AC circuits

Alternating current

In alternating current, or AC, voltage and current vary periodically in time. A common sinusoidal voltage is

$V(t)=V_0cos(omega t)$

where $V_0$ is peak voltage and $omega$ is angular frequency.

AC is used in power distribution because transformers can easily raise and lower AC voltages, reducing transmission losses and making delivery practical.

RMS values

For sinusoidal AC, average voltage over a cycle is zero, but useful power is not zero. We use root-mean-square values:

V_{rms}=rac{V_0}{sqrt{2}}

I_{rms}=rac{I_0}{sqrt{2}}

Power formulas using RMS values resemble DC formulas.

Resistor in AC

For a resistor,

$V_R=IR$

Voltage and current are in phase. If current is

$I(t)=I_0cos(omega t)$

then voltage is

$V_R(t)=I_0Rcos(omega t)$

The average power dissipated is

$P_{avg}=I_{rms}^2R$

Capacitor in AC

For a capacitor,

$Q=CV$

and current is

I=rac{dQ}{dt}=Crac{dV}{dt}

Current leads voltage by $pi/2$ radians in a capacitor. The capacitive reactance is

X_C=rac{1}{omega C}

At high frequency, a capacitor has smaller reactance. At low frequency, it has larger reactance.

Inductor in AC

For an inductor,

V_L=Lrac{dI}{dt}

Voltage leads current by $pi/2$ radians, or equivalently current lags voltage. The inductive reactance is

$X_L=omega L$

At high frequency, an inductor has larger reactance. At low frequency, it has smaller reactance.

Impedance

In AC circuits, resistance generalizes to impedance. For a series RLC circuit, impedance magnitude is

$Z=sqrt{R^2+(X_L-X_C)^2}$

The current amplitude is

I_0=rac{V_0}{Z}

The phase angle satisfies

anphi=rac{X_L-X_C}{R}

If $X_L>X_C$, the circuit is more inductive and current lags. If $X_C>X_L$, it is more capacitive and current leads.

Resonance in RLC circuits

Resonance occurs when

$X_L=X_C$

so

omega L=rac{1}{omega C}

The resonant angular frequency is

omega_0=rac{1}{sqrt{LC}}

At resonance, impedance is minimized in a series RLC circuit and equals $R$. Current is maximum.

Average power

Average power in an AC circuit is

$P_{avg}=V_{rms}I_{rms}cosphi$

The factor $cosphi$ is the power factor. Only the in-phase part of voltage and current contributes to average power transfer. Ideal capacitors and inductors store and return energy but dissipate no average power.

The big idea

AC circuits depend on frequency and phase. Resistors dissipate energy, capacitors oppose low-frequency changes, and inductors oppose high-frequency changes. Impedance combines resistance and reactance, while resonance occurs when inductive and capacitive reactances cancel. These ideas underlie power systems, filters, radios, and signal processing.