Leo Lam © Signals and Systems EE235

Today’s menu Leo Lam © Laplace Transform

Laplace Stability Conditions Leo Lam © 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 j-axis LTIC system is if marginally stable : poles in the LHP + unrepeated poles on the jaxis.

Laplace Stability Conditions Leo Lam © Generally: system H(s) stability conditions: The system’s ROC includes the jaxis Stable? Causal? σ jωjω x x x Stable+CausalUnstable+Causal σ jωjω x x x x σ jωjω x x x Stable+Noncausal

Laplace: Poles and Zeroes Leo Lam © Given: Roots are poles: Roots are zeroes: Only poles affect stability Example:

Laplace Stability Example: Leo Lam © Is this stable?

Laplace Stability Example: Leo Lam © Is this stable?

Standard Laplace question Find the Laplace Transform, stating the ROC. So: Leo Lam © ROC extends from to the right of the most right pole ROC xxo

Inverse Laplace Example (2 methods!) Find z(t) given the Laplace Transform: So: Leo Lam ©

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 ©

Inverse Laplace Example (Diffy-Q) Find the differential equation relating y(t) to x(t), given: Leo Lam ©

Laplace for Circuits! Don’t worry, it’s actually still the same routine! Leo Lam © 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 © R L +-+- Given: input voltage And i(0)=0 Step 1: represent the whole circuit in Laplace 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 © LTIC

A “strange signal” example Find the Laplace transform of this signal: What is x(t)? We know these pairs: So: Leo Lam © x(t)

One last bit: Parseval’s Theorem Leo Lam ©

And that’s it! House cleaning –Sample Final –Review sessions –Class review (now online) Leo Lam ©