Laplace Transform in Circuit Analysis

Transcription

1 Laplace Transform in Circuit Analysis Laplace Transforms The Laplace transform* is a technique for analyzing linear time-invariant systems such as electrical circuits It provides an alternative functional description that often simplifies: The process of analyzing the behaviour of the system The synthesis of a new system based on a set of specifications * After Pierre-Simon Laplace ( )

2 Mechatronic System Mechanical System Sensors Actuators Input Signal Conditioning & Interfacing Control Architecture Operator Output Signal Conditioning & Interfacing Display System

3 Introduction to Transformations A mathematical transformation employs rules to change the form of data without altering its meaning Popular transformations used in signals Fourier (suited to solving problems where input domain is either repetitive or if the input is on a loop) Z (suited for problems where the input is discrete instead of continuous) Laplace (suited to solving problems with known initial values) ImplantDefribillator.wmv

4 Laplace Transform A powerful tool for circuit analysis The steps involved are A set of differential equations describing a circuit converted to the complex frequency domain The variables of interest are solved Convert from frequency domain back to time domain

5 Implant Defibrillator Problem Implant defibrillator manufacturer Guidant found that the close spacing between a wire and device component could potentially arc between them and cause a short circuit In March 2005, a 21-year-old college student who had a Guidant defibrillator implanted in his chest died suddenly The type of defibrillator in his death was short-circuiting at a rate of about once a month from 2003 to 2004; but this finding was not reported until February 2005

6 Breadboard (protoboard) A breadboard (protoboard) is a construction base for a one-of-a-kind electronic circuit, a prototype. Because the solderless breadboard does not require soldering, it is reusable, and thus can be used for temporary prototypes and experimenting with circuit design more easily. A breadboard with a completed circuit

7 Printed Circuit Boards Printed circuit boards (PCBs) are used to mechanically support and electrically connect electronic components using conductive pathways, or traces, etched from copper sheets laminated onto a non-conductive substrate. PCB for mobile phones

8 Printed Circuit Board Design Printed circuit board designs are normally very complex. Hence, this is normally done on computer software developed for this purpose. Most such software are able to perform auto-routing. Screenshot of PCB design software DNA_Circuits.wmv

9 Soldering Electrical components need to be physically attached to the right locations on the printed circuit board. This is accomplished using soldering.

10 Surface Mount Technology Surface mount technology (SMT) is a method for constructing electronic circuits in which the components are mounted directly onto the surface of printed circuit boards (PCBs). Electronic devices so made are called surface-mount devices or SMDs. In the industry it has largely replaced the previous construction method of fitting components with wire leads into holes in the circuit board (also called through-hole technology) WedgeBonding.wmv BallBonding.wmv

11 Definition of Laplace Transform Laplace Transform is defined as L f st ( t) = F ( s) = f ( t) e dt 0 s is a complex variable given by s = σ + jω The inverse Laplace transform is defined as 1 2πj [ F ( s) ] = f ( t) = α+ j 1 L F ( s) α+ j A list of Laplace transform pairs e st ds Uniqueness of Laplace Transform enables us to avoid the complex integration

12 Laplace Transform Properties (1) Linearity: Scaling:

13 Laplace Transform Properties (2) Time Shift: u(t-a) = 0 for t<a and u(t-a) = 1 for t>a Frequency Shift:

14 Laplace Transform Properties (3) Time Differentiation: Integrating by parts With one more differentiation In the general case

15 Laplace Transform Properties (4) Time Integration: Integrating by parts The first term is zero

16 Laplace Transform Properties (5) Frequency Differentiation: Taking derivative wrt x

17 Laplace Transform Properties (6) Time Periodicity: With the time-shift property Periodic function Using the identity The transform of a periodic function is the transform of the first period of the function divided by 1 e -Ts Decomposition of periodic function

18 Laplace Transform Properties Summary The properties of the Laplace Transform allow us to obtain transform properties without performing the integral.

19 Laplace Transform of Circuit Elements Voltage Source Capacitor Inductor 1 V ( s) = Resistor Z R s R = 1 F s L t ( ) v( t) = f ( t) i( t) dt = C 0 0 s 1 1 V ( s) 1 V ( s) = I ( s) Z s C s C ( ) = = I ( s) Cs d d V ( t) = L i( t) L f ( t) = sf( s) F(0) dt dt V ( s) = sli ( s) V ( s) Z L ( s) = = I( s) Ls

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21 Transfer Function For excitation X(s) and response Y(s) in the complex frequency domain. The transfer function is given by H ( s) = Y ( s) X ( s) The transfer function of a circuit describes how the output behaves with respect to the input. It also indicates how a signal is processed as it passes through a network SpaceShuttleAirlock.mpeg SolarBlast.wmv