1Basic Concepts & Definitions
Fundamental Quantities
Charge (\(Q\)): \( Q = \int i \, dt \) (Coulombs, C)
Current (\(i\)): \( i = \frac{dQ}{dt} \) (Amperes, A)
Voltage (\(V\)): \( V = \frac{dW}{dQ} \) (Volts, V)
Power (\(P\)): \( P = V \times i = i^2R = \frac{V^2}{R} \) (Watts, W)
Energy (\(W\)): \( W = \int P \, dt = V \times Q \) (Joules, J)
Active vs Passive Sign Convention:
• Active: Current flows from - to + (supplying power)
• Passive: Current flows from + to - (absorbing power)
• Active: Current flows from - to + (supplying power)
• Passive: Current flows from + to - (absorbing power)
Circuit Elements
| Element | V-I Relationship | Power |
|---|---|---|
| Resistor (\(R\)) | \( V = iR \) | \( P = i^2R = \frac{V^2}{R} \) |
| Inductor (\(L\)) | \( V = L\frac{di}{dt} \) | \( W = \frac{1}{2}Li^2 \) |
| Capacitor (\(C\)) | \( i = C\frac{dV}{dt} \) | \( W = \frac{1}{2}CV^2 \) |
2Fundamental Circuit Laws
Ohm's Law
\( V = I \times R \)
The voltage across a resistor is directly proportional to the current through it.
Kirchhoff's Current Law (KCL)
\( \sum i_{\text{entering}} = \sum i_{\text{leaving}} \) (at any node)
\( \sum i = 0 \) (algebraic sum at a node)
The sum of currents entering a node equals the sum leaving.
Kirchhoff's Voltage Law (KVL)
\( \sum V = 0 \) (around any closed loop)
The algebraic sum of voltages around any closed loop is zero.
3Series & Parallel Combinations
Series Connection
\( R_{\text{eq}} = R_1 + R_2 + R_3 + \dots \)
\( \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots \)
\( L_{\text{eq}} = L_1 + L_2 + L_3 + \dots \)
• Same current through all elements
• Voltages divide proportionally
• Voltages divide proportionally
Parallel Connection
\( \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \)
\( C_{\text{eq}} = C_1 + C_2 + C_3 + \dots \)
\( \frac{1}{L_{\text{eq}}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + \dots \)
• Same voltage across all elements
• Currents divide proportionally
• Currents divide proportionally
Voltage & Current Dividers
Voltage Divider: \( V_x = V_{\text{total}} \times \frac{R_x}{R_{\text{total}}} \)
Current Divider: \( I_x = I_{\text{total}} \times \frac{R_{\text{other}}}{R_{\text{total}}} \)
4Network Analysis Theorems
Nodal Analysis
Steps:
1. Select a reference node (ground)
2. Assign voltages to other nodes
3. Apply KCL at each node
4. Solve simultaneous equations
1. Select a reference node (ground)
2. Assign voltages to other nodes
3. Apply KCL at each node
4. Solve simultaneous equations
\( \sum \frac{V_{\text{node}} - V_{\text{adjacent}}}{R} = 0 \)
Mesh Analysis
Steps:
1. Identify mesh currents
2. Apply KVL to each mesh
3. Express voltages in terms of mesh currents
4. Solve simultaneous equations
1. Identify mesh currents
2. Apply KVL to each mesh
3. Express voltages in terms of mesh currents
4. Solve simultaneous equations
\( \sum V_{\text{drops}} - \sum V_{\text{rises}} = 0 \)
Superposition Theorem
In a linear circuit with multiple sources, the response is the sum of responses due to each source acting alone.
📌 Short-circuit voltage sources, open-circuit current sources when deactivating.
Thévenin's Theorem
\( V_{\text{th}} = \) Open circuit voltage at terminals
\( R_{\text{th}} = \frac{V_{\text{oc}}}{I_{\text{sc}}} = R_{\text{eq}} \) (with sources deactivated)
Any linear two-terminal network can be replaced by \( V_{\text{th}} \) in series with \( R_{\text{th}} \).
