Page 120 - Schaum's Outline of Theory and Problems of Electric Circuits

WAVEFORMS AND SIGNALS
CHAP. 6]
109
0 for t < 0
(d) v 4 ðtÞ¼ (24)
sin !t for t > 0

0 for t < 0
(e) v 5 ðtÞ¼ t= (25)
e cos !t for t > 0
t=
( f ) v 6 ðtÞ¼ e for all t (26)
(g) v 7 ðtÞ¼ e ajtj for all t (27)
(h) v 8 ðtÞ¼ e ajtj cos !t for all t (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
uðtÞ¼ ð29Þ
1 for t > 0
The function is graphed in Fig. 6-7. 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 v AB ¼ V 0 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 t 0 . Express v AB using the
step function.
The appearance of V 0 across A-B is delayed until t ¼ t 0 . Replace the argument t in the step function by t t 0
and so we have v AB ¼ V 0 uðt t 0 Þ: