1 CHAPTER 14 LAPLACE TRANSFORMS

Transcription

1 CHAPTER 4 LAPLACE TRANSFORMS 4 nroducion f x) i a funcion of x, where x lie in he range o, hen he funcion p), defined by p) px e x) dx, 4 i called he Laplace ranform of x) However, in hi chaper, where we hall be applying Laplace ranform o elecrical circui, y will mo ofen be a volage or curren ha i varying wih ime raher han wih "x" Thu hall ue a our variable raher han x, and hall ue raher han p (alhough i will be noed ha, a ye, have given no paricular phyical meaning o eiher p or o ) Thu hall define he Laplace ranform wih he noaion ) e ) d, 4 i being underood ha lie in he range o For hor, could wrie hi a ) L ) 4 When we fir learned differenial calculu, we oon learned ha here were ju a few funcion whoe derivaive i wa worh commiing o memory Thu we learned he derivaive n x ofx, in x, e and a very few more We found ha we could readily find he derivaive of more complicaed funcion by mean of a few imple rule, uch a how o differeniae a produc of wo funcion, or a funcion of a funcion, and o on Likewie, we have o know only a very few baic Laplace ranform; here are a few imple rule ha will enable u o calculae more complicaed one Afer we had learned differenial calculu, we came acro inegral calculu Thi wa he invere proce from differeniaion We had o ak: Wha funcion would we have had o differeniae in order o arrive a hi funcion? wa a hough we were given he anwer o a problem, and had o deduce wha he queion wa will be a imilar iuaion wih Laplace ranform We hall ofen be given a funcion ), and we hall wan o know: wha funcion ) i hi he Laplace ranform of? n oher word, we hall need o know he invere Laplace ranform: ) L ) 44 We hall find ha faciliy in calculaing Laplace ranform and heir invere lead o very quick way of olving ome ype of differenial equaion in paricular he ype of differenial equaion ha arie in elecrical heory We can ue Laplace ranform o ee he relaion beween varying curren and volage in circui conaining reiance, capaciance and inducance

2 However, hee mehod are quick and convenien only if we are in conan daily pracice in dealing wih Laplace ranform wih eay familiariy Few of u, unforunaely, have he luxury of calculaing Laplace ranform and heir invere on a daily bai, and hey loe many of heir advanage if we have o refreh our memorie and regain our kill every ime we may wan o ue hem may herefore be aked: Since we already know perfecly well how o do AC calculaion uing complex number, i here any poin in learning wha ju amoun o anoher way of doing he ame hing? There i an anwer o ha The heory of AC circui ha we developed in Chaper uing complex number o find he relaion beween curren and volage deal primarily wih eady ae condiion, in which volage and curren were varying inuoidally did no deal wih he ranien effec ha migh happen in he fir few momen afer we wich on an elecrical circui, or iuaion where he ime variaion are no inuoidal The Laplace ranform approach will deal equally well wih eady ae, inuoidal, non-inuoidal and ranien iuaion 4 Table of Laplace Tranform i eay, by uing equaion 4, o derive all of he ranform hown in he following able, in which > (Do i!) ) ) / / n ( n )! in a co a inha coh a / n a + a + a a a a e a a

3 Thi able can, of coure, be ued o find invere Laplace ranform a well a direc ranform Thu, for example, L e n pracice, you may find ha you are uing i more ofen o find invere ranform han direc ranform Thee are really all he ranform ha i i neceary o know and hey need no be commied o memory if hi able i handy For more complicaed funcion, here are rule for finding he ranform, a we hall ee in he following ecion, which inroduce a number of heorem Alhough hall derive ome of hee heorem, hall merely ae oher, hough perhap wih an example Many (no all) of hem are raighforward o prove, bu in any cae am more anxiou o inroduce heir applicaion o circui heory han o wrie a formal coure on he mahemaic of Laplace ranform Afer you have underood ome of hee heorem, you may well wan o apply hem o a number of funcion and hence grealy expand your able of Laplace ranform wih reul ha you will dicover on applicaion of he heorem 4 The Fir negraion Theorem ) The heorem i: L x) dx 4 Before deriving hi heorem, here' a quick example o how wha i mean The heorem i mo ) ueful, a in hi example, for finding an invere Laplace ranform e L ( ) y x dx Calculae L ( a) Soluion From he able, we ee ha L e a The inegraion heorem ell u ha a ax a L e dx ( e ) / a ( a) You hould now verify ha hi i he correc anwer by ubiuing hi in equaion 4 and inegraing or (and!) uing he able of Laplace ranform The proof of he heorem i ju a maer of inegraing by par Thu L x) dx x) dx e d x) dx d ( e ) e x) dx + e ) d