Norton's Theorem
\( I_N = \) Short circuit current at terminals
\( R_N = R_{\text{th}} = \frac{V_{\text{oc}}}{I_{\text{sc}}} \)
Any linear two-terminal network can be replaced by \( I_N \) in parallel with \( R_N \).
Conversion: \( V_{\text{th}} = I_N \times R_N \)
Maximum Power Transfer
\( R_L = R_{\text{th}} \) (for maximum power transfer)
\( P_{\text{max}} = \frac{V_{\text{th}}^2}{4R_{\text{th}}} \)
5AC Circuit Analysis
Sinusoidal Sources
\( v(t) = V_m \sin(\omega t + \phi) = V_m \cos(\omega t + \phi - 90^\circ) \)
\( \omega = 2\pi f \) (rad/s), \( T = \frac{1}{f} = \frac{2\pi}{\omega} \)
\( V_{\text{rms}} = \frac{V_m}{\sqrt{2}} \), \( I_{\text{rms}} = \frac{I_m}{\sqrt{2}} \)
Average Power: \( P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos(\phi) \)
Phasors
Phasor representation: Time-domain to frequency-domain
\( V_m \cos(\omega t + \phi) \longleftrightarrow V = V_m\angle\phi = V_m e^{j\phi} \)
Rectangular: \( V = a + jb \), Polar: \( V = |V|\angle\phi \)
\( |V| = \sqrt{a^2 + b^2} \), \( \phi = \tan^{-1}\left(\frac{b}{a}\right) \)
Impedance & Admittance
| Element | Impedance (\(Z\)) | Admittance (\(Y\)) |
|---|---|---|
| Resistor | \( Z = R \) | \( Y = \frac{1}{R} \) |
| Inductor | \( Z = j\omega L = jX_L \) | \( Y = \frac{1}{j\omega L} = -\frac{j}{\omega L} \) |
| Capacitor | \( Z = \frac{1}{j\omega C} = -jX_C \) | \( Y = j\omega C \) |
\( Z = R + jX \) (\( R \): resistance, \( X \): reactance)
\( |Z| = \sqrt{R^2 + X^2} \), \( \phi = \tan^{-1}\left(\frac{X}{R}\right) \)
\( Y = \frac{1}{Z} = G + jB \) (\( G \): conductance, \( B \): susceptance)
Series RLC Circuit
\( Z = R + j(X_L - X_C) = R + j(\omega L - \frac{1}{\omega C}) \)
Resonant frequency: \( \omega_0 = \frac{1}{\sqrt{LC}} \)
Quality factor: \( Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 R C} \)
Bandwidth: \( BW = \frac{\omega_0}{Q} = \frac{R}{L} \)
Parallel RLC Circuit
\( Y = G + j(B_C - B_L) = \frac{1}{R} + j(\omega C - \frac{1}{\omega L}) \)
Resonant frequency: \( \omega_0 = \frac{1}{\sqrt{LC}} \)
Quality factor: \( Q = \omega_0 R C = \frac{R}{\omega_0 L} \)
6AC Power Analysis
Power Components
Instantaneous Power: \( p(t) = v(t) \times i(t) \)
Average/Real Power: \( P = V_{\text{rms}} I_{\text{rms}} \cos(\phi) \) (Watts, W)
Reactive Power: \( Q = V_{\text{rms}} I_{\text{rms}} \sin(\phi) \) (VAR)
Apparent Power: \( S = V_{\text{rms}} I_{\text{rms}} = \sqrt{P^2 + Q^2} \) (VA)
Complex Power: \( S = P + jQ = V \times I^* \)
Power Factor
\( \text{pf} = \cos(\phi) = \frac{P}{S} = \frac{R}{|Z|} \)
• Leading pf: Capacitive load (current leads voltage)
• Lagging pf: Inductive load (current lags voltage)
• Unity pf: Resistive load (\( \phi = 0 \))
• Lagging pf: Inductive load (current lags voltage)
• Unity pf: Resistive load (\( \phi = 0 \))
