Page 120

CHAP. 6]

WAVEFORMS AND SIGNALS



0 sin !t

for t < 0 for t > 0

109

(24)

(d)

v4 ðtÞ ¼

(e)

v5 ðtÞ ¼

(f)

v6 ðtÞ ¼ et=

for all t

(26)

(g)

v7 ðtÞ ¼ eajtj

for all t

(27)

(h)

ajtj



v8 ðtÞ ¼ e

0 et= cos !t

cos !t

for t < 0 for t > 0

for all t

(25)

(28)

Several of these functions are used as mathematical models and building blocks for actual signals in analysis and design of circuits. Examples are discussed in the following sections.

6.8

THE UNIT STEP FUNCTION The dimensionless unit step function, is defined by  0 for t < 0 uðtÞ ¼ 1 for t > 0 The function is graphed in Fig. 6-7.

ð29Þ

Note that the function is undefined at t ¼ 0.

Fig. 6-7

To illustrate the use of uðtÞ, assume the switch S in the circuit of Fig. 6-8(a) has been in position 1 for t < 0 and is moved to position 2 at t ¼ 0. The voltage across A-B may be expressed by vAB ¼ V0 uðtÞ. The equivalent circuit for the voltage step is shown in Fig. 6-8(b).

Fig. 6-8

EXAMPLE 6.14 The switch in the circuit of Fig. 6-8(a) is moved to position 2 at t ¼ t0 . Express vAB using the step function. The appearance of V0 across A-B is delayed until t ¼ t0 . Replace the argument t in the step function by t  t0 and so we have vAB ¼ V0 uðt  t0 Þ:

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