Laplace Transformation M d Q Tavares , TB425 dqtm@zhaw.ch
Applied Mathematics (SiSy) SiSy Overview Signals step / impulse / rect / sinc continuous / discret periodic / aperiodic deterministic / random representation in time / frequency domains power and energy in freq domain Systems linearisation feedback and stability discretisation Transforms FR ; FT ; DFT/FFT : Laplace : Z-Transformation : DCT : Applications Biomedical / Operational Research/ Sensorics & Messtechnik Automation / Location & Telecommunication Systems Data Compression & Cryptography / Image Processing / Audio Synthesis & Analysis Statistical Signal Processing 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) ω t f S-Plane Control (RT) Telecomm (NTM) SigProc (DSV, ASV) Z-Plane n Applied Mathematics (SiSy) x y Mathematics
Laplace Transformation 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“
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.
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)
Laplace Transformation: Properties Linearity Time Derivatives (Differentiationsregel) Time Integration (Integrationsregel) Proof: vide script
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
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
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)
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
Laplace Transformation Transform Table (Script pg 95-99) …
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
Laplace Transformation : S-Plane Pole Location and corresponding Exponential Functions (Zusammenhang S-Ebene Pol-Stelle und Zeitfunktionen)
Laplace Transformation : S-Plane Pole Location and corresponding Exponential Functions (Zusammenhang S-Ebene Pol-Stelle und Zeitfunktionen)
Laplace Transformation : S-Plane Pole Location and corresponding Exponential Functions (Zusammenhang S-Ebene Pol-Stelle und Zeitfunktionen)
Laplace Transformation Comparison to Fouriertransformation
Laplace Transformation Comparison to Fouriertransformation