Power Factor Correction
\( Q_C = P(\tan \phi_1 - \tan \phi_2) \)
\( C = \frac{Q_C}{\omega V^2} \)
📌 Capacitors in parallel improve lagging power factor
7Three-Phase Systems
Balanced Three-Phase
Phase sequence: abc (positive) or acb (negative)
Phase difference: 120° between phases
Phase difference: 120° between phases
Y (Star) Connection
\( V_L = \sqrt{3} V_{\text{ph}} \)
\( I_L = I_{\text{ph}} \)
\( Z_Y = \frac{V_{\text{ph}}}{I_{\text{ph}}} \)
Δ (Delta) Connection
\( V_L = V_{\text{ph}} \)
\( I_L = \sqrt{3} I_{\text{ph}} \)
\( Z_\Delta = \frac{V_{\text{ph}}}{I_{\text{ph}}} \)
Three-Phase Power
\( P = 3V_{\text{ph}} I_{\text{ph}} \cos(\phi) = \sqrt{3} V_L I_L \cos(\phi) \)
\( Q = 3V_{\text{ph}} I_{\text{ph}} \sin(\phi) = \sqrt{3} V_L I_L \sin(\phi) \)
\( S = 3V_{\text{ph}} I_{\text{ph}} = \sqrt{3} V_L I_L \)
Y-Δ Conversion
\( Z_Y = \frac{Z_\Delta}{3} \)
\( Z_\Delta = 3Z_Y \)
8Transient Response
First-Order RC Circuit
Time constant: \( \tau = RC \)
\( v(t) = V_f + (V_i - V_f)e^{-t/\tau} \)
Natural response: \( v(t) = V_0 e^{-t/RC} \)
• 63.2% of final value at \( t = \tau \)
• 99.3% of final value at \( t = 5\tau \)
• 99.3% of final value at \( t = 5\tau \)
First-Order RL Circuit
Time constant: \( \tau = \frac{L}{R} \)
\( i(t) = I_f + (I_i - I_f)e^{-t/\tau} \)
Natural response: \( i(t) = I_0 e^{-Rt/L} \)
Second-Order RLC Circuit
Characteristic equation: \( s^2 + \frac{R}{L}s + \frac{1}{LC} = 0 \)
Natural frequency: \( \omega_0 = \frac{1}{\sqrt{LC}} \)
Damping ratio: \( \zeta = \frac{R}{2\sqrt{L/C}} = \frac{R}{2\omega_0 L} \)
Response types:
• Overdamped: \( \zeta > 1 \)
• Critically damped: \( \zeta = 1 \)
• Underdamped: \( \zeta < 1 \)
• Overdamped: \( \zeta > 1 \)
• Critically damped: \( \zeta = 1 \)
• Underdamped: \( \zeta < 1 \)
Damped frequency: \( \omega_d = \omega_0 \sqrt{1 - \zeta^2} \)
9Two-Port Networks
Z-Parameters (Impedance)
\( V_1 = Z_{11}I_1 + Z_{12}I_2 \)
\( V_2 = Z_{21}I_1 + Z_{22}I_2 \)
Y-Parameters (Admittance)
\( I_1 = Y_{11}V_1 + Y_{12}V_2 \)
\( I_2 = Y_{21}V_1 + Y_{22}V_2 \)
h-Parameters (Hybrid)
\( V_1 = h_{11}I_1 + h_{12}V_2 \)
\( I_2 = h_{21}I_1 + h_{22}V_2 \)
📌 Widely used in transistor analysis
ABCD Parameters (Transmission)
\( V_1 = AV_2 + BI_2 \)
\( I_1 = CV_2 + DI_2 \)
📌 Useful for cascaded networks
10Frequency Response & Filters
Transfer Function
\( H(j\omega) = \frac{V_{\text{out}}}{V_{\text{in}}} = |H(j\omega)| \angle \phi(\omega) \)
Magnitude: \( |H(j\omega)| = \frac{|V_{\text{out}}|}{|V_{\text{in}}|} \)
Phase: \( \phi(\omega) = \angle V_{\text{out}} - \angle V_{\text{in}} \)
Bode Plots
Magnitude (dB): \( 20 \log_{10}|H(j\omega)| \)
Phase: \( \phi(\omega) \) in degrees or radians
Asymptotic approximations:
• Zero: +20 dB/decade slope
• Pole: -20 dB/decade slope
• Corner frequency: \( \omega = \frac{1}{\tau} \)
