Transient Response - Basics

• Definition: Response during transition from one steady state to another

• Complete Response: \(i(t) = i_{natural}(t) + i_{forced}(t)\)

• Natural Response: Solution of homogeneous equation

• Forced Response: Particular solution due to input

Key Steps:

1. Find initial conditions: \(i(0^+)\), \(v(0^+)\)

2. Write differential equation

3. Solve for natural + forced response

4. Apply initial conditions to find constants

Important: \(i_L(0^+) = i_L(0^-)\), \(v_C(0^+) = v_C(0^-)\)

First Order Circuits (RC & RL)

RC Circuit: \(\tau = RC\) (time constant)

RL Circuit: \(\tau = L/R\)

Key Points:

• At \(t = \tau\): 63.2% of final value

• At \(t = 5\tau\): 99.3% of final value (practically steady)

• Energy stored: \(W_C = \dfrac{1}{2}CV^2\), \(W_L = \dfrac{1}{2}LI^2\)

Second Order Circuits (RLC)

Characteristic Equation: \(s^2 + 2\alpha s + \omega_0^2 = 0\)

Where: \(\alpha = \dfrac{R}{2L}\), \(\omega_0 = \dfrac{1}{\sqrt{LC}}\)

Three Cases:

• Overdamped (\(\alpha > \omega_0\)): Two real roots \(s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}\)

• Critically Damped (\(\alpha = \omega_0\)): One repeated root \(s = -\alpha\)

• Underdamped (\(\alpha < \omega_0\)): Complex conjugate roots \(s = -\alpha \pm j\omega_d\)

Underdamped Response:

Damping Ratio: \(\zeta = \dfrac{\alpha}{\omega_0} = \dfrac{R}{2}\sqrt{\dfrac{C}{L}}\)

DC Network Analysis

Key Laws:

• KCL: \(\sum I_{in} = \sum I_{out}\) (at any node)

• KVL: \(\sum V_{drops} = \sum V_{rises}\) (in any loop)

• Ohm’s Law: \(V = IR\)

Network Theorems:

• Superposition: Linear combination of individual source responses

• Thevenin: \(V_{th} = V_{oc}\), \(R_{th} = R_{eq}\) (all sources deactivated)

• Norton: \(I_N = I_{sc}\), \(R_N = R_{th}\)

• Maximum Power Transfer: \(R_L = R_{th}\) for max power to load

• Reciprocity: \(\dfrac{V_2}{I_1} = \dfrac{V_1}{I_2}\) (linear bilateral networks)

Analysis Methods:

• Nodal Analysis: Apply KCL at nodes (good for parallel circuits)

• Mesh Analysis: Apply KVL in loops (good for series circuits)

Phasor Representation

• Sinusoidal Signal: \(v(t) = V_m \cos(\omega t + \phi)\)

• Phasor: \(\mathbf{V} = V_m \angle \phi\) or \(\mathbf{V} = V_m e^{j\phi}\)

• RMS Value: \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\) (for sinusoidal signals)

• Impedances:

  \[\begin{aligned} Z_R &= R \angle 0° \\ Z_L &= j\omega L = \omega L \angle 90° \\ Z_C &= \dfrac{1}{j\omega C} = \dfrac{1}{\omega C} \angle -90° \end{aligned}\]

• Admittances: \(Y = \dfrac{1}{Z}\)

  \[\begin{aligned} Y_R &= \dfrac{1}{R}, \quad Y_L = \dfrac{1}{j\omega L}, \quad Y_C = j\omega C \end{aligned}\]

AC Circuit Analysis

• Kirchhoff’s Laws in Phasor Form:

  • KCL: \(\sum \mathbf{I} = 0\)

  • KVL: \(\sum \mathbf{V} = 0\)

• Series Impedance: \(Z_{eq} = Z_1 + Z_2 + Z_3 + ...\)

