Study Notes/Electrical Engineering/Circuit Analysis

Circuit Analysis

DC/AC circuits, network theorems, and transient analysis

1. Basic Circuit Laws

Ohm's Law

The fundamental relationship between voltage, current, and resistance in a circuit.

V = IR

V = Voltage (Volts), I = Current (Amperes), R = Resistance (Ohms)

Power and Energy

Power (P)

P = VI = I²R = V²/R

Unit: Watt (W)

Energy (W)

W = Pt

Unit: Joule (J) or kWh

Kirchhoff's Laws

KCL (Current Law)

ΣI_in = ΣI_out

Sum of currents entering a node equals sum of currents leaving

Based on conservation of charge

KVL (Voltage Law)

ΣV = 0

Sum of voltage drops around a closed loop equals zero

Based on conservation of energy

Series and Parallel Circuits

Parameter	Series	Parallel
Resistance	R_T = R₁ + R₂ + R₃...	1/R_T = 1/R₁ + 1/R₂ + 1/R₃...
Current	I_T = I₁ = I₂ = I₃...	I_T = I₁ + I₂ + I₃...
Voltage	V_T = V₁ + V₂ + V₃...	V_T = V₁ = V₂ = V₃...
Inductance	L_T = L₁ + L₂ + L₃...	1/L_T = 1/L₁ + 1/L₂ + 1/L₃...
Capacitance	1/C_T = 1/C₁ + 1/C₂ + 1/C₃...	C_T = C₁ + C₂ + C₃...

2. Network Theorems

Circuit Analysis Techniques

Methods to simplify complex circuit analysis and solve for unknown values.

Superposition Theorem

In a linear circuit with multiple sources, the response equals the algebraic sum of responses due to each source acting alone.

• Voltage sources → replaced with short circuit
• Current sources → replaced with open circuit
• Only works for linear circuits
• Cannot be used for power calculations directly

Thevenin's Theorem

Any linear two-terminal circuit can be replaced by an equivalent circuit with a voltage source (V_TH) in series with a resistance (R_TH).

V_TH (Thevenin Voltage)

Open-circuit voltage across terminals

R_TH (Thevenin Resistance)

Equivalent resistance seen from terminals (sources deactivated)

Norton's Theorem

Any linear two-terminal circuit can be replaced by an equivalent circuit with a current source (I_N) in parallel with a resistance (R_N).

I_N (Norton Current)

Short-circuit current through terminals

R_N (Norton Resistance)

R_N = R_TH

V_TH = I_N × R_N

Maximum Power Transfer

R_L = R_TH

Maximum power is transferred when load resistance equals source resistance

P_max = V_TH² / (4R_TH)

Efficiency at max power transfer = 50%

Other Analysis Methods

Mesh Analysis

Based on KVL, assign mesh currents to loops

Best for circuits with multiple voltage sources

Nodal Analysis

Based on KCL, assign voltages to nodes

Best for circuits with multiple current sources

Delta-Wye (Δ-Y) Transformation

Δ to Y Conversion

R₁ = R_abR_ca / (R_ab + R_bc + R_ca)

R₂ = R_abR_bc / (R_ab + R_bc + R_ca)

R₃ = R_bcR_ca / (R_ab + R_bc + R_ca)

Y to Δ Conversion

R_ab = (R₁R₂ + R₂R₃ + R₃R₁) / R₃

R_bc = (R₁R₂ + R₂R₃ + R₃R₁) / R₁

R_ca = (R₁R₂ + R₂R₃ + R₃R₁) / R₂

3. AC Circuit Fundamentals

Sinusoidal Waveforms

AC quantities vary sinusoidally with time, characterized by amplitude, frequency, and phase.

v(t) = V_m sin(ωt + φ)

V_m = peak/maximum value, ω = angular frequency (rad/s)

φ = phase angle (radians or degrees)

AC Values

RMS Value

V_rms = V_m/√2 = 0.707V_m

Root Mean Square - effective value

Used in power calculations

Average Value

V_avg = 2V_m/π = 0.637V_m

Average of half-cycle

Full-cycle average = 0

Form Factor & Crest Factor

Form Factor = V_rms/V_avg = 1.11

Crest Factor = V_m/V_rms = √2 = 1.414

Frequency and Period

Period (T)

T = 1/f

Time for one cycle (seconds)

Frequency (f)

f = 1/T

Cycles per second (Hz)

Angular Frequency (ω)

ω = 2πf

Radians per second

Phasor Representation

Phasors represent sinusoidal quantities as complex numbers, simplifying AC analysis.

