Network theorems

Network theorems let you analyse a complicated circuit without writing a full set of simultaneous equations. Superposition handles several sources, Thevenin and Norton reduce a network to a single source and resistance, and maximum power transfer tells you what load gets the most power. The second half of the topic covers AC circuits with R, L and C: impedance, phase angle, resonance and the three kinds of power (active, reactive, apparent). These are among the most frequently examined ideas in the electrical chapter.

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, others are "killed": ideal voltage sources are replaced by a short circuit, ideal current sources by an open circuit. Dependent sources are left in place. Superposition does not apply to power directly (power is not linear in current).
• 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 killed.
• Norton's theorem: The same network can be replaced by a current source $I_N$ in parallel with $R_N$. $I_N$ = short-circuit current; $R_N = R_{th}$. Conversion: $V_{th} = I_N R_{th}$.
• Maximum power transfer: A DC load receives maximum power when $R_L = R_{th}$. Efficiency at that point is only 50%. For AC, the load must be the complex conjugate of the source impedance, $Z_L = Z_{th}^*$.
• R-L circuit: Current lags voltage by $\phi = \tan^{-1}(X_L/R)$, between 0° and 90°. Time constant $\tau = L/R$.
• R-C circuit: Current leads voltage by $\phi = \tan^{-1}(X_C/R)$. Time constant $\tau = RC$. A capacitor charges to 63.2% of the final value in one $\tau$ and is considered fully charged after about $5\tau$.
• R-L-C series circuit: $Z = R + j(X_L - X_C)$. Inductive if $X_L > X_C$, capacitive if $X_C > X_L$.
• Series resonance: When $X_L = X_C$. Impedance is minimum ($Z = R$), current is maximum, power factor is unity, and the voltage across L or C can be $Q$ times the supply voltage (voltage magnification). Called an acceptor circuit.
• Parallel resonance: Impedance is maximum (dynamic impedance $Z_D = L/(CR)$ for a coil with resistance R in parallel with C), line current is minimum, and circulating current in the tank is $Q$ times the line current (current magnification). Called a rejector circuit.
• Power in AC:

Quantity	Symbol	Formula	Unit
Active (real) power	P	$VI\cos\phi$	W
Reactive power	Q	$VI\sin\phi$	VAR
Apparent power	S	$VI$	VA

$S^2 = P^2 + Q^2$, and power factor $= \cos\phi = P/S = R/Z$. Inductive loads absorb reactive power; capacitors supply it, which is why capacitors are used for power factor correction.

High-yield points

• Maximum power: $R_L = R_{th}$, $P_{max} = V_{th}^2/(4R_{th})$, efficiency 50%.
• $R_N = R_{th}$ and $V_{th} = I_N R_{th}$.
• Killing sources: voltage source → short circuit, current source → open circuit.
• Series resonance: Z = R (minimum), unity pf, acceptor. Parallel resonance: Z maximum, rejector.
• $f_r = 1/(2\pi\sqrt{LC})$ and $BW = f_r/Q$.
• RC circuit reaches 63.2% of final value in one $\tau$; about $5\tau$ for full charge.

Formulas

$X_L = 2\pi f L, \qquad X_C = \frac{1}{2\pi f C}$ $f$ in Hz, $L$ in henry, $C$ in farad, reactances in Ω.

$Z = \sqrt{R^2 + (X_L - X_C)^2}$

$f_r = \frac{1}{2\pi\sqrt{LC}}$ (resonant frequency, Hz; same for ideal series and ideal parallel LC.)

$Q = \frac{X_L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}, \qquad BW = \frac{f_r}{Q}$ $Q$ = quality factor (dimensionless), $BW$ = bandwidth between half-power points (Hz).

$P_{max} = \frac{V_{th}^2}{4R_{th}}$

Worked example

A network has $V_{th} = 20$ V and $R_{th} = 5$ Ω. Find the load for maximum power and the maximum power.

1. $R_L = R_{th} = 5$ Ω.
2. Current $I = 20/(5+5) = 2$ A.
3. $P = I^2R_L = 4\times5 = 20$ W. Check: $V_{th}^2/(4R_{th}) = 400/20 = 20$ W. ✓

Second check — series RLC: $L = 0.1$ H, $C = 10\ \mu$F. $f_r = 1/(2\pi\sqrt{0.1\times10^{-5}}) = 1/(2\pi\times 10^{-3}) \approx 159$ Hz.

Exam tips

• To find $R_{th}$: voltage source → short, current source → open. Swapping these is the classic trap.
• Maximum power transfer gives 50% efficiency, not maximum efficiency. Power systems are not run at this point.
• Series resonance: Z minimum, I maximum. Parallel resonance: Z maximum, I minimum.
• Below resonance a series RLC is capacitive; above resonance it is inductive.
• Higher Q means sharper resonance and narrower bandwidth.

Exam-style questions

From Calmon's question bank, which has 24 questions on this topic.

1. A 10 V source in series with a 5 Ω resistor feeds node A. A 5 Ω resistor connects node A to the reference (source negative), and an independent 2 A current source drives current from the reference into node A. Using superposition, the voltage of node A is:

• A. 15 V
• B. 20 V
• C. 10 V
• D. 5 V

Answer: C. Voltage source alone (current source open): $V_A = 10 \times \frac{5}{10} = 5$ V. Current source alone (voltage source shorted): $V_A = 2 \times (5\parallel5) = 5$ V. Total $= 10$ V.

2. A network has Thevenin equivalent $V_{th} = 10$ V and $R_{th} = 5$ Ω. Its Norton equivalent is:

• A. 50 A in parallel with 5 Ω
• B. 2 A in series with 5 Ω
• C. 2 A in parallel with 5 Ω
• D. 0.5 A in parallel with 5 Ω

Answer: C. $I_N = V_{th}/R_{th} = 10/5 = 2$ A and $R_N = R_{th} = 5$ Ω, in parallel.

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