Calculating Power in Electrical Circuits: Help Needed

[chrisking2021]

i am just about to start an electrical course at college and I've been reading up on some mathematics i will need when i start my problem is i don't know much about compound angles and trig and i want to complete some examples within a book the following problem is the one i need help with

the power P in an electrical circuit is given by P=V²/R

find the power in terms of V, R and cos2t when v= Vcost

any help would be appreciated

chris

 

[HallsofIvy]

First, clarify your notation. Is the "V" in P= V²r the "V" or "v" in v= V cos t?

If it is really P= V²/R, there is nothing to be done, P is already in terms of V and R (and cos 2t is irrelevant).

If it is really P= v²/R, then v= Vcos t so v²= V² cos² t and you can use the trigonometric identity cos 2t= cos² t- sin<sup>2 t= cos2 t- (1- cos2 t)= 2cos2 t- 1 so that 2cos2 t= cos 2t+ 1 and then
cos2t= (cos(2t)+ 1)/2.

v2= V2cos2 t= V2(cos(2t)+ 1)/2.

Now, if P= v2/R, then P= V2(cos(2t)+ 1)/(2R).

A third, though unlikely, possibility is that P= V2/R and v= Vcos t but you meant "in terms of v, R, and cos 2t. In that case, V= v/cos t so cos2 t belongs in the denominator.
In that case, P= 2v2/(R(cos(2t)+ 1)).</sup>

 

[uart]

Yes Chis your notation is a little confused, but I suppose that’s to be expect since you are also a little confused by the area you’re studying. Still what about some punctuation, there’s no excuse for that!

The general statement for power in a resistive load is that the instantaneous power is,

$$P = v^2 / R$$

For the “DC case” where the voltage is a constant, $$v = V$$, so $$P=V^2/R$$


For the “AC case” where the voltage is sinusoidal, $$v = V cos(t)$$, so

$$P(t) = V^2 cos^2(t) / R$$

Now often we are interested primarily in just the average value of the power. In this case it is convenient to use the trig identity,

$$2 \cos^2(t) = 1 + \cos(2t)$$,

to write the AC power as,

$$P(t) = \frac{V^2}{2R} \left( 1 + \cos(2t) \right)$$

Note here that the cos(2t) part has zero average. Hence for the AC case the average power is,

$$P_{av} = V^2 / 2R$$

 

[chrisking2021]

Thankyou Hallsofivy

I did think i should be using trig identities . I was a little confused because the question was taken directly from a book it is written exactly as i wrote it in my request.

P.S sorry about my grammer as i was rushing before work.

Thanks again

Chris