PE Exam Exam > PE Exam Notes > Electrical & Computer Engineering for PE > Cheatsheet: AC Circuits
Cheatsheet: AC Circuits
1. AC Fundamentals
1.1 Sinusoidal Waveform
| Parameter | Formula/Definition |
|---|---|
| Instantaneous Voltage | v(t) = Vm sin(ωt + θ) where Vm = peak amplitude, ω = angular frequency, θ = phase angle |
| Angular Frequency | ω = 2πf (rad/s), f = frequency (Hz) |
| Period | T = 1/f = 2π/ω (seconds) |
| RMS Value | Vrms = Vm/√2 = 0.707Vm |
| Average Value | Vavg = 2Vm/π = 0.637Vm (for half-wave rectified) |
| Form Factor | FF = Vrms/Vavg = 1.11 (for sinusoid) |
| Crest Factor | CF = Vm/Vrms = √2 = 1.414 (for sinusoid) |
1.2 Phase Relationships
- Leading: waveform reaches peak earlier (positive phase angle)
- Lagging: waveform reaches peak later (negative phase angle)
- In phase: θ = 0°, waveforms aligned
- Out of phase: θ = 180°, waveforms opposite
- Quadrature: θ = 90°, waveforms perpendicular
2. Phasor Analysis
2.1 Phasor Representation
| Form | Representation |
|---|---|
| Rectangular | V = a + jb where a = real part, b = imaginary part |
| Polar | V = V∠θ where V = magnitude, θ = angle |
| Exponential | V = Vejθ |
2.2 Phasor Conversions
| Conversion | Formula |
|---|---|
| Rectangular to Polar | V = √(a² + b²), θ = tan⁻¹(b/a) |
| Polar to Rectangular | a = V cos(θ), b = V sin(θ) |
| Multiplication | (V₁∠θ₁)(V₂∠θ₂) = V₁V₂∠(θ₁ + θ₂) |
| Division | (V₁∠θ₁)/(V₂∠θ₂) = (V₁/V₂)∠(θ₁ - θ₂) |
2.3 Complex Operators
- j = √(-1), j² = -1, j³ = -j, j⁴ = 1
- 1/j = -j
- Euler's Identity: ejθ = cos(θ) + j sin(θ)
3. Impedance and Admittance
3.1 Impedance (Z)
| Element | Impedance |
|---|---|
| Resistor | ZR = R (purely real) |
| Inductor | ZL = jωL = jXL where XL = ωL = 2πfL |
| Capacitor | ZC = 1/(jωC) = -j/(ωC) = -jXC where XC = 1/(ωC) = 1/(2πfC) |
| General Form | Z = R + jX where R = resistance, X = reactance |
| Magnitude | |Z| = √(R² + X²) |
| Phase Angle | θ = tan⁻¹(X/R) |
3.2 Admittance (Y)
| Parameter | Formula |
|---|---|
| Admittance | Y = 1/Z = G + jB where G = conductance, B = susceptance |
| Conductance | G = R/(R² + X²) |
| Susceptance | B = -X/(R² + X²) |
| Magnitude | |Y| = 1/|Z| |
3.3 Series Impedance
- Ztotal = Z₁ + Z₂ + Z₃ + ...
- Series RL: Z = R + jωL
- Series RC: Z = R - j/(ωC)
- Series RLC: Z = R + j(ωL - 1/(ωC))
3.4 Parallel Impedance
- 1/Ztotal = 1/Z₁ + 1/Z₂ + 1/Z₃ + ...
- Two elements: Ztotal = (Z₁Z₂)/(Z₁ + Z₂)
- Parallel admittances: Ytotal = Y₁ + Y₂ + Y₃ + ...