Rectangular Form

V = a + jb

a = real part, b = imaginary part

Polar Form

V = |V|∠θ

|V| = magnitude, θ = angle

|V| = √(a² + b²)

θ = tan⁻¹(b/a)

4. Impedance and Admittance

Component Impedances

Resistor (R)

Z_R = R

Voltage and current in phase

Phase angle: 0°

Inductor (L)

Z_L = jωL = jX_L

X_L = 2πfL

V leads I by 90° (ELI)

Capacitor (C)

Z_C = 1/jωC = -jX_C

X_C = 1/(2πfC)

I leads V by 90° (ICE)

Memory Aid: ELI the ICE man

ELI: E (voltage) Leads I (current) in an L (inductor)
ICE: I (current) leads in a C (capacitor) before E (voltage)

Total Impedance

Z = R + jX = |Z|∠θ

|Z| = √(R² + X²)

θ = tan⁻¹(X/R)

Where X = X_L - X_C (net reactance)

Admittance

Y = 1/Z = G + jB

Admittance (Y)

Unit: Siemens (S)

Conductance (G)

G = R/|Z|²

Susceptance (B)

B = -X/|Z|²

Ohm's Law for AC

V = IZ

I = V/Z = VY

All quantities are phasors (complex numbers)

5. AC Power Analysis

Power Triangle

In AC circuits, power has three components forming a right triangle.

Real Power (P)

P = VI cos θ = I²R

Unit: Watt (W)

Actual power consumed

Reactive Power (Q)

Q = VI sin θ = I²X

Unit: VAR

Power stored/released by L & C

Apparent Power (S)

S = VI = I²|Z|

Unit: VA

Product of V and I

Power Relationships

S² = P² + Q²

S = P + jQ

S = VI* (complex conjugate of current)

Power Factor

pf = cos θ = P/S = R/|Z|

Lagging Power Factor

Inductive load (current lags voltage)

θ positive, Q positive

Leading Power Factor

Capacitive load (current leads voltage)

θ negative, Q negative

Power Factor Correction

Adding capacitors in parallel to improve lagging power factor:

Q_c = P(tan θ₁ - tan θ₂)

C = Q_c/(2πfV²)

θ₁ = original angle, θ₂ = desired angle

Complex Power

S = VI* = P + jQ

For voltage V = V∠α and current I = I∠β:
S = VI∠(α-β) = VI cos(α-β) + jVI sin(α-β)

6. Resonance

Resonant Frequency

Condition when X_L = X_C, resulting in purely resistive impedance.

f_r = 1/(2π√LC)

ω_r = 1/√LC

Series Resonance (RLC)

At Resonance:

• X_L = X_C
• Z = R (minimum)
• I = V/R (maximum)
• pf = 1 (unity)

Quality Factor (Q):

Q = ω_rL/R = 1/(ω_rCR)

Q = V_L/V_s = V_C/V_s

Higher Q = sharper resonance

Parallel Resonance (RLC)

At Resonance:

• B_L = B_C
• Z = R (maximum)
• I_s = V/R (minimum)
• pf = 1 (unity)

Quality Factor (Q):

Q = R/(ω_rL) = ω_rCR

Also called anti-resonance

Tank circuit in oscillators

Bandwidth

BW = f_r/Q = f₂ - f₁

Where f₁ and f₂ are half-power frequencies (-3dB points)

f₁ × f₂ = f_r² (geometric mean)

7. Transient Analysis

Time Constants

Transient response describes circuit behavior during switching transitions.

