Find the average power over one cycle

[s3a]

Homework Statement

The problem and its solution are attached as TheProblemAndSolution.png.

Homework Equations

Integration
P = VI
V = RI

The Attempt at a Solution

I think Wolfram Alpha “says” ( http://www.wolframalpha.com/input/?i=integrate+900sin^2(377t)+dt+from+0+to+pi ) that the equation should be the way it is with a dt rather than a d(377t).

I think I get why the limits of integration are 0 to π rather than 0 to 2π (I'm guessing it's because we're taking the average – which is (0+2π)/2 = π) but, I'm having trouble with the following.:
(1) How am I supposed to know to multiply by 1/π?
(2) Is the p_(avg) equation supposed to have dt instead of d(377t)?
(3) Is it a fluke that the limits of integration have been from 0 to 2π and then divided by 2 to get the same answer?

I might have more questions that arise as I get these answered but, I am not sure yet.

Any help would be greatly appreciated!

 

[gneill]

(1) ##\pi## is the angular "duration" of the portion of the function that is being integrated (see (2)).

(2) The solution method is integrating over the angle rather than the time. This is valid because no matter what the actual frequency or period of the sinewave, you want to average over a cycle. Since both half cycles are symmetrical they will have the same power, so you really only need to integrate over a half cycle to find the average. For a half cycle the angle goes from 0 to ##\pi##.

(3) The chosen limits of integration are 0 to ##\pi##. That's a half cycle. If you were to integrate over 0 to ##2\pi## (a whole cycle) then you should divide by the 'period' of the cycle, which is ##2\pi##.

 

[s3a]

Sorry if I'm being repetitive but I'd like to reiterate and add upon what you said to make sure I fully get what's going on.:

(1) Okay so, is it correct to say that a step was skipped in that all 377t “occurences” in the integral should be replaced by θ prior to substituting the limits of the integral? In other words, should the sin^2 (377t) part be replaced by sin^2(θ) and should the d(377t) be replaced by dθ?

What I was doing before was moving the 377 outside of the integral and integrating with 377t within the squared sine function and with a dt instead of a d(377t).

(2) The book says “[. . .] is taken over one-half cycle”. Was the integral taken over one-half cycle just to simplify the math (which is the impression I get by reading your reply) or is there more to it than that?

 

[gneill]

Sorry if I'm being repetitive but I'd like to reiterate and add upon what you said to make sure I fully get what's going on.:

(1) Okay so, is it correct to say that a step was skipped in that all 377t “occurences” in the integral should be replaced by θ prior to substituting the limits of the integral? In other words, should the sin^2 (377t) part be replaced by sin^2(θ) and should the d(377t) be replaced by dθ?

What I was doing before was moving the 377 outside of the integral and integrating with 377t within the squared sine function and with a dt instead of a d(377t).

Yes, you can think of the substitution of θ for 377t as a change of variable (i.e. a typical integration technique).

(2) The book says “[. . .] is taken over one-half cycle”. Was the integral taken over one-half cycle just to simplify the math (which is the impression I get by reading your reply) or is there more to it than that?

No, that's all there is to it.

 

[s3a]

Thanks a lot. :)