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Chapter 9
The RLC Circuit
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an RLC circuit hasboth an inductor and a
capacitor
these circuits have a wide range ofapplications, including oscillators and
frequency filters
they also can model automobile suspensionsystems, temperature controllers, airplane
responses, and so forth
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Apply KCL and
differentiate to show:
Cd2
v
dt2 1
R
dv
dt 1
Lv 0
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To solve, assume v=Aest .
The solution must then satisfy
which is called the characteristic equation.
Ifs1ands2are the solutions, then the natural response is
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Cs2 1Rs1
L 0
v(t)A1es1t A2e
s2t
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The solutions to the characteristic equation are
Define 0
the resonant frequency:
and the damping coefficient:
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12RC
12RC
2
1LC
0
1LC
1
2RC
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With these definitions, the solutions can be
expressed as:
The constantsA1andA2are determined by the
initial conditions.
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s1 2 0
2
s2 2 0
2
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If >0the solutions are
real, unequal and the
response is termedoverdamped.
If
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Show that v(t) = 84(et e6t) when i(0+)=10 A
and v(0+)=0 V.
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Show that vC(t) = 80e50,000t 20e200,000t V for t>0.
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FindR1such that the circuit is critically damped
for t>0 andR2so that v(0)=2 V.
Answer: R1= 31.63 k, R2=0.4
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If
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Show for t>0
iL= e1.2t (2.027 cos 4.75t + 2.561 sin 4.75t)
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For the series RLC circuit,
This circuit is the dual of the parallel RLC
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Ld2i
dt2R
di
dt1
Ci 0
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The characteristic equation is
and the solution is
where
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Ls
2
Rs
1
C 0
v(t)A1es1t A2e
s2t
s1,s2 R
2L
R
2L
2
1
LC
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Define and
Then if>0 (overdamped):
=0 (critically damped):
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The response of RLC circuits with dc sources
and switches will consist of the natural
response and the forced response:
v(t) = vf(t)+vn(t)
The complete response must satisfy both the
initial conditions and the final conditions or
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Find the labeled voltages and currents at t=0-and
t=0+.
Answer:iR(0
) = 5 A vR(0) = 150 V iR(0
+) = 1 A vR(0+) = 30 V
iL(0) = 5 A vL(0
) = 0 V iL(0+) = 5 A vL(0
+) = 120 V
iC(0) = 0 A vC(0
) = 150 V iC(0+) = 4 A vC(0
+) = 150 V
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Find the first derivatives of the labeled voltages
and currents at t=0+.
Answer:diR/dt(0
+) = 40 A/s dvR/dt(0+) = -1200 V/s
diL/dt(0+) = 40 A/s dvL/dt(0
+) = -1092 V/s
diC/dt(0+) = -40 A/s dvC/dt(0
+) = 108 V/s
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Show that for t>0
vC(t) = 150 + 13.5(et e9t)volts
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The resistor in the RLC circuit serves to
dissipate initial stored energy.
When this resistor becomes 0 in the seriesRLC or infinite in the parallel RLC, the circuit
will oscillate.
Example: for t>0,
v(t) =2 sin 3t
ifi(0)=-1/6 Aandv(0)=0 V
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