Discover the Analytical Calculation for RMS Voltage from v(t) = Vm cos(wt+theta)

[blanik]

The question is "starting from v(t) = Vm cos(wt+theta), show analytically that the RMS value of v(t) is v(t)=Vm/sqrt(2) for any w or theta."

I'm not really sure how to begin this. Do I start by taking the integral?

 

[antonantal]

http://en.wikipedia.org/wiki/Root_mean_square

 

[ravenprp]

Yes, you can start by taking the integral of v(t) = Vm cos(wt+theta) over a period T. This will give you the average value of the function over that period. However, since we are interested in the RMS value, we need to square the function before taking the integral. This will give us:

v^2(t) = Vm^2 cos^2(wt+theta)

Next, we can use the trigonometric identity cos^2(x) = (1+cos(2x))/2 to simplify the equation:

v^2(t) = Vm^2 (1+cos(2wt+2theta))/2

Now, we can take the integral of this equation over a period T and divide by T to get the average value:

1/T ∫v^2(t)dt = Vm^2/2 ∫(1+cos(2wt+2theta))dt

Since the integral of cos(2wt+2theta) over a period T is 0, we are left with:

1/T ∫v^2(t)dt = Vm^2/2

Now, we know that the RMS value is equal to the square root of the average value. So, we can take the square root of both sides to get:

RMS value = sqrt(1/T ∫v^2(t)dt) = sqrt(Vm^2/2) = Vm/sqrt(2)

Therefore, we have shown analytically that the RMS value of v(t) is v(t)=Vm/sqrt(2) for any w or theta. This result is important because it allows us to calculate the effective or average value of a sinusoidal function, which is useful in various applications such as AC circuits.