4 The expreion in bracke i zero a boh limi, and herefore he heorem i proved 4 44 The Second negraion Theorem (Dividing a Funcion by ) Thi heorem look very like he fir inegraion heorem, bu "he oher way round" i ) L x) dx 44 'll leave i for he reader o derive he heorem Here ju give an example of i ue Wherea he fir inegraion heorem i mo ueful in finding invere ranform, he econd inegraion heorem i more ueful for finding direc ranform in a Example: Calculae L in a Thi mean calculae e d While hi inegral can no doub be done, you may find i a bi dauning, and he econd inegraion heorem provide an alernaive way of doing i, reuling in an eaier inegral Noe ha he righ hand ide of equaion 44 i a funcion of, no of x, which i ju a dummy variable The funcion x) i he Laplace ranform, wih x a argumen, of ) n our paricular cae, ) i in a, o ha, from he able, y ( a x ) a + x The econd inegraion heorem, hen, in a a ell u ha L dx Thi i a much eaier inegral i a + x x π a an an an You may wan o add hi reul o your able of a a Laplace inegral ndeed, you may already wan o expand he able coniderably by applying boh inegraion heorem o everal funcion 45 Shifing Theorem Thi i a very ueful heorem, and one ha i almo rivial o prove (Try i!) i ( e a ) ) + a) L 45 For example, from he able, we have L( ) / The hifing heorem ell u ha

5 ( e a ) /( + a) 5 L 'm ure you will now wan o expand your able even more Or you may wan o go he oher way, and cu down he able a bi! Afer all, you know ha L() / The hifing heorem, hen, ell you ha L(e a ) /( a), o ha enry in he able i uperfluou! Noe ha you can ue he heorem o deduce eiher direc or invere ranform 46 A Funcion Time n 'll ju give hi one wih ou proof: n n d y For n a poiive ineger, L ( y) ( ) 46 n d Example: Wha i L( e )? Anwer: For, /( + ) y e y L ( e ) /( + ) Before proceeding furher, rongly recommend ha you now apply heorem 4, 44, 45 and 46 o he everal enrie in your exiing able of Laplace ranform and grealy expand your able of Laplace ranform For example, you can already add a (in a) /, e and e o he li of funcion for which you have calculaed he Laplace ranform 47 Differeniaion Theorem n n n d y n n n dy n d y d y d y L y y K K 47 n n n d d d d d Thi look formidable, and you will be emped o kip i bu don', becaue i i eenial! However, o make i more palaable, 'll poin ou ha one rarely, if ever, need derivaive higher han he econd, o 'll re-wrie hi for he fir and econd derivaive, and hey will look much le frighening L y& y 47 y and L & y y y y& 47 Here, he ubcrip zero mean "evaluaed a " Equaion 47 i eaily proved by inegraion by par: y [ ye ] + e dy L y ye d yde

6 6 y + yd & y + L& y 474 L y& y 475 y From hi, L & y y& y& Ly& y& yy ) y& y y & 476 ( y Apply hi over and over again, and you arrive a equaion A Fir Order Differenial Equaion Solve y & + y e, wih iniial condiion y f you are in good pracice wih olving hi ype of equaion, you will probably muliply i hrough by e, o ha i become d d ( ye ) e, from which y e ( ) + Ce (You can now ubiue hi back ino he original differenial equaion, o verify ha i i indeed he correc oluion) Wih he given iniial condiion, i i quickly found ha C, o ha he oluion i y e e + e Now, here' he ame oluion, uing Laplace ranform We ake he Laplace ranform of boh ide of he original differenial equaion: y + y L( e ) ( ) Thu y ( + )( ) Parial fracion: y + + ( )

