ESE 271 / Spring 2013 / Lecture 17
Solution of Midterm Exam 2.
ESE 271 / Spring 2013 / Lecture 17
Solution of Midterm Exam 2.
ESE 271 / Spring 2013 / Lecture 17
Solution of Midterm Exam 2.
ESE 271 / Spring 2013 / Lecture 17
Solution of Midterm Exam 2.
ESE 271 / Spring 2013 / Lecture 17
Revisit charging capacitor by practical voltage source
It is easy to find solution if Vs is step function.
Wh i V i
What is Vs is more complicated?
li
d?
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ESE 271 / Spring 2013 / Lecture 17
Series RLC circuit
This is second order equation and it is not easy even for step function Vs …. What should we do?
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ESE 271 / Spring 2013 / Lecture 17
Series RLC circuit under harmonic excitation (AC steady‐state)
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ESE 271 / Spring 2013 / Lecture 17
Series RLC circuit under harmonic excitation (AC steady‐state)
Phasor analysis – method to solve linear differential equations with s e a e sou ces
sine wave sources
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ESE 271 / Spring 2013 / Lecture 17
Back to simple RC circuit with sources of different frequencies
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ESE 271 / Spring 2013 / Lecture 17
Back to simple RC circuit with sources of different frequencies
For each frequency we can use phasors
‐ Transfer function
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ESE 271 / Spring 2013 / Lecture 17
Laplace Transform
Function in time‐domain
One sided Laplace transform of V(t)
Function in s‐domain
allowed
not allowed
not allowed
Exists if
is piecewise continuous
is of exponential order
is of exponential order
exists
Why we talk about Laplace transform?
Why
we talk about Laplace transform?
– because it is method to solve differential equations.
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ESE 271 / Spring 2013 / Lecture 17
Laplace Transform of Derivative
Differentiation in time‐domain is replaces with multiplication in s‐domain
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ESE 271 / Spring 2013 / Lecture 17
Laplace Transform of Integral
Integration in time‐domain is replaces with division in s‐domain
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ESE 271 / Spring 2013 / Lecture 17
Let’s use Laplace Transform to solve RC circuit
Apply Laplace Transform method
Laplace transform is linear operation
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ESE 271 / Spring 2013 / Lecture 17
Let’s use Laplace Transform to solve RC circuit
‐ Assume that C was initially discharged
‐ Impedance of the capacitor in s‐domain
Impedance of the capacitor in s domain
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ESE 271 / Spring 2013 / Lecture 17
Let’s use Laplace Transform to solve RLC circuit
‐ Assume that C was initially discharged
Looks very similar to phasor y
p
method with s in place of j∙
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RC circuit
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ESE 271 / Spring 2013 / Lecture 17
How to get back to time‐domain from s‐domain
1. Use Laplace inversion integral:
any within region of convergence
2. Use partial fraction expansion and utilize table:
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RC circuit
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RC circuit
We were able to get this without help from Laplace transform
bl
h
h
h l f
l
f
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RLC circuit
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RLC circuit
Option 1:
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RLC circuit
Option 1:
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RLC circuit
Option 2:
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RLC circuit
Option 2:
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RLC circuit
Option 3:
Can not find A and B
Need to find function in t‐domain that corresponds to
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ESE 271 / Spring 2013 / Lecture 17
T‐multiplication
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ESE 271 / Spring 2013 / Lecture 17
Back to step response of RLC circuit
Option 3:
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