Laplace Transformation M d Q Tavares , TB425 dqtm@zhaw.ch

1. Laplace Transformation M d Q Tavares , TB425 dqtm@zhaw.ch

2. x n t f ω S-Plane Z-Plane y SiSy Overview LTI DGl; BSB; ZVD; h(t); g(t); G(ω); G(s) u(t) U(ω) u[n] U(z) y(t) Y(ω) y[n] Y(z) LTD DzGl; BSB; ZVD; g[n]; G(ω);G(z) Control (RT) Telecomm (NTM) SigProc (DSV, ASV) Applied Mathematics (SiSy) Mathematics

3. Inhalt • Laplace Transformation • Definition • Properties • Examples • Comparison to Fouriertransformation • References: • Laplace Transformation: Skript Kapitel 7 • Comparison LTI views: Skript Kapitel 6 • Cha P., Rosenberg J., Dym C., „Fundamentals of Modeling and Analyzing Engineering Systems“

4. Laplace Transformation Notation one-sided transform x(t) X(s) original function transform (Bildfunktion) Definition - Laplace Transformation obs.: x(t) transformable if: - Inverse Laplace Transformation (contour integration) obs: not used. Alternative partial fraction method and Laplace transform tables.

5. Laplace Transformation Application Solution of linear, time-invariant, ordinary differential equations: - allow resolution through algebraic manipulations - many transform tables already available (less work) - homogeneous and particular solutions obtained simultaneously (solution for transient and steady-state system response) - evaluate system stability (poles of the transfer function)

6. Laplace Transformation: Properties Linearity Time Derivatives (Differentiationsregel) Time Integration (Integrationsregel) Proof: vide script

7. Laplace Transformation: Properties Shift in Time Shift in s Time Scaling Start Value Theorem (Anfangswertsatz) Final Value Theorem (Endwertsatz) Only valid if poles on left-halft s-plane

8. Laplace Transformation Examples A - Laplace transform of the unit step function ε(t) B - Laplace transform of the impulse function δ(t) C - Laplace transform of the unit ramp function x(t) = t ; t≥0 D - Laplace transform of an exponential function x(t)=exp(-at) ; t≥0 E - Laplace transform of sinusoidal functions x(t)=cos(ω0t) ; t≥0 x(t)=sin(ω0t) ; t≥0

9. Inverse Laplace Transformation Method: Partial Fraction Expansion (Partialbruchzerlegung) Laplace Transform of function x(t) with: m < n Factorise the numerator and denominator: zi : zeros of X(s) pi : poles of X(s) k : gain Calculate the residues αi (alpha-i)

10. Inverse Laplace Transformation Method: Partial Fractions (Partialbruchzerlegung) Inverse Laplace Transformation for exponential function Obs.: pi roots can be real (single/multiple) or complex conjugate Im{s} S-Plane (S-Ebene) PN-Map (Pol- und Nullstelle) X Re{s} Pole Zero

11. Laplace Transformation Transform Table (Script pg 95-99) …

12. Inverse Laplace Transformation Examples A – Function with distinct real poles (unterschiedliche reelle Polstelle) B – Function with real and complex conjugate poles C – Function with repeated poles D – Response of First-Order System : free and step response E – Response of Second-Order System : free and step response

13. Laplace Transformation : S-Plane Pole Location and corresponding Exponential Functions (Zusammenhang S-Ebene Pol-Stelle und Zeitfunktionen)

14. Laplace Transformation : S-Plane Pole Location and corresponding Exponential Functions (Zusammenhang S-Ebene Pol-Stelle und Zeitfunktionen)

15. Laplace Transformation : S-Plane Pole Location and corresponding Exponential Functions (Zusammenhang S-Ebene Pol-Stelle und Zeitfunktionen)

16. Laplace Transformation Comparison to Fouriertransformation

17. Laplace Transformation Comparison to Fouriertransformation