Page 130 - Schaum's Outline of Theory and Problems of Electric Circuits

WAVEFORMS AND SIGNALS
CHAP. 6]
6.10 Find the average and effective value of v 2 ðtÞ in Fig. 6-1(b) for V 1 ¼ 2, V 2 ¼ 1, T ¼ 4T 1 . 119

V 1 T 1 V 2 ðT T 1 Þ V 1 3V 2
V 2;avg ¼ ¼ ¼ 0:25
T 4
2 2 p ffiffiffi
2 V 1 T 1 þ V 2 ðT T 1 Þ 7
V 2;eff ¼ ¼ or V 2;eff ¼ 7=2 ¼ 1:32
T 4
6.11 Find V 3;avg and V 3;eff in Fig. 6-1(c) for T ¼ 100T 1 .
2
2
From Fig. 6-1(c), V 3;avg ¼ 0. To find V 3;eff , observe that the integral of v 3 over one period is V 0 T 1 =2.
2
The average of v 3 over T ¼ 100T 1 is therefore
p ffiffiffi
2
2
2
2
hv 3 ðtÞi ¼ V 3;eff ¼ V 0 T 1 =200T 1 ¼ V 0 =200 or V 3;eff ¼ V 0 2=20 ¼ 0:0707V 0
p ffiffiffiffiffiffiffiffiffiffiffiffi
The effective value of the tone burst is reduced by the factor T=T 1 ¼ 10.
6.12 Referring to Fig. 6-1(d), let T ¼ 6 and let the areas under the positive and negative sections of
v 4 ðtÞ be þ5 and 3, respectively. Find the average and effective values of v 4 ðtÞ.
V 4;avg ¼ð5 3Þ=6 ¼ 1=3
The effective value cannot be determined from the given data.

6.13 Find the average and effective value of the half-rectified cosine wave v 1 ðtÞ shown in Fig. 6-17(a).
ð
T=4 T=4
V m 2 t V m T 2 t V m
V 1;avg ¼ cos dt ¼ sin ¼
T T=4 T 2 T T T=4
ð ð
2 T=4 2 T=4
2 V m 2 2 t V m 4 t
V 1;eff ¼ cos dt ¼ 1 þ cos dt
T T=4 T 2T T=4 T

2 T=4 2 2
V m T 4 t V m T T V m
¼ t þ sin ¼ þ ¼
2T 4 T 2T 4 4 4
T=4
from which V 1;eff ¼ V m =2.
6.14 Find the average and effective value of the full-rectified cosine wave v 2 ðtÞ¼ V m j cos 2 t=Tj shown
in Fig. 6-17(b).

Fig. 6-17

Use the results of Problems 6.3 and 6.13 to find V 2;avg . Thus,
v 2 ðtÞ¼ v 1 ðtÞþ v 1 ðt T=2Þ and V 2;avg ¼ V 1;avg þ V 1;avg ¼ 2V 1;avg ¼ 2V m =
Use the results of Problems 6.5 and 6.13 to find V 2;eff . And so,
p ffiffiffi
2
2
2
2
2
V 2;eff ¼ V 1;eff þ V 1;eff ¼ 2V 1;eff ¼ V m =2 or V 2;eff ¼ V m = 2