Page 106 - Circuit Analysis II with MATLAB Applications

Chapter 3 Elementary Signals

3.3 The Unit Ramp Function u t
1

The unit ramp function, denoted as u t , is defined as
1
t
d
u t = ³ u W W (3.23)
0
1
– f
where is a dummy variable.
W
We can evaluate the integral of (3.23) by considering the area under the unit step function u t from
0
– f to t as shown in Figure 3.18.

Area = 1 W = W = t
u
1

t
W

Figure 3.18. Area under the unit step function from f to t–

Therefore, we define u t as
1
­ 0 t 0
u t = ® (3.24)
1
¯ t t t 0

Since u t is the integral of u t , then u t must be the derivative of u t , i.e.,
1
0
1
0
d
-----u t = u t (3.25)
dt 1 0
t
Higher order functions of can be generated by repeated integration of the unit step function. For
example, integrating u t twice and multiplying by 2, we define u t as
0
2
­ 0 t 0 t
d
u t = ® 2 or u t = 2 ³ u W W (3.26)
2
2
1
t ¯ t t 0 – f
Similarly,
­ 0 t 0 t
d
u t = ® 3 or u t = 3 ³ u W W (3.27)
2
3
3
t ¯ t t 0 – f
and in general,

3-10 Circuit Analysis II with MATLAB Applications

Orchard Publications