• Parallel Impedance: \(\dfrac{1}{Z_{eq}} = \dfrac{1}{Z_1} + \dfrac{1}{Z_2} + \dfrac{1}{Z_3} + ...\)

• Voltage Divider: \(\mathbf{V_2} = \mathbf{V_s} \dfrac{Z_2}{Z_1 + Z_2}\)

• Current Divider: \(\mathbf{I_2} = \mathbf{I_s} \dfrac{Z_1}{Z_1 + Z_2}\)

• AC Network Theorems: All DC theorems apply with impedances replacing resistances

Series RLC Resonance

Resonant Frequency: \(\omega_0 = \dfrac{1}{\sqrt{LC}}\), \(f_0 = \dfrac{1}{2\pi\sqrt{LC}}\)

At resonance (\(\omega = \omega_0\)):

• \(X_L = X_C\) (reactances cancel)

• \(Z = R\) (minimum impedance)

• Current is maximum: \(I = \dfrac{V}{R}\)

• Power factor = 1 (purely resistive)

• Phase angle \(\phi = 0°\)

Quality Factor: \(Q = \dfrac{\omega_0 L}{R} = \dfrac{1}{\omega_0 RC} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}\)

Bandwidth: \(BW = \dfrac{\omega_0}{Q} = \dfrac{R}{L}\)

Half-power frequencies: \(\omega_1 = \omega_0 - \dfrac{BW}{2}\), \(\omega_2 = \omega_0 + \dfrac{BW}{2}\)

Selectivity: Higher Q \(\Rightarrow\) sharper resonance, better selectivity

Parallel RLC Resonance

Resonant Frequency: \(\omega_0 = \dfrac{1}{\sqrt{LC}}\) (same as series)

At resonance:

• \(B_L = B_C\) (susceptances cancel)

• \(Y = G\) (minimum admittance, maximum impedance)

• Voltage is maximum

• Power factor = 1

• Current from source is minimum

Quality Factor: \(Q = \omega_0 RC = \dfrac{R}{\omega_0 L}\)

Practical Parallel Circuit: For coil with resistance \(r\):

Comparison:

• Series: Current max, Impedance min, Voltage across L,C can exceed source

• Parallel: Voltage max, Impedance max, Current in L,C can exceed source

Two-Port Parameters

Impedance (Z) Parameters:

Admittance (Y) Parameters:

Hybrid (h) Parameters:

Transmission (ABCD) Parameters:

Two-Port Network Properties

Reciprocal Networks: \(Z_{12} = Z_{21}\), \(Y_{12} = Y_{21}\), \(h_{12} = -h_{21}\), \(AD - BC = 1\)

Symmetric Networks: \(Z_{11} = Z_{22}\), \(Y_{11} = Y_{22}\), \(h_{11}h_{22} - h_{12}h_{21} = 1\), \(A = D\)

Interconnections:

• Series: \([Z] = [Z_1] + [Z_2]\)

• Parallel: \([Y] = [Y_1] + [Y_2]\)

• Series-Parallel: \([h] = [h_1] + [h_2]\)

• Cascade: \([ABCD] = [ABCD_1][ABCD_2]\)

Parameter Conversions:

• \([Y] = [Z]^{-1}\)

• \(Z_{11} = \dfrac{\Delta Y}{Y_{22}}\), \(Z_{12} = \dfrac{-Y_{12}}{Y_{22}}\), etc.

• \(\Delta Y = Y_{11}Y_{22} - Y_{12}Y_{21}\)

Three-Phase Systems

Balanced Three-Phase:

• Equal magnitudes, 120° phase difference

• \(\mathbf{V_a} = V \angle 0°\), \(\mathbf{V_b} = V \angle -120°\), \(\mathbf{V_c} = V \angle 120°\)

• Sum of phasors = 0: \(\mathbf{V_a} + \mathbf{V_b} + \mathbf{V_c} = 0\)

Phase Sequence: Order in which phases reach maximum

• Positive (abc): \(a \rightarrow b \rightarrow c\)

