Electric Circuits - Electrical Engineering (GATE EE)

GATE EE syllabus for electric circuits, with KCL and KVL, network theorems, maximum power transfer, transients, sinusoidal steady state, resonance, two-port networks, three-phase circuits and complex power, with solved examples.

Syllabus

Network elements and sources (independent, dependent); KCL, KVL, node and mesh analysis; Thevenin's, Norton's, superposition and maximum power transfer theorems; transient response of DC and AC networks; sinusoidal steady state; resonance; two-port networks; balanced three-phase circuits; star-delta transformation; complex power and power factor.

Key concepts and formulas

Network theorems. Thevenin: any linear network seen from two terminals is a source V_th in series with R_th; Norton: I_N = V_th/R_th in parallel with R_th. Maximum power transfer: R_L = R_th gives P_max = V_th²/(4R_th) (for AC, Z_L = Z_th*). Star-delta: R_Y = R_Δ/3 for equal resistors.

Transients. First-order circuits: x(t) = x(∞) + (x(0⁺) − x(∞))e^−t/τ, with τ = RC or L/R. A capacitor voltage and an inductor current cannot change instantly.

AC steady state. Impedances Z_R = R, Z_L = jωL, Z_C = 1/(jωC). Complex power S = VI* = P + jQ, with P = VI cos φ and Q = VI sin φ; power factor = cos φ.

Resonance. f₀ = 1/(2π√(LC)). Series RLC: Q = (1/R)√(L/C), bandwidth = f₀/Q, and the impedance is minimum (= R) at resonance. Parallel RLC: Q = R√(C/L).

Three-phase (balanced). Star: V_L = √3V_ph, I_L = I_ph. Delta: I_L = √3I_ph, V_L = V_ph. Power P = √3V_LI_L cos φ. Two-wattmeter method: P = W₁ + W₂ and tan φ = √3(W₁ − W₂)/(W₁ + W₂).

Two-port networks. Z, Y, h and ABCD parameters. Reciprocal: z₁₂ = z₂₁ (and AD − BC = 1); symmetrical: z₁₁ = z₂₂ (and A = D).

Worked example

A 12 V source with internal resistance 4 Ω feeds a network that reduces to R_th = 6 Ω, V_th = 9 V at the load terminals. Find the maximum power. Find f₀, Q and the bandwidth of a series RLC with R = 10 Ω, L = 10 mH and C = 1 μF. Then find the power factor from wattmeter readings of 6 kW and 2 kW.

Python

Open in playground

import math

Vth, Rth = 9, 6
print(f"Pmax = {Vth ** 2 / (4 * Rth):.4f} W at RL = {Rth} ohm")
R, L, C = 10, 10e-3, 1e-6
f0 = 1 / (2 * math.pi * math.sqrt(L * C))
Q = math.sqrt(L / C) / R
print(f"f0 = {f0:.2f} Hz, Q = {Q:.2f}, bandwidth = {f0 / Q:.2f} Hz")
W1, W2 = 6, 2
phi = math.atan(math.sqrt(3) * (W1 - W2) / (W1 + W2))
print(f"P = {W1 + W2} kW, power factor = {math.cos(phi):.4f}")

Output

Pmax = 3.3750 W at RL = 6 ohm
f0 = 1591.55 Hz, Q = 10.00, bandwidth = 159.15 Hz
P = 8 kW, power factor = 0.7559