4. AC Power
4.1 Power Definitions
| Type | Formula |
|---|---|
| Instantaneous Power | p(t) = v(t) × i(t) |
| Real Power (P) | P = VrmsIrms cos(θ) = I²R (Watts) |
| Reactive Power (Q) | Q = VrmsIrms sin(θ) = I²X (VAR) |
| Apparent Power (S) | S = VrmsIrms = |S| (VA) |
| Complex Power | S = P + jQ = VI* where I* = conjugate of current phasor |
| Power Triangle | |S|² = P² + Q² |
4.2 Power Factor
| Parameter | Definition |
|---|---|
| Power Factor (PF) | PF = cos(θ) = P/S = R/|Z| |
| Leading PF | Current leads voltage (capacitive load, Q negative) |
| Lagging PF | Current lags voltage (inductive load, Q positive) |
| Unity PF | θ = 0°, purely resistive (PF = 1) |
4.3 Power Factor Correction
- Add capacitor in parallel to correct lagging PF
- Required capacitance: QC = P(tan(θ₁) - tan(θ₂)) where θ₁ = original angle, θ₂ = desired angle
- C = QC/(ωV²) = QC/(2πfV²)
4.4 Maximum Power Transfer
- Maximum power to load when ZL = Zs* (conjugate match)
- For purely resistive source: RL = Rs
- Pmax = |Vth|²/(4Rs)
5. Resonance
5.1 Series Resonance
| Parameter | Formula/Value |
|---|---|
| Resonant Frequency | f₀ = 1/(2π√(LC)), ω₀ = 1/√(LC) |
| Impedance at Resonance | Z = R (minimum, purely resistive) |
| Current at Resonance | Imax = V/R (maximum) |
| Quality Factor | Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C) |
| Bandwidth | BW = f₀/Q = R/(2πL) |
| Half-Power Frequencies | f₁ = f₀ - BW/2, f₂ = f₀ + BW/2 |
| Voltage Magnification | VL = VC = QV at resonance |
5.2 Parallel Resonance
| Parameter | Formula/Value |
|---|---|
| Resonant Frequency | f₀ = 1/(2π√(LC)) (ideal case) |
| Impedance at Resonance | Zmax = L/(CR) (maximum, purely resistive) |
| Current at Resonance | Imin = V/Zmax (minimum from source) |
| Quality Factor | Q = R√(C/L) = R/(ω₀L) |
| Bandwidth | BW = f₀/Q |
| Current Magnification | IL = IC = QI at resonance |
6. AC Circuit Analysis Methods
6.1 Nodal Analysis
- Apply KCL at each node using phasor voltages and currents
- Express branch currents as I = (V₁ - V₂)/Z
- Solve system of complex equations for node voltages
- Reference node (ground) voltage = 0
6.2 Mesh Analysis
- Apply KVL around each mesh using phasor voltages
- Express voltages as V = IZ
- Solve system of complex equations for mesh currents
- Impedances add when in same mesh, subtract when shared
6.3 Superposition
- Analyze each source separately (different frequencies require separate analysis)
- Voltage sources: short circuit when deactivated
- Current sources: open circuit when deactivated
- Sum phasor responses from sources at same frequency
- Cannot superpose power directly
6.4 Thevenin and Norton Equivalents
| Parameter | Definition |
|---|---|
| Thevenin Voltage | Vth = open-circuit voltage at terminals |
| Thevenin Impedance | Zth = Voc/Isc or impedance with sources deactivated |
| Norton Current | IN = short-circuit current at terminals |
| Norton Impedance | ZN = Zth |
| Conversion | Vth = INZN, IN = Vth/Zth |
6.5 Source Transformation
- Voltage source Vs in series with Z ↔ Current source Is = Vs/Z in parallel with Z
- Valid only for phasor (frequency domain) circuits at single frequency
7. Three-Phase Circuits
7.1 Three-Phase Configurations
| Configuration | Relationships |
|---|---|
| Wye (Y) Connection | VL = √3 VP, IL = IP |
| Delta (Δ) Connection | VL = VP, IL = √3 IP |
