I've dipped my toe into the math and am immediately confused. I'm struggling to understand:

(a) why the DC voltage equivalent of an AC current isn't |avg voltage of 1/2 AC sine wave| -- or stated differently, the sum of the absolute values of the average voltage of both +ve and -ve, all divided by 2 --- which is (0.637*Vpk); and,

(b) why the work done by an AC current isn't directly proportional to its avg voltage value

But VAC avg isn't used to calculate the "effective value" of an AC current source (which I take to mean its DC equivalent). Instead the effective value = VAC rms = 0.707 * VAC pk.

I'm clearly missing something - but when i search for answers online, all I come across are explanations to the effect that the average value of a sine wave is 0 b/c its +ve and -ve cancel each other out -- which is obviously true, but I would have thought that currents of the same absolute magnitude do the same work, regardless of polarity (whether "pushing" or "pulling"). So the "cancels out" explanation doesn't answer my problem.

What do I need to read up on to figure this out? Once upon a time I was comfortable with the integration/differentiation of simple x/y functions-- if this requires something more complicated, I may have to put figuring this out on hold (perhaps permanently

).

Many thanks,

Derek