21.1 Characteristics of alternating currents

An alternating voltage or current varies periodically, with period T, frequency f=1/T, angular frequency ω=2πf and peak magnitude I₀ or V₀.

Read peak from the centre line to an extreme, not peak-to-peak, and distinguish the waveform’s cycle time from its angular frequency.

A 50 Hz mains waveform has T=20 ms and ω=100π rad s⁻¹.

The mean of a symmetric AC waveform can be zero while its heating effect is not zero.

A sinusoidal alternating quantity is modelled by x=x₀sin(ωt+φ), where x₀ is peak value and φ sets the phase.

Use the initial value and slope to choose phase; differentiate or inspect the graph to identify when the signal is increasing or decreasing.

A signal starting at zero and rising has φ=0 in x=x₀sinωt.

The sine expression gives instantaneous value, not rms value or average magnitude.

With i=I₀sinωt and fixed resistance R, instantaneous power i²R varies from zero to I₀²R; its cycle average is ½I₀²R.

Use average over a complete cycle and distinguish peak power from mean heating power.

If peak current doubles, mean resistive power quadruples because it depends on I₀².

A zero mean current does not imply zero mean power, since power depends on current squared.

For a sinusoidal AC signal, rms current or voltage produces the same mean power in a resistor as a DC value: I_rms=I₀/√2 and V_rms=V₀/√2.

Use rms values in P=VI or P=I²R for resistive loads, and specify whether a quoted AC value is rms or peak.

A 10 A peak sinusoidal current has I_rms≈7.07 A.

RMS is not the arithmetic average of a symmetric waveform, which is zero.