Page 99 - Circuit Analysis II with MATLAB Applications

The Unit Step Function

­ 0 t t 0
u t – t 0 = ® (3.5)

0
¯ 1 t ! t 0
1
If the unit step function changes abruptly from to at t = t – 0 , it is denoted as u t + 0 t 0 . Its
0
waveform and definition are as shown in Figure 3.5 and relation (3.6).

1 u t + 0 t 0

t 0 0 t

Figure 3.5. Waveform for u t + t 0

0
­ 0 t t – 0
u t + 0 t 0 = ® (3.6)
¯ 1 t ! t – 0

Example 3.1

Consider the network of Figure 3.6, where the switch is closed at time t = . T

R

v S t = T +
+ v out open terminals

Figure 3.6. Network for Example 3.1

Express the output voltage v out as a function of the unit step function, and sketch the appropriate
waveform.

Solution:
For this example, the output voltage v out = 0 for t T , and v out = v S for t ! T . Therefore,

=
v out v u t – T (3.7)
S 0
and the waveform is shown in Figure 3.7.

Circuit Analysis II with MATLAB Applications 3-3
Orchard Publications