Electricity and magnetism

Alternating current

Alternating current varies periodically and reverses direction; in a pure resistor, $i=I_0\cos(\omega t+\varphi)$ .

i(t)=I_0\cos(\omega t+\varphi),\qquad I_{\mathrm{eff}}=\frac{I_0}{\sqrt{2}}

Definition: RMS value

A sinusoidal current reverses every half-cycle; $T=1/f$ and $\omega=2\pi f$ . In a pure resistor, voltage and current are in phase. The RMS value is the DC current producing the same average heating in a resistor: $I_{ m rms}=I_0/\sqrt{2}$ and $U_{ m rms}=U_0/\sqrt{2}$ .

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Reading the model

Example: Worked example

An AC source has peak voltage $U_0=170$ V. Find $U_{rms}$ .

Solution

$U_{rms}=U_0/\sqrt{2}\approx120$ V.

For a sinusoidal current, $i=I_0\cos(\omega t+\varphi)$ has period $T=2\pi/\omega$ and frequency $f=1/T$ . RMS is defined to produce the same average power in a resistor as an equivalent DC current: $I_{rms}=I_0/\sqrt2$ . A 50 Hz supply has period 0.020 s; an RMS voltage of 220 V corresponds to a peak of about 311 V. Thus the voltage quoted for a mains outlet is not its peak value.

Over a full cycle, sinusoidal current has zero average because the two half-cycles cancel, yet its RMS value is nonzero and it still heats a resistor. For $i=I_0\cos\omega t$ , average power in a pure resistor is $P=I_{rms}^2R=I_0^2R/2$ .

When voltage and current are in phase across a resistor, average power is the product of their RMS values. A phase difference in other loads introduces the power factor $\cos\varphi$ .

Quick check

For a sinusoidal current, the RMS value is:

If the frequency of a sinusoidal current doubles, its period:

References

Young, Freedman (2019). University Physics