Page 137 - Schaum's Outline of Theory and Problems of Electric Circuits

WAVEFORMS AND SIGNALS
126

Fig. 6-25 [CHAP. 6

6.37 Find an impulse voltage which creates a 1-A current jump at t ¼ 0 when applied across a 10-mH inductor.
2
Ans: vðtÞ¼ 10 ðtÞ (V)

6.38 (a) Given v 1 ¼ cos t, v 2 ¼ cos ðt þ 308Þ and v ¼ v 1 þ v 2 , write v in the form of a single cosine function
2
2
2
v ¼ A cosðt þ Þ.(b) Find effective values of v 1 , v 2 , and v. Discuss why V eff > ðV 1;eff þ V 2;eff Þ.
Ans.(a) v ¼ 1:93 cos ðt þ 158Þ; ðbÞ V 1;eff ¼ V 2;eff ¼ 0:707; V eff ¼ 1:366 V eff is found from the following
derivation
2
2
2
2
2
2
2
V eff ¼hv i¼hðv 1 þ v 2 Þ i¼hv 1 þ v 2 þ 2v 1 v 2 i¼hv 1 iþhv 2 iþ 2hv 1 v 2 i
p ffiffiffi
1
Since v 1 and v 2 have the same frequency and are 308 out of phase, we get hV 1 V 2 i¼ cos 308 ¼ 3=4,
2
2
2
2
which is positive. Therefore, V eff > ðV 1;eff þ V 2;eff Þ:
p ffiffiffi p ffiffiffi
6.39 (a) Show that v 1 ¼ cos t þ cos 2t is not periodic. (b) Replace 2 by 1.4 and then show that
p ffiffiffi
v 2 ¼ cos t þ cos 1:4t is periodic and find its period T 2 . (c) Replace 2 by 1.41 and find the period T 3 of
p ffiffiffi
v 3 ¼ cos t þ cos 1:41t. (d) Replace 2 by 1.4142 and find the period T 4 of v 4 ¼ cos t þ cos 1:4142t.
p ffiffiffi
Ans.(a) 2 is not a rational number. Therefore, v 1 is not periodic. (b) T 2 ¼ 10 s. (c) T 3 ¼ 200 s.
(d) T 4 ¼ 10 000 s.
6.40 A random signal sðtÞ with an rms value of 5 V has a dc value of 2 V. Find the rms value of s 0 ðtÞ¼ sðtÞ 2,
ffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ffiffiffiffiffi
p
2
that is, when the dc component is removed. Ans: S 0;eff ¼ 5 4 ¼ 21 ¼ 4:58 V