Single-Phase AC Circuits

Unit III: Single-Phase AC Circuits

Duration: 5 Hours | Credit: ELX 133.3

Introduction to AC Circuits

Alternating Current (AC) circuits involve voltages and currents that vary sinusoidally with time. AC circuits are fundamental to power distribution, telecommunications, and most household appliances.

Sinusoidal Signals

The standard form of a sinusoidal voltage or current is:

v(t) = V_m sin(ωt + φ)

Parameters of Sinusoidal Signals

V_m or I_m: Peak (maximum) amplitude
ω: Angular frequency in rad/s (ω = 2πf)
f: Frequency in Hz (cycles per second)
T: Period in seconds (T = 1/f)
φ: Phase angle in degrees or radians
RMS Value: V_rms = V_m / √2 ≈ 0.707 V_m

Standard Frequency: In most countries, AC power is 50 Hz or 60 Hz. In the US and many other countries, it's 60 Hz (ω = 2π × 60 = 377 rad/s)

Phasors and Phasor Representation

A phasor is a complex number that represents the magnitude and phase of a sinusoidal signal. It simplifies AC circuit analysis by converting differential equations to algebraic equations.

Conversion from Time Domain to Phasor Domain

Time Domain: v(t) = V_m cos(ωt + φ)

Phasor Domain: V = V_rms ∠ φ or V = V_m ∠ φ

Phasor Operations

Addition: V₁ + V₂ = (R₁ + R₂) + j(X₁ + X₂)
Multiplication: V₁ × V₂ = |V₁||V₂| ∠(φ₁ + φ₂)
Division: V₁ / V₂ = |V₁| / |V₂| ∠(φ₁ - φ₂)

Impedance and Admittance

Impedance (Z) is the AC equivalent of resistance. It includes both resistance and reactance.

Impedance Formula

Z = R + jX

Z (magnitude) = √(R² + X²)

Phase angle φ = arctan(X/R)

Reactance Types

Inductive Reactance: X_L = ωL (capacitor blocks DC, passes AC)
Capacitive Reactance: X_C = 1/(ωC)
Total Reactance: X = X_L - X_C

Power in AC Circuits

Power in AC circuits has three components:

Three Types of AC Power

Real Power (P): P = VI cos(φ) = I²R (measured in Watts, W)
Reactive Power (Q): Q = VI sin(φ) = I²X (measured in VAR, Volt-Ampere Reactive)
Apparent Power (S): S = VI (measured in VA, Volt-Ampere)

Power Factor

Power Factor (PF) = cos(φ) = P/S

Ranges from 0 to 1
PF = 1: Purely resistive circuit (ideal)
PF < 1: Presence of reactance (inductive or capacitive)
Lagging PF: Inductive circuit (current lags voltage)
Leading PF: Capacitive circuit (current leads voltage)

Resonance in AC Circuits

Resonance occurs when the inductive and capacitive reactances are equal (X_L = X_C), resulting in impedance being purely resistive.

Resonant Frequency

f₀ = 1 / (2π√LC)

Characteristics at Resonance (Series RLC)

Impedance is minimum (Z = R)
Current is maximum (I = V/R)
Voltage across R is maximum
Voltages across L and C are equal and opposite
Power factor = 1 (unity)
No reactive power (Q = 0)

Quality Factor (Q)

Q = ω₀L / R = 1 / (ω₀RC) = f₀ / BW

Where BW is the bandwidth (frequency range at which power is ≥ 50% of maximum)

AC Circuit Analysis Methods

The same methods used for DC circuits (Nodal, Mesh, Superposition, Thevenin, Norton) apply to AC circuits using phasors and impedances.

AC Equivalent Impedances

Series Impedance: Z_total = Z₁ + Z₂ + ... + Z_n
Parallel Impedance: 1/Z_total = 1/Z₁ + 1/Z₂ + ... + 1/Z_n

Frequency Response

Frequency response shows how circuit behavior changes with frequency. It is typically plotted as a Bode plot (magnitude and phase vs. frequency).

Key Points for Reactive Elements

Inductor: Impedance increases with frequency (X_L = ωL)
Capacitor: Impedance decreases with frequency (X_C = 1/ωC)

RMS (Root Mean Square) Values

RMS values represent the DC equivalent in terms of power delivery. Most AC measurements and calculations use RMS values.

RMS Value: V_rms = V_m / √2 ≈ 0.707 V_m

Instantaneous Power

p(t) = v(t) × i(t)

Instantaneous power varies with time. For a sinusoidal circuit:

p(t) = VI cos(φ) + VI cos(2ωt + φ) [Average Power + Oscillating Power]

Key Takeaways

AC signals are represented as sinusoids with amplitude, frequency, and phase
Phasors simplify AC analysis by converting it to the complex plane
Impedance combines resistance and reactance
Power has three components: real, reactive, and apparent
Resonance occurs when X_L = X_C
Quality factor Q determines the sharpness of resonance
RMS value represents DC equivalent for power calculations