AC Circuit Analysis: Phasors, Impedance & RLC Circuits (2026 Guide)

Quick Answer: In AC analysis, replace resistance R with complex impedance Z = R + j(X_L − X_C), where X_L = 2πfL (inductive) and X_C = 1/(2πfC) (capacitive). Apply Ohm's Law and Kirchhoff's Laws using phasor (complex) arithmetic. Real power P = V_rms × I_rms × cosθ; reactive power Q = V_rms × I_rms × sinθ; apparent power S = V_rms × I_rms. Resonance occurs at f₀ = 1/(2π√LC).

Alternating Current (AC) circuit analysis extends DC circuit principles to time-varying sinusoidal voltages and currents. For U.S. electrical work, the same phasor framework shows up in 120/240V split-phase troubleshooting, 208Y/120V and 480Y/277V distribution review, motor circuits, filters, and power-quality checks.

Introduction to AC Circuits

Sinusoidal Waveforms

General Sinusoidal Function: v(t) = V_m sin(ωt + φ)

Where:

• V_m = Peak amplitude (maximum value)
• ω = Angular frequency (rad/s)
• t = Time (seconds)
• φ = Phase angle (radians or degrees)

Key Parameters:

• Period (T): Time for one complete cycle (seconds)
• Frequency (f): Cycles per second (Hz), f = 1/T
• Angular Frequency (ω): ω = 2πf (rad/s)
• RMS Value: V_rms = V_m/√2 (effective value)

AC vs DC Comparison

DC Characteristics:

• Constant magnitude and polarity
• Simple resistive analysis
• Power = VI (always positive)
• No frequency considerations

AC Characteristics:

• Time-varying magnitude and polarity
• Requires complex analysis methods
• Power calculations more complex
• Frequency-dependent behavior

Where AC Analysis Shows Up in U.S. Work

Common U.S. applications:

• 120/240V split-phase branch circuits, MWBC troubleshooting, and service checks
• 208Y/120V and 480Y/277V three-phase distribution review
• Transformer, motor, VFD, and power-factor workflows
• Control-power, filter, and power-quality screening at 60 Hz

Phasor Representation

Concept of Phasors

Phasor Definition: A complex number representing the magnitude and phase of a sinusoidal quantity.

Mathematical Representation: V = V_m ∠φ = V_m e^(jφ) = V_m cos(φ) + jV_m sin(φ)

Phasor Diagram:

• Magnitude represented by vector length
• Phase angle measured from positive real axis
• Rotation at angular frequency ω (implied)

Phasor Arithmetic

Addition and Subtraction: Convert to rectangular form, perform operation, convert back if needed.

Example 1: Phasor Addition V₁ = 10∠30° V V₂ = 8∠-45° V

Rectangular Form: V₁ = 10 cos(30°) + j10 sin(30°) = 8.66 + j5.00 V₂ = 8 cos(-45°) + j8 sin(-45°) = 5.66 - j5.66

Sum: V_total = V₁ + V₂ = (8.66 + 5.66) + j(5.00 - 5.66) = 14.32 - j0.66

Polar Form: |V_total| = √(14.32² + 0.66²) = 14.34 V ∠V_total = arctan(-0.66/14.32) = -2.64° V_total = 14.34∠-2.64° V

Time Domain to Phasor Conversion

Conversion Rules:

• v(t) = V_m cos(ωt + φ) ↔ V = V_m∠φ
• i(t) = I_m cos(ωt + φ) ↔ I = I_m∠φ
• Use RMS values for power calculations

Example 2: Time to Phasor v(t) = 120√2 cos(377t + 30°) V

Phasor representation: V = 120∠30° V (RMS)

Impedance and Reactance

Impedance Concept

Impedance Definition: The total opposition to AC current flow, combining resistance and reactance.

Mathematical Expression: Z = R + jX = |Z|∠θ

Where:

• R = Resistance (real part)
• X = Reactance (imaginary part, positive for inductive, negative for capacitive)
• |Z| = Magnitude of impedance = √(R² + X²)
• θ = Phase angle = arctan(X/R)

Resistive Impedance

Pure Resistance:

• Z_R = R + j0 = R∠0°
• Voltage and current in phase
• No energy storage
• Power dissipated as heat

Ohm's Law for AC: V = I × Z_R = I × R

Inductive Reactance

Inductor Impedance: Z_L = jωL = ωL∠90°

Inductive Reactance: X_L = ωL = 2πfL

Characteristics:

• Current lags voltage by 90° (mnemonic: ELI — voltage E leads current I in an inductor L)
• Reactance increases linearly with frequency
• Energy stored in magnetic field
• No power dissipation (ideal inductor)

Example 3: Inductive Reactance L = 50 mH, f = 60 Hz X_L = 2π × 60 × 0.05 = 18.85 Ω Z_L = 18.85∠90° Ω

Capacitive Reactance

Capacitor Impedance: Z_C = −j/(ωC) = (1/ωC)∠−90°

Capacitive Reactance: X_C = 1/(ωC) = 1/(2πfC)

Characteristics:

• Current leads voltage by 90° (mnemonic: ICE — current I leads voltage E in a capacitor C)
• Reactance decreases with increasing frequency (capacitor becomes a short at very high f)
• Energy stored in electric field
• No power dissipation (ideal capacitor)

Example 4: Capacitive Reactance C = 100 μF, f = 60 Hz X_C = 1/(2π × 60 × 100×10⁻⁶) = 26.53 Ω Z_C = 26.53∠−90° Ω

Impedance Reference Table at 60 Hz (US Power Frequency)

X_L = 2π × 60 × L; X_C = 1/(2π × 60 × C).

RLC Circuit Analysis

Series RLC Circuit

Total Impedance: Z = R + jωL - j/(ωC) = R + j(X_L - X_C)

Magnitude and Phase: |Z| = √[R² + (X_L - X_C)²] θ = arctan[(X_L - X_C)/R]

Example 5: Series RLC R = 10 Ω, L = 20 mH, C = 50 μF, f = 100 Hz

Calculate Reactances: X_L = 2π × 100 × 0.02 = 12.57 Ω X_C = 1/(2π × 100 × 50×10⁻⁶) = 31.83 Ω

Total Impedance: Z = 10 + j(12.57 - 31.83) = 10 - j19.26 Ω |Z| = √(10² + 19.26²) = 21.73 Ω θ = arctan(-19.26/10) = -62.6°

Circuit Behavior: Since X_C > X_L, circuit is capacitive (current leads voltage).

Parallel RLC Circuit

Admittance Approach: Y = G + jB = 1/R + j(ωC - 1/ωL)

Where:

• G = Conductance = 1/R
• B = Susceptance = (ωC - 1/ωL)

Total Impedance: Z = 1/Y

Example 6: Parallel RLC Same components as Example 5 in parallel:

Calculate Susceptances: B_L = -1/X_L = -1/12.57 = -0.0796 S B_C = 1/X_C = 1/31.83 = 0.0314 S

Total Admittance: Y = 0.1 + j(0.0314 - 0.0796) = 0.1 - j0.0482 S |Y| = √(0.1² + 0.0482²) = 0.111 S

Total Impedance: Z = 1/Y = 9.01∠25.8° Ω

AC Power Analysis

Types of AC Power

Instantaneous Power: p(t) = v(t) × i(t)

Average (Real) Power: P = V_rms × I_rms × cos(θ) (Watts)

Reactive Power: Q = V_rms × I_rms × sin(θ) (VARs)

Apparent Power: S = V_rms × I_rms (VA)

Power Triangle: S² = P² + Q²

AC Power Types — Reference Table

Leading vs Lagging Power Factor:

• Lagging PF (most common): Inductive loads (motors, transformers). Current lags voltage. Q > 0.
• Leading PF: Capacitive loads (capacitor banks, lightly-loaded cables). Current leads voltage. Q < 0.
• Unity PF: Purely resistive. θ = 0. All apparent power is real power.

Power Factor

Definition: Power Factor (PF) = cos(θ) = P/S

Types:

• Leading PF: Capacitive load (current leads voltage)
• Lagging PF: Inductive load (current lags voltage)
• Unity PF: Resistive load (current in phase with voltage)

Example 7: Power Calculations V = 120∠0° V, I = 5∠-30° A

Power Calculations: P = 120 × 5 × cos(30°) = 519.6 W Q = 120 × 5 × sin(30°) = 300 VAR (inductive, positive Q since θ = −30°, current lags) S = 120 × 5 = 600 VA PF = cos(30°) = 0.866 lagging

Complex Power

Complex Power Definition: S = P + jQ = V_rms × I*_rms

Where I* is the complex conjugate of current.