• Negative (acb): \(a \rightarrow c \rightarrow b\)

Star (Y) Connection:

• \(V_L = \sqrt{3} V_P\) (line voltage = \(\sqrt{3} \times\) phase voltage)

• \(I_L = I_P\) (line current = phase current)

• \(V_L\) leads \(V_P\) by 30°

Delta (\(\Delta\)) Connection:

• \(V_L = V_P\) (line voltage = phase voltage)

• \(I_L = \sqrt{3} I_P\) (line current = \(\sqrt{3} \times\) phase current)

• \(I_L\) lags \(I_P\) by 30°

Star-Delta Transformation

Delta to Star:

Star to Delta:

For Equal Resistances: \(R_Y = \dfrac{R_\Delta}{3}\), \(R_\Delta = 3R_Y\)

Memory Aid: Star value = (Product of adjacent deltas)/(Sum of all deltas)

Three-Phase Power

Balanced Load Power:

Where \(\phi\) is the phase angle between phase voltage and phase current.

Per-Phase Analysis: For balanced systems, analyze one phase and multiply by 3

Neutral Current: In balanced Y-connected load: \(I_N = 0\)

Power Measurement:

• One-wattmeter method: For balanced load with accessible neutral

• Two-wattmeter method: \(P = W_1 + W_2\), \(Q = \sqrt{3}(W_1 - W_2)\)

• Three-wattmeter method: When neutral is available

Complex Power

Complex Power: \(\mathbf{S} = \mathbf{V} \mathbf{I}^* = P + jQ\)

Power Components:

• Real Power: \(P = V I \cos\phi\) (W) - actual power consumed

• Reactive Power: \(Q = V I \sin\phi\) (VAR) - power exchanged

• Apparent Power: \(S = V I = |\mathbf{S}|\) (VA) - total power

Power Triangle: \(S^2 = P^2 + Q^2\)

Power Factor: \(pf = \cos\phi = \dfrac{P}{S}\)

• Leading pf: Capacitive load (\(\phi < 0°\), \(Q < 0\))

• Lagging pf: Inductive load (\(\phi > 0°\), \(Q > 0\))

• Unity pf: Resistive load (\(\phi = 0°\), \(Q = 0\))

Conservation: \(P_{total} = \sum P_i\), \(Q_{total} = \sum Q_i\), \(S_{total} \neq \sum S_i\)

Power Factor Correction

Need: Improve power factor to reduce reactive power and line losses

Method: Add capacitor in parallel with inductive load

Capacitor Value:

Where:

• \(\phi_1\): Original power factor angle

• \(\phi_2\): Desired power factor angle

• \(P\): Real power (remains constant)

Alternative Formula:

Benefits:

• Reduced line losses (\(I^2R\) losses decrease)

• Improved voltage regulation

• Reduced kVA demand charges

• Better utilization of electrical equipment

Frequency Response & Filters

Transfer Function: \(H(j\omega) = \dfrac{V_{out}}{V_{in}}\)

Bode Plots:

• Magnitude: \(20\log|H(j\omega)|\) dB vs \(\log\omega\)

• Phase: \(\angle H(j\omega)\) vs \(\log\omega\)

Basic Filters:

• Low-pass RC: \(H(j\omega) = \dfrac{1}{1 + j\omega RC}\), \(f_c = \dfrac{1}{2\pi RC}\)

• High-pass RC: \(H(j\omega) = \dfrac{j\omega RC}{1 + j\omega RC}\)

• Low-pass RL: \(H(j\omega) = \dfrac{R}{R + j\omega L}\), \(f_c = \dfrac{R}{2\pi L}\)

• High-pass RL: \(H(j\omega) = \dfrac{j\omega L}{R + j\omega L}\)

Cutoff Frequency: Frequency at which \(|H(j\omega)| = \dfrac{1}{\sqrt{2}} = 0.707\) (-3dB point)

Roll-off Rate: First-order: ±20 dB/decade, Second-order: ±40 dB/decade

Time Constants & Energy

Time Constants:

