3.7.5.5 - Alternating Currents

Cycles, period and frequency

An alternating current repeatedly reverses direction. In this topic, voltages and currents are sinusoidal and centred on zero. Their sign specifies a chosen polarity or current direction; a negative current is flow in the opposite direction, not a negative amount of charge.

A sinusoidal voltage can be written $V=V_0\sin(2\pi ft)$. The period $T$ is the time between points at the same stage of successive cycles: for example, one rising zero crossing and the next rising zero crossing. A rising zero crossing followed by a falling one spans only half a cycle.

[DIAGRAM: asset_name: 7.5.5 - Alternating Currents - Diagram 1; asset_slug: 7.5.5 - Alternating Currents - Diagram 1; recommended_method: local_png; caption: One complete sinusoidal voltage cycle: the period spans identical rising zero crossings, and peak and peak-to-peak measure different vertical distances.; description: One complete sinusoidal voltage cycle: the period spans identical rising zero crossings, and peak and peak-to-peak measure different vertical distances.]

There are two direction reversals per cycle, so a $50\,\mathrm{Hz}$ sinusoidal current has period $20\,\mathrm{ms}$ and reverses $100$ times per second. A generator's sinusoidal voltage produces a sinusoidal current in a fixed resistor, with current and voltage in phase.

Peak values and the heating meaning of rms

The peak value is the maximum magnitude from zero to a crest, $V_0$ or $I_0$. For a sinusoid centred on zero, the peak-to-peak value spans trough to crest: $V_{\mathrm{pp}}=2V_0$. Neither quantity is the arithmetic mean, which is zero over a complete cycle.

The root mean square (rms) value is the steady dc value producing the same mean power in the same fixed resistor. The resistor heats in both half-cycles because instantaneous power is $P=I^2R=V^2/R$, which is never negative.

To understand the factor, square $I=I_0\sin(\omega t)$. The mean of $\sin^2(\omega t)$ over a complete cycle is $1/2$, so $\overline{I^2}=I_0^2/2$. Taking its square root gives:

For a fixed resistor, mean power is $I_{\mathrm{rms}}^2R=V_{\mathrm{rms}}^2/R=V_{\mathrm{rms}}I_{\mathrm{rms}}$. The last product assumes voltage and current are in phase. Do not extend it without that assumption to an arbitrary ac circuit. The peak-to-rms factor $\sqrt2$ here is specifically for a sinusoid; the specification does not require other waveform shapes.

For example, $I_0=2.0\,\mathrm A$ through $5.0\,\Omega$ gives $I_{\mathrm{rms}}=1.41\,\mathrm A$, mean power $10\,\mathrm W$, and peak instantaneous power $20\,\mathrm W$. Zero mean current does not imply zero energy transfer.

Mains rms, peak and component ratings

The nominal UK mains values used in calculations are $230\,\mathrm V$ rms and $50\,\mathrm{Hz}$. For a sinusoid,

to three significant figures. Keeping unrounded intermediate values avoids a final rounding discrepancy; about $650\,\mathrm V$ is sufficient to two significant figures.