Advantages:

• Combines all power information
• Simplifies power calculations
• Enables power balance analysis
• Useful for system design

Frequency Response

Frequency-Dependent Behavior

Impedance vs Frequency:

• Resistive impedance: Constant with frequency
• Inductive impedance: Increases with frequency
• Capacitive impedance: Decreases with frequency

Transfer Functions: H(jω) = V_out/V_in = Output phasor/Input phasor

Resonance in RLC Circuits

Series Resonance: Occurs when X_L = X_C

Resonant Frequency: f₀ = 1/(2π√LC)

At Resonance:

• Impedance is minimum (Z = R)
• Current is maximum
• Voltage across L and C can exceed source voltage (by factor Q)
• Power factor = 1 (unity)

Example 8: Series Resonance L = 10 mH, C = 100 μF

f₀ = 1/(2π√(0.01 × 100×10⁻⁶)) = 1/(2π × 0.001) = 159.2 Hz

Parallel Resonance:

• Impedance is maximum (tank circuit)
• Current from source is minimum
• Used in tuned circuits and bandpass/bandstop filters

Series vs Parallel RLC Resonance Comparison

Resonant frequency f₀ = 1/(2π√LC) is the same formula for both series and parallel RLC circuits (assuming ideal L and C).

Quality Factor (Q)

Definition: Q = ω₀L/R = 1/(ω₀RC) = (1/R)√(L/C)

Significance:

• Measures sharpness of resonance peak
• Higher Q = narrower bandwidth = more selective
• Important in filter design and oscillator stability
• Q < 0.5 produces overdamped (non-oscillatory) response

Bandwidth: BW = f₀/Q

Filter Circuits

Low-Pass Filters

RC Low-Pass: H(jω) = 1/(1 + jωRC)

Cutoff Frequency: f_c = 1/(2πRC)

Characteristics:

• Passes low frequencies
• Attenuates high frequencies
• -3dB at cutoff frequency
• -20dB/decade rolloff

High-Pass Filters

RC High-Pass: H(jω) = jωRC/(1 + jωRC)

Cutoff Frequency: f_c = 1/(2πRC)

Characteristics:

• Passes high frequencies
• Attenuates low frequencies
• +20dB/decade rolloff below cutoff

Band-Pass and Band-Stop Filters

Band-Pass:

• Passes frequencies within a band
• Rejects frequencies outside the band
• Can be implemented with RLC circuits
• Used in communication systems

Band-Stop (Notch):

• Rejects frequencies within a band
• Passes frequencies outside the band
• Used to eliminate interference
• Power line filter applications

Three-Phase Systems

Three-Phase Basics

Advantages:

• More efficient power transmission
• Constant instantaneous power
• Smaller conductor requirements
• Better motor performance

Phase Relationships:

• 120° phase separation
• Balanced system analysis
• Line and phase quantities
• Wye and delta connections

Balanced Three-Phase Analysis

Wye Connection:

• V_line = √3 × V_phase
• I_line = I_phase
• Neutral current = 0 (balanced)

Delta Connection:

• V_line = V_phase
• I_line = √3 × I_phase
• No neutral connection

Power Calculations: P_total = 3 × P_phase = √3 × V_line × I_line × cos(θ)

Practical Applications

Power Systems

Transmission Lines:

• Impedance matching
• Power transfer efficiency
• Voltage regulation
• Fault analysis

Transformers:

• Equivalent circuits
• Impedance transformation
• Efficiency calculations
• Regulation analysis

Electronic Circuits

Amplifiers:

• Frequency response analysis
• Input/output impedance
• Gain and phase calculations
• Stability analysis

Filters:

• Active and passive designs
• Frequency selectivity
• Phase response
• Group delay considerations

Motor Drives

AC Motors:

• Equivalent circuits
• Torque-speed characteristics
• Power factor correction
• Variable frequency drives

Summary

AC circuit analysis extends DC principles to sinusoidal systems:

1. Phasor Representation: Simplifies sinusoidal analysis using complex numbers
2. Impedance Concept: Combines resistance and reactance for AC analysis
3. RLC Circuits: Series and parallel combinations with frequency dependence
4. AC Power: Real, reactive, and apparent power relationships
5. Frequency Response: Circuit behavior varies with frequency
6. Practical Applications: Power systems, electronics, and motor drives

Understanding AC circuit analysis is fundamental to all areas of electrical engineering.

Next Steps

Continue your AC circuit education with these related topics:

• Resonance and Filters: Detailed frequency-selective circuit design
• Three-Phase Power Systems: Balanced and unbalanced system analysis
• Transformer Analysis: Magnetic circuit principles and applications
• Motor Control Systems: AC motor analysis and control techniques

Mastering AC circuit analysis opens the door to advanced electrical engineering applications in power systems, electronics, and control systems.