Laplace Transforms Definition Region of convergence Useful - PowerPoint PPT Presentation

Laplace Transforms • Definition • Region of convergence • Useful properties • Inverse & partial fraction expansion • Distinct, complex, & repeated poles • Applied to linear constant-coefficient ODE’s J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 1

Laplace Transform Motivation v s ( t ) + Linear v s ( t ) v o Circuit t - • In ECE 221, you learned – 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 2

Laplace Transform Motivation Continued Why are we studying the Laplace transform? • Makes analysis of circuits – Easier than working with multiple differential equations – More general than the types of analysis we discussed in ECE 221 • Used extensively in – Controls (ECE 311) – Communications – Signal Processing – Analog circuits (ECE 32X sequence) • Expected to know for interviews • Gives you insight in circuit analysis and design J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 3

Laplace Transform Analysis Illustration 1 k Ω t = 0 + sin(1000 t ) 1 µ F v o - Given v o (0) = 0 , solve for v o ( t ) for t ≥ 0 . 2 e − t/ 0 . 001 + 1 1 2 sin(1000 t − 45 ◦ ) v o ( t ) = √ = v tr ( t ) + v ss ( t ) 2 e − t/ 0 . 001 1 v tr ( t ) = 1 2 sin(1000 t − 45 ◦ ) v ss ( t ) = √ J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 4

Laplace Transform Analysis Illustration Continued Total 1 Transient Steady State 0.8 0.6 v o ( t ) (V) 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0 5 10 15 20 25 Time (ms) 2 e − t/ 0 . 001 + 1 1 2 sin(1000 t − 45 ◦ ) v o ( t ) = √ = v tr ( t ) + v ss ( t ) J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 5

Laplace Transform for ODE’s + N M d k y ( t ) d k x ( t ) Linear x ( t ) � � y ( t ) = a k b k Circuit d t k d t k k =0 k =0 - • Relationship of a voltage (or current) in a linear circuit to any other voltage (or current) is defined by a linear, time-invariant constant-coefficient ordinary differential equation (ODE) • Describes the behavior of many types of systems: Electrical, Mechanical, Chemical, Biological, etc. • Laplace transform is an easier approach than applying standard techniques of differential equations or convolution J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 6

Approach • We will begin with a thorough discussion of the Laplace transform • The elegance and simplicity of using this approach for circuit analysis will not become apparent for several lectures • We will spend a lot of time on this topic • Bear with me J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 7

Laplace Transform Definition � ∞ 0 − x ( t )e − st d t L { x ( t ) } = X ( s ) � • Transform will be written with an upper-case letter • Defined from 0 − to include impulses at t = 0 • s = σ + jω is a complex variable • 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 8

Laplace Transform Convergence • The Laplace transform does not converge to a finite value for all signals and all values of s • 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

Example 1: Laplace Transform of x ( t ) Find the Laplace transform of x ( t ) = u ( t ) . What is the region of convergence? What is the transform of x ( t ) = 1 ? J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 10

Example 2: Laplace Transform of x ( t ) Find the Laplace transform of x ( t ) = e − at u ( t ) . What is the region of convergence? What is the transform of x ( t ) = e − at ? J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 11

Example 3: Laplace Transform of x ( t ) Find the Laplace transform of x ( t ) = δ ( t ) . What is the region of convergence? J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 12

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

Example 4: Workspace J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 14

Laplace Transform Properties • The Laplace transform has many important properties • We need to know these for at least three reasons – Improves our understanding of the transform – Enables us to find the transform more easily – Enables us to find the inverse transform more easily • Will use the following notation for Laplace transform pairs L x ( t ) u ( t ) ⇐ ⇒ X ( s ) X ( s ) = L { x ( t ) } L − 1 { X ( s ) } x ( t ) u ( t ) = J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 15

Linearity X 1 ( s ) = L { x 1 ( t ) } X 2 ( s ) = L { x 2 ( t ) } then you should be able to show that L [ a 1 x 1 ( t ) + a 2 x 2 ( t )] u ( t ) ⇐ ⇒ a 1 X 1 ( s ) + a 2 X 2 ( s ) Example: Find the Laplace transform of x ( t ) = 5 δ ( t ) − 2 cos 5 t . L { δ ( t ) } = 1 s L { cos ωt } = s 2 + ω 2 � � 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 16

Scaling Given X ( s ) = L { x ( t ) } , what is L { x ( at ) } for a > 0 ? � ∞ 0 − x ( at )e − st d t L { x ( at ) } = t = τ τ = at a d t = 1 d τ = a d t a d τ � ∞ L { x ( at ) } = 1 0 − x ( τ )e − s τ a d τ a � ∞ = 1 0 − x ( τ )e − ( s a ) τ d τ a ⇒ 1 � s � L x ( at ) u ( t ) ⇐ aX a J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 17

Translation in Time Given X ( s ) = L { x ( t ) } , what is L { x ( t − t 0 ) u ( t − t 0 ) } for t 0 > 0 ? � ∞ 0 − x ( t − t 0 ) u ( t − t 0 )e − st d t L { x ( t − t 0 ) u ( t − t 0 ) } = τ = t − t 0 d τ = d t t = τ + t 0 � ∞ x ( τ ) u ( τ )e − s ( τ + t 0 ) d τ L { x ( t − t 0 ) u ( t − t 0 ) } = − t 0 � ∞ 0 − x ( τ ) u ( τ )e − s ( τ + t 0 ) d τ = � ∞ 0 − x ( τ )e − sτ d τ e − st 0 = 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 18

Translation in Frequency Given X ( s ) = L { x ( t ) } , what is the inverse Laplace transform of X ( s + s 0 ) ? � ∞ 0 − x ( t ) e − ( s + s 0 ) t d t X ( s + s 0 ) = � ∞ e − st d t x ( t ) e − s 0 t � � = 0 − e − s 0 t x ( t ) � � = L L − 1 { X ( s + s 0 ) } e − s 0 t x ( t ) u ( t ) = L e − s 0 t x ( t ) u ( t ) ⇐ ⇒ X ( s + s 0 ) J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 19

Time Differentiation Given X ( s ) = L { x ( t ) } , what is the Laplace transform of ˙ x ( t ) ? � ∞ � d x ( t ) � d x ( t ) d t e − st d t L = d t 0 − d u = − s e − st d t u = e − st d v = d x ( t ) d t d t v = x ( t ) � ∞ � ∞ � d x ( t ) � 0 − u d v = uv | ∞ L = 0 − − 0 − v d u d t � ∞ � ∞ = e − st x ( t ) − s e − st � � � 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 20

Time Differentiation Continued In general d n x ( t ) s n X ( s ) − s n − 1 x (0 − ) − · · · − s 0 d n − 1 x ( t ) � L � u ( t ) ⇐ ⇒ � d t n d t n − 1 � t =0 − If all of the initial conditions are zero, d n x ( t ) L s n X ( s ) u ( t ) ⇐ ⇒ d t J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.73 21