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by: Judd Okuneva

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# System Modeling and Control ECE 360

Judd Okuneva
BSU
GPA 3.91

Staff

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COURSE
PROF.
Staff
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Class Notes
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13
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KARMA
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## Popular in Engineering Electrical & Compu

This 13 page Class Notes was uploaded by Judd Okuneva on Saturday October 3, 2015. The Class Notes belongs to ECE 360 at Boise State University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/217983/ece-360-boise-state-university in Engineering Electrical & Compu at Boise State University.

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Date Created: 10/03/15
Boise State University Department of Electrical and Computer Engineering ECE 360 7 System Modeling and Control The Laplace Transform I Reading Assignment Read Sec 23 Lecture Objectives 1 To de ne the Laplace transform 2 To review some properties of Laplace transforms 3 To derive the Laplace transforms of elementary time functions Example of a Transform The Logarithm Transform Real Domain a Real Domain a A m lna b 1L y lnb l l cabelnz e il z my lnalnb lnab De nition of the One Sided Laplace Transform was iofte dt 1 0 Notes 1 The Laplace integral is integrated over time starting shortly time t 0 This is indicated by the notation 07 which means 0 7 e 6 being an arbitrarily small number The reason for using 07 instead of 0 will become apparent later on to The variable 5 is a complex variable Thus the Laplace domain or s domain represents functions of a complex number s a jw 03 The Laplace integral de ned above will converge for a particular 5 if lfte l dt T W In particular7 a function that does not grow faster than an exponential7 701 dt lt 00 F m lt Me for allt is Laplacetransformable since 00 00 may dt lt Me tdt lt ooif a gt a 07 07 01 All functions encountered in electrical engineering have Laplace transforms O for tlt0 ut l for tgt0 J O t Example The unit step function 0fortlt0 t 1fort0 ut Amlxe dt 57100 timid Properties of the Laplace Transform 1 was Amme dt Gltsgt ago 0 glttgtestdt Property 1 Linearity E aft 3905 aFs BG5 for all real or complex a 3 Proof am 6905 amen dt was dt 04000 fte st dt 6000 gte 9 dt a ft gt aFs BGM Special Cases ftgt Fltsgtaltsgt for a 1 aft 7 aFs for arbitrary Oz and B 0 Example Kut K ut Property 2 Frequency Shift Property r 50705 Fs a Loci 450705 e at t fst dt fte 9 gt dt fte 9tdt my Fsa Am A 0 0 Example 1 was e atut Direct Proof CO CO e atut 5quot X 6757 dt eiltsa dt 0 0 1 50 57mm 57saoo 57sa0 1 7 7sa50 7sa77sa 7 50 f t yo 0 t O r t Property 3 TimeShift Property UN 7 e STFM Loci mam Am twe s dt Oft7739e 5tdt 0 m 7 e s ew dt 7 T e STOooft e 5tdt EMF was E ut77 57M Property 4 Multiplication by Time was Ammwdt 1 aw 7 Loci ms Amme wt imam i gmw dt itfte tdt Amt m stdt 7 tft amt HWQZ mum MW 7amp6 7 712 52 t2ulttgt 33 EWw 25 amt m i Vigil s31 elttgt sia eattut m e tzulttgt sins