7.17 Can the instantaneous power output of an ac source ever be negative? Can the average power output be negative?
Let the applied e.m.f. =E=E0sin(ωt) I=I0sin(ωt−ϕ) Instantaneous power output of ac source P=EI =E0sinωtI0sin(ωt−ϕ) =E0I0sinωt[sinωtcosϕ−cosωtsinϕ] =E0I0[sin2ωtcosϕ−sinωtcosωtsinϕ] =E0I0[2(1−cos2ωt)cosϕ−21sin2ωtsinϕ] =2E0I0[cosϕ−cos2ωtcosϕ−sin2ωtsinϕ] =2E0I0[cosϕ−(cos2ωtcosϕ+sin2ωtsinϕ] P=2E0I0[cosϕ−cos(2ωt−ϕ)] Taken phase angle ϕ,±ve. Instantaneous Power P=2E0I0[cosϕ±cos(2ωt−ϕ)] as cosϕ=ZR,R and Z can never be negative and value of cosθ(θ=2ωt±ϕ) can vary from (1 to 0 to −1) in any case P can never be negative. We know that average power of LCR series ac circuit is PaV=2E0I0cosϕ Again as cosϕ=ZR is always positive, because R and Z, the reactances are always positive. So Pav can never be negative.
Let the applied e.m.f. =E=E0sin(ωt) I=I0sin(ωt−ϕ) Instantaneous power output of ac source P=EI =E0sinωtI0sin(ωt−ϕ) =E0I0sinωt[sinωtcosϕ−cosωtsinϕ] =E0I0[sin2ωtcosϕ−sinωtcosωtsinϕ] =E0I0[2(1−cos2ωt)cosϕ−21sin2ωtsinϕ] =2E0I0[cosϕ−cos2ωtcosϕ−sin2ωtsinϕ] =2E0I0[cosϕ−(cos2ωtcosϕ+sin2ωtsinϕ] P=2E0I0[cosϕ−cos(2ωt−ϕ)] Taken phase angle ϕ,±ve. Instantaneous Power P=2E0I0[cosϕ±cos(2ωt−ϕ)] as cosϕ=ZR,R and Z can never be negative and value of cosθ(θ=2ωt±ϕ) can vary from (1 to 0 to −1) in any case P can never be negative.
