Transient Response
Transient Response - Basics
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Definition: Response during transition from one steady state to another
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Complete Response: \(i(t) = i_{natural}(t) + i_{forced}(t)\)
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Natural Response: Solution of homogeneous equation
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Forced Response: Particular solution due to input
Key Steps:
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Find initial conditions: \(i(0^+)\), \(v(0^+)\)
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Write differential equation
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Solve for natural + forced response
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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:
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At \(t = \tau\): 63.2% of final value
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At \(t = 5\tau\): 99.3% of final value (practically steady)
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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:
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Overdamped (\(\alpha > \omega_0\)): Two real roots \(s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}\)
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Critically Damped (\(\alpha = \omega_0\)): One repeated root \(s = -\alpha\)
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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 Networks
DC Network Analysis
Key Laws:
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KCL: \(\sum I_{in} = \sum I_{out}\) (at any node)
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KVL: \(\sum V_{drops} = \sum V_{rises}\) (in any loop)
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Ohm’s Law: \(V = IR\)
Network Theorems:
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Superposition: Linear combination of individual source responses
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Thevenin: \(V_{th} = V_{oc}\), \(R_{th} = R_{eq}\) (all sources deactivated)
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Norton: \(I_N = I_{sc}\), \(R_N = R_{th}\)
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Maximum Power Transfer: \(R_L = R_{th}\) for max power to load
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Reciprocity: \(\dfrac{V_2}{I_1} = \dfrac{V_1}{I_2}\) (linear bilateral networks)
Analysis Methods:
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Nodal Analysis: Apply KCL at nodes (good for parallel circuits)
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Mesh Analysis: Apply KVL in loops (good for series circuits)
Sinusoidal Steady State & Phasor Analysis
Phasor Representation
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Sinusoidal Signal: \(v(t) = V_m \cos(\omega t + \phi)\)
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Phasor: \(\mathbf{V} = V_m \angle \phi\) or \(\mathbf{V} = V_m e^{j\phi}\)
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RMS Value: \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\) (for sinusoidal signals)
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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
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Kirchhoff’s Laws in Phasor Form:
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KCL: \(\sum \mathbf{I} = 0\)
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KVL: \(\sum \mathbf{V} = 0\)
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Series Impedance: \(Z_{eq} = Z_1 + Z_2 + Z_3 + ...\)
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Parallel Impedance: \(\dfrac{1}{Z_{eq}} = \dfrac{1}{Z_1} + \dfrac{1}{Z_2} + \dfrac{1}{Z_3} + ...\)
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Voltage Divider: \(\mathbf{V_2} = \mathbf{V_s} \dfrac{Z_2}{Z_1 + Z_2}\)
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Current Divider: \(\mathbf{I_2} = \mathbf{I_s} \dfrac{Z_1}{Z_1 + Z_2}\)
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AC Network Theorems: All DC theorems apply with impedances replacing resistances
Resonance
Series RLC Resonance
Resonant Frequency: \(\omega_0 = \dfrac{1}{\sqrt{LC}}\), \(f_0 = \dfrac{1}{2\pi\sqrt{LC}}\)
At resonance (\(\omega = \omega_0\)):
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\(X_L = X_C\) (reactances cancel)
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\(Z = R\) (minimum impedance)
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Current is maximum: \(I = \dfrac{V}{R}\)
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Power factor = 1 (purely resistive)
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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:
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\(B_L = B_C\) (susceptances cancel)
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\(Y = G\) (minimum admittance, maximum impedance)
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Voltage is maximum
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Power factor = 1
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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:
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Series: Current max, Impedance min, Voltage across L,C can exceed source
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Parallel: Voltage max, Impedance max, Current in L,C can exceed source
Two-Port Networks
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:
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Series: \([Z] = [Z_1] + [Z_2]\)
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Parallel: \([Y] = [Y_1] + [Y_2]\)
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Series-Parallel: \([h] = [h_1] + [h_2]\)
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Cascade: \([ABCD] = [ABCD_1][ABCD_2]\)
Parameter Conversions:
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\([Y] = [Z]^{-1}\)
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\(Z_{11} = \dfrac{\Delta Y}{Y_{22}}\), \(Z_{12} = \dfrac{-Y_{12}}{Y_{22}}\), etc.
