Alternating current and its true value

Every source so far has pushed charge one way, a direct current that a battery supplies and that flows from plus to minus without ever reversing. The electricity in the walls of a building does something stranger: it swings back and forth many times a second, pushing the charge one way and then the other, never settling. This is alternating current, AC, and it is not a curiosity but the form in which nearly all electrical power is generated and delivered. Understanding why, and how to describe a quantity that is never still, is the doorway to the last ideas of the subject.

The shape of the swing

An alternating voltage is described by a sine wave, rising smoothly to a positive peak, falling back through zero to a negative peak, and returning, over and over:

V (t) = V_{peak} \sin (ω t)

Two numbers fix it. The peak value $V_{peak}$ is the height of the swing. The frequency $f$ is how many complete cycles happen each second, measured in hertz; mains electricity runs at 50 or 60 hertz depending on the country. It is often more convenient to use the angular frequency $ω = 2 π f$ , in radians per second, because the sine function naturally speaks in radians, and $ω$ appears throughout the mathematics of the next lesson. A 50 hertz supply has an angular frequency of $2 π \times 50 = 314$ radians per second. The sine shape is not arbitrary: it is the shape a spinning coil in a magnetic field naturally produces, so it falls out of the simplest possible generator for free, which is a large part of why the world runs on it.

Why alternating, and not direct

If direct current is simpler, why did the grid choose alternating? The decisive reason is the transformer, a device that steps an alternating voltage up or down efficiently, and which works only on a changing current, because it relies on the changing magnetic field of the last lesson to induce a voltage in a second coil. This matters because of the power lesson's lesson: transmitting power over long distances wastes far less as $I^{2} R$ heat when done at high voltage and low current, so power is generated at a modest voltage, transformed up to hundreds of thousands of volts to cross the country, then transformed back down for use. Direct current cannot be stepped between voltages so simply, so the ability to transform, which alternating current has and direct current lacks, is what settled the matter. Alternating current is also what a rotating generator produces without any effort to straighten it, so it is cheap at both the generating and the transmitting end.

The trouble with an average, and the true value

A quantity that is positive as often as it is negative has an average of zero, so the simple average is useless for describing how much an alternating voltage amounts to: the mains averages to nothing yet plainly does work. What is wanted is a single steady value that captures the wave's real effect, and the right one comes from the power lesson. A resistor heats at a rate proportional to the square of the voltage, and the square of a sine wave is always positive, so it has a genuine, non-zero average even though the voltage does not. The root-mean-square value, the RMS, is defined as the steady voltage that would dissipate the same average power in a resistor as the alternating one does. It is found by squaring the wave, averaging, and taking the square root, which for a sine wave gives a clean result:

V_{RMS} = \frac{V_{peak}}{\sqrt{2}} \approx 0.707 V_{peak}

The RMS value is the honest value of an alternating supply, the one quoted on every specification and read by every meter. When a supply is called 230 volts, that is its RMS value; its peak is higher, $230 \times \sqrt{2} = 325$ volts, which is why the insulation must withstand more than the nameplate suggests. The RMS value lets alternating and direct current be compared directly: a 230 volt RMS supply delivers the same average heating power to a heater as a steady 230 volt direct supply would, which is exactly the property it was defined to have.

Example. A country's mains is quoted as 120 volts. What is the peak voltage of the wave?

The quoted value is RMS, so $V_{peak} = V_{RMS} \times \sqrt{2} = 120 \times 1.414 = 170$ volts. The wave swings up to 170 volts and down to minus 170, and its heating effect equals that of a steady 120 volts.

Now you. An alternating supply has a peak voltage of 12 volts. What is its RMS value?

Answer

$V_{RMS} = 12 / \sqrt{2} = 8.49$ volts.

What changes when the source alternates

With an alternating source, the circuit's behaviour changes qualitatively, because the components that care about change, the capacitor and the inductor, are now driven by a voltage and current that never stop changing. A capacitor, which blocked a steady voltage, now passes an alternating one, charging and discharging every cycle; an inductor, which passed a steady current, now opposes an alternating one, fighting the reversal every cycle. Each still obeys its own law, $I = C d V / d t$ for the capacitor and $V = L d I / d t$ for the inductor, but now those rates of change are never zero, so the components are always active. The opposition each offers to an alternating current is not a plain resistance, because it depends on how fast the current alternates, on the frequency, and it shifts the timing between voltage and current rather than simply scaling it. Describing that frequency-dependent, timing-shifting opposition needs one more idea, impedance, and it is the final step that unites the resistor, the capacitor and the inductor into a single framework for alternating current, which the last lesson builds.