| Phase Sequence | ABC (positive) or ACB (negative), 120° separation |
7.2 Balanced Three-Phase Power
| Power Type | Formula |
|---|---|
| Total Real Power | P3φ = 3VPIP cos(θ) = √3 VLIL cos(θ) |
| Total Reactive Power | Q3φ = 3VPIP sin(θ) = √3 VLIL sin(θ) |
| Total Apparent Power | S3φ = 3VPIP = √3 VLIL |
| Per-Phase Power | Pφ = VPIP cos(θ) |
7.3 Y-Δ Conversions
| Conversion | Formula |
|---|---|
| Y to Δ | ZΔ = 3ZY |
| Δ to Y | ZY = ZΔ/3 |
7.4 Sequence Components
- Positive sequence: phases ABC, normal rotation, 120° apart
- Negative sequence: phases ACB, reverse rotation, 120° apart
- Zero sequence: all phases in phase, equal magnitude and angle
- Used for unbalanced fault analysis
8. Transformers in AC Circuits
8.1 Ideal Transformer
| Parameter | Formula |
|---|---|
| Turns Ratio | a = N₁/N₂ where N₁ = primary turns, N₂ = secondary turns |
| Voltage Ratio | V₁/V₂ = N₁/N₂ = a |
| Current Ratio | I₁/I₂ = N₂/N₁ = 1/a |
| Impedance Transformation | Z₁ = a²Z₂ (referred to primary) |
| Power Conservation | P₁ = P₂, S₁ = S₂ |
8.2 Practical Transformer Model
- Core losses: represented by resistance Rc in parallel
- Magnetizing reactance: Xm in parallel with Rc
- Winding resistance: R₁ (primary), R₂ (secondary)
- Leakage reactance: X₁ (primary), X₂ (secondary)
- Efficiency: η = Pout/Pin = P₂/(P₂ + Plosses)
8.3 Per-Unit System
| Quantity | Formula |
|---|---|
| Per-Unit Value | pu = Actual Value / Base Value |
| Base Power | Sbase (VA) |
| Base Voltage | Vbase (V) |
| Base Current | Ibase = Sbase/Vbase |
| Base Impedance | Zbase = Vbase²/Sbase |
9. Frequency Response and Filters
9.1 Transfer Function
| Parameter | Definition |
|---|---|
| Transfer Function | H(jω) = Vout/Vin or Iout/Iin |
| Magnitude | |H(jω)| = |Vout|/|Vin| |
| Phase | ∠H(jω) = ∠Vout - ∠Vin |
| Decibel Scale | |H|dB = 20 log₁₀(|H|) |
| Cutoff Frequency | fc where |H| = |H|max/√2 or -3 dB point |
9.2 First-Order Filters
| Filter Type | Transfer Function |
|---|---|
| RC Low-Pass | H(jω) = 1/(1 + jωRC), fc = 1/(2πRC) |
| RC High-Pass | H(jω) = jωRC/(1 + jωRC), fc = 1/(2πRC) |
| RL Low-Pass | H(jω) = R/(R + jωL), fc = R/(2πL) |
| RL High-Pass | H(jω) = jωL/(R + jωL), fc = R/(2πL) |
9.3 Second-Order RLC Filters
- Series RLC bandpass: H(jω) = (jωRC)/(1 - ω²LC + jωRC)
- Parallel RLC bandpass: H(jω) = (jω/RC)/(1/LC - ω² + jω/RC)
- Center frequency: f₀ = 1/(2π√(LC))
- Quality factor Q determines selectivity and bandwidth
9.4 Bode Plots
- Magnitude plot: 20 log₁₀(|H|) vs. log(f)
- Phase plot: ∠H vs. log(f)
- First-order slope: ±20 dB/decade
- Second-order slope: ±40 dB/decade
- Decade: 10× frequency change
- Octave: 2× frequency change
10. Mutual Inductance and Coupled Circuits
10.1 Mutual Inductance
| Parameter | Formula |
|---|---|
| Mutual Inductance | M = k√(L₁L₂) where k = coupling coefficient (0 ≤ k ≤ 1) |
| Induced Voltage | V₁₂ = jωMI₂, V₂₁ = jωMI₁ |
| Dot Convention | Current entering dot → positive mutual voltage at other dot |
| Series Aiding | Ltotal = L₁ + L₂ + 2M |
| Series Opposing | Ltotal = L₁ + L₂ - 2M |
10.2 Coupled Circuit Analysis
- KVL with mutual terms: V₁ = jωL₁I₁ ± jωMI₂
- Sign determined by dot convention and current direction
- For perfect coupling: k = 1, M = √(L₁L₂)
- Energy stored: W = ½L₁I₁² + ½L₂I₂² ± MI₁I₂
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Apr 20, 2026 Last updated
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