Question about finding av. Power from V(t) and I(t)

[Master1022]

Homework Statement

We know that V(t) = f(t) and I(t) = g(t), which have been found by Fourier Series analysis and some approximations.

The next part of the problem is about finding the average power disappated by the system. I was wondering whether I would be able to take averages of both of the functions independently and then multiply them, or whether I just need to multiply everything out?

So perhaps to put it concisely, does: $$P_{av} = V_{av} \times I_{av} ?$$

Homework Equations

f_{av} = \frac{1}{T} \int_0^T f(t) \, dt

The Attempt at a Solution

I would just rather not multiply the expressions out and deal with all the extra arithmetic if possible. However, if I can integrate them separately and multiply, I feel as if I have divided by T^2 as opposed to just T.

Thanks in advance.

 

[haruspex]

if I can integrate them separately and multiply

That is not going to work. Consider two sequences of numbers, both 1, 2, 3. The average product is 14/3, but the product of the averages is 4.

 

[gneill]

You could test your theory by considering a simple circuit with a 1 V voltage source, a switch, and a 1 Ω load. Say the switch is open initially and closes at time t = 1 s. The cycle is complete at time t = 2 s when the switch opens again. You have a period of T = 2 s for the cycle.

Calculate the average voltage over the period. Calculate the average current over the period. Calculate the average power over the period. Does the product $V_{av} \cdot I_{av} = P_{av}$?

Spoiler: No.

 

[rude man]

Yes, you have to integrate the product, not multiply the separate integrals.

Another example: Sine voltage across an inductor. The product of separate voltage and current averages calculates to a non-zero average power which you know is incorrect.