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\(\Delta Y = Y_{11}Y_{22} - Y_{12}Y_{21}\)
Three-Phase Circuits
Three-Phase Systems
Balanced Three-Phase:
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Equal magnitudes, 120° phase difference
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\(\mathbf{V_a} = V \angle 0°\), \(\mathbf{V_b} = V \angle -120°\), \(\mathbf{V_c} = V \angle 120°\)
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Sum of phasors = 0: \(\mathbf{V_a} + \mathbf{V_b} + \mathbf{V_c} = 0\)
Phase Sequence: Order in which phases reach maximum
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Positive (abc): \(a \rightarrow b \rightarrow c\)
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Negative (acb): \(a \rightarrow c \rightarrow b\)
Star (Y) Connection:
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\(V_L = \sqrt{3} V_P\) (line voltage = \(\sqrt{3} \times\) phase voltage)
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\(I_L = I_P\) (line current = phase current)
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\(V_L\) leads \(V_P\) by 30°
Delta (\(\Delta\)) Connection:
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\(V_L = V_P\) (line voltage = phase voltage)
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\(I_L = \sqrt{3} I_P\) (line current = \(\sqrt{3} \times\) phase current)
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\(I_L\) 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:
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One-wattmeter method: For balanced load with accessible neutral
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Two-wattmeter method: \(P = W_1 + W_2\), \(Q = \sqrt{3}(W_1 - W_2)\)
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Three-wattmeter method: When neutral is available
Complex Power & Power Factor
Complex Power
Complex Power: \(\mathbf{S} = \mathbf{V} \mathbf{I}^* = P + jQ\)
Power Components:
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Real Power: \(P = V I \cos\phi\) (W) - actual power consumed
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Reactive Power: \(Q = V I \sin\phi\) (VAR) - power exchanged
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Apparent Power: \(S = V I = |\mathbf{S}|\) (VA) - total power
Power Triangle: \(S^2 = P^2 + Q^2\)
Power Factor: \(pf = \cos\phi = \dfrac{P}{S}\)
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Leading pf: Capacitive load (\(\phi < 0°\), \(Q < 0\))
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Lagging pf: Inductive load (\(\phi > 0°\), \(Q > 0\))
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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:
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\(\phi_1\): Original power factor angle
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\(\phi_2\): Desired power factor angle
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\(P\): Real power (remains constant)
Alternative Formula:
Benefits:
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Reduced line losses (\(I^2R\) losses decrease)
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Improved voltage regulation
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Reduced kVA demand charges
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Better utilization of electrical equipment
Additional Important Topics
Frequency Response & Filters
Transfer Function: \(H(j\omega) = \dfrac{V_{out}}{V_{in}}\)
Bode Plots:
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Magnitude: \(20\log|H(j\omega)|\) dB vs \(\log\omega\)
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Phase: \(\angle H(j\omega)\) vs \(\log\omega\)
Basic Filters:
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Low-pass RC: \(H(j\omega) = \dfrac{1}{1 + j\omega RC}\), \(f_c = \dfrac{1}{2\pi RC}\)
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High-pass RC: \(H(j\omega) = \dfrac{j\omega RC}{1 + j\omega RC}\)
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Low-pass RL: \(H(j\omega) = \dfrac{R}{R + j\omega L}\), \(f_c = \dfrac{R}{2\pi L}\)
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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
Important Formulas Summary
Time Constants & Energy
Time Constants:
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RC: \(\tau = RC\)
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RL: \(\tau = L/R\)
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RLC: \(\omega_0 = 1/\sqrt{LC}\), \(\alpha = R/2L\), \(\zeta = \dfrac{R}{2}\sqrt{\dfrac{C}{L}}\)
Energy Storage:
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Capacitor: \(W_C = \dfrac{1}{2}CV^2 = \dfrac{1}{2}QV = \dfrac{Q^2}{2C}\)
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Inductor: \(W_L = \dfrac{1}{2}LI^2 = \dfrac{1}{2}\Phi I = \dfrac{\Phi^2}{2L}\)
Power Relations:
