Biomedical Engineering Reference
In-Depth Information
EXAMPLE PROBLEM 9.13
Find
i
for the following circuit.
i
v
(V)
1
+
−
v
2 F
0
1
2
t
(s)
Solution
For
t
<
0 and
t
>
2,
v
¼
0 V, and therefore
i
¼
0 in this interval
:
For nonzero values, the voltage
waveform is described with two different functions:
v
¼
t
V for 0
t
1, and
v
¼
t
ð
2
Þ
V
for 1
<
t
2
:
Equation (9.24) is used to determine the current for each interval as follows.
For 0 < t < 1
i
¼
C
dv
d
dt
dt
¼
2
ðÞ¼
2A
For 1 t 2
i
¼
C
dv
d
dt
¼
2
dt
t
ð
ð
2
Þ
Þ ¼
2A
Voltage cannot change instantaneously across a capacitor. To have a step change in volt-
age across a capacitor, an infinite current must flow through the capacitor, and that is not
physically possible. Of course, this is mathematically possible using a Dirac delta function.
Equation (9.24) defines the current through a capacitor for a given voltage. Suppose one
is given a current through a capacitor and asked to find the voltage. To find the voltage, we
start from Eq. (9.24) by multiplying both sides by
dt,
giving
i
ðÞ
dt
¼
Cdv
Integrating both sides yields
Z
t
Z
v
ð
t
Þ
i
ðÞ
d
l ¼
C
dv
t
o
v
ð
t
o
Þ