rlc circuit transient response solutions
TRANSCRIPT
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EE305 Lecture 6
Aaron Mueller
February 11, 2013
Aaron Mueller EE305 Lecture 6 February 11, 2013 1 / 15
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Roots for various damping ratios
s 1,2 = (−ζ ± ζ 2−1)ω 0
(−ζ ± j
1−ζ 2)ω 0 = σ ± j ω d if ζ
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Location of roots in complex plane
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Location of roots in complex plane
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Total responses for various damping ratios
x (t ) =
( s 2s 1−s 2
)e s 1t + ( s 1s 2−s 1
)e s 2t + 1 if ζ > 1,
−(ω 0t + 1)e −ω 0t + 1 if ζ = 1,−e −σ t (cos(ω d t ) +
σ ω d
sin(ω d t )) + 1 if ζ
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Total responses for various damping ratios
x (t ) =
( s 2s 1−s 2
)e s 1t + ( s 1s 2−s 1
)e s 2t + 1 if ζ > 1,
−(ω 0t + 1)e −ω 0t + 1 if ζ = 1,−e −σ t (cos(ω d t ) +
σ ω d
sin(ω d t )) + 1 if ζ
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General procedure for solving for second order responses
1 Find the homogeneous solution (natural response) to the differential
equation. This will generally contain two unknown coefficients.
x n (t ) = A1x n ,1(t ) +A2x n ,2(t )
2 Find the particular solution (forced response) to the differential equation.
This can often be done using the method of undetermined coefficients.
For a constant forcing function (DC voltage and/or current source), this
solution will be a constant. x F (t ) = x f 3 Construct the total solution, which is the sum of the homogeneous and
particular solutions.
x (t ) = x n (t ) + x F (t ) = A1x n ,1(t ) +A2x n ,2(t ) + x f
4 Solve for the unknown coefficients (A1 and A2) by, for example, using
known the known values for x (t ) and/or its derivative x (t ) at a given timeor times. If these values are not directly given. They can usually be
obtained by using the original (integro-)differential equation and/or other
circuit relations.Aaron Mueller EE305 Lecture 6 February 11, 2013 7 / 15
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Overdamped CaseThe natural response is of the form x n (t ) = A1e
s 1t +A2e s 2t . Assume
x F (t ) = x f , x (0) = x 0, and x (0) = x 1 (all constant) are given. Then we can
solve for A1
and A2
.
x (t ) = A1e s 1t +A2e
s 2t +x f
x (0) = A1 +A2 +x f = x 0
x
(t ) = A1s 1e s 1t
+A2s 2e s 2t
x (0) = A1s 1 +A2s 2 = x 1
We have two equations with two unknowns. Solving, we get
A1 = x 1 + (x f −x 0)s 2
s 1− s 2 , A2 =
x 1 + (x f − x 0)s 1s 1−s 1
General overdamped solution
x (t ) = x 1+(x f −x 0)s 2s 1−s 2
e s 1t + x 1+(x f −x 0)s 1s 1−s 1
e s 2t
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Critically Damped CaseThe natural response is of the form x n (t ) = (A3 +A4t )e
st . Assume x F (t ) = x f ,x (0) = x 0, and x
(0) = x 1 (all constant) are given. Then we can solve for A3and A
4.
x (t ) = (A3 +A4t )e st +x f
x (0) = A3 +x f = x 0
x
(t ) = e st
[s (A3 +A4t ) +A4]
x (0) = A3s +A4 = x 1
We have two equations with two unknowns. Solving, we get
A3 = x 0−x f , A4 = x 1− (x 0−x f )s
General critically damped solution
x (t ) = {(x 0− x f ) + [x 1− (x 0−x f )s ]t }e st +x f
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Underdamped CaseThe natural response is of the form x n (t ) = e
−σ t [A5cos(ω d t ) +A6sin(ω d t )].Assume x F (t ) = x f , x (0) = x 0, and x
(0) = x 1 (all constant) are given. Thenwe can solve for A
5 and A
6.
x (t ) = e −σ t [A5cos(ω d t ) +A6sin(ω d t )] +x f
x (0) = [A5 + 0] + x f = x 0
x (t ) = e −σ t [(A6ω d −A5σ )cos(ω d t )− (A5ω d +A6σ )sin(ω d t )]
x (0) = A6ω d −A5σ = x 1
We have two equations with two unknowns. Solving, we get
A5 = x 0−x f , A6 = x 1 + σ (x 0− x f )
ω d
General underdamped solution
x (t ) = e −σ t [(x 0−x f )cos(ω d t ) + x 1+σ (x 0−x f )
ω d sin(ω d t )] + x f
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Problem 7.101 (1)
Given the circuit below, solve for v 0(t ) for t > 0.
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Problem 7.101 (2)
t = 0−
By inspection,
i L(0−) = 0
v C (0−) = −12V
t > 0
We can construct a Thévenin equivalent for the
portion of the circuit to the right of the inductor.
Here
V OC = 4V , R Th = 4k Ω
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Problem 7.101 (3)The integro-differential equation that governs the circuit behavior for t > 0 is(using KCL)
(R 2 +R Th )i (t ) +Ldi dt
+ 1C
t
0i (t )dt +v C (0) = V OC .
Differentiating to get the purely differential form,
d 2i
dt 2 + (
R 2 +R Th L )
di
dt +
1
LC i = 0.
The characteristic equation is
s 2 + (R 2 +R Th
L)s +
1
LC = 0, or
s 2 + 4×106s + 3×1012 = 0
⇒ s 1 = −1×106,s 2 = −3×10
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Problem 7.101 (4)
i (t ) = A1e s 1t +A2e s 2t + i (∞)
Since the capacitor acts as an open circuit at steady state, i (∞) = v 0(∞) = 0.We know that i (0+) = i L(0
−) = 0 = A1 +A2, but how do we get a value fordi dt t =0 (so that we can use
di dt
= A1s 1e s 1t +A2s 2e
s 2t )?
Use the original (integro-differential) equation at t = 0:
(R 2 +R Th )i (0) +Ldi dt
t =0
+v C (0) = V OC
i (0) = V OC −v C (0)L
= (4− (−12))/(2.5×10−3) = 6400 A/sA1 +A2 = 0 & (−1×10
6A1−3×106A2 = 6400)
⇒ A1 = 0.0032,A2 = −0.0032⇒ i (t ) = 3.2(e −1×10
6t −e 3×106t ) mA
⇒ v 0(t ) = R 2i (t ) = 19.2(e −1×106t −e 3×10
6t ) V
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