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\(P = VI\cos\phi = I^2 R = \dfrac{V^2}{R}\) (resistive)
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\(Q = VI\sin\phi = I^2 X = \dfrac{V^2}{X}\) (reactive)
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\(S = VI = \sqrt{P^2 + Q^2}\) (apparent)
Three-Phase & AC Circuit Formulas
Three-Phase Relations:
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Star: \(V_L = \sqrt{3}V_P\), \(I_L = I_P\)
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Delta: \(V_L = V_P\), \(I_L = \sqrt{3}I_P\)
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Power: \(P = \sqrt{3}V_L I_L \cos\phi = 3V_P I_P \cos\phi\)
Resonance:
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Frequency: \(f_0 = \dfrac{1}{2\pi\sqrt{LC}}\)
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Quality factor: \(Q = \dfrac{\omega_0 L}{R} = \dfrac{1}{\omega_0 RC}\) (series)
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Bandwidth: \(BW = \dfrac{f_0}{Q}\)
RMS & Average:
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Sinusoidal: \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\), \(V_{avg} = 0\)
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Rectified sine: \(V_{avg} = \dfrac{2V_m}{\pi}\), \(V_{rms} = \dfrac{V_m}{\sqrt{2}}\)
GATE-Specific Tips
Key Points for GATE
Always Remember:
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Initial conditions: \(i_L(0^+) = i_L(0^-)\), \(v_C(0^+) = v_C(0^-)\)
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At steady state DC: Capacitor = open circuit, Inductor = short circuit
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At steady state AC: Use impedance analysis
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Resonance condition: \(X_L = X_C\) or \(\omega L = \dfrac{1}{\omega C}\)
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Power factor correction always uses capacitors (for inductive loads)
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In balanced 3-phase: Neutral current = 0, line quantities relate by \(\sqrt{3}\)
Sign Conventions:
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Passive sign convention: \(P = VI\) (+ if absorbing power)
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Current direction: Assumed direction for analysis
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Inductive reactance: \(+jX_L\) (lagging), Capacitive: \(-jX_C\) (leading)
Common GATE Mistakes to Avoid
Transient Analysis:
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Forgetting to check initial conditions properly
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Mixing up \(0^-\) and \(0^+\) values
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Not considering switching action effects
AC Analysis:
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Confusing impedance magnitude with resistance
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Wrong signs for reactive power (Q)
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Mixing up RMS and peak values
Three-Phase:
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Confusing line and phase quantities
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Wrong \(\sqrt{3}\) factor application
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Forgetting \(30^{\circ}\) phase shifts
Resonance:
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Mixing up series and parallel resonance conditions
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Forgetting that impedance is minimum (series) or maximum (parallel)
Quick Problem-Solving Strategy
For Transient Problems:
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Identify the type of circuit (RC, RL, RLC)
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Find initial conditions at \(t = 0^+\)
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Find final steady-state values
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Apply appropriate exponential formula
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Check units and signs
For AC Problems:
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Convert to phasor domain
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Find equivalent impedance
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Apply KVL/KCL or network theorems
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Calculate power if required
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Convert back to time domain if needed
For Three-Phase:
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Check if balanced or unbalanced
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Identify connection type (Y or \(\Delta\))
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Use per-phase analysis for balanced systems
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Apply appropriate voltage/current relations
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Calculate total power