Page 117 - Schaum's Outline of Theory and Problems of Electric Circuits

WAVEFORMS AND SIGNALS
106
6.5 COMBINATIONS OF PERIODIC FUNCTIONS [CHAP. 6

The sum of two periodic functions with respective periods T 1 and T 2 is a periodic function if a
common period T ¼ n 1 T 1 ¼ n 2 T 2 , where n 1 and n 2 are integers, can be found. This requires
T 1 =T 2 ¼ n 2 =n 1 to be a rational number. Otherwise, the sum is not a periodic function.
EXAMPLE 6.5 Find the period of vðtÞ¼ cos 5t þ 3 sinð3t þ 458Þ.
The period of cos 5t is T 1 ¼ 2 =5 and the period of 3 sinð3t þ 458Þ is T 2 ¼ 2 =3. Take T ¼ 2 ¼ 5T 1 ¼ 3T 2
which is the smallest common integral multiple of T 1 and T 2 . Observe that vðt þ TÞ¼ vðtÞ since
vðt þ TÞ¼ cos 5ðt þ 2 Þþ 3 sin½3ðt þ 2 Þþ 458Š¼ cos 5t þ 3 sinð3t þ 458Þ¼ vðtÞ
Therefore, the period of vðtÞ is 2 .

EXAMPLE 6.6 Is vðtÞ¼ cos t þ cos 2 t periodic? Discuss.
The period of cos t is T 1 ¼ 2 . The period of cos 2 t is T 2 ¼ 1. No common period T ¼ n 1 T 1 ¼ n 2 T 2 exists
because T 1 =T 2 ¼ 2 is not a rational number. Therefore, vðtÞ is not periodic.
EXAMPLE 6.7 Given p ¼ 3:14, find the period of vðtÞ¼ cos t þ cos 2pt.
The period of cos t is T 1 ¼ 2 and the period of cos 2pt is T 2 ¼ =3:14. The ratio T 1 =T 2 ¼ 6:28 is a rational
number. The integer pair n 1 ¼ 25 and n 2 ¼ 157 satisfies the relation n 2 =n 1 ¼ T 1 =T 2 ¼ 628=100 ¼ 157=25. There-
fore, vðtÞ is periodic with period T ¼ n 1 T 1 ¼ n 2 T 2 ¼ 50 s.

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

Table 6-1

sin a ¼ sinð aÞ (5a)
cos a ¼ cos ð aÞ (5b)
sin a ¼ cos ða 908Þ (5c)
cos a ¼ sinða þ 908Þ (5d)
sin 2a ¼ 2 sin a cos a (6a)
2
2
2
2
cos 2a ¼ cos a sin a ¼ 2 cos a 1 ¼ 1 2 sin a (6b)
1 cos 2a (7a)
2
sin a ¼
2
1 þ cos 2a (7b)
2
cos a ¼
2
sinða þ bÞ¼ sin a cos b þ cos a sin b (8a)
cosða þ bÞ¼ cos a cos b sin a sin b (8b)
1
1
sin a sin b ¼ cos ða bÞ cos ða þ bÞ (9a)
2 2
1
1
sin a cos b ¼ sin ða þ bÞþ sin ða bÞ (9b)
2 2
1
1
cos a cos b ¼ cos ða þ bÞþ cos ða bÞ (9c)
2 2
1
1
sin a þ sin b ¼ 2 sin ða þ bÞ cos ða bÞ (10a)
2 2
1
1
cos a þ cos b ¼ 2 cos ða þ bÞ cos ða bÞ (10b)
2 2
EXAMPLE 6.8 Express vðtÞ¼ cos 5t sinð3t þ 458Þ as the sum of two cosine functions and find its period.
vðtÞ¼ cos 5t sinð3t þ 458Þ¼½sinð8t þ 458Þ sinð2t 458ފ=2 [Eq. ð9bފ
¼½cos ð8t 458Þþ cos ð2t þ 458ފ=2 [Eq. (5cފ
The period of vðtÞ is .