Compound angles 175
2
(^0) t (seconds)
v
p
i
p
i
v
1
2
Figure 17.10
p
i
v
v
p
i
0
1
2
^2 t (seconds)
Figure 17.11
Substitutingωt=Aand(ωt+φ)=Bgives:
power, p=VmIm{−^12 [cos(ωt+ωt+φ)
−cos(ωt−(ωt+φ))]}
i.e. p=^12 VmIm[cos(−φ)−cos( 2 ωt+φ)]
However, cos(−φ)=cosφ
Thusp= 21 VmIm[cosφ−cos(2ωt+φ)]
The instantaneous powerpthus consists of
(i) a sinusoidal term,−^12 VmImcos( 2 ωt+φ)which
has a mean value over a cycle of zero, and
(ii) a constant term,^12 VmImcosφ(sinceφis constant
for a particular circuit).
Thus the average value of power,P=^12 VmImcosφ.
SinceVm=
√
2 VandIm=
√
2 I, average power,
P=^12 (
√
2 V)(
√
2 I)cosφ
i.e. P=VIcosφ
The waveforms ofv,iandp, are shown in Fig. 17.11
for anR–Lcircuit. The waveform of power is seen to