• Zero: +20 dB/decade slope
• Pole: -20 dB/decade slope
• Corner frequency: \( \omega = \frac{1}{\tau} \)
Filter Types
| Filter | Cutoff Frequency | Characteristic |
|---|---|---|
| Low-Pass | \( \omega_c = \frac{1}{RC} \) or \( \frac{R}{L} \) | Passes low frequencies |
| High-Pass | \( \omega_c = \frac{1}{RC} \) or \( \frac{R}{L} \) | Passes high frequencies |
| Band-Pass | \( \omega_0 = \frac{1}{\sqrt{LC}} \) | Passes band of frequencies |
| Band-Stop | \( \omega_0 = \frac{1}{\sqrt{LC}} \) | Rejects band of frequencies |
Cutoff frequency: \( |H(j\omega_c)| = 0.707 |H(j\omega)|_{\text{max}} = -3 \) dB
11Magnetically Coupled Circuits
Mutual Inductance
\( v_1 = L_1 \frac{di_1}{dt} \pm M \frac{di_2}{dt} \)
\( v_2 = L_2 \frac{di_2}{dt} \pm M \frac{di_1}{dt} \)
\( M = k\sqrt{L_1 L_2} \), where \( 0 \leq k \leq 1 \) (coupling coefficient)
📌 Use dot convention: + if currents both enter or leave dots
Transformer Equations
Turns ratio: \( n = \frac{N_1}{N_2} \)
Ideal Transformer: \( \frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1} \)
Reflected impedance: \( Z_{\text{reflected}} = n^2 Z_L \)
For ideal transformer: \( k = 1 \), \( L_1 \) and \( L_2 \to \infty \)
12Laplace Transform in Circuit Analysis
Basic Transforms
| Time Domain \( f(t) \) | Laplace Domain \( F(s) \) |
|---|---|
| \( \delta(t) \) | 1 |
| \( u(t) \) | \( \frac{1}{s} \) |
| \( e^{-at} \) | \( \frac{1}{s+a} \) |
| \( \sin(\omega t) \) | \( \frac{\omega}{s^2 + \omega^2} \) |
| \( \cos(\omega t) \) | \( \frac{s}{s^2 + \omega^2} \) |
| \( t^n \) | \( \frac{n!}{s^{n+1}} \) |
Circuit Elements in s-Domain
| Element | Impedance \( Z(s) \) | Initial Condition |
|---|---|---|
| Resistor | \( R \) | - |
| Inductor | \( sL \) | \( Li(0^-) \) voltage source |
| Capacitor | \( \frac{1}{sC} \) | \( \frac{v(0^-)}{s} \) voltage source |
Important Properties
Differentiation: \( \mathcal{L}\left\{\frac{df}{dt}\right\} = sF(s) - f(0^-) \)
Integration: \( \mathcal{L}\left\{\int f(t)dt\right\} = \frac{F(s)}{s} \)
Initial Value Theorem: \( f(0^+) = \lim_{s \to \infty} sF(s) \)
Final Value Theorem: \( f(\infty) = \lim_{s \to 0} sF(s) \)
13Fourier Series Analysis
Trigonometric Form
\( f(t) = a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right] \)
\( a_0 = \frac{1}{T} \int_{0}^{T} f(t)dt \) (DC component)
\( a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(n\omega_0 t)dt \)
\( b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(n\omega_0 t)dt \)
\( \omega_0 = \frac{2\pi}{T} \) (fundamental frequency)
Exponential Form
\( f(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t} \)
\( c_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-jn\omega_0 t}dt \)
Symmetry Properties
Even function: \( f(t) = f(-t) \to b_n = 0 \)
Odd function: \( f(t) = -f(-t) \to a_n = 0, a_0 = 0 \)
