Laplace Transforms Laplace Transform Motivation Definition v s ( t - PowerPoint PPT Presentation

Laplace Transforms Laplace Transform Motivation • Definition v s ( t ) + • Region of convergence Linear v s ( t ) v o • Useful properties Circuit t - • Inverse & partial fraction expansion • Distinct, complex, & repeated poles • In ECE 221, you learned • Applied to linear constant-coefficient ODE’s – DC circuit analysis – Transient response (limited to simple RL & RC circuits) – Sinusoidal steady-state response (Phasors) • We did not learn how to find the total response (transient and steady-state) to an arbitrary waveform • The Laplace transform enables us to do this • Circuit elements limited to resistors, capacitors, inductors, transformers, op amps, and ideal sources until ECE 321 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 1 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 2 Laplace Transform Motivation Continued Laplace Transform Analysis Illustration Why are we studying the Laplace transform? • Makes analysis of circuits 1 k Ω t = 0 + – Easier than working with multiple differential equations sin(1000 t ) 1 µ F v o – More general than the types of analysis we discussed in - ECE 221 • Used extensively in Given v o (0) = 0 , solve for v o ( t ) for t ≥ 0 . – Controls (ECE 311) 2 e − t/ 0 . 001 + 1 1 2 sin(1000 t − 45 ◦ ) v o ( t ) = – Communications √ – Signal Processing = v tr ( t ) + v ss ( t ) – Analog circuits (ECE 32X sequence) 2 e − t/ 0 . 001 1 v tr ( t ) = • Expected to know for interviews 1 2 sin(1000 t − 45 ◦ ) v ss ( t ) = √ • Gives you insight in circuit analysis and design J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 3 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 4

Laplace Transform Analysis Illustration Continued Laplace Transform for ODE’s Total 1 + Transient N M Steady State 0.8 d k y ( t ) d k x ( t ) Linear x ( t ) y ( t ) � � a k = b k 0.6 Circuit d t k d t k v o ( t ) (V) 0.4 - k =0 k =0 0.2 0 • Relationship of a voltage (or current) in a linear circuit to any −0.2 −0.4 other voltage (or current) is defined by a linear, time-invariant −0.6 constant-coefficient ordinary differential equation (ODE) −0.8 0 5 10 15 20 25 • Describes the behavior of many types of systems: Electrical, Time (ms) Mechanical, Chemical, Biological, etc. • Laplace transform is an easier approach than applying standard 2 e − t/ 0 . 001 + 1 1 2 sin(1000 t − 45 ◦ ) v o ( t ) = √ techniques of differential equations or convolution = v tr ( t ) + v ss ( t ) J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 5 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 6 Approach Laplace Transform Definition � ∞ • We will begin with a thorough discussion of the Laplace transform 0 − x ( t )e − st d t L { x ( t ) } = X ( s ) � • The elegance and simplicity of using this approach for circuit • Transform will be written with an upper-case letter analysis will not become apparent for several lectures • Defined from 0 − to include impulses at t = 0 • We will spend a lot of time on this topic • s = σ + jω is a complex variable • Bear with me • s has units of inverse seconds (s − 1 ) • Known as the one-sided (unilateral) Laplace transform • There is also a two-sided (bilateral) version: � + ∞ −∞ x ( t )e − st d t X ( s ) = • We will only work with the one-sided version + Easier to obtain the transient response + Consistent with common practice – Ignores x ( t ) for t < 0 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 7 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 8

Laplace Transform Convergence Example 1: Laplace Transform of x ( t ) • The Laplace transform does not converge to a finite value for all Find the Laplace transform of x ( t ) = u ( t ) . What is the region of signals and all values of s convergence? What is the transform of x ( t ) = 1 ? • Does converge for all signals we will be interested in – Sinusoids – e − at u ( t ) for any real | a | < ∞ – δ ( t ) • The values of s for which the Laplace transform converges is called the region of convergence (ROC) • Will not discuss in detail this term, but may see this in other classes on linear systems • See Signals and Systems chapter for more information J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 9 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 10 Example 2: Laplace Transform of x ( t ) Example 3: Laplace Transform of x ( t ) Find the Laplace transform of x ( t ) = e − at u ( t ) . What is the region of Find the Laplace transform of x ( t ) = δ ( t ) . What is the region of convergence? What is the transform of x ( t ) = e − at ? convergence? J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 11 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 12

