Introduction to Electric Circuits

Example 4.2

4.1 Alternating quantities 69

Four sinusoidally alternating quantities are represented by:

a = 5 sin oot; b = 15 sin (~ot - 30~ c = 10 sin (~ot + 60~ d = 5 sin 2~ot

If w = 314 rad/s:

(1) comment on the relative magnitudes and frequencies of these quantities;

(2) determine the frequency of quantity d;

(3) state the period of quantity b;

(4) state the phase relationship of

(a) a with respect to b

(b) a with respect to c

(c) b with respect to c.

Solution

1 From Equation (4.2) we see that the coefficient of the sine function

represents the magnitude of the quantity. Thus the magnitude of d is the

same as that of a (5 units); b is three times as big as a (15 units); c is twice as

big as a (10 units). From Equation (4.3) we see that f= ~o/27r so the

frequency of quantities a, b and c is the same at oJ/2rr, whereas that of

quantity d is double at 2oJ/2rr.

2 The frequency of d is (2 • 314)/(2 x 3.14) = 100 Hz.

3 The period (T) of quantity b is the reciprocal of its frequency (i.e. 1~If),

which is half that of quantity d at 50 Hz. Therefore T - 1/50 = 0.02 s.

4 Taking quantity a as the reference, we see from the sine functions that (a) a

leads b by 30~ (b) a lags c by 60~ (c) b lags c by 30 ~ + 60 ~ = 90 ~

Phasorial representation of sinusoidal quantities

In Fig. 4.3 the line OP is considered to be rotating in an anticlockwise direction

with a constant angular velocity of 6o radians per second. Starting from the

93

~P2

O~ 02 x/2

Figure 4.3