128-129 - Alternating Currents and RMS Values

Reading an AC Waveform

A direct current has one direction through a component. An alternating current repeatedly reverses direction. In a simple sinusoidal AC supply, the potential difference and current rise, fall, pass through zero, reverse, and repeat.

That reversing story is why a time graph matters. At one instant the potential difference may be positive; half a cycle later it may have the same size but the opposite sign. The sign tells you the direction of the potential difference or current, not whether the supply is "stronger" or "weaker".

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Four quantities describe the waveform:

Frequency and period are linked by:

So a supply with $T = 0.020\,\mathrm{s}$ has:

The peak value is not the same as the peak-to-peak value. Peak is measured from the zero line to one crest. Peak-to-peak is measured from the positive crest to the negative crest, so for a symmetric sine wave:

For Edexcel “determine” questions using a graph or oscilloscope trace, show where the reading came from: one complete cycle for $T$, zero-to-crest for peak, and sensible unit conversion such as $\mathrm{ms}$ to $\mathrm{s}$.

Why RMS Is Needed

If you average a perfectly symmetric AC current over one complete cycle, the signed average is zero. The positive half-cycle and negative half-cycle cancel.

That does not mean the AC supply transfers no energy. A lamp filament or heater still gets hot. In a resistor, the rate of heating depends on $I^{2}R$, and squaring the current makes both halves of the cycle contribute positively:

• positive current gives positive heating
• negative current gives positive heating too, because $(-I)^{2} = I^{2}$

This is the physical reason for root-mean-square values.

The name describes the operation:

1. Square the instantaneous values so negative parts do not cancel.
2. Find the mean of those squared values over a cycle.
3. Take the square root to return to the original unit, such as volts or amps.

You do not need calculus for this topic. The important model is that rms is an effective value, not an ordinary signed average.

Common mistake: saying "the average value of AC is zero, so the rms value is zero." The signed average can be zero, but rms is based on the average of squared values, so it is positive for a non-zero alternating waveform.

Using Peak and RMS Values

For sinusoidal alternating currents and potential differences:

The rms value is about $0.707$ of the peak value because $1 / \sqrt{2} = 0.707\ldots$.

Worked example: a signal generator produces a sinusoidal alternating p.d. with peak value $18.0\,\mathrm{V}$. The peak current in a connected resistor is $0.640\,\mathrm{A}$. Calculate the rms p.d. and rms current.

The fork in the road is whether the numbers are peak or rms. The question says peak, so divide by $\sqrt{2}$.

The answers are smaller than the peak values, which makes sense because a sine wave is below its peak for most of the cycle.

You may also need the reverse route. If a mains supply is quoted as $230\,\mathrm{V}\ \text{rms}$, its peak value is:

That does not mean a "230 V" AC supply is harmless below 230 V or only sometimes 230 V. It means its effective heating value is $230\,\mathrm{V}$; its instantaneous value varies between about $+325\,\mathrm{V}$ and $-325\,\mathrm{V}$ for a sinusoidal supply.

The $\sqrt{2}$ link is for sinusoidal waveforms. Do not assume every possible alternating waveform has the same rms-to-peak ratio.