Electrical current comes out of our wall plugs (well, in New Zealand anyway) as alternating current (AC), which means that the current and the voltage vary like a sine wave, with a root mean square voltage of 230 V. But what does the root mean square (RMS) voltage mean? In this exercise we take a brief look at how and why RMS voltage is calculated.

Let’s start with a voltage function, V(t), that is a sinusoidal function of time;

ω is not a w, although it looks like one. It’s the Greek letter omega. It’s commonly used as the symbol for the angular frequency of a periodic function.

V(t) = V0 sin(ωt).

a. What is the average value of V over the interval [0,T]?

Call this average value V¯(T). Note that it’s a function of T.

b. Show that limT→∞ V¯(T) = 0.

Since this limit is zero, this means that we cannot use V¯ as a measure of how much power is in the electrical current. Instead, we define the RMS of V:

In other words, we square V , take its mean, and then take its square root, to get the root mean square. RMS is Vrms = pretty much just what it says it is. lim T→∞ s 1

T

∫ T 0

(V(t))2 dt.

This gets around the problem of all the positive and negative bits cancelling out to give a zero average. If V is squared then there are no negative bits at all in the integral. However, one then has to take the square root once the average is calculated.

c. To calculate Vrms, first show that

d. Use the result of the previous integration to show that Vrms = V0 √
2 .

e. Finally, since we know that, in New Zealand, Vrms = 230 V, what is the amplitude of the AC voltage? In other words, what is V0 in New Zealand (and many other countries too)?