Question

Maybe an odd/slightly difficult question, but would appreciate any explanation given...

Why does RSM of a sine wave = peak x sqrt 2...?

Answer 1

RSM?

Answer 2

RMS value i think

Answer 3

ugh, my bad, root mean squared

Answer 4

RMS using the continuous definition?

Answer 5

yeah, i guess?
all i understand is how to find the RMS value of a sine wave... not the why it is so...

Answer 6

this whole q is screwy, sorry, peak = rms x sqrt 2
my bad ... again

Answer 7

Start from definition of RMS \[
f_{RMS}=\lim_{x\to \infty}\sqrt{\int_0^x[f(t)]^2\;dt}
\]

Answer 8

So we'll start with something as simple as \[
\int_0^x\sin^2(t)\;dt
\]

Answer 9

Use \(\sin^2(t)=\frac 12[1-\cos(2t)]\)

Answer 10

This doesn't look good because the limit won't converge at infinity.

Answer 11

So instead just do something like the period of the sine wave, I suppose. It is perfectly representative.

Answer 12

The period of \(\sin^2(t)\) is \(\pi\).

Answer 13

So I guess it is good enough to just do: \[
\sqrt{\frac 1\pi \int_0^\pi \sin^2(t)\;dt}
\]

Answer 14

@Jack1 Does that make sense so far?

Answer 15

yeah, im with you so far

Answer 16

\[
\int \sin^2(t)\;dt=\frac 12\int 1-\cos (2t)\;dt=\frac 12 t-\frac 14\sin(2t)+C
\]

Answer 17

So \[
\int _0^\pi \sin^2 (t)\;dt = \frac 12 t-\frac 14 \sin(2t)\Bigg|_0^\pi=\frac \pi2
\]

Answer 18

Putting it back in: \[
\sqrt{\frac 1\pi \times \frac \pi2} = \sqrt\frac 12
\]

Answer 19

Peak x, is really just the amplitude.

Answer 20

In this case amplitude was 1.

Answer 21

brilliant, but my integration is a bit off, so why did we take the 1/pi out in the first place again...?

Answer 22

Because it is in the definition.

Answer 23

If you're doing RMS over an interval, you have \[
\sqrt{\frac 1{b-a}\int _a^b[f(t)]^2\;dt}
\]

Answer 24

ahh, with u again, cheers

Answer 25

For a periodic function of period \(p\), you can just let \(a=0, b=p\)

Answer 26

For non periodic functions, it would seem you'd do something like \(a=0\, b\to \infty\)

Answer 27

thanks for that man, makes sense now
cheers @wio