Page 105 - Circuit Analysis II with MATLAB Applications

The Unit Step Function

Express the capacitor voltage v t as a function of the unit step.
C
Solution:

The current through the capacitor is i t = i = constan t , and the capacitor voltage v t is
S
C
C
1 t *
v t = ---- ³ i W Wd (3.19)
C
C
C
– f
where is a dummy variable.
W
Since the switch closes at t = 0 , we can express the current i t as
C
i t = i u t (3.20)
S
0
C
and assuming that v t = 0 for t 0 , we can write (3.19) as
C
i S 0
1 t ---- ³ u W Wd i S t
0
v t = ---- ³ i u W Wd = C ­ ° – f ° ® ° ° ¯ + ---- ³ u W Wd (3.21)
S
C
0
0
C
C
–
f
0 0
or
i
S
v t = ----- tu t (3.22)
C
0
C
Therefore, we see that when a capacitor is charged with a constant current, the voltage across it is a
linear function and forms a ramp with slope i e C as shown in Figure 3.17.
S
v t
C
slope = i e C
S
t
0

Figure 3.17. Voltage across a capacitor when charged with a constant current source.

–
* Since the initial condition for the capacitor voltage was not specified, we express this integral with f at the
lower limit of integration so that any non-zero value prior to t 0 would be included in the integration.

Circuit Analysis II with MATLAB Applications 3-9
Orchard Publications