126

WAVEFORMS AND SIGNALS

[CHAP. 6

Fig. 6-25 6.37

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

6.38

(a) Given v1 ¼ cos t, v2 ¼ cos ðt þ 308Þ and v ¼ v1 þ v2 , write v in the form of a single cosine function 2 2 2 v ¼ A cosðt þ Þ. (b) Find eﬀective values of v1 , v2 , and v. Discuss why Veff > ðV1;eff þ V2;eff Þ. Ans. (a) v ¼ 1:93 cos ðt þ 158Þ; ðbÞ V1;eff ¼ V2;eff ¼ 0:707; Veff ¼ 1:366 Veff is found from the following derivation 2 ¼ hv2 i ¼ hðv1 þ v2 Þ2 i ¼ hv21 þ v22 þ 2v1 v2 i ¼ hv21 i þ hv22 i þ 2hv1 v2 i Veff

Since v1 and v2 have the same frequency and are 308 out of phase, we get hV1 V2 i ¼ 12 cos 308 ¼ 2 2 2 which is positive. Therefore, Veff > ðV1;eff þ V2;eff Þ:

6.39

6.40

pﬃﬃﬃ 3=4,

pﬃﬃﬃ pﬃﬃﬃ (a) Show that v1 ¼ cos t þ cos 2t is not periodic. (b) Replacepﬃﬃﬃ 2 by 1.4 and then show that v2 ¼ cos t þ cos 1:4t is periodic and ﬁnd pﬃﬃﬃ its period T2 . (c) Replace 2 by 1.41 and ﬁnd the period T3 of v3 ¼ cos t þ cos 1:41t. (d) Replace 2 by 1.4142 and ﬁnd the period T4 of v4 ¼ cos t þ cos 1:4142t. pﬃﬃﬃ Ans. (a) 2 is not a rational number. Therefore, v1 is not periodic. (b) T2 ¼ 10 s. (c) T3 ¼ 200 s. (d) T4 ¼ 10 000 s.

A random signal sðtÞ with an rms value of 5 V has a dc value of p 2 V. Findp the ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ rms value of s0 ðtÞ ¼ sðtÞ  2, that is, when the dc component is removed. Ans: S0;eff ¼ 52  4 ¼ 21 ¼ 4:58 V

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An