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2.LaplaceTransformx4

# 2.LaplaceTransformx4 - Overview of Laplace Transform Topics...

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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 Gives you insight in circuit analysis and design Used extensively in Controls (ECE 311) Communications Signal Processing Analog circuits (ECE 32X sequence) Expected to know for interviews J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.75 3 Overview of Laplace Transform Topics 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. 1.75 1 Laplace Transform Analysis Illustration t = 0 v o - + 1 k Ω sin(1000 t ) 1 μ F Given v o (0) = 0 , solve for v o ( t ) for t 0 . v o ( t ) = 1 2 e t/ 0 . 001 + 1 2 sin(1000 t 45 ) = v tr ( t ) + v ss ( t ) v tr ( t ) = 1 2 e t/ 0 . 001 v ss ( t ) = 1 2 sin(1000 t 45 ) J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.75 4 Laplace Transform Motivation t Linear Circuit v s ( t ) v o ( t ) - + v s ( 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. 1.75 2

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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. 1.75 7 Laplace Transform Analysis Illustration Continued 0 5 10 15 20 25 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total Transient Steady State Time (ms) v o ( t ) (V) v o ( t ) = 1 2 e t/ 0 . 001 + 1 2 sin(1000 t 45 ) = v tr ( t ) + v ss ( t ) J. McNames Portland State University ECE 222 Laplace Transform Ver. 1.75 5 Laplace Transform Definition L { x ( t ) } = X ( s ) 0 - x ( t )e st d t Transform will be written with an upper-case letter Defined from 0 to include impulses at t = 0 s = σ + 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 ( s ) = + −∞ x ( t )e st d t 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. 1.75 8 Laplace Transform for ODE’s x ( t ) Linear Circuit y ( t ) - + N k =0 a k d k y ( t ) d t k = M k =0 b k d k x ( t ) d t k Relationship of a voltage (or current) in a linear circuit to any
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2.LaplaceTransformx4 - Overview of Laplace Transform Topics...

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