a. Find the mathematical expression for the voltage across the capacitor
once the switch is closed.
b. Find the mathematical expression for the current during the transient
period.
c. Sketch the waveform for each from initial value to final value.
60
11
Example 3Cont’d
61
•
Solution:
a. Substituting the opencircuit equivalent for the capacitor will result in a final or
steadystate voltage
v
C
of 24 V
.
The time constant is determined by:
And:
Example 3Cont’d
62
b. Since the voltage across the capacitor is constant at 4 V prior to the closing of
the switch, the current (whose level is sensitive only to changes in voltage across
the capacitor) must have an initial value of 0 mA. At the instant the switch is
closed, the voltage across the capacitor cannot change instantaneously, so the
voltage across the resistive elements at this instant is the applied voltage less the
initial voltage across the capacitor. The resulting peak current is:
The current will then decay (with the same time constant as the voltage
v
C
)
to zero
because the capacitor is approaching its
open circuit equivalence. The equation
for
i
C
is therefore
:
Example 3Cont’d
63
c.
Example 4
•
For the circuit shown below:
a. Find the mathematical expression for the transient behaviour of the
voltage
v
C
and the current
i
C
following the closing of the switch
(position 1 at t=0 s).
b. Find the mathematical expression for the voltage
v
C
and current
i
C
as a function of time if the switch is thrown into position 2 at
t=
9 ms.
c. Draw the resultant waveforms of parts (a) and (b) on the same time
axis.
64
Example 4Cont’d
65
a. Applying Thévenin’s theorem to the 0.2mF capacitor, we obtain:
Example 4Cont’d
66
12
Example 4Cont’d
67
The resultant Thévenin equivalent circuit with the capacitor replaced is
shown in Fig.
For the current:
Example 4Cont’d
68
b.
Ans:
c.
Prove
Interpretation of
τ
•
In general, you should note that an equation of the form:
means
exponential growth (!)
. The time constant,
τ,
is the amount of
time necessary for an exponential to grow to
0.63
of its final value.
/
(1
)
t
y
Y
e
τ

=

t
y
0
Y
t=
τ
0.63Y
69
Charging
Interpretation of
τ
Cont’d
•
An equation of the form:
means
exponential decay
. The time constant,
τ,
is the amount of time
necessary for an exponential to decay to
36.7%
of its initial value.
/
t
y
Ye
τ

=
t
y
0
Y
t=
τ
0.37Y
70
Discharging
Example 1
•
Find the mathematical expressions for the transient behaviour of
i
L
and v
L
for the circuit below after the closing of the
switch. Sketch the resulting
curves.
71
Example 1Cont’d
•
Solution:
72
13
Example 2
The inductor of the figure below has an initial current level of 4 mA in the
direction shown.
a.
Find the mathematical expression for the current through the coil once the
switch is closed.
b.
Find the mathematical expression for the voltage across the coil during the
same transient period.
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