Introduction to Electric Circuits

4.7 Power in single-phase a.c. circuits 97

2 We have already calculated the phase angle of the inductive branch to be

63.4 ~ and this is of course lagging. The phase angle of the circuit as a whole

is given by Equation (4.36) to be

4)- tan-1 [(IL sin 4~- IC)/(IL COS 4~)] = tan -~ (2.96/3.99) = 36.5 ~

Because IL sin L > Ic then the circuit is predominantly inductive and so the

phase angle is lagging.

4.7 POWER IN SINGLE-PHASE A.C. CIRCUITS

We saw in Chapter 3 that in d.c. circuits, power (P) is the product of voltage (V)
and current (I)"

P- VI watts (4.38)

In a.c. circuits, where the voltage and current are both changing from instant to
instant, the instantaneous power (p) is the product of the instantaneous voltage
(v) and the instantaneous current (i) i.e. p - vi.

Purely resistive circuits

For the purely resistive circuit shown in Fig. 4.6 the waveforms of voltage,

current and power are given in Fig. 4.37. Note that the power waveform never

goes negative (the product vi is always positive) and that its frequency is twice

that of the voltage and current waveforms.

v,i,p P P

0 r

t

Figure 4.37

If v - Vm sin wt then i - I m sin ~ot and since p - vi,
P - Vmlm sin2~ot (4.39)
The average power is obtained by determining the mean value of the waveform
shown, which is given by
27r~ ~o 27r oJ
P - (w/ZTr) f VmI m sin 2 tot dt - (w/ZTr)VmI m f sin 2 tot dt
0 0
Using the identity sin 2 0 - (1_ - cos 20)/2 we have
2rr/w
P = (~o//27r)Vmlm - f [(1 - cos 2o)0)2 ] at - (Vm[m~)/47r[t - (sin 2o)t/2m)]2'; ~~
0