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Transcript of Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of...
![Page 1: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/1.jpg)
Leo Lam © 2010-2011
Signals and Systems
EE235
![Page 2: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/2.jpg)
Leo Lam © 2010-2011
Laplace Examples
A bunch of them
![Page 3: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/3.jpg)
Leo Lam © 2010-2011
Standard Laplace question
• Find the Laplace Transform, stating the ROC.
• So:
ROC extends from to the right of the most right pole
ROCxxo
![Page 4: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/4.jpg)
Leo Lam © 2010-2011
Inverse Laplace Example (2 methods!)
• Find z(t) given the Laplace Transform:
• So:
![Page 5: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/5.jpg)
Leo Lam © 2010-2011
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:
![Page 6: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/6.jpg)
Leo Lam © 2010-2011
Inverse Laplace Example (Diffy-Q)
• Find the differential equation relating y(t) to x(t), given:
![Page 7: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/7.jpg)
Leo Lam © 2010-2011
Laplace for Circuits!
• Don’t worry, it’s actually still the same routine!
Time domain
inductor
resistor
capacitor
Laplace domain
Impedance!
![Page 8: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/8.jpg)
Laplace for Circuits!
• Find the output current i(t) of this ugly circuit!
• Then KVL: • Solve for I(s):
• Partial Fractions:• Invert:
Leo Lam © 2010-2011
RL
+-
Given: input voltage
And i(0)=0
Step 1: representthe whole circuit inLaplace domain.
![Page 9: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/9.jpg)
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 © 2010-2011
LTIC
)(
)()(
sX
sYsH
LTIC
![Page 10: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Laplace Examples A bunch of them.](https://reader038.fdocuments.net/reader038/viewer/2022102909/56649f1c5503460f94c32bc4/html5/thumbnails/10.jpg)
A “strange signal” example
• Find the Laplace transform of this signal:
• What is x(t)?
• We know these pairs:
• So:
Leo Lam © 2010-2011
x(t)
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2
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