Question

Can the instantaneous power output of an ac source ever be negative? Can the average power output be negative?

Answer 1

Let the applied e.m.f  E = E₀ sin (ωt)

I = I₀ sin (ωt – `phi)`

Instantaneous power output of ac source

P = EI

= E₀ sin ωtI₀ sin(ωt – `phi`)

= E₀I₀ sin ωt [sin ωt cos `phi` – cos ωt sin `phi`]

= E₀I₀ [sin² ωt cos `phi` – sin ωt cos ωt sin `phi`]

= E₀I₀ `[((1 - cos 2ωt))/2 cos phi - 1/2 sin 2ωt sin phi]`

= `(E_0I_0)/2 [cos phi - cos  2ωt cos phi - sin 2ωt sin phi]`

= `(E_0I_0)/2 [cos phi - (cos 2ωt cos phi + sin 2ωt sin phi]`

P = `(E_0I_0)/2 [cos phi - cos(2ωt - phi)]`

Taken phase angle `phi`, ± ve.

Instantaneous Power P = `(E_0I_0)/2 [cos phi +- cos (2ωt - phi)]` as cos `phi = R/Z`, R and Z can never be negative and value of `cos phi(phi = 2ωt +- phi)` can vary from (1 to 0 to – 1) in any case P can never ne negative.

We know that average power of LCR series ac circuit is P_av = `(E_0I_0)/2 cos phi`

Again as `cos phi = R/Z` is always positive, because R and Z, the reactances are always positive.

So P_av can never be negative.