Transients in DC Circuits II
1- Natural Behavior of Coil-Capacitor-Resistor Circuit Natural Response of RLC Circuit
The study of the natural discharge of the circuit means finding the
relationship of the voltage on the components of the connected circuit in
parallel when the energy stored in the coil or capacitor is released. The
circuit shown in Figure (1) can be used, where the initial value of the voltage
on the capacitor V0 expresses the energy stored in the capacitor, and the
initial current I0 in the coil expresses the energy stored in the coil.
Figure (1)
The natural behavior will be determined by finding the differential
equations of voltage v by aggregating the current at the top point and
calculating the current in terms of voltage as follows:
(1)
(2)
(3)
2- Calculating the temporal disposition of the state of sudden change of a resistor-coil-capacitor circuit in parallel
Step Response of Parallel RLC Circuit
This part involves finding the voltage in the case of a sudden
application of a voltage source. Figure (2) shows the circuit used in the
analysis.
Figure (2)
The focus will be on the current in the file iL as it represents
the final value of the current drawn from the source, and it will be assumed
that there is no energy stored in the circuit. A quadratic differential
equation needs to be solved and by applying Kirchhoff's law of current, we get:
(4)
or
(5)
(6)
(7)
(8)
(9)
Solving the equation:
The equation can be solved in terms of the
initial values of the function iL(0) and
diL(0)/dt, and thus we get a solution
that includes two parts: the natural behavior and the imposed behavior.
i = If
+ function of the same form as the natural response (10)
or
v = Vf + function of the same form
as the natural response (11)
where if and vf are the final values of the
function.
3- Calculate the natural time disposition of the coil-capacitor-resistor circuit respectively
The natural responses of a series RLC circuits
The normal temporal behavior can be calculated
by analyzing the circuit shown in Figure (3), and we start by calculating the
voltage in the closed circuit as follows:
(12)
Differentiating with respect to time, we
get:
(13)
(14)
Figure (3)
The special equation for the department can be
found as follows:
(15)
The roots of the equation are as follows:
(16)
or
(17)
The Nieper frequency (α) can be found as follows:
(18)
And the frequency of ringing:
(19)
There are three solutions for the current:
(overdamped)
(20)
(underdamped) (21)
(critically
damped) (22)
4- Calculating the temporal disposition of the state of sudden change of a resistor-coil-capacitor circuit respectively
Step Response of Series RLC Circuit
The solution method is similar to the circuit in
parallel, but the solution is done in relation to the voltage on the capacitor,
and the circuit shown in Figure (4) will be used in the analysis with the
assumption that the energy initially stored is equal to zero.
Figure (4)
Applying Kirchhoff's law of effort yields:
(23)
The current i is related to the voltage on the capacitor vC in
relation to:
(24)
(25)
(26)
There are three solutions:
(overdamped) (27)
(underdamped) (28)
(critically damped) (29)
where Vf is the final value of the voltage on the capacitor vC, which as shown in Figure 5 will be equal to
the source voltage V.
Example (5):
The initial energy
stored in the circuit is zero. At t = 0, a dc current source of 24 mA is
applied to the circuit. The value of the resistor is 400 Ω.
1) What is the initial value of iL?
2) What is the initial value of diL/dt?
3) What are the roots of the
characteristic equation?
4) What is the numerical expression
for iL(t) when 
?
Solution:
1) Since the initial current in the
inductor is zero, the inductor prohibits an instantaneous change in its
current. Therefore IL(0) = 0 immediately after the switch has been
opened.
2) The voltage at the capacitor is
zero at t = 0. Because v = L diL/dt, then:
3) From the circuit elements, we
obtain:
and
Because
, the roots of the characteristic equation are real and distinct, thus:
4) The inductor current response
will be overdamped. Thus:
Hence, from this solution, the two simultaneous equations which determine
and
are:
Solving for
and
gives:
and
The numerical solution for iL(t)
is
Example (6):
No energy is stored in the inductor or the capacitor when the switch is closed. Find vC(t) for
.
Solution:
The roots of the
characteristic equation are
The roots are complex,
so the voltage response is underdamped. Thus
No energy is stored in
the circuit initially, so both vC(0) and dvC(0+)/dt
are zero. Then:
and
Solving these two
equations:
and
Therefore the solution
for vC(t) is
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