Applications of Laplace Transforms Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Contents Introduction Circuit Element Models Circuit Analysis Transfer Functions State Variables Network Stability Summary

Introduction To learn how easy it is to work with circuits in the s domain To learn the concept of modeling circuits in the s domain To learn the concept of transfer function in the s domain To learn how to apply the state variable method for analyzing linear systems with multiple inputs and multiple outputs To learn how the Laplace transform can be used in stability analysis

Circuit Element Models Steps in applying the Laplace transform: –Transform the circuit from the time domain to the s domain (a new step to be discussed later) –Solve the circuit using circuit analysis technique (nodal/mesh analysis, source transformation, etc.) –Take the inverse Laplace transform of the solution and thus obtain the solution in the time domain

s-Domain Models for R and L Time domains domain

s-Domain Model for C Time domains domain

Summary For inductor:For capacitor:

Summary ElementZ(s)Z(s) ResistorR InductorsL Capacitor1/sC *Assuming zero initial conditions Impedance in the s domain – Z(s)=V(s)/I(s) Admittance in the s domain – Y(s)=1/Z(s)=V(s)/I(s)

Example 1

Example 2

Example 2 (Cont’d)

Example 3

Circuit Analysis Operators (derivatives and integrals) into simple multipliers of s and 1/s Use algebra to solve the circuit equations All of the circuit theorems and relationships developed for dc circuits are perfectly valid in the s domain

Example 1

Example 2

Example 3 Assume that no initial energy is stored. (a)Find V o (s) using Thevenin’s theorem. (b)Find v o (0 + ) and v o (  ) by apply the initial- and final-value theorems. (c) Determine v o (t). =10u(t)

Example 3: (a)

Example 3: (b), (c)

Transfer Functions The transfer function H(s) is the ratio of the output response Y(s) to the input excitation X(s), assuming all initial conditions are zero.

Transfer Functions (Cont’d) Two ways to find H(s) –Assume an input and find the output –Assume an output and find the input (the ladder method: Ohm’s law + KCL) Four kinds of transfer functions

Example 1

Example 2 Find H(s)=V 0 (s)/I 0 (s).

Example 2 (The Ladder Method)

Example 3 Find (a) H(s) = V o /V i, (b) the impulse response, (c) the response when v i (t) = u(t) V, (d) the response when v i (t) = 8cos2t V.

Example 3: (a), (b)

Example 3: (c), (d)

State Variables The state variables are those variables which, if known, allow all other system parameters to be determined by using only algebraic equations. In an electric circuit, the state variables are the inductor current and the capacitor voltage since they collectively describe the energy state of the system.

State Variable Method

State Variable Method (Cont’d)

How to Apply State Variable Method Steps to apply the state variable method to circuit analysis: –Select the inductor current i and capacitor voltage v as the state variables (define vector x, z) –Apply KCL and KVL to obtain a set of first-order differential equations (find matrix A, B) –Obtain the output equation and put the final result in state-space representaion (find matrix C) –H(s)=C(sI-A) -1 B

Network Stability A circuit is stable if its impulse response h(t) is bounded as t approaches  ; it is unstable if h(t) grows without bound as t approaches . Two requirements for stability –Degree of N(s) < Degree of D(s) –All the poles must lie in the left half of the s plane

Network Stability (Cont’d) A circuit is stable when all the poles of its transfer function H(s) lie in the left half of the s plane. Circuits composed of passive elements (R, L, and C) and independent sources either are stable or have poles with zero real parts. Active circuits or passive circuits with controlled sources can supply energy, and can be unstable.

Example 1

Example 2 Find k for a stable circuit.

Summary The methodology of circuit analysis using Laplace transform –Convert each element to its s-domain model –Obtain the s-domin solution –Apply the inverse Laplace transform to obtain the t-domain solution

Summary The transfer function H(s) of a network is the Laplace transform of the impulse response h(t) A circuit is stable when all the poles of its transfer function H(s) lie in the left half of the s plane.