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Waveforms and Signals
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6.1 INTRODUCTION The voltages and currents in electric circuits are described by three classes of time functions: (i) (ii) (iii) Periodic functions Nonperiodic functions Random functions
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In this chapter the time domain of all functions is 1 < t < 1 and the terms function, waveform, and signal are used interchangeably.
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PERIODIC FUNCTIONS A signal v t is periodic with period T if v t v t T for all t
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Four types of periodic functions which are speci ed for one period T and corresponding graphs are as follows: (a) Sine wave: v1 t V0 sin 2t=T See Fig. 6-1(a). 1
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Fig. 6-1(a)
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WAVEFORMS AND SIGNALS
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(b) Periodic pulse:  v2 t See Fig. 6-1(b). V1 V2 for 0 < t < T1 for T1 < t < T 2
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Fig. 6-1(b)
(c) Periodic tone burst:  v3 t where T k and k is an integer. V0 sin 2t= 0 See Fig. 6-1(c). for 0 < t < T1 for T1 < t < T 3
Fig. 6-1(c)
(d) Repetition of a recording every T seconds: v4 t See Fig. 6-1(d). 4
Fig. 6-1(d)
Periodic signals may be very complex. However, as will be seen in 17, they may be represented by a sum of sinusoids. This type of function will be developed in the following sections.
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WAVEFORMS AND SIGNALS
SINUSOIDAL FUNCTIONS A sinusoidal voltage v t is given by v t V0 cos !t 
where V0 is the amplitude, ! is the angular velocity, or angular frequency, and  is the phase angle. The angular velocity ! may be expressed in terms of the period T or the frequency f, where f  1=T. The frequency is given in hertz, Hz, or cycles/s. Since cos !t cos t !t 2 , ! and T are related by !T 2. And since it takes T seconds for v t to return to its original value, it goes through 1=T cycles in one second. In summary, for sinusoidal functions we have ! 2=T 2f f 1=T !=2 T 1=f 2=!
EXAMPLE 6.1 Graph each of the following functions and specify period and frequency. a v1 t cos t b v2 t sin t c v3 t 2 cos 2t
d v4 t 2 cos t=4 458 2 cos t=4 =4 2 cos  t 1 =4 e v5 t 5 cos 10t 608 5 cos 10t =3 5 cos 10 t =30 (a) See Fig. 6-2(a). (b) See Fig. 6-2(b). (c) (e) See Fig. 6-2(c). See Fig. 6-2(e). (d) See Fig. 6-2(d). T 2 6:2832 s and f 0:159 Hz. T 2 6:2832 s and f 0:159 Hz. T 1 s and f 1 Hz. T 8 s and f 0:125 Hz. T 0:2 0:62832 s and f 1:59 Hz.
EXAMPLE 6.2 Plot v t 5 cos !t versus !t. See Fig. 6.3.
TIME SHIFT AND PHASE SHIFT
If the function v t cos !t is delayed by  seconds, we get v t  cos ! t  cos !t  , where  !. The delay shifts the graph of v t to the right by an amount of  seconds, which corresponds to a phase lag of  ! 2f . A time shift of  seconds to the left on the graph produces v t  , resulting in a leading phase angle called an advance. Conversely, a phase shift of  corresponds to a time shift of . Therefore, for a given phase shift the higher is the frequency, the smaller is the required time shift.
EXAMPLE 6.3 Plot v t 5 cos t=6 308 versus t and t=6. Rewrite the given as v t 5 cos t=6 =6 5 cos  t 1 =6 This is a cosine function with period of 12 s, which is advanced in time by 1 s. the left by 1 s or 308 as shown in Fig. 6-4. In other words, the graph is shifted to
EXAMPLE 6.4 Consider a linear circuit with the following input-output pair valid for all ! and A: Input: vi t A cos !t Output: v0 t A cos !t 
Given vi t cos !1 t cos !2 t, nd v0 t when (a)  10 6 ! [phase shift is proportional to frequency, Fig. 6-5(a)] (b)  10 6 [phase shift is constant, Fig. 6-5(b)] The output is v0 t cos !1 t 1 cos !2 t 2 .
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