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Transcript of Leo Lam © 2010-2012 Signals and Systems EE235. Today’s menu Leo Lam © 2010-2012 Almost done!...
![Page 1: Leo Lam © 2010-2012 Signals and Systems EE235. Today’s menu Leo Lam © 2010-2012 Almost done! Laplace Transform.](https://reader038.fdocuments.net/reader038/viewer/2022110206/56649f4a5503460f94c6c48a/html5/thumbnails/1.jpg)
Leo Lam © 2010-2012
Signals and Systems
EE235
![Page 2: Leo Lam © 2010-2012 Signals and Systems EE235. Today’s menu Leo Lam © 2010-2012 Almost done! Laplace Transform.](https://reader038.fdocuments.net/reader038/viewer/2022110206/56649f4a5503460f94c6c48a/html5/thumbnails/2.jpg)
Leo Lam © 2010-2012
Today’s menu
• Almost done!• Laplace Transform
![Page 3: Leo Lam © 2010-2012 Signals and Systems EE235. Today’s menu Leo Lam © 2010-2012 Almost done! Laplace Transform.](https://reader038.fdocuments.net/reader038/viewer/2022110206/56649f4a5503460f94c6c48a/html5/thumbnails/3.jpg)
Leo Lam © 2010-2012
Laplace & LTI Systems
• If:
• Then
LTI
LTI
Laplace of the zero-state (zero initialconditions) response
Laplace of the input
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Leo Lam © 2010-2012
Laplace & Differential Equations
• Given:
• In Laplace:– where
• So:
• Characteristic Eq:– The roots are the poles in s-domain, the “power” in time domain.
012
2
012
2
)(
)(
bsbsbsP
asasasQ
0)( sQ
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Leo Lam © 2010-2012
Laplace & Differential Equations
• Example (causal LTIC):
• Cross Multiply and inverse Laplace:
![Page 6: Leo Lam © 2010-2012 Signals and Systems EE235. Today’s menu Leo Lam © 2010-2012 Almost done! Laplace Transform.](https://reader038.fdocuments.net/reader038/viewer/2022110206/56649f4a5503460f94c6c48a/html5/thumbnails/6.jpg)
Leo Lam © 2010-2012
Laplace Stability Conditions
• LTI – Causal system H(s) stability conditions:• LTIC system is stable : all poles are in the LHP• LTIC system is unstable : one of its poles is in the RHP• LTIC system is unstable : repeated poles on the jw-axis• LTIC system is if marginally stable : poles in the LHP +
unrepeated poles on the j -w axis.
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Leo Lam © 2010-2012
Laplace Stability Conditions
• Generally: system H(s) stability conditions:• The system’s ROC includes the j -w axis• Stable? Causal?
σ
jω
x
x
x
Stable+Causal Unstable+Causal
σ
jω
x
xx
xσ
jω
x
x
x
Stable+Noncausal
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Leo Lam © 2010-2012
Laplace: Poles and Zeroes
• Given:
• Roots are poles:
• Roots are zeroes:
• Only poles affect stability
• Example:
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Leo Lam © 2010-2012
Laplace Stability Example:
• Is this stable?
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Leo Lam © 2010-2012
Laplace Stability Example:
• Is this stable?
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Leo Lam © 2010-2012
Standard Laplace question
• Find the Laplace Transform, stating the ROC.
• So:
ROC extends from to the right of the most right pole
ROCxxo
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Leo Lam © 2010-2012
Inverse Laplace Example (2 methods!)
• Find z(t) given the Laplace Transform:
• So:
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Leo Lam © 2010-2012
Inverse Laplace Example (2 methods!)
• Find z(t) given the Laplace Transform (alternative method):
• Re-write it as:
• Then:
• Substituting back in to z(t) and you get the same answer as before:
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Leo Lam © 2010-2012
Inverse Laplace Example (Diffy-Q)
• Find the differential equation relating y(t) to x(t), given:
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Leo Lam © 2010-2012
Laplace for Circuits!
• Don’t worry, it’s actually still the same routine!
Time domain
inductor
resistor
capacitor
Laplace domain
Impedance!
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Laplace for Circuits!
• Find the output current i(t) of this ugly circuit!
• Then KVL: • Solve for I(s):
• Partial Fractions:• Invert:
Leo Lam © 2010-2012
RL
+-
Given: input voltage
And i(0)=0
Step 1: representthe whole circuit inLaplace domain.
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Step response example
• Find the transfer function H(s) of this system:
• We know that:
• We just need to convert both the input and the output and divide!
Leo Lam © 2010-2012
LTIC
)(
)()(
sX
sYsH
LTIC
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A “strange signal” example
• Find the Laplace transform of this signal:
• What is x(t)?
• We know these pairs:
• So:
Leo Lam © 2010-2012
x(t)
1 2 3
2
1