Network Theorems

Thevenin, Norton, superposition, source transformation and maximum power transfer — the shortcuts that reduce a complex linear network to something you can reason about in one step.

Real circuits can have dozens of components, but you often only care about what happens at one pair of terminals — say, the load. Network theorems let you replace everything behind those terminals with a single source and a single resistance, without changing anything the load experiences.

These theorems apply to linear circuits (resistors, linear sources, and — in AC — impedances). They are the basis of how engineers model batteries, amplifiers, sensors, power grids and communication links.

An analogy

Thevenin's theorem is like describing a whole city's water network, as seen from your tap, as 'one tank at a certain height behind one narrow pipe'. For anything connected at the tap, that simple model behaves exactly like the real network.

Thevenin's theorem

Any linear two-terminal network is equivalent to a voltage source V_th in series with a resistance R_th.

• V_th: the open-circuit voltage at the terminals (remove the load, measure the voltage).
• R_th: turn off all independent sources (voltage sources → short, current sources → open) and find the resistance seen from the terminals.
• Alternatively, R_th = V_oc / I_sc (open-circuit voltage divided by short-circuit current).

Norton's theorem

The same network is also equivalent to a current source I_N (the short-circuit current) in parallel with R_N = R_th. Thevenin and Norton forms convert into each other by source transformation.

Superposition

In a linear circuit with several independent sources, find the response to each source alone (others turned off) and add the results. It works for voltages and currents but not for power, since power is non-linear (I²R).

Maximum power transfer

Power into R_L peaks when R_L = R_th, and then only half the source's power reaches the load (50 % efficiency). Matching is used in RF and antenna systems, where signal power matters. Power systems deliberately avoid it to keep efficiency high.

AC and dependent sources

In AC, the theorems work with impedances; maximum power transfer then requires the conjugate match Z_L = Z_th*. With dependent sources (transistor models), R_th must be found by applying a test source at the terminals, since you can't simply switch the dependent source off.

Millman, reciprocity and substitution

Millman's theorem gives the common voltage of parallel branches directly: V = Σ(Vₖ/Rₖ) / Σ(1/Rₖ). Reciprocity says a passive linear network's transfer from A to B equals that from B to A. Together with substitution, these speed up bridge and network problems.

Formulas

• Thevenin ↔ Norton: V_th = I_N · R_th, R_N = R_th
• Load current: I_L = V_th / (R_th + R_L)
• Maximum power transfer: R_L = R_th → P_max = V_th² / (4 R_th)
• Superposition: response = Σ (response to each source alone)

Worked example: Thevenin equivalent and matched load

Given: 12 V source, R₁ = 6 Ω in series; R₂ = 3 Ω from the output node to ground; Load connected across R₂

Find: V_th, R_th, and the power in a matched load.

• V_th = open-circuit voltage = 12 × 3/(6 + 3) = 4 V
• R_th: short the source → R₁ ∥ R₂ = 6 × 3/9 = 2 Ω
• Matched load R_L = 2 Ω → I = 4/(2 + 2) = 1 A
• P_L = 1² × 2 = 2 W = V_th²/(4R_th) ✓; Norton I_N = 4/2 = 2 A

Answer: V_th = 4 V, R_th = 2 Ω, P_max = 2 W

Worked example: Superposition with two sources

Given: 10 V source through R₁ = 5 Ω to node A; R₂ = 5 Ω from A to ground; 2 A current source injecting into A

Find: V_A.

• Voltage source alone (current source open): V_A' = 10 × 5/10 = 5 V
• Current source alone (voltage source shorted): R₁ ∥ R₂ = 2.5 Ω → V_A'' = 2 × 2.5 = 5 V
• V_A = 5 + 5 = 10 V
• Check with KCL: (10 − 10)/5 + 10/5 = 0 + 2 = 2 A injected ✓

Answer: V_A = 10 V

Common mistakes

• Wrong: Leaving a voltage source in place when finding R_th. — Right: Replace independent voltage sources with shorts and current sources with opens.
• Wrong: Adding powers from each source in superposition. — Right: Superpose voltages or currents, then compute power from the total.

Applications

• Battery and sensor models: A battery is modelled as a Thevenin source (EMF + internal resistance); a sensor as a source with an output impedance that sets how much the ADC loads it.
• Fault current: Utilities represent the grid at a substation as a Thevenin source; the short-circuit current I_sc = V_th / Z_th sets breaker ratings.
• RF matching: Antennas, amplifiers and cables are matched to 50 Ω so maximum signal power transfers and reflections are minimised.

Key takeaways

• Any linear network looks like V_th + R_th (or I_N ∥ R_th) from two terminals.
• R_th = V_oc / I_sc, or the resistance with independent sources turned off.
• Superposition adds responses to each source; never add powers.
• Max power transfer at R_L = R_th, at only 50 % efficiency.

Going further

• Thevenin in AC power systems: Grid 'strength' at a connection point is expressed as short-circuit ratio SCR = S_sc / P_rated. Weak grids (low SCR, high Z_th) make inverter-based renewables hard to control, a key issue in modern grid-connection studies.
• Two-port networks: Extending Thevenin/Norton to two pairs of terminals gives Z-, Y-, h- and ABCD-parameters. Transmission lines use ABCD parameters; transistors use h-parameters; RF uses S-parameters.
• Linearisation: Non-linear devices get Thevenin/Norton small-signal models around an operating point: a diode becomes its dynamic resistance, a transistor becomes g_m and r_o. All amplifier design builds on that step.