271 2013 lecture 18 last time: laplace transformoe/leon/ese271s13/lecture18.pdfese 271 / spring 2013...
TRANSCRIPT
ESE 271 / Spring 2013 / Lecture 18
Last time: Laplace Transform
Function in time‐domain
Function in s‐domain
One sided Laplace transform of V(t)
Exists ifallowed
not allowed
is piecewise continuous
is of exponential order exists
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ESE 271 / Spring 2013 / Lecture 18
Laplace Transform of voltage step
time‐domains‐domain
i ll d “ l ” fHere is so called “pole” of
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ESE 271 / Spring 2013 / Lecture 18
Why do we care about Laplace Transform?h l l d ff l !Because it helps to solve differential equations !
Example – finding response of RC‐integrator
Perform Laplace transformation
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Answer in s‐domain:
ESE 271 / Spring 2013 / Lecture 18
What we should do to apply Laplace transform technique:
1. Find the Laplace transforms of an important functions and build a table for future reference.
2. Develop technique to go from s‐domain back to time‐domain.
3. Develop circuit analysis techniques in s‐domain.3. Develop circuit analysis techniques in s domain.
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ESE 271 / Spring 2013 / Lecture 18
Laplace transform of
when
when
PoleRegion of convergence ofPole convergence of
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ESE 271 / Spring 2013 / Lecture 18
Laplace transform of
Poles
Observe:
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Region of convergence:
ESE 271 / Spring 2013 / Lecture 18
Laplace transform of sine wave voltage.
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ESE 271 / Spring 2013 / Lecture 18
Laplace transform of
Poles:
Region of convergence:
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ESE 271 / Spring 2013 / Lecture 18
Laplace transform of time‐shifted waveform
hifshift
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ESE 271 / Spring 2013 / Lecture 18
Example 1.
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ESE 271 / Spring 2013 / Lecture 18
Example 2.
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ESE 271 / Spring 2013 / Lecture 18
Frequency shift.
What happens with when ?
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ESE 271 / Spring 2013 / Lecture 18
Example.
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ESE 271 / Spring 2013 / Lecture 18
Laplace transform of damped cosine.
Recall that:
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ESE 271 / Spring 2013 / Lecture 18
Time and frequency scaling.
and now
Change variable
f
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Time compression factor
ESE 271 / Spring 2013 / Lecture 18
Example.
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ESE 271 / Spring 2013 / Lecture 18
Delta function.
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Generalized function
ESE 271 / Spring 2013 / Lecture 18
Singularity functions.
Unit step function (Heaviside function)
Unit impulse function (‐function)
Unit ramp function
And all higher orders of integration and differentiation
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ESE 271 / Spring 2013 / Lecture 18
Sifting property of ‐functions.
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ESE 271 / Spring 2013 / Lecture 18
Laplace transforms of singularity functions.
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ESE 271 / Spring 2013 / Lecture 18
Example.
KVL:
Perform Laplace transform to solve integral differential equationPerform Laplace transform to solve integral‐differential equation
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ESE 271 / Spring 2013 / Lecture 18
Example ‐ cont.
KVL:
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ESE 271 / Spring 2013 / Lecture 18
Example ‐ cont.
KVL:
Damped sine
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ESE 271 / Spring 2013 / Lecture 18
Pair of complex poles.
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ESE 271 / Spring 2013 / Lecture 18
Circuits in s‐domain.
Laplace transform
Laplace transform
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ESE 271 / Spring 2013 / Lecture 18
Resistor in s‐domain.
time‐domain
s‐domain
Impedance in s‐domain
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ESE 271 / Spring 2013 / Lecture 18
Capacitor in s‐domain.
time‐domain
Initial condition
s‐domain
Impedance in
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Impedance in s‐domain
ESE 271 / Spring 2013 / Lecture 18
Inductor in s‐domain.
time‐domain
Initial acondition
s‐domain
Impedance in
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s‐domain
ESE 271 / Spring 2013 / Lecture 18
Example.Find equivalent Impedance in s domainFind equivalent Impedance in s‐domain
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ESE 271 / Spring 2013 / Lecture 18
Admittance in s‐domain.
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ESE 271 / Spring 2013 / Lecture 18
Example.Thevenin/Norton equivalents in s domainThevenin/Norton equivalents in s‐domain
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