118

WAVEFORMS AND SIGNALS

hcos ð2f1 t þ 1 Þ cos ð2f2 t þ 2 Þi ¼

2 ¼ 12 ðV12 þ V22 Þ and Veff ¼ Therefore, Veff

6.7

[CHAP. 6

1 hcos ½2ð f1 þ f2 Þt þ ð1 þ 2 Þi 2 1 þ hcos ½2ð f1  f2 Þt þ ð1  2 Þi ¼ 0 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 2 2 ðV1 þ V2 Þ:

The signal vðtÞ in Fig. 6-16 is sinusoidal. Find its period and frequency. vðtÞ ¼ A þ B cos ð!t þ Þ and ﬁnd its average and rms values.

Express it in the form

Fig. 6-16 The time between two positive peaks, T ¼ 20 s, is one period corresponding to a frequency f ¼ 0:05 Hz. The signal is a cosine function with amplitude B added to a constant value A. B ¼ 12 ðVmax  Vmin Þ ¼ 12 ð8 þ 4Þ ¼ 6

A ¼ Vmax  B ¼ Vmin þ B ¼ 2

The cosine is shifted by 2 s to the right, which corresponds to a phase lag of ð2=20Þ3608 ¼ 368. Therefore, the signal is expressed by   t  368 vðtÞ ¼ 2 þ 6 cos 10 The average and eﬀective values are found from A and B: Vavg ¼ A ¼ 2;

6.8

2 Veff ¼ A2 þ B2 =2 ¼ 22 þ 62 =2 ¼ 22

or

Veff ¼

pﬃﬃﬃﬃﬃ 22 ¼ 4:69

Let v1 ¼ cos 200t and v2 ¼ cos 202t. Show that v ¼ v1 þ v2 is periodic. Find its period, Vmax , and the times when v attains its maximum value. The periods of v1 and v2 are T1 ¼ 1=100 s and T2 ¼ 1=101 s, respectively. The period of v ¼ v1 þ v2 is the smallest common multiple of T1 and T2 , which is T ¼ 100T1 ¼ 101T2 ¼ 1 s. The maximum of v occurs at t ¼ k with k an integer when v1 and v2 are at their maxima and Vmax ¼ 2.

6.9

Convert vðtÞ ¼ 3 cos 100t þ 4 sin 100t to A sinð100t þ Þ.

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Note that 3= 32 þ 42 ¼ 3=5 ¼ sin 36:878 and 4= 32 þ 42 ¼ 4=5 ¼ cos 36:878.

Then,

vðtÞ ¼ 3 cos 100t þ 4 sin 100t ¼ 5ð0:6 cos 100t þ 0:8 sin 100tÞ ¼ 5ðsin 36:878 cos 100t þ cos 36:878 sin 100tÞ ¼ 5 sinð100t þ 36:878Þ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An