Introduction to Electric Circuits

68 Single-phase a.c. circuits

means of a machine called a transformer which, having no moving parts, is
extremely efficient. Now the emf induced in transformer windings is propor-
tional to d~/dt, the rate of change of magnetic flux linking them (i.e.
differentiation is involved). The only a.c. waveform which when differentiated
(or integrated) gives the same waveshape is the sine wave. Others become
progressively more distorted with each subsequent differentiation, leading to
harmonics and reduced efficiency and performance. For this reason the
sinusoidal waveform in the most commonly encountered waveform in electrical
engineering.

V,I

V m _Em

I "~ 2~/(0

/ ",, I/

r

t

Figure 4.2

Fig. 4.2 shows a sinusoidal voltage waveform of maximum value Vm. This may

be represented mathematically by

v = Vm sin wt (4.2)

where w is the angular frequency measured in radians per second, related to the

frequency f (Hz) by

= 2wf (4.3)

The time axis may be converted into an angle axis simply by multiplying by w.

Phase difference

The second sinusoidal waveform (shown dashed) in Fig. 4.2 shows a current of

maximum value Ira. This waveform is described mathematically by

i - Im sin (~0t -- ~h) (4.4)

and is said to lag the voltage waveform by an angle + because its peak value

occurs r seconds after that of the voltage wave. Alternatively we could say

that the voltage waveform leads the current waveform by an angle 4~ (i.e. by

4~/oJ seconds). There is said to be a phase difference between the two

waveforms.