Example 4: Workspace Example 4: Laplace Transform of x ( t ) Find the Laplace transform of x ( t ) = cos( ωt ) u ( t ) . What is the region of convergence? What is the Laplace transform of cos( ωt ) ? J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 13 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 14 Laplace Transform Properties Linearity • The Laplace transform has many important properties X 1 ( s ) = L { x 1 ( t ) } • We need to know these for at least three reasons X 2 ( s ) = L { x 2 ( t ) } – Improves our understanding of the transform then you should be able to show that – Enables us to find the transform more easily L [ a 1 x 1 ( t ) + a 2 x 2 ( t )] u ( t ) ⇐ ⇒ a 1 X 1 ( s ) + a 2 X 2 ( s ) – Enables us to find the inverse transform more easily • Will use the following notation for Laplace transform pairs Example: Find the Laplace transform of x ( t ) = 5 δ ( t ) − 2 cos 5 t . L x ( t ) u ( t ) ⇐ ⇒ X ( s ) L { δ ( t ) } = 1 s X ( s ) = L { x ( t ) } L { cos ωt } = s 2 + ω 2 L − 1 { X ( s ) } x ( t ) u ( t ) = � � s L { 5 δ ( t ) − 2 cos 5 t } = 5(1) − 2 s 2 + 5 2 2 s = 5 − s 2 + 25 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 15 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 16

Scaling Translation in Time Given X ( s ) = L { x ( t ) } , what is L { x ( t − t 0 ) u ( t − t 0 ) } for t 0 > 0 ? Given X ( s ) = L { x ( t ) } , what is L { x ( at ) } for a > 0 ? � ∞ � ∞ 0 − x ( t − t 0 ) u ( t − t 0 )e − st d t 0 − x ( at )e − st d t L { x ( t − t 0 ) u ( t − t 0 ) } = L { x ( at ) } = t = τ τ = t − t 0 τ = at a d τ = d t d t = 1 t = τ + t 0 d τ = a d t a d τ � ∞ � ∞ x ( τ ) u ( τ )e − s ( τ + t 0 ) d τ L { x ( at ) } = 1 0 − x ( τ )e − s τ L { x ( t − t 0 ) u ( t − t 0 ) } = a d τ − t 0 a � ∞ � ∞ 0 − x ( τ ) u ( τ )e − s ( τ + t 0 ) d τ = 1 0 − x ( τ )e − ( s = a ) τ d τ a � ∞ � s 0 − x ( τ )e − sτ d τ ⇒ 1 e − st 0 � = L x ( at ) u ( t ) ⇐ aX a L e − st 0 X ( s ) x ( t − t 0 ) u ( t − t 0 ) ⇐ ⇒ J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 17 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 18 Translation in Frequency Time Differentiation Given X ( s ) = L { x ( t ) } , what is the Laplace transform of ˙ x ( t ) ? Given X ( s ) = L { x ( t ) } , what is the inverse Laplace transform of � ∞ X ( s + s 0 ) ? � d x ( t ) � d x ( t ) d t e − st d t L = � ∞ d t 0 − 0 − x ( t ) e − ( s + s 0 ) t d t X ( s + s 0 ) = d u = − s e − st d t u = e − st � ∞ d v = d x ( t ) e − st d t x ( t ) e − s 0 t � � = d t d t v = x ( t ) 0 − � ∞ � ∞ e − s 0 t x ( t ) � � = L � d x ( t ) � 0 − u d v = uv | ∞ L = 0 − − 0 − v d u L − 1 { X ( s + s 0 ) } e − s 0 t x ( t ) u ( t ) d t = � ∞ L � ∞ e − s 0 t x ( t ) u ( t ) = e − st x ( t ) � − s e − st � ⇐ ⇒ X ( s + s 0 ) � 0 − − 0 − x ( t ) d t � ∞ 0 − x ( t ) e − st d t 0 − x (0 − ) � � = + s d x ( t ) L ⇒ sX ( s ) − x (0 − ) u ( t ) ⇐ d t J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 19 J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 20