7 7 nvere ranform: y e e + e You will probably admi ha you can follow hi, bu will ay ha you can do hi a peed only afer a grea deal of pracice wih many imilar equaion Bu hi i equally rue of he fir mehod, oo 49 A Second Order Differenial Equaion Solve & y 4& y + y e wih iniial condiion y, y& You probably already know ome mehod for olving hi equaion, o pleae go ahead and do i Then, when you have finihed, look a he oluion by Laplace ranform Laplace ranform: y + 4( y ) + y / ( + ) (My! Wan' ha fa!) A lile algebra: y 5 + ( )( )( + ) ( )( ) Parial fracion: y, or y nvere ranform: y 8 e + 4 e 8 e, and you can verify ha hi i correc by ubiuion in he original differenial equaion So: We have found a new way of olving differenial equaion f (bu only if) we have a lo of pracice in manipulaing Laplace ranform, and have ued he variou manipulaion o prepare a lighly larger able of ranform from he baic able given above, and we can go from o and from o wih equal faciliy, we can believe ha our new mehod can be boh fa and eay Bu, wha ha hi o do wih elecrical circui? Read on

8 8 4 Generalized mpedance We have deal in Chaper wih a inuoidally varying volage applied o an inducance, a reiance and a capaciance in erie The equaion ha govern he relaion beween volage and curren i L& + R + Q /C 4 f we muliply by C, differeniae wih repec o ime, and wrie for Q &, hi become ju C & LC&& + RC& + 4 f we uppoe ha he applied volage i varying inuoidally (ha i, ˆ jω e, or, if you prefer, ˆ in ω ), hen he operaor d d /, or "double do", i equivalen o muliplying by ω, and he operaor d / d, or "do", i equivalen o muliplying by jω Thu equaion 4 i equivalen o jωc LCω + jrcω + 4 Tha i, [ R + jlω + / jcω)] 44 The complex expreion inide he bracke i he now familiar impedance Z, and we can wrie Z 45 Bu wha if i no varying inuoidally? Suppoe ha i varying in ome oher manner, perhap no even periodically? Thi migh include, a one poible example, he iuaion where i conan and no varying wih ime a all Bu wheher or no varying wih ime, equaion 4 i ill valid excep ha, unle he ime variaion i inuoidally, we canno ubiue jω for d/d We are faced wih having o olve he differenial equaion 4 Bu we have ju learned a nea new way of olving differenial equaion of hi ype We can ake he Laplace ranform of each ide of he equaion Thu C & LC&& + RC& + 46 Now we are going o make ue of he differeniaion heorem, equaion 47 and 47 C ) LC( & ) + RC( ) + 47 ( Le u uppoe ha, a, and are boh zero ie before a wich wa open, and we cloe he wich a Furhermore, ince he circui conain inducance, he curren canno

9 9 change inananeouly, and, ince i conain capaciance, he volage canno change inananeouly, o he equaion become ( R + L + / C) 48 Thi i o regardle of he form of he variaion of : i could be inuoidal, i could be conan, or i could be omehing quie differen Thi i a generalized Ohm' law The generalized impedance of he circui i R + L + Recall ha in he complex number reamen of a C eady-ae inuoidal volage, he complex impedance wa R + jlω + jcω To find ou how he curren varie, all we have o do i o ake he invere Laplace ranform of R + L + /( C) 49 We look a a couple of example in he nex ecion 4 RLC Serie Tranien A baery of conan EMF i conneced o a wich, and an R, L and C in erie The wich i cloed a ime We'll fir olve hi problem by "convenional" mehod; hen by Laplace ranform The reader who i familiar wih he mechanic of damped ocillaory moion, uch a i deal wih in Chaper of he Claical Mechanic noe of hi erie, may have an advanage over he reader for whom hi opic i new hough no necearily o! "Ohm' law" i Q / C + R + L&, 4 or LC Q & + RCQ& + Q C 4 Thoe who are familiar wih hi ype of equaion will recognize ha he general oluion (complemenary funcion plu paricular inegral) i λ λ Q Ae + Be + C, 4 R R where λ + λ L 4 L LC and R R L 4L LC 44 (Thoe who are no familiar wih he oluion of differenial equaion of hi ype hould no give up here Ju go on o he par where we do hi by Laplace ranform You'll oon be reaking ahead of your more learned colleague, who will be ruggling for a while)

