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Fig. 6-2

CHAP. 6]

Fig. 6-3

Fig. 6-4

Fig. 6-5

(a) 1 10 6 !1 , 2 10 6 !2 . Then v0 t cos !1 t 10 6 !1 cos !2 t 10 6 !2 cos !1 t 10 6 cos !2 t 10 6 vi t 10 6 vi t  where  10 6 s 1 ms. Thus a phase shift proportional to ! [Fig. 6-5(a)] delays all frequency components of the input signal by 1 ms. The output follows the input with no distortion. (b) 1 2 10 6 . Then v0 t cos !1 t 10 6 cos !2 t 10 6 cos !1 t 10 6 =!1 cos !2 t 10 6 =!2 A constant phase shift [Fig. 6-5(b)] delays di erent frequency components of the input signal by di erent amounts. The output is a distorted form of the input.

The sum of two periodic functions with respective periods T1 and T2 is a periodic function if a common period T n1 T1 n2 T2 , where n1 and n2 are integers, can be found. This requires T1 =T2 n2 =n1 to be a rational number. Otherwise, the sum is not a periodic function.

EXAMPLE 6.5 Find the period of v t cos 5t 3 sin 3t 458 . The period of cos 5t is T1 2=5 and the period of 3 sin 3t 458 is T2 2=3. Take T 2 5T1 3T2 which is the smallest common integral multiple of T1 and T2 . Observe that v t T v t since v t T cos 5 t 2 3 sin 3 t 2 458 cos 5t 3 sin 3t 458 v t Therefore, the period of v t is 2. EXAMPLE 6.6 Is v t cos t cos 2t periodic Discuss. The period of cos t is T1 2. The period of cos 2t is T2 1. No common period T n1 T1 n2 T2 exists because T1 =T2 2 is not a rational number. Therefore, v t is not periodic. EXAMPLE 6.7 Given p 3:14, nd the period of v t cos t cos 2pt. The period of cos t is T1 2 and the period of cos 2pt is T2 =3:14. The ratio T1 =T2 6:28 is a rational number. The integer pair n1 25 and n2 157 satis es the relation n2 =n1 T1 =T2 628=100 157=25. Therefore, v t is periodic with period T n1 T1 n2 T2 50 s.

Trigonometric Identities The trigonometric identities in Table 6-1 are useful in the study of circuit analysis.

Table 6-1 sin a sin a cos a cos a sin a cos a 908 cos a sin a 908 sin 2a 2 sin a cos a cos 2a cos2 a sin2 a 2 cos2 a 1 1 2 sin2 a 1 cos 2a sin2 a 2 1 cos 2a 2 cos a 2 sin a b sin a cos b cos a sin b cos a b cos a cos b sin a sin b sin a sin b 1 cos a b 1 cos a b 2 2 sin a cos b 1 sin a b 1 sin a b 2 2 cos a cos b 1 cos a b 1 cos a b 2 2 sin a sin b 2 sin 1 a b cos 1 a b 2 2 cos a cos b 2 cos 1 a b cos 1 a b 2 2 (5a) (5b) (5c) (5d) (6a) (6b) (7a) (7b) (8a) (8b) (9a) (9b) (9c) (10a) (10b)

EXAMPLE 6.8 Express v t cos 5t sin 3t 458 as the sum of two cosine functions and nd its period. v t cos 5t sin 3t 458 sin 8t 458 sin 2t 458 =2 [Eq. 9b cos 8t 458 cos 2t 458 =2 [Eq. (5c The period of v t is .

CHAP. 6]