Introduction to Electric Circuits

74 Single-phase a.c. circuits

vt9

1

L

Figure 4.8

according to Faraday's law, an emf will be induced in it, given by

e = -L(di/dt). Thus e - -Ld(Im sin o~t)/dt = -L(Imo9 cos o90, so

e = - wLIm cos oJt (4.9)

The maximum value of this waveform is oJLIm and as it is a minus cosine it is
zr/2 (90 ~ behind the current wave. The direction of this emf is such as to
oppose the current in accordance with Lenz's law, so the supply voltage V must
be equal and opposite to E.
Thus the supply voltage is represented by v - wLIm cos ~ot, which is zr/2 (90 ~
ahead of the current wave (cos 00t = sin (ox + zr/2). In a purely inductive circuit
therefore the current lags the supply voltage by zr/2 (90~ The waveforms and
the phasor diagram are shown in Fig. 4.9(a) and (b), respectively.

v,i

e v

r

t

E~

(a) Waveforms (b) Phasor diagram

Figure 4.9

The maximum value of the voltage wave is Vm = wLIm. Dividing Vm and Im by

~/2 gives V- wLI, where V and I are now the rms values. Dividing both sides

by I we have

VII = ~oL (4.10)

Now ~oL is written XL and is called the inductive reactance of the inductor. Its

unit is the volt per ampere (the ohm) and because XL = o~L = 2zrfL it varies

with frequency. The graphs of Fig. 4.10 show how the inductive reactance and

the current vary with frequency in the circuit of Fig. 4.8.

Note that as

f~O so XL~O and I~