Euler s Identity 079 0080 j 81110 0080 079 5 792 5 79 0080 7 j 81110 81110 579 7 5 792j Sine and Cosine Fun0tion8 ejwteijwt 2 1 at 7 JLth e 2 6 111 2573111 25jw 1sjwsijw h E 008 wt 1Hw1H s 2sjwsijw 7 52w2 jwti ijwt 8inwt J 1 1 Z 07 t727j 0 Wt 1 1 1 1 Qijsijw72ij5jw 1sjwisjw i w 27ltsijsijwgt 82w Damped Sine and Cosine Fun0tion8 e at cos 1140 e sin 1140 MW Table 1 Laplace Transform Properties Time Function Laplace Transform t F8 f0 1 05 dt 905 98 IS 905 dt a t 696 aF8 BGM ms got Fltsgt0ltsgt swim 1 a we 7 WM mt 7 was HMS Ht sFltsgt e W Table 2 Laplace Transforms Time Function Laplace Transform 6t 1 W5 was 2525 5211 eiatut in cos wtut Su sin wtut shim wt coswtut e at sin wtut MW 2Ke 0t coswt 1 2Kte coswt gt KZqS Kliqs 9447771102 sajw2 2045 cos wt 7 235 sin wt 04 73919 axial saijw sajw 20m cos wt 7 23hr sin wt 04739 Drij 9447771102 sajw2 Boise State University Department of Electrical and Computer Engineering ECE 360 7 System Modeling and Control The Laplace Transform IV Lecture Objectives 1 To discuss the unit impulse or Dirac delta function and its applications 2 To review important inverse Laplace transforms 1 58t ugm 8 Jdt l 05 i o i i o i 2 2 2 S Q 0 S Q 0 5t ut dt 1 O O l e l 6 6575 27105 774057 675 7 li165t EH 1 5552 57552 5552 757552 A45 7lt 7 7 e s 5 65 was 335 e 5592 7 57552 82ees2 7 78267552 mum L605 7 3 5 7 5 Property 1 30 6t dt all 6t dt 1 fj ti dt 1 PropertyQ ff m6tftdt f j6tf0dt f0 gt ffooo6t7739ftdt f7 Property3 6t 0036te 9tdt 5 90 1 gt 6t77 5 97 Example 1 t ut ut 7 3 ut 7 6 7 3ut 7 9 F5 l i 8 8 8 Indirect Method f t 6t 6t 7 3 6t 7 6 7 36t 7 9 1 1 5 39 5 69 7 3595 Fs E 1 5 39 5 69 7 35799 8 Example 2 f t Direct Method ft 2tut 7 ut717t71ut717 2ut 7 2 7 t 7 2ut 7 2 2 e79 679 26725 6729 Indirect Method f t 2m 7 ut 7 1 7 6t 7 1 7 ut 7 2 7 26t 7 2 2 679 7 e729 7 f t g7 S 7597 S 72529 ft 2 57s 57s 5729 25729 F 7 7 7 7 7 7 7 7 7 S 8 82 82 8 82 8 Example 3 fquot t 725t71 725t73 f39 t 2 1 3 4 J 11 1 2 l l t 72 ft 3 2 1 0 1 2 3 4 t Direct Method ft 2tut 7 2t71ut717 2t 7 3ut 7 3 2t 7 4ut 7 4 2 25 s 25739 25749 8 82 82 Indirect Method f t 26t 7 26t 7 1 7 26t 7 3 26t 7 4 f t 2 7 25 s 7 2535 2549 g 09 2 259 2539 2549 Ewe75 s E7 s 7 s s g 1 2 259 2539 2549 F f 52 t 52 Important Inverse Laplace Transforms Ns F Poles 07 7a 7a 70 ijw7 70 i jw Partial Fraction Expansion 8 7 5 50 sa2 saijw sajw E E swim Haw rig Am 5 71 B Be atut 50 C 71 fat E 7sa2 Cte ut 7 7 7 7 7 71 134 D I 71DZ jDM 80397jw sayw 80397jw sayw 7 D e z eiwijw 7 D ei z eiw mt 51mm 7 ef wtw f DEWquot coswt gt Q D e d 2 D 13 4 7 4 71 71Ma J 80397jw sayw 80397jw sayw a j6e ltquot 39 gt a 7 j6e quotquot gt jwt ijwt jwt 7 ijwt 2045 i 7 255th 2045 cos wt 7 235 sin wt 2 2 701 0 7 2 04 56 Wcoswt W 2 D g7 cosztcoswt 7 sin gtsinwt 2D gidcoswt gt sin wt y y E saijw2 E saijw2 E 8ayw2 E s a jw2 1 EW va WWW swww E ej teiwijmt E e j bte WJrjmt 51mm ei wtw f 2 E te coswt gt Q E te 71 a j aim 8aijw2 8UJ w2 a j te ltquot 39 gt a 7 j6gtte ltquotquot gt jwt ijwt 5 5 J i 2ozte 01 cos wt 7 2 te sin wt egwt 7 eigwt Qate d 2j 2 2 7 t 0 7 2 Oz t5 7 Wooswt W 2 E te d cosztcos wt 7 sin gtsin wt 2 E te coswt gt J sinwt

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