Laplace Transforms. Euler s Formula = cos θ + j sin θ e = cos θ - j sin θ

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1 Laplace Transforms ENGI 252 e jθ Euler s Formula = cos θ + j sin θ -jθ e = cos θ - j sin θ Euler s Formula jθ -jθ ( e + e ) jθ -jθ ( ) < a > < b > Now add < a > and < b > and solve for cos θ 1 cos θ = 2 2 cos θ = e + e Subtract < b > from < a > and solve for sin θ 1 sin θ = ( jθ -jθ e ) 2j - e 20 March 2004 ENGI Laplace Transforms 2 Laplace Transforms 1

2 Laplace Transforms The Laplace Transform is a technique that transforms circuit analysis from the time domain into the frequency domain The mathematical solution in the frequency domain is easier For most circuits or systems whose behavior is described by a 2 nd Order or higher Differential Equation, solving for the output may be either impossible or very difficult It is easier to solve algebraic equations rather than Differential Equations The Laplace Transform is: a powerful mathematical tool It will allow us to transform differential equations into algebraic equations We can then use all the circuit analysis techniques that we developed for passive circuits It allows us to find the complete time response for any circuit with any input 20 March 2004 ENGI Laplace Transforms 3 Methods for Solving Differential Equations x(t) Laplace Transform X(s) Time Domain Time Domain Circuit Circuit L 1 L s-domain Circuit Y(s) s = α + jω = Complex Frequency 2 Types of s-domain Circuits With and Without Initial Conditions y(t) Inverse Laplace Transform 20 March 2004 ENGI Laplace Transforms 4 Laplace Transforms 2

3 The Laplace transform analysis provides insight into transient phenomena in the circuit Laplace transform methodology allows us to analyze much more complex circuits than is possible with SS methods L{f(t)} = -st f(t) e dt 0 The power of e must be dimensionless. Therefore s must have the dimensions of 1/t, or frequency Thus the s-domain is a frequency domain We refer to the Laplace transform of a function f(t) as F(s), or: L{f(t)} = F(s) 20 March 2004 ENGI Laplace Transforms 5 The Laplace Transform of f(t) = F(s) L f(t) = F(s) = f(t) e dt -st [ ] 0 We will use tables to find F(s) from f(t) Since the power of e must be dimensionless, then s must have the dimensions of 1/t, or frequency. Thus the s- domain is a frequency domain. Using Laplace transforms can greatly ease the solution of many time-domain electrical circuit problems. Integral and differential equations in the time domain are reduced to algebraic equations in the s-domain, making their solutions much easier. 20 March 2004 ENGI Laplace Transforms 6 Laplace Transforms 3

4 Once a problem is solved in the s-domain, we can use the inverse Laplace transform to determine the time-domain solution (similar to phasor analysis) The Inverse Laplace Transform of F(s) = f(t) 1 L [ ] F(s) = f(t) = 1 j 2 π α ω α + jω st F(s) e ds - j We will use either tables of Inverse Laplace Transform or Partial Fraction Expansion to find f(t) from F(s) 20 March 2004 ENGI Laplace Transforms 7 In this course, we will use the Laplace Transform as a unilateral transform - it will be valid only for t > 0 Not all functions have a Laplace Transform We normally use 0 as the lower limit of the integral The upper limit of the integral is, which means that some transforms diverge most problems in electrical engineering will have a convergent solution There are two types of Laplace Transforms: Functional transforms Laplace Transforms of a specific mathematical function (sine or cosine, exponentials, algebraic expressions, etc) Operational transforms Laplace Transforms of mathematical operations (derivative, integral, product, etc.) Now we need an review of some mathematical functions that are extremely important 20 March 2004 ENGI Laplace Transforms 8 Laplace Transforms 4

5 Representation of the Resistor A resistor represented: (a) in the time domain (b) in the frequency domain using the Laplace transform 20 March 2004 ENGI Laplace Transforms 9 Representation of the Capacitor A capacitor represented: (a) in the time domain (b) in the frequency domain using the Laplace Transform (c) in an alternate frequency-domain representation 20 March 2004 ENGI Laplace Transforms 10 Laplace Transforms 5

6 Representation of the Inductor An inductor represented: (a) in the time domain (b) in the frequency domain using the Laplace Transform (c) in an alternate frequency-domain representation 20 March 2004 ENGI Laplace Transforms 11 Poles and Zeroes in the s-domain 20 March 2004 ENGI Laplace Transforms 12 Laplace Transforms 6

7 Plot of Poles and Zeroes on the s Plane Example of a Typical Pole Zero S-Plane Plot 20 March 2004 ENGI Laplace Transforms 13 Laplace Transforms 7