Page 107 - Circuit Analysis II with MATLAB Applications

The Unit Ramp Function

­ 0 t 0 t
u t = ® n or u t = 3 ³ u n – 1 W W (3.28)
d
n
n
t ¯ t t 0 – f
Also,
1 d
-
u n – 1 t = -------u t (3.29)
n
ndt
Example 3.7

In the network of Figure 3.19, the switch is closed at time t = 0 and i t = 0 for t . 0
L

R t = 0

i S +

i t ` v t
L
L
L

Figure 3.19. Network for Example 3.7
Express the inductor current i t in terms of the unit step function.
L
Solution:

The voltage across the inductor is

di
v t = L------- L (3.30)
L
dt
and since the switch closes at t = , 0
i t = i u t (3.31)
0
L
S
Therefore, we can write (3.30) as

d
v t = Li -----u t (3.32)
L
S
0
dt
1
0
But, as we know, u t is constant ( or ) for all time except at t = 0 where it is discontinuous.
0
Since the derivative of any constant is zero, the derivative of the unit step u t has a non-zero value
0
only at t = 0 . The derivative of the unit step function is defined in the next section.

Circuit Analysis II with MATLAB Applications 3-11
Orchard Publications