Leo Lam © 2010-2011

Signals and Systems

EE235

Leo Lam © 2010-2011

Laplace Examples

A bunch of them

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

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Inverse Laplace Example (2 methods!)

• Find z(t) given the Laplace Transform:

• So:

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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:

Leo Lam © 2010-2011

Inverse Laplace Example (Diffy-Q)

• Find the differential equation relating y(t) to x(t), given:

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Laplace for Circuits!

• Don’t worry, it’s actually still the same routine!

Time domain

inductor

resistor

capacitor

Laplace domain

Impedance!

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.

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

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)

1 2 3

2

1