1. Laplace Transformation
M d Q Tavares , TB425

2. 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

3. 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“

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