AC electric circuits

AC (alternating current) circuits are circuits driven by a current or voltage that varies sinusoidally with time, alternating between positive and negative values. The standard mathematical form is

$v(t)=V_m\cos (\omega t+\varphi )$

where $V_m$ is the amplitude, $\omega$ is the angular frequency (radians/second), and $\varphi$ is the phase angle.

In contrast, a DC (direct current) source is constant in time.

Sinusoidal sources

A sinusoid is a signal of the form $A\cos (\omega t+\varphi )$ or $A\sin (\omega t+\varphi )$ — equivalent up to a phase shift.

Key parameters:

• Amplitude $A$: peak value.
• Angular frequency $\omega$ in rad/s. Related to ordinary frequency by $\omega =2\pi f$.
• Frequency $f$ in hertz (Hz).
• Period $T=1/f=2\pi /\omega$.
• Phase $\varphi$: shift from the reference sinusoid.

For North American power: $f=60$ Hz, $\omega =120\pi \approx 377$ rad/s. For Europe: $f=50$ Hz.

RMS value

The root-mean-square (RMS) value of a sinusoid is its DC-equivalent power-delivering value:

$V_{\text{RMS}}=\frac{V_m}{\sqrt{2}}$

When you say “120 V AC outlet,” you mean 120 V RMS — the amplitude is actually $120\sqrt{2}\approx 170$ V.

RMS values multiply with current RMS values to give average power, just as DC values do. That’s the whole point of using RMS rather than peak.

Why sinusoids matter

Three reasons sinusoids dominate AC analysis:

1. Generated naturally: rotating generators produce approximately sinusoidal voltages by Faraday’s law applied to a uniformly rotating coil in a magnetic field. Real generators add slot harmonics, saturation effects, and small distortions from non-ideal pole geometry, so the actual waveform is the fundamental sinusoid plus low-amplitude harmonics. Power systems are designed around the fundamental and rely on filtering and rotor inertia to keep harmonic content low.

2. Linear circuits preserve frequency: the steady-state response of a linear circuit (resistors, capacitors, inductors) to a sinusoid is a sinusoid of the same frequency. Only the amplitude and phase change.

3. Fourier decomposition: any periodic waveform can be decomposed into a sum of sinusoids. Analyzing one sinusoid at a time, then summing — the principle of frequency-domain analysis.

Phasor analysis

For sinusoidal steady-state, the most efficient analysis technique is the phasor method — represent each sinusoid as a complex number (phasor) encoding amplitude and phase. Then circuit equations become algebraic equations in complex variables, with no time dependence.

See Phasor for the representation and Phasor relationships for circuit elements for how R, L, C transform to the frequency domain.

Impedance

A generalized “resistance” for AC circuits, written $Z$. Includes both magnitude and phase:

• Resistor: $Z_R=R$ (real, no phase shift).
• Inductor: $Z_L=j\omega L$ (positive imaginary, voltage leads current by 90°).
• Capacitor: $Z_C=1/(j\omega C)=-j/(\omega C)$ (negative imaginary, voltage lags current by 90°).

Series combination: $Z_{\text{tot}}=Z_1+Z_2+\dots$ (just like resistors).

Parallel combination: $1/Z_{\text{tot}}=1/Z_1+1/Z_2+\dots$ (just like resistors).

Ohm’s law for AC: $V=IZ$ (phasor voltage = phasor current × impedance).

Power in AC circuits

Three power quantities:

• Average (real) power $P=V_{\text{RMS}}{I}_{\text{RMS}}\cos \theta$ where $\theta$ is the phase angle between voltage and current. Units: watts.
• Reactive power $Q=V_{\text{RMS}}{I}_{\text{RMS}}\sin \theta$. Units: VAR (volt-ampere reactive).
• Apparent power $S=V_{\text{RMS}}{I}_{\text{RMS}}$. Units: VA.

Power factor: $\cos \theta$. Equal to 1 for purely resistive loads; less for loads with reactive components. Utilities charge industrial customers for low power factor because reactive power increases line currents without delivering useful power.

Where AC is used

• Power distribution: AC is easier to step up and down with transformers than DC, making long-distance transmission efficient. The reason most of the world uses AC for power.
• Audio/RF signals: speech, music, radio waves are all AC.
• Communications: modulated AC carriers (AM, FM).

For specific circuit elements’ phasor relationships, see Phasor relationships for circuit elements. For the underlying complex-number machinery, see Polar representation of complex numbers.