BLM1612 Circuit Theory
The RLC Circuit
1
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The RLC Circuit
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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The Source-Free Parallel Circuit
Apply KCL and
differentiate to show:
2
d v 1 dv 1
C 2+
+ v =0
dt
R dt L
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Solving the Differential Equation
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
s1 t
s2 t
1
2
v(t) = A e + A e
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Exploring the Solution
The solutions to the characteristic equation are
 1 
1
1
−
± 
 −
 2RC  LC
2RC
2
Define ω0 the resonant frequency:
and α the damping coefficient:
ω0 = 1
LC
1
α=
2RC
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Exploring the Solution
With these definitions, the solutions can be
expressed as:
2
2
0
2
2
0
s1 = −α + α − ω
s2 = −α − α − ω
The constants A1 and A2 are determined by the
initial conditions.
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Types of Responses
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.
2
2
0
2
2
0
s1 = −α + α − ω
s2 = −α − α − ω
[If α=ω0 the solutions are
real and equal and the
response is termed
critically damped. ]
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Overdamped Parallel RLC
Show that v(t) = 84(e−t − e−6t ) when i(0+)=10 A and
v(0+)=0 V.
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Graphing the Response
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Example 9.2: RLC Circuit
Show that vC(t) = 80e−50,000t − 20e−200,000t V for t>0.
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Critical Damping Example 9.5
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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The Underdamped Response
If α<ω0, define
s1 = −α + α 2 − ω 02
2
ωd = ω0 − α
s2 = −α − α 2 − ω 02
2
and the solution is
v(t) = e
−αt
(A e
1
jω d t
+ A2e
−ω d t
)
or equivalently
v(t) = e
−αt
(B cos(ω t )+ B sin(ω t))
1
d
2
d
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Euler’s Identity and Exponential Form
Euler’s Identity
if we have a complex number which is given in
rectangular form,
– and wish to express it in exponential form,
Example: Underdamped Response
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Comparing the Responses
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Determining the Response: Example 9.6
Show for t>0
iL = e−1.2t (2.027 cos 4.75t + 2.561 sin 4.75t)
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Source-Free Series RLC Circuit
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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Series RLC Differential Equation
The characteristic equation is
1
Ls + Rs + = 0
C
2
and the solution is
s1 t
v(t) = A1e + A2e
where
s2 t
R
R
1
s1,s2 = −
±   −
 2L  LC
2L
2
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Series RLC Circuit Solution
Define ω 0 = 1
s1 = −α + α 2 − ω 02
R
α=
2L
and
LC
Then if
α>ω0 (overdamped):
s2 = −α − α 2 − ω 02
s1 t
v(t) = A1e + A2e
α=ω0 (critically damped):
v(t) = e
(
−αt
s2 t
(A1t + A2 )
)
v(t) = e −αt B1 cos(ω d t )+ B2 sin(ω d t )
α<ω0 (underdamped):
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Summary of Relevant Equations for Source-Free RLC Circuits
The Complete Response
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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Example: Initial Conditions
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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Example: Initial Slopes
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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Example: Complete Response
Show that for t>0
vC(t) = 150 + 13.5(e−t − e−9t ) volts
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Lecture 9 Summary
When you are presented with a DC-switched or DC-pulsed RLC circuit and asked
to solve for v(t) or i(t) somewhere in the circuit, for all time
(1)
Identify the circuit as containing two (and only two) energy storage
elements (i.e. a single Leq and a single Ceq).
(2)
Determine 1 initial condition just before the switch/pulse,e.g. v(0–)
(3)
Determine 2 conditions just after the switch/pulse, e.g. v(0+) and
d/dt{v(0+)} , with the knowledge that vC(0–) = vC(0+) , iL(0–) = iL(0+) ,
iC(0+) = C d/dt{vC(0+)}, vL(0+) = L d/dt{iL(0+)} .
(4)
Determine final conditions (a long time after switch/pulse),e.g. v(∞) .
(5)
Identify the circuit as a series or parallel RLC circuit and find the Thevenin
equivalent resistance seen by the LC pair, Rth .
(6)
Calculate α and ω0 and determine the circuit damping (over, under, critical) .
(7)
Assume the appropriate solution form and use α, ω0 , initial conditions, &
the final condition to determine the complete (particular) solution.