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

Explanation

Let the applied emf,
`E=E_m sin omegat` and current developed is,
`I=I_m sin (omegatpmphi)` (Here E is taken instead of V)
Instantaneous power output of the AC source,
P=EI
`=E_m sin omegat xx I_m sin (omegat + phi)`
`=(E_m I_m)/2 [2 sin omegat sin (omegat+phi)]`
`=(E_m I_m)/2 [cos phi - cos (2omegat+phi)]` .....(1)
[`because` 2sinAsinB=cos(A-B) -cos(A+B)]
(i) from equation (1),
If cos `phi lt cos (2omegat+phi)`, then
P `lt` 0 mean negative power
`therefore` Yes, the instantaneous power output of an AC source can be negative .
`therefore` Average power `lt P gt =V_m/sqrt2 I_m/sqrt2 cos phi`
`lt P gt =V_"rms" I_"rms" cos phi`...(2)
where `phi` is phase difference .
(ii) and from equation (2) `lt P gt gt 0`
because cos`phi=R/Z gt 0`
No, the average power output of an AC source cannot be negative .