Network Analysis (GATE EC)

GATE EC syllabus for network analysis, with node and mesh analysis, network theorems, phasors and complex power, transients in RL, RC and RLC circuits, Laplace-domain analysis and two-port parameters, with solved examples.

Syllabus

Node and mesh analysis, superposition, Thevenin's and Norton's theorems, reciprocity; sinusoidal steady state, phasors, complex power, maximum power transfer; time and frequency domain analysis of RL, RC and RLC circuits; solution of network equations using the Laplace transform; two-port network parameters; wye-delta transformation.

Key concepts and formulas

Theorems. Thevenin/Norton equivalents (R_th found with independent sources off; with dependent sources, use a test source: R_th = V_test/I_test). Maximum power transfer: R_L = R_th (P_max = V_th²/(4R_th)), or Z_L = Z_th* for AC. Reciprocity holds for networks of R, L, C and transformers without dependent sources.

First-order transients. x(t) = x(∞) + (x(0⁺) − x(∞))e^−t/τ, τ = RC or L/R. At t = 0⁺ a capacitor keeps its voltage and an inductor keeps its current; at DC steady state a capacitor is an open circuit and an inductor a short circuit.

Second-order (series RLC). Characteristic equation s² + (R/L)s + 1/(LC) = 0, with ω₀ = 1/√(LC) and damping ratio ζ = (R/2)√(C/L). Overdamped for ζ > 1, critically damped at ζ = 1 (R = 2√(L/C)), underdamped for ζ < 1.

Laplace domain. Impedances R, sL and 1/(sC); initial conditions appear as sources: an inductor with i(0) is sL in series with a source L i(0); a capacitor with v(0) is 1/(sC) in series with a source v(0)/s.

Phasors and power. S = V_rmsI_rms* = P + jQ. Power factor correction: the capacitor needs Q_C = P(tan φ₁ − tan φ₂).

Two-port parameters.

Parameters	Equations	Reciprocal	Symmetric
Z	V₁ = z₁₁I₁ + z₁₂I₂, V₂ = z₂₁I₁ + z₂₂I₂	z₁₂ = z₂₁	z₁₁ = z₂₂
Y	I₁ = y₁₁V₁ + y₁₂V₂, I₂ = y₂₁V₁ + y₂₂V₂	y₁₂ = y₂₁	y₁₁ = y₂₂
h	V₁ = h₁₁I₁ + h₁₂V₂, I₂ = h₂₁I₁ + h₂₂V₂	h₁₂ = −h₂₁	Δh = 1
ABCD	V₁ = AV₂ − BI₂, I₁ = CV₂ − DI₂	AD − BC = 1	A = D

Cascaded two-ports multiply their ABCD matrices; series connections add Z matrices; parallel connections add Y matrices.

Worked example

Find the Z parameters of a T network with series arms 2 Ω and 4 Ω and a shunt arm of 6 Ω, and check reciprocity. For a series RLC with L = 1 H and C = 0.25 F, find the resistance for critical damping. Then find the capacitor current 1 s after a 10 V step is applied to an RC circuit with R = 2 Ω and C = 0.5 F.

Python

Open in playground

import math

Za, Zb, Zc = 2, 4, 6       # series left, series right, shunt
z11, z12, z21, z22 = Za + Zc, Zc, Zc, Zb + Zc
print(f"Z = [[{z11}, {z12}], [{z21}, {z22}]], reciprocal: {z12 == z21}, symmetric: {z11 == z22}")
L, C = 1, 0.25
print(f"critical damping R = 2 sqrt(L/C) = {2 * math.sqrt(L / C):.1f} ohm")
V, R, Cap, t = 10, 2, 0.5, 1
tau = R * Cap
print(f"tau = {tau} s, i(1 s) = {V / R * math.exp(-t / tau):.4f} A, vC(1 s) = {V * (1 - math.exp(-t / tau)):.4f} V")

Output

Z = [[8, 6], [6, 10]], reciprocal: True, symmetric: False
critical damping R = 2 sqrt(L/C) = 4.0 ohm
tau = 1.0 s, i(1 s) = 1.8394 A, vC(1 s) = 6.3212 V