any load can be calculated in one line: \(i_L = V_{\text{Th}}/(R_{\text{Th}} + R_L)\). Analysing a circuit for ten different loads becomes ten one-line calculations instead of ten full analyses.

Norton's theorem

Applying a source transformation to the Thévenin equivalent gives the Norton equivalent: a current source \(I_N\) in parallel with a resistor \(R_N\). Edward Norton of Bell Labs described it in 1926. The values are

For the example above, \(I_N = 8/4 = 2\) A (matching the short-circuit current found in the check) in parallel with 4 Ω. Norton equivalents are natural when loads are connected in parallel, and they are the standard model for transistor outputs and for current-output sensors.

Thévenin resistance with a dependent source

Maximum power transfer

A source with Thévenin resistance \(R_{\text{Th}}\) delivers power to a load \(R_L\):

If \(R_L\) is very small, the current is large but the load voltage is nearly zero; if \(R_L\) is very large, the voltage is large but the current nearly zero. In between there is a maximum. Setting \(dp_L/dR_L = 0\) gives the maximum power transfer theorem: maximum power is delivered when

For the example, \(R_L = 4\) Ω receives \(8^2/(4 \times 4) = 4\) W, the most any load can draw from that circuit.

At maximum power transfer, the load and \(R_{\text{Th}}\) carry the same current and have the same resistance, so they dissipate equal power: only 50% of the power taken from the Thévenin source reaches the load. That is acceptable, even desirable, where the signal is weak and every microwatt matters: antennas, radio receivers, sensor interfaces and audio amplifiers are matched to their loads. It is never acceptable in power systems, where efficiency is everything. A utility generator or a data centre UPS is designed with a Thévenin (internal) resistance far smaller than the load, so that nearly all the power reaches the load and the voltage stays steady as the load changes.

Measuring a Thévenin equivalent

Thévenin's theorem also works experimentally, with no circuit diagram at all. Measure the open-circuit voltage with a high-resistance voltmeter: that is \(V_{\text{Th}}\). Then connect a known load resistor and measure the voltage across it; the drop tells you \(R_{\text{Th}}\) from the divider relation \(v_L = V_{\text{Th}}R_L/(R_{\text{Th}} + R_L)\). This is how the internal resistance of a battery is measured, and the same idea, applied to a power network, gives the source impedance that sets fault levels at a building's service entrance (DCE-310).