AC Theory

Part of Electrical Theory

How alternating current works mathematically and physically, including frequency, phase, RMS values, and the behavior of reactive components.

Why This Matters

Nearly all large-scale electrical power generation produces alternating current. Generators, by their nature, produce voltage that alternates direction as the rotor spins. The transformer—which makes efficient long-distance power transmission possible—only works with alternating current. Understanding AC theory is not optional for anyone building or operating a generator-based power system.

AC theory is also more complex than DC theory. While DC circuits require only Ohm’s Law and Kirchhoff’s Laws, AC circuits involve additional phenomena: inductors and capacitors exhibit frequency-dependent behavior, voltage and current may be out of phase with each other, and power calculation requires care. These phenomena can make AC circuits perform in ways that seem counterintuitive until the theory is clear.

The practical payoff is substantial: a person who understands AC theory can design transformer windings, match generator output to loads, understand power factor, and diagnose motor and generator problems that arise from reactive effects.

The AC Waveform

Alternating current reverses direction periodically. The standard waveform is sinusoidal—it follows the mathematical sine function—because a generator with a rotating coil in a uniform magnetic field produces exactly this shape.

The AC waveform is described by: v(t) = V_peak × sin(2π × f × t + φ)

Where:

• V_peak = the maximum (peak) voltage
• f = frequency in hertz (cycles per second)
• t = time in seconds
• φ = phase angle (position in the cycle at t=0)

Key terms:

• Period (T): Time for one complete cycle. T = 1/f
• Frequency (f): Cycles per second. f = 1/T
• Peak voltage (V_pk): Maximum instantaneous voltage
• Peak-to-peak voltage (V_pp): Voltage from most positive to most negative = 2 × V_pk
• RMS voltage (V_rms): The effective voltage for power calculations (explained below)

Standard power frequencies:

• North America: 60 Hz (period = 16.67 ms)
• Europe, Africa, Asia, Australia: 50 Hz (period = 20 ms)
• Aircraft: 400 Hz (smaller transformers due to higher frequency)

The choice of 50 or 60 Hz was made by early power companies in the 1890s. For a rebuilding community, either is workable; 60 Hz is slightly preferable because motors run a bit faster and transformers are somewhat smaller at 60 Hz than at 50 Hz.

RMS Values

The peak voltage of household AC in North America is approximately 170V, yet we call it 120V. The 120V is the RMS (Root Mean Square) value—the equivalent DC voltage that would deliver the same power to a resistive load.

Relationship between peak and RMS for a sine wave: V_rms = V_peak / √2 ≈ V_peak × 0.707 V_peak = V_rms × √2 ≈ V_rms × 1.414

Similarly for current: I_rms = I_peak / √2

Why RMS matters: The power delivered to a resistive load is P = V × I. For AC, both V and I are continuously varying. The power averaged over a complete cycle equals V_rms × I_rms. All power calculations in AC circuits use RMS values.

Measuring RMS: Most analog meters read average, not RMS, and are calibrated with a correction factor assuming a sine wave. They give correct RMS readings only for sine waves. Non-sinusoidal waveforms (from switching power supplies, chopped motor drives) require a true-RMS meter.

Frequency and Angular Frequency

Frequency f (in Hz) is the practical measure of AC cycles per second. Angular frequency ω (in radians per second) appears in mathematical derivations:

ω = 2π × f

At 50 Hz: ω = 314 rad/s At 60 Hz: ω = 377 rad/s

Phase and Phase Difference

Two AC quantities at the same frequency may not reach their peaks at the same time. The phase angle (φ) describes this time offset.

In-phase: Voltage and current reach their peaks simultaneously. Occurs in purely resistive circuits.

Current leading voltage (capacitive circuit): Current peaks before voltage. In a capacitor, current flows as voltage builds; current is maximum when voltage is changing fastest (at zero crossing), not at the voltage peak.

Current lagging voltage (inductive circuit): Voltage peaks before current. An inductor resists changes in current; the magnetic field must build before current reaches maximum.

Phasor representation: Because AC quantities are sinusoids with the same frequency, they can be represented as rotating vectors (phasors) in a plane. The length of the phasor represents the magnitude (usually RMS); the angle represents the phase. Phasor addition is equivalent to adding two sinusoids of the same frequency—far simpler arithmetic than adding time-domain sinusoid equations.

Reactance: The AC Equivalent of Resistance

Inductors and capacitors oppose current in AC circuits, but unlike resistors they do not dissipate energy—they store and return it. Their opposition is called reactance (X), measured in ohms.

Inductive reactance: X_L = ω × L = 2π × f × L

Where L is inductance in henries. Inductive reactance increases with frequency—an inductor is a short circuit at DC and increasingly opposes AC at higher frequencies.

Capacitive reactance: X_C = 1/(ω × C) = 1/(2π × f × C)

Where C is capacitance in farads. Capacitive reactance decreases with frequency—a capacitor is an open circuit at DC and increasingly passes AC at higher frequencies.

Impedance: The combined opposition of resistance and reactance in an AC circuit is impedance (Z):

For a series R-L circuit: Z = √(R² + X_L²) For a series R-C circuit: Z = √(R² + X_C²) For a series R-L-C circuit: Z = √(R² + (X_L - X_C)²)

Ohm’s Law in AC form: V = I × Z (using RMS values; Z in ohms)

Resonance

When inductive and capacitive reactances are equal (X_L = X_C), they cancel. The circuit impedance is purely resistive—at minimum, equal to just the resistance. This is resonance.

Resonant frequency: f_resonant = 1 / (2π × √(L × C))

At resonance in a series circuit:

• Impedance is minimum (= R only)
• Current is maximum
• Voltage across L and C individually can be much larger than the supply voltage

At resonance in a parallel circuit:

• Impedance is maximum
• Current from the supply is minimum
• The LC tank circuit oscillates internally at large amplitude

Resonance is the basis of radio tuning: adjusting C or L to make the resonant frequency match the desired radio station frequency, at which point that frequency has maximum response.

Power in AC Circuits

In a purely resistive AC circuit, all power is dissipated as heat (real power). In a purely reactive circuit (ideal inductor or capacitor), power is stored and returned each cycle—no net dissipation (reactive power).

Power factor (cos φ): Real power (P, in watts) = V_rms × I_rms × cos φ Reactive power (Q, in vars) = V_rms × I_rms × sin φ Apparent power (S, in volt-amps) = V_rms × I_rms

P² + Q² = S²

The power factor (cos φ) is the ratio of real to apparent power. A power factor of 1 means all supplied power does useful work. A power factor of 0.7 means only 70% does useful work; 30% is reactive.

Practical consequence: A motor with power factor 0.7 drawing 10A at 240V consumes 240 × 10 × 0.7 = 1,680W useful power, but the supply wires must carry 10A—sized for 2,400VA apparent power. Low power factor wastes wire capacity and generator capacity.

Correcting power factor: Adding capacitors in parallel with inductive loads (motors) corrects the power factor toward 1. The capacitor supplies reactive current locally, reducing the reactive current the generator must supply.