Chapter 9
The RLC Circuit
1
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an RLC circuit has both an inductor and a
capacitor
these circuits have a wide range of
applications, including oscillators and
frequency filters
they also can model automobile suspension
systems, temperature controllers, airplane
responses, and so forth
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2
Apply KCL and
differentiate to show:
2
d v 1 dv 1
C 2
 v 0
dt
R dt L
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3
To solve, assume v=Aest .
The solution must then satisfy
1
1
Cs  s   0
R
L
2
which is called the characteristic equation.
If s1 and s2 are the solutions, then the natural response is

v(t)  A1e  A2e
s1 t
s2 t
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4
The solutions to the characteristic equation are
 1 
1
1

 
 
2RC  LC
2RC
2
Define ω0 the resonant frequency:  0  1
andα the damping coefficient:
LC
1

2RC

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5
With these definitions, the solutions can be
expressed as:
s1     2   02
s2     2   02
The constants A1 and A2 are determined by the
initial conditions.

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6
If α>ω0 the solutions are
real, unequal and the
response is termed
overdamped.
If α<ω0 the solutionsare
complex conjugates and
the response is termed
underdamped.
s1      
2
2
0
s2      
2
2
0
[If α=ω0 the solutions are
real and equal and the
response is termed
critically damped. ]
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7
Show that v(t) = 84(e−t − e−6t ) when i(0+)=10 A
and v(0+)=0 V.
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8
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9
Show that vC(t) = 80e−50,000t − 20e−200,000t V for t>0.
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10
Find R1 such that the circuit is critically damped
for t>0 and R2 so that v(0)=2 V.
Answer: R1 = 31.63 kΩ, R2=0.4Ω
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11
s1      
2
If α<ω0, define
d  0  
2
s2      
2
2
and the solution is
v(t)  e
t
v(t)  e
A e
1
or equivalently
t
2
0
j d t
 A2e
 d t
2
0

B cos t B sin  t
1
d
2
d
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13
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14
Show for t>0
iL = e−1.2t (2.027 cos 4.75t + 2.561 sin 4.75t)
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15
For the series RLC circuit,
2
di
di 1
L 2 R  i 0
dt
dt C
This circuit is the dual of the parallel RLC
circuit.
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16
The characteristic equation is
1
2
Ls  Rs   0
C
and the solution is
v(t)  A1e  A2e
s1 t
where 

s2 t
 R 
R
1
s1,s2  
   
2L  LC
2L
2
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Define  0  1
R
and  
LC
2L
Then if
α>ω0 (overdamped):


α=ω0 (critically damped):
α<ω0 (underdamped):

s1     2   02
s2     2   02
v(t)  A1e  A2e
s1 t
v(t)  e
t
s2 t
A1t  A2 


v(t)  e t B1 cos d t  B2 sin  d t 
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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
the forced response.
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19
Find the labeled voltages and currents at t=0- and
t=0+.
Answer:
iR(0−) = −5 A vR(0−) = −150 V
iL (0−) = 5 A
vL (0−) = 0 V
iC(0−) = 0 A
vC(0−) = 150 V
iR(0+) = −1 A
iL (0+) = 5 A
iC(0+) = 4 A
vR(0+) = −30 V
vL (0+) = 120 V
vC(0+) = 150 V
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20
Find the first derivatives of the labeled voltages
and currents at t=0+.
Answer:
diR/dt(0+) = −40 A/s
diL /dt(0+) = 40 A/s
diC/dt(0+) = -40 A/s
dvR/dt(0+) = -1200 V/s
dvL /dt(0+) = -1092 V/s
dvC/dt(0+) = 108 V/s
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21
Show that for t>0
vC(t) = 150 + 13.5(e−t − e−9t ) volts
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22
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The resistor in the RLC circuit serves to
dissipate initial stored energy.
When this resistor becomes 0 in the series
RLC or infinite in the parallel RLC, the circuit
will oscillate.
Example: for t>0,
v(t) =2 sin 3t
if i(0)=-1/6 A and v(0)=0 V
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