All the Laplace Transform you will encounter has the following form: Rational function X(s)

Transcription

1 EE G Note: Chpter Itructor: Cheug Pge - - Iverio of Rtiol Fuctio All the Lplce Trform you will ecouter h the followig form: m m m m e τ Rtiol fuctio Dely Why? Rtiol fuctio come out turlly from the Lplce trform of oriry ifferetil equtio, expoetil, coie ie fuctio. Our trtegy: rekow geerl rtiol fuctio ito impler frctio polyomil whoe ivere trform hve lrey ee compute i Tle -. How? I. Covert o-proper rtiol fuctio ito proper rtiol fuctio No-proper: egree of umertor egree of eomitor Approch: Log Diviio Exmple: Polyomil Proper frctio

2 EE G Note: Chpter Itructor: Cheug It i ey to fi the Lplce trform of polyomil: δ t L L L δ t δ t II. Fctorize the eomitor polyomil: Exmple: Fctorize ito REAL lier irreucile qurtic fctor. Further rek ow REAL irreucile qurtic fctor ito cougte root. Notice ome fctor my repet. No lyticl formul for polyomil with egree or other. Nee to rely o umericl metho. >> root[ ] i i >> fctorym'^*^*^*' ^*^ III. Brek ow the proper rtiol fuctio ito imple frctio. Thi techique i clle Prtil Frctio Expio. Exmple: Firt, write ow the full prtil frctio expio with eough umer of ukow to repreet ll poile umertor: Pge -

3 EE G Note: Chpter Itructor: Cheug Four imple rule to write ow ukow:... k.... for rel root k k.... for complex cougte root k k k k k k k IV. Solve for the ukow. My metho exit. I will tlk out two:. Compre coefficiet The Dum Wy Exmple: Re 8Re Im 8Re 8Im 8Im Notice tht whe you exp it out, o complex coefficiet remi you c ALWAYS o tht if you follow the four imple rule ove. You c k Mtl to o thi too: >> xym'//^ri*i/ir-i*i/-i'; >> x implifyx % Multiply out x 8*i**i*^**^**^**^8*r*^8*r* *r*^8*i/^/^ >> x ortcollectx %collect like term i eceig power x *r*^*8*r*i*^*8*r8*i*8*i/ ^/^ Pge -

4 EE G Note: Chpter Itructor: Cheug Sice we re give, we kow tht the umertor polyomil i ut. By comprig coefficiet, we get the followig et of equtio: Re 8Re Im 8Re 8Im 8Im To olve thi ytem of equtio umericlly, rewrite it i mtrix form: Re 0 8 Im 0 The ue mtl: >> A [ 0 0; 8 ; 0 8 8; 0 8]; >> [0;0;;0]; >> A\ % Solutio to Ax or 0.8,.6, Re 0., Im 0. Pge -

5 EE G Note: Chpter Itructor: Cheug Pge -6. Heviie Theorem: Altertively, you c ue the Heviie Theorem: Exmple/ Multiply oth ie y, Note tht every term o the right h fctor. Sutitute - the root for : or A imilr pproch c olve for well. Multiply oth ie y : Sutitutig -: 6. However, we cot ue thi pproch fi. If we multiple to oth ie, we hve A i repete fctor, we till hve oe left fter the multiplictio. We c t utitute - here it will tur oth ie to ifiity. Ite, we till multiply ut the we tke the erivtive with repect to : 8 Tkig the erivtive kill expoe while ll the remiig term will till hve fctor of. Sutitute -, we get -0.8.

6 EE G Note: Chpter Itructor: Cheug The me pproch c e ue for fctor with multiplicity >. Exmple: Fi the prtil frctio expio of Agi, we firt write ow the prtil frctio expio Here i the geerl rule: if the eomitor of h fctor -p the the coefficiet m for the prtil frctio /-p m for m< i for m m! [ p ] p m m m [ p ] p Both metho rrive t the me wer: The previou four tep: covertig to proper polyomil, fctorizig eomitor polyomil, writig ow prtil frctio expio, olvig for the ukow coefficiet c ll oe y imple MATLAB comm: reiue From the Mtl Help-pge: RESIDUE Prtil-frctio expio reiue. [R,P,K] RESIDUEB,A fi the reiue, pole irect term of prtil frctio expio of the rtio of two polyomil B/A. If there re o multiple root, B R R R K A - P - P - P. If P... Pm- i pole of multiplicity m, the the expio iclue term of the form Pge -7

7 EE G Note: Chpter Itructor: Cheug R R Rm P - P^ - P^m Uig the mple exmple: 9 9 By h, thi i the wer we get: Firt, we efie the rtiol fuctio y writig the coefficiet of the eomitor umertor i DESCENDING ORDER without y SKIP OF POWER: >> e [ ]; >> um [ 9 9 ]; Ruig the reiue comm: >> [r,p,k] reiueum,e r % coefficiet of t fctor % coefficiet of fctor i % coefficiet of r fctor i % coefficiet of th fctor p % firt fctor: % eco fctor: ^ i % thir fctor: i % fourth fctor: k % leig polyomil: Pge -8

8 EE G Note: Chpter Itructor: Cheug V. Compute the ivere Lplce trform for ech term. Comie cougte root, eprte frctio for ech umertor term: Write every term i the form whoe Lplce trform we kow: δ t δ t 0.8 t 0.8 e.6 t.6 te cot i t For geerl qurtic term, we ee to complete the qure / - / ee the exmple property : complex frequecy hift theorem. Ue lierity to comie them you re oe!! t t x t δ t δ t 0.8e.6te 0.8cot 0.6i t u t Pge -9