• RC: \(\tau = RC\)

• RL: \(\tau = L/R\)

• RLC: \(\omega_0 = 1/\sqrt{LC}\), \(\alpha = R/2L\), \(\zeta = \dfrac{R}{2}\sqrt{\dfrac{C}{L}}\)

Energy Storage:

• Capacitor: \(W_C = \dfrac{1}{2}CV^2 = \dfrac{1}{2}QV = \dfrac{Q^2}{2C}\)

• Inductor: \(W_L = \dfrac{1}{2}LI^2 = \dfrac{1}{2}\Phi I = \dfrac{\Phi^2}{2L}\)

Power Relations:

• \(P = VI\cos\phi = I^2 R = \dfrac{V^2}{R}\) (resistive)

• \(Q = VI\sin\phi = I^2 X = \dfrac{V^2}{X}\) (reactive)

• \(S = VI = \sqrt{P^2 + Q^2}\) (apparent)

Three-Phase & AC Circuit Formulas

Three-Phase Relations:

• Star: \(V_L = \sqrt{3}V_P\), \(I_L = I_P\)

• Delta: \(V_L = V_P\), \(I_L = \sqrt{3}I_P\)

• Power: \(P = \sqrt{3}V_L I_L \cos\phi = 3V_P I_P \cos\phi\)

Resonance:

• Frequency: \(f_0 = \dfrac{1}{2\pi\sqrt{LC}}\)

• Quality factor: \(Q = \dfrac{\omega_0 L}{R} = \dfrac{1}{\omega_0 RC}\) (series)

• Bandwidth: \(BW = \dfrac{f_0}{Q}\)

RMS & Average:

• Sinusoidal: \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\), \(V_{avg} = 0\)

• Rectified sine: \(V_{avg} = \dfrac{2V_m}{\pi}\), \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\)

Key Points for GATE

Always Remember:

• Initial conditions: \(i_L(0^+) = i_L(0^-)\), \(v_C(0^+) = v_C(0^-)\)

• At steady state DC: Capacitor = open circuit, Inductor = short circuit

• At steady state AC: Use impedance analysis

• Resonance condition: \(X_L = X_C\) or \(\omega L = \dfrac{1}{\omega C}\)

• Power factor correction always uses capacitors (for inductive loads)

• In balanced 3-phase: Neutral current = 0, line quantities relate by \(\sqrt{3}\)

Sign Conventions:

• Passive sign convention: \(P = VI\) (+ if absorbing power)

• Current direction: Assumed direction for analysis

• Inductive reactance: \(+jX_L\) (lagging), Capacitive: \(-jX_C\) (leading)

Common GATE Mistakes to Avoid

Transient Analysis:

• Forgetting to check initial conditions properly

• Mixing up \(0^-\) and \(0^+\) values

• Not considering switching action effects

AC Analysis:

• Confusing impedance magnitude with resistance

• Wrong signs for reactive power (Q)

• Mixing up RMS and peak values

Three-Phase:

• Confusing line and phase quantities

• Wrong \(\sqrt{3}\) factor application

• Forgetting \(30^{\circ}\) phase shifts

Resonance:

• Mixing up series and parallel resonance conditions

• Forgetting that impedance is minimum (series) or maximum (parallel)

Quick Problem-Solving Strategy

For Transient Problems:

1. Identify the type of circuit (RC, RL, RLC)

2. Find initial conditions at \(t = 0^+\)

3. Find final steady-state values

4. Apply appropriate exponential formula

5. Check units and signs

For AC Problems:

1. Convert to phasor domain

2. Find equivalent impedance

3. Apply KVL/KCL or network theorems

4. Calculate power if required

5. Convert back to time domain if needed

For Three-Phase:

1. Check if balanced or unbalanced

2. Identify connection type (Y or \(\Delta\))

3. Use per-phase analysis for balanced systems

4. Apply appropriate voltage/current relations

5. Calculate total power