10 Cae R 4L i poiive For hor 'm going o wrie equaion 44 a LC λ a + k and λ a k 45 a k a k Then Q Ae ( ) + + Be ( ) + C 46 and, by differeniaion wih repec o ime, ( a k) ( a + k) A( a k) e B( a + k) e 47 A, Q and are boh zero, from which we find ha A ( a + k ) C ( a k) C and B 48 k k Thu Q a + k e k ( a k ) a k + e k ( a + k ) + C 49 a k k ( a k ) ( a + k ) and ( e e ) C 4 On recalling he meaning of a and k and he inh funcion, and a lile algebra, we obain Lk e a inh k 4 Exercie: erify ha hi equaion i dimenionally correc Draw a graph of : The curren i, of coure, zero a and Wha i he maximum curren, and when doe i occur? R Cae i zero n hi cae, hoe who are in pracice wih differenial equaion 4L LC will obain for he general oluion Q e λ ( A + B ) + C, 4 where λ R / ( L), 4 λ from which A B e λ λ( + ) + Be 44 Afer applying he iniial condiion ha Q and are iniially zero, we obain

11 R R /(L) Q C e 45 L and L R /( L e ) 46 A in cae, hi ar and end a zero and goe hrough a maximum, and you may wih o calculae wha he maximum curren i and when i occur Cae R 4L i negaive n hi cae, am going o wrie equaion 44 a LC λ a + jω and λ a jω, 47 where a R R and ω 48 L LC 4L All ha i neceary, hen, i o repea he analyi for Cae, bu o ubiue ω for k and jω for k, and, provided ha you know ha inh jω j in ω, you finih wih L e a in ω ω 49 Thi i lighly damped ocillaory moion Now le u ry he ame problem uing Laplace ranform Recall ha we have a in erie wih an R, L and C, and ha iniially Q, and & are all zero (The circui conain capaciance, o Q canno change inananeouly; i conain inducance, o canno change inananeouly) mmediaely, auomaically and wih carcely a hough, our fir line i he generalized Ohm' law, wih he Laplace ranform of and and he generalized impedance: [ R + L /( C) ] + 4 Since i conan, reference o he very fir enry in your able of ranform how ha /, and o, L( + b + c) 4 [ R + L + /( C) ] where b R / L and c / ( LC) 4

12 Cae b > 4c ( )( ) L α β L α β α β 4 Here, of coure, α b + b 4c and β b b 4c 44 On aking he invere ranform, we find ha β ( e α e ) L ) 45 α β From here i i a maer of rouine algebra (do i!) o how ha hi i exacly he ame a equaion 4 n order o arrive a hi reul, i wan' a all neceary o know how o olve differenial equaion All ha wa neceary wa o underand generalized impedance and o look up a able of Laplace ranform Cae b 4c n hi cae, equaion 4 i of he form L ( α), 46 where α b f you have duifully expanded your original able of Laplace ranform, a uggeed, you will probably already have an enry for he invere ranform of he righ hand ide f no, you know ha he Laplace ranform of i /, o you can ju apply he hifing heorem o α ee ha he Laplace ranform of e i / ( α) Thu L e α 47 which i he ame a equaion 46 [Goh wha could be quicker and eaier han ha!?] Cae b < 4c Thi ime, we'll complee he quare in he denominaor of equaion 4 :

13 ω, L ( + b ) + ( c b ) L ( + b ) ω + ω 4 48 where have inroduced ω wih obviou noaion On aking he invere ranform (from our able, wih a lile help from he hifing heorem) we obain b e in ω, 49 Lω which i he ame a equaion 49 Wih hi brief inroducory chaper o he applicaion of Laplace ranform o elecrical circuiry, we have ju opened a door by a iny crack o glimpe he poenial grea power of hi mehod Wih pracice, i can be ued o olve complicaed problem of many or wih grea rapidiy All we have o far i a iny glimpe hall end hi chaper wih ju one more example, in he hope ha hi hor inroducion will whe he reader' appeie o learn more abou hi echnique 4 Anoher Example R R + + C Q C Q + FGURE X The circui in figure X conain wo equal reiance, wo equal capaciance, and a baery The baery i conneced a ime Find he charge held by he capacior afer ime Apply Kirchhoff econd rule o each half: Q & + Q & ) RC + Q C, 4 ( and Q & RC + Q Q 4 Eliminae Q : Tranform, wih Q and Q & iniially zero: R C Q & + RCQ + Q C 4

14 4 C R C + RC + ) Q 44 ( e R CQ, ( + a + a ) 45 where a /( RC) 46 Tha i R CQ ( + 68a )( + 8a) Parial fracion: R CQ + 68a 8a a Tha i, Q + C 68a 8a /( RC) 8 /( RC) nvere ranform: Q [ + 78e 78e ] 49 The curren can be found by differeniaion leave i o he reader o eliminae Q from equaion 4 and and hence o how ha Q 68 /( RC) 8 /( RC) [ 764e 76e ] 4