RC Circuit

τ = RC (time constant)

Charging:

v_C(t) = V(1 - e^-t/τ)

i(t) = (V/R)e^-t/τ

Discharging:

v_C(t) = V₀e^-t/τ

i(t) = -(V₀/R)e^-t/τ

RL Circuit

τ = L/R (time constant)

Current Build-up:

i(t) = (V/R)(1 - e^-t/τ)

v_L(t) = Ve^-t/τ

Current Decay:

i(t) = I₀e^-t/τ

v_L(t) = -I₀Re^-t/τ

Time Constant Values

Time (τ)	% of Final Value	% Remaining
1τ	63.2%	36.8%
2τ	86.5%	13.5%
3τ	95.0%	5.0%
4τ	98.2%	1.8%
5τ	99.3%	0.7%

Circuit reaches ~99% of steady state after 5τ

RLC Circuit Transients

α = R/(2L) (damping factor)

ω₀ = 1/√LC (natural frequency)

Overdamped: α > ω₀ - No oscillation

Critically damped: α = ω₀ - Fastest non-oscillatory response

Underdamped: α < ω₀ - Oscillation with decay

8. Coupled Circuits and Filters

Mutual Inductance

M = k√(L₁L₂)

k = coefficient of coupling (0 ≤ k ≤ 1)

Series Aiding:

L_T = L₁ + L₂ + 2M

Series Opposing:

L_T = L₁ + L₂ - 2M

Filter Circuits

Low-Pass Filter

Passes low frequencies, attenuates high

f_c = 1/(2πRC)

RC or RL configuration

High-Pass Filter

Passes high frequencies, attenuates low

f_c = 1/(2πRC)

CR or LR configuration

Band-Pass Filter

Passes band of frequencies

BW = f_H - f_L

Center: f₀ = √(f_Hf_L)

Band-Stop (Notch) Filter

Rejects band of frequencies

Passes frequencies outside band

Used to eliminate specific noise

Cutoff Frequency (-3dB point)

At cutoff frequency:

• Output power = 50% of input (or -3dB)
• Output voltage = 70.7% of maximum (1/√2)
• Phase shift = 45°

Key Takeaways for EE Board Exam

Must-Know Formulas

✓ V = IR, P = VI = I²R = V²/R
✓ Z = R + jX, |Z| = √(R² + X²)
✓ X_L = 2πfL, X_C = 1/(2πfC)
✓ P = VI cos θ, Q = VI sin θ, S = VI
✓ f_r = 1/(2π√LC)
✓ τ = RC, τ = L/R
✓ Maximum power: R_L = R_TH

Critical Concepts

✓ KCL: ΣI = 0 at node
✓ KVL: ΣV = 0 around loop
✓ ELI the ICE man (phase relationships)
✓ pf = cos θ = P/S = R/|Z|
✓ Series vs Parallel combinations
✓ Thevenin/Norton equivalents
✓ 5τ ≈ 99% steady state

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1. Basic Circuit Laws

Ohm's Law

Power and Energy

Power (P)

Energy (W)

Kirchhoff's Laws

KCL (Current Law)

KVL (Voltage Law)

Series and Parallel Circuits

2. Network Theorems

Circuit Analysis Techniques

Superposition Theorem

Thevenin's Theorem

VTH (Thevenin Voltage)

RTH (Thevenin Resistance)

Norton's Theorem

IN (Norton Current)

RN (Norton Resistance)

Maximum Power Transfer

Other Analysis Methods

Mesh Analysis

Nodal Analysis

Delta-Wye (Δ-Y) Transformation

Δ to Y Conversion

Y to Δ Conversion

3. AC Circuit Fundamentals

Sinusoidal Waveforms

AC Values

RMS Value

Average Value

Form Factor & Crest Factor

Frequency and Period

Period (T)

Frequency (f)

Angular Frequency (ω)

Phasor Representation

Rectangular Form

Polar Form

4. Impedance and Admittance

Component Impedances

Resistor (R)

Inductor (L)

Capacitor (C)

Memory Aid: ELI the ICE man

Total Impedance

Admittance

Ohm's Law for AC

5. AC Power Analysis

Power Triangle

Real Power (P)

Reactive Power (Q)

Apparent Power (S)

Power Relationships

Power Factor

Lagging Power Factor

Leading Power Factor

Power Factor Correction

Complex Power

6. Resonance

Resonant Frequency

Series Resonance (RLC)

At Resonance:

Quality Factor (Q):

Parallel Resonance (RLC)

At Resonance:

Quality Factor (Q):

Bandwidth

7. Transient Analysis

Time Constants

RC Circuit

Charging:

Discharging:

RL Circuit

Current Build-up:

Current Decay:

Time Constant Values

RLC Circuit Transients

8. Coupled Circuits and Filters

Mutual Inductance

Series Aiding:

V_TH (Thevenin Voltage)

R_TH (Thevenin Resistance)

I_N (Norton Current)

R_N (Norton Resistance)