GATE EE

Transient Response in Electrical Circuits – GATE EE Notes

Lecture Notes

SEC 01

Transient Response

1Transient 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: \(i_L(0^+) = i_L(0^-)\), \(v_C(0^+) = v_C(0^-)\)

1First Order Circuits (RC & RL)

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

\[\begin{aligned} v_C(t) &= V_f + (V_0 - V_f)e^{-t/\tau} \\ i_C(t) &= \dfrac{V_f - V_0}{R}e^{-t/\tau} \end{aligned}\]

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

\[\begin{aligned} i_L(t) &= I_f + (I_0 - I_f)e^{-t/\tau} \\ v_L(t) &= (I_f - I_0)Re^{-t/\tau} \end{aligned}\]

Key Points:

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

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

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

1Second 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:

\[v(t) = e^{-\alpha t}(A\cos\omega_d t + B\sin\omega_d t)\]
where \(\omega_d = \sqrt{\omega_0^2 - \alpha^2}\) (damped frequency)

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

SEC 02

DC Networks

1DC 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)

SEC 03

Sinusoidal Steady State & Phasor Analysis

1Phasor 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: \(Y = \dfrac{1}{Z}\)

    \[\begin{aligned} Y_R &= \dfrac{1}{R}, \quad Y_L = \dfrac{1}{j\omega L}, \quad Y_C = j\omega C \end{aligned}\]
    • 1AC Circuit Analysis
    SEC 04

    Resonance

    1Series RLC Resonance

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

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

    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

    1Parallel RLC Resonance

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

    At resonance:

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

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

    \[\omega_0 = \dfrac{1}{\sqrt{LC}}\sqrt{1 - \dfrac{r^2 C}{L}}\]

    Comparison:

    SEC 05

    Two-Port Networks

    1Two-Port Parameters

    Impedance (Z) Parameters:

    \[\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}\]

    Admittance (Y) Parameters:

    \[\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}\]

    Hybrid (h) Parameters:

    \[\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}\]

    Transmission (ABCD) Parameters:

    \[\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}\]
    1Two-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:

    Parameter Conversions:

    SEC 06

    Three-Phase Circuits

    1Three-Phase Systems

    Balanced Three-Phase:

    Phase Sequence: Order in which phases reach maximum

    Star (Y) Connection:

    Delta (\(\Delta\)) Connection:

    1Star-Delta Transformation

    Delta to Star:

    \[\begin{aligned} R_1 &= \dfrac{R_a R_c}{R_a + R_b + R_c} \\ R_2 &= \dfrac{R_a R_b}{R_a + R_b + R_c} \\ R_3 &= \dfrac{R_b R_c}{R_a + R_b + R_c} \end{aligned}\]

    Star to Delta:

    \[\begin{aligned} R_a &= \dfrac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_3} \\ R_b &= \dfrac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_1} \\ R_c &= \dfrac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_2} \end{aligned}\]

    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)

    1Three-Phase Power

    Balanced Load Power:

    \[\begin{aligned} P &= \sqrt{3} V_L I_L \cos\phi \\ Q &= \sqrt{3} V_L I_L \sin\phi \\ S &= \sqrt{3} V_L I_L \end{aligned}\]

    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:

    SEC 07

    Complex Power & Power Factor

    1Complex Power

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

    Power Components:

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

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

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

    1Power Factor Correction

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

    Method: Add capacitor in parallel with inductive load

    Capacitor Value:

    \[C = \dfrac{P(\tan\phi_1 - \tan\phi_2)}{\omega V^2}\]

    Where:

    Alternative Formula:

    \[Q_C = P(\tan\phi_1 - \tan\phi_2)\]

    Benefits:

    SEC 08

    Additional Important Topics

    1Frequency Response & Filters

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

    Bode Plots:

    Basic Filters:

    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

    SEC 09

    Important Formulas Summary

    1Time Constants & Energy

    Time Constants:

    Energy Storage:

    Power Relations:

    1Three-Phase & AC Circuit Formulas

    Three-Phase Relations:

    Resonance:

    RMS & Average:

    SEC 10

    GATE-Specific Tips

    1Key Points for GATE

    Always Remember:

    Sign Conventions:

    1Common GATE Mistakes to Avoid

    Transient Analysis:

    AC Analysis:

    Three-Phase:

    Resonance:

    1Quick 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