Network theorems

Network theorems let us replace a complicated circuit with a simpler equivalent, so we can find one current or voltage without solving the whole network. This topic also covers AC circuits containing R, L and C, the phenomenon of resonance in series and parallel circuits, and the meaning of active and reactive power. In the NEC exam, expect questions on what each theorem requires (linearity, independent sources), the maximum power transfer condition, resonance frequency, and the Q-factor.

Key concepts

Superposition theorem

• In a linear, bilateral network with several independent sources, the response (current or voltage) in any element equals the algebraic sum of responses due to each source acting alone.
• While one source acts, other ideal voltage sources are short-circuited and ideal current sources are open-circuited (internal resistances retained). Dependent sources are kept active.
• It does not apply directly to power, because power is proportional to $I^2$ (non-linear).

Thevenin's theorem

• Any linear two-terminal network can be replaced by a voltage source $V_{th}$ in series with a resistance $R_{th}$.
• $V_{th}$ = open-circuit voltage at the terminals; $R_{th}$ = resistance seen from the terminals with independent sources deactivated (or $R_{th} = V_{oc}/I_{sc}$).

Norton's theorem

• The same network can be replaced by a current source $I_N$ (short-circuit current) in parallel with $R_N = R_{th}$.
• Conversion: $V_{th} = I_N R_{th}$. Thevenin and Norton are duals.

Maximum power transfer theorem

• A DC source delivers maximum power to a load when $R_L = R_{th}$. Efficiency at that point is only 50%.
• For AC: maximum power when $Z_L = Z_{th}^*$ (complex conjugate). If only the load resistance can vary, $R_L = |Z_{th}|$.
• Used in communication circuits (matching), not in power systems where efficiency matters.

R-L, R-C, R-L-C AC circuits

Element	Impedance	Phase of current
R	$R$	in phase with V
L	$jX_L = j\omega L$	lags V by 90°
C	$-jX_C = 1/(j\omega C)$	leads V by 90°

• Series RL: $Z = \sqrt{R^2 + X_L^2}$, current lags, $\phi = \tan^{-1}(X_L/R)$.
• Series RC: $Z = \sqrt{R^2+X_C^2}$, current leads.
• Series RLC: $Z = \sqrt{R^2+(X_L-X_C)^2}$; inductive if $X_L > X_C$, capacitive if $X_C > X_L$.
• Power factor $= \cos\phi = R/Z$.

Resonance

• Series resonance: $X_L = X_C$, impedance is minimum (= R), current is maximum, power factor is unity. Called an acceptor circuit. Voltage across L or C can be Q times the supply voltage (voltage magnification).
• Parallel resonance (anti-resonance): admittance is minimum, impedance is maximum ($Z_d = L/(CR)$, dynamic impedance), line current is minimum. Called a rejector circuit. Shows current magnification.
• Bandwidth $= f_0/Q$; half-power frequencies $f_1, f_2$ with $f_0 = \sqrt{f_1f_2}$ (geometric mean).

Active, reactive and apparent power

• Active (real) power $P = VI\cos\phi$ in W — actually converted to work/heat.
• Reactive power $Q = VI\sin\phi$ in VAR — oscillates between source and L/C fields; inductors absorb, capacitors supply.
• Apparent power $S = VI$ in VA; $S^2 = P^2 + Q^2$ (power triangle).

High-yield points

• Maximum power transfer: $R_L = R_{th}$ (DC), $Z_L = Z_{th}^*$ (AC); efficiency at that point is 50%.
• Series resonance: Z = R (minimum), current maximum, pf = 1 (acceptor); parallel resonance: Z maximum $= L/(CR)$ (rejector).
• $f_0 = 1/(2\pi\sqrt{LC})$ and $BW = f_0/Q$.
• Superposition requires a linear network and cannot be applied to power.
• $S^2 = P^2 + Q^2$: P in W, Q in VAR, S in VA.

Formulas

$f_0 = \frac{1}{2\pi\sqrt{LC}}\ \text{(Hz)}$

$Q_{series} = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}},\qquad BW = \frac{R}{2\pi L}\ \text{(Hz)}$

Parallel coil (resistance R) and capacitor: $f_0 = \frac{1}{2\pi}\sqrt{\frac{1}{LC} - \frac{R^2}{L^2}}$, $Z_d = \frac{L}{CR}$.

Maximum power: $P_{max} = \frac{V_{th}^2}{4R_{th}}$ (W).

Worked example

A series RLC circuit has R = 10 Ω, L = 0.1 H, C = 10 μF on a 100 V variable-frequency supply. Find $f_0$, current at resonance and Q.

1. $LC = 0.1 \times 10\times10^{-6} = 10^{-6}$, so $\sqrt{LC} = 10^{-3}$ s.
2. $f_0 = 1/(2\pi\times10^{-3}) = 159.2$ Hz.
3. At resonance Z = R = 10 Ω, so $I = 100/10 = 10$ A.
4. $Q = \frac{1}{R}\sqrt{L/C} = \frac{1}{10}\sqrt{0.1/10^{-5}} = \frac{1}{10}\times100 = 10$.
5. Voltage across L (or C) $= QV = 1000$ V — far above supply.

Exam tips

• Maximum power transfer efficiency = 50%; P_max = $V^2/4R$.
• Superposition: voltage source → short, current source → open. It cannot be used for power.
• Series resonance = minimum Z, maximum I (acceptor); parallel = maximum Z, minimum I (rejector).
• Below resonance a series RLC circuit is capacitive; above resonance it is inductive.
• Higher Q means sharper selectivity and narrower bandwidth.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. In a purely inductive ac circuit, the current:

• A. Leads the voltage by 90°
• B. Is in phase with the voltage
• C. Lags the voltage by 45°
• D. Lags the voltage by 90°

Answer: D. For an inductor $v = L\,di/dt$, so the current lags the applied voltage by 90°.

2. A series R-C circuit with $R = 8\ \Omega$ and $X_C = 6\ \Omega$ is connected to a 200 V ac supply. What active power does it consume?

• A. 5000 W
• B. 3200 W
• C. 4000 W
• D. 2400 W

Answer: B. $Z = \sqrt{8^2+6^2} = 10\ \Omega$, $I = 20$ A, $P = I^2R = 400 \times 8 = 3200$ W.

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