Introduction to Electric Circuits

98 Single-phase a.c. circuits

Note that this indicates a frequency of twice that of the supply voltage,
confirming the evidence of the waveforms of Fig. 4.37.
Putting in the limits, this reduces to Vmlmw/47r[27r/oo] = Vmlm/2. Rewriting
this as (Vm/V'2)(Im/~/2) we convert the maximum values to rms values and we
have for the mean power

P = VI (4.38 bis)

Since V = IR we may also write

p = I2R (4.40)

and

p = V2//R (4.41)

Purely inductive circuits

For the purely inductive circuit of Fig. 4.8 the voltage, current and power

waveforms are given in Fig. 4.38. Note that the power waveform is sinusoidal

and has equal positive and negative half cycles. The average is therefore zero.

v,i,p P

0 ~-- t

Figure 4.38

If the voltage is represented by v = Vm sin ~ot then the current will be given by

i = --Im COS wt since it is 90 ~ lagging the voltage. The instantaneous power is

then given by

p = vi - -Vml m sin wt cos ~ot

27"r: to

The average power is P = -(~o/27r) f (VmI m cos ~ot sin o~t)dt.

0

(4.42)

Using the identity sin 20 - 2 sin 0 cos 0 we get

27/',,,'0)

P - -(to/ZTr)VmI m f [(sin 2oJt)/Z]dt - -(og/4"n-)Vmlm[-COS 2~o/2~O]2o ~ = 0

0

Again the power frequency is twice the supply frequency.