Page 103 - Circuit Analysis II with MATLAB Applications

The Unit Step Function

2 vt 2
--- t + 1 1 – --- t + 1
T T
{ |
t
– T2 0 T2
e
e
Figure 3.13. Equations for the linear segments of Figure 3.12
For line segment {,

·
--- t +
v t = § © 2 1 u 0 § © t + T · --- – u t (3.15)
1
0
¹
¹
T
2
and for line segment |,
·
v t = § © – 2 1 u t – u 0 § © t – T · --- ¹ (3.16)
--- t +
2
0
¹
T
2
Combining (3.15) and (3.16), we get
vt = v t + v t
2
1
2
·
·
= § --- t + 1 u § t + T · --- – u t + § – 2 1 u t – u § t – T · --- (3.17)
--- t +
© T ¹ 0 © 2 ¹ 0 © T ¹ 0 0 © 2 ¹
Example 3.5
Express the waveform of Figure 3.14 as a sum of unit step functions.
vt
3

2

1

t
0 1 2 3

Figure 3.14. Waveform for Example 3.5.
Solution:

As in the previous example, we first find the equations of the linear segments { and | shown in Fig-
ure 3.15.

Circuit Analysis II with MATLAB Applications 3-7
Orchard Publications