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 et=

for all t

(26)

(g)

v7 ðtÞ ¼ eajtj

for all t

(27)

(h)

ajtj



v8 ðtÞ ¼ e

0 et= 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:

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An