Page 132 - Schaum's Outline of Theory and Problems of Electric Circuits

WAVEFORMS AND SIGNALS
121
CHAP. 6]
6.16 A radar signal sðtÞ, with amplitude V m ¼ 100 V, consists of repeated tone bursts. Each tone
burst lasts T b ¼ 50 ms. The bursts are repeated every T s ¼ 10 ms. Find S eff and the average
power in sðtÞ.
p ffiffiffi
Let V eff ¼ V m 2 be the effective value of the sinusoid within a burst. The energy contained in a single
2
2
burst is W b ¼ T b V eff . The energy contained in one period of sðtÞ is W s ¼ T s S eff . Since W b ¼ W s ¼ W,we
obtain
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2 2
T b V eff ¼ T s S eff S eff ¼ðT b =T s ÞV eff S eff ¼ T b =T s V eff ð40Þ
Substituting the values of T b , T s , and V eff into (40), we obtain
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffi
3
6
S eff ¼ ð50 10 Þ=ð10 10 Þ ð100= 2Þ¼ 5V
2
Then W ¼ 10 ð25Þ¼ 0:25 J. The average power in sðtÞ is
2
2
P ¼ W=T s ¼ T s S eff =T s ¼ S eff ¼ 25 W
2
2
The average power of sðtÞ is represented by S eff and its peak power by V eff . The ratio of peak power to
ffiffiffiffiffiffiffiffiffiffiffiffiffi
p
average power is T s =T b . In this example the average power and the peak power are 25 W and 5000 W,
respectively.
6.17 An appliance uses V eff ¼ 120 V at 60 Hz and draws I eff ¼ 10 A with a phase lag of 608. Express
v, i, and p ¼ vi as functions of time and show that power is periodic with a dc value. Find the
frequency, and the average, maximum, and minimum values of p.
p ffiffiffi p ffiffiffi
v ¼ 120 2 cos !t i ¼ 10 2 cosð!t 608Þ
p ¼ vi ¼ 2400 cos !t cos ð!t 608Þ¼ 1200 cos 608 þ 1200 cos ð2!t 608Þ¼ 600 þ 1200 cos ð2!t 608Þ
The power function is periodic. The frequency f ¼ 2 60 ¼ 120 Hz and P avg ¼ 600 W, p max ¼ 600 þ
1200 ¼ 1800 W, p min ¼ 600 1200 ¼ 600 W.

6.18 A narrow pulse i s of 1-A amplitude and 1-ms duration enters a 1-mF capacitor at t ¼ 0, as shown
in Fig. 6-19. The capacitor is initially uncharged. Find the voltage across the capacitor.

Fig. 6-19

The voltage across the capacitor is
8
ð < 0 for t < 0
1 t 6
V C ¼ idt ¼ 10 t ðVÞ for 0 < t < 1 ms (charging period)
C 1 : 1 V for t > 1 ms

If the same amount of charge were deposited on the capacitor in zero time, then we would have v ¼ uðtÞ
6
(V) and iðtÞ¼ 10 ðtÞ (A).

6.19 The narrow pulse i s of Problem 6.18 enters a parallel combination of a 1-mF capacitor and a
1-M
resistor (Fig. 6-20). Assume the pulse ends at t ¼ 0 and that the capacitor is initially
uncharged. Find the voltage across the parallel RC combination.