Half-wave symmetry: \( f(t) = -f(t + \frac{T}{2}) \to \) even harmonics = 0
Odd function: \( f(t) = -f(-t) \to a_n = 0, a_0 = 0 \)
Half-wave symmetry: \( f(t) = -f(t + \frac{T}{2}) \to \) even harmonics = 0
14Quick Reference Formulas
Star-Delta Transformation
\( R_1 = \frac{R_a R_b + R_b R_c + R_c R_a}{R_a} \)
\( R_a = \frac{R_1 R_2}{R_1 + R_2 + R_3} \)
Parallel Resistor Shortcuts
Two resistors: \( R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2} \)
\( n \) equal resistors: \( R_{\text{eq}} = \frac{R}{n} \)
Complex Numbers
\( e^{j\theta} = \cos(\theta) + j \sin(\theta) \) (Euler's formula)
\( (a + jb)(c + jd) = (ac - bd) + j(ad + bc) \)
\( \frac{a + jb}{c + jd} = \frac{(ac + bd) + j(bc - ad)}{c^2 + d^2} \)
Capacitor & Inductor Energy
\( W_C = \frac{1}{2} C V^2 = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} Q V \)
\( W_L = \frac{1}{2} L i^2 = \frac{1}{2} \frac{\Phi^2}{L} = \frac{1}{2} \Phi i \)
RMS Values
\( V_{\text{rms}} = \sqrt{\frac{1}{T} \int_{0}^{T} v^2(t)dt} \)
• Sinusoidal: \( V_{\text{rms}} = \frac{V_m}{\sqrt{2}} = 0.707 V_m \)
• Square wave: \( V_{\text{rms}} = V_m \)
• Triangle: \( V_{\text{rms}} = \frac{V_m}{\sqrt{3}} \)
• Square wave: \( V_{\text{rms}} = V_m \)
• Triangle: \( V_{\text{rms}} = \frac{V_m}{\sqrt{3}} \)
15Problem-Solving Strategy
General Approach
- Identify: Circuit type, known/unknown quantities
- Choose method: KVL/KCL, nodal, mesh, or theorem
- Simplify: Combine elements, use transformations
- Apply: Relevant equations and laws
- Solve: Algebraically or using matrices
- Verify: Check using KVL/KCL or power conservation
Common Pitfalls to Avoid
⚠️ Sign conventions for voltage and current
⚠️ Forgetting initial conditions in transients
⚠️ Mixing time and frequency domains
⚠️ Incorrect phasor conversions
⚠️ Wrong dot convention in coupled circuits
⚠️ Confusing line and phase values in 3-phase
⚠️ Forgetting initial conditions in transients
⚠️ Mixing time and frequency domains
⚠️ Incorrect phasor conversions
⚠️ Wrong dot convention in coupled circuits
⚠️ Confusing line and phase values in 3-phase
When to Use Which Method
Nodal Analysis: Many nodes, voltage sources
Mesh Analysis: Many meshes, current sources
Thévenin/Norton: Single load, variable load
Superposition: Multiple independent sources
Source transformation: Simplify series-parallel combinations
Mesh Analysis: Many meshes, current sources
Thévenin/Norton: Single load, variable load
Superposition: Multiple independent sources
Source transformation: Simplify series-parallel combinations
📚 Study Tips
Practice numerical problems regularly • Master sign conventions early •
Draw neat circuit diagrams • Verify answers using alternative methods •
Understand physical meaning behind formulas
Good luck with your revision! ⚡