Hi, I am confused over the term rms.

From an AC waveform, the average should be zero.

how is rms value derived from the AC waveform?

I know that in order to obtain rms, it is just dividing

the max value by square root of 2, BUT WHY??!

Any help is appreciated.

It's not simple to look at an AC waveform (like a sine wave, for

example) and "see" how it compares to DC. It's clear, though, that DC

will heat a resistor by delivering power to it. It's also clear that

AC, too, will heat a resistor. POWER is what heats a resistor. So, if

a given AC waveform heats a resistor exactly as much as a particular

DC voltage, then we can say that their "heating effect" is the same.

An AC voltage's RMS value is often referred to as its heating effect.

When a DC voltage heats a resistor to some temperature, it does it by

delivering power to the resistor continuously. When an AC voltage

heats the same resistor to the same temperature, it does it by

delivering the same AVERAGE POWER to the resistor. (This is true

because power is what heats the resistor. An AC waveform's

"instanteous" power--the power delivered at a particular instant of

time--varies with time because its voltage varies with time. But the

resistor's temperature tends to vary slowly, much slower than the

variation of the AC waveform. So it's AVERAGE POWER from the AC

waveform that we care about.)

As John Popelish explained, "RMS stands for the three mathematical

operations (Root Mean Square) needed to calculate the" AC voltage that

is equivalent to a DC voltage. R, M, and S are the mathematical

operations in reverse. First, square the voltage. This makes sense

because voltage squared is used to calculate power.

Next, take the mean (average) of the squared voltage. That is,

calculate the average power.

Finally, take the square root. This makes sense because the square

root of power can be used to calculate voltage.

So, by operating on an AC waveform, first by squaring, then averaging,

then by square-rooting, you convert the AC voltage to an equivalent DC

voltage.

Here's an example:

A sine wave has a peak value of 10 volts. What is its RMS (or

equivalent DC) value?

V = 10 * sin(2 * pi * f * t)

SQUARE:

V^2 = 100 [sin(2 * pi * f * t)]^2

but [sin(x)]^2 = 0.5 [1 - cos(2x)] (Check a trig reference if you're

not familiar with trigonometry.)

so V^2 = 50 - 50 * cos(4 * pi * f * t)

MEAN:

Ave(V^2) = 50 - 50 * Ave[cos(4 * pi * f * t)]

Since the average of the cosine function over a full period is zero,

then

Ave(V^2) = 50 - 0 = 50

ROOT:

SquareRoot[Ave(V^2)] = SquareRoot[50] = 7.07, approximately.

So, a sine wave with a peak value of 10 has an RMS value of 7.07. This

means, of course, that a sine wave AC voltage with a peak value of 10

volts will heat a given resistor exactly as much as a DC voltage of

7.07 volts. For a sine wave, then, to find the RMS value, just divide

the peak value by the square root of 2: 10 / sqrt(2) = 7.07.