Mdellng Physcal Systems The Transer Functn
Derental Equatns U Plant Y In the plant shwn, the nput u aects the respnse the utput y. In general, the dynamcs ths respnse can be descrbed by a derental equatn the rm a n d y dt a n m m d y dy d u d u du a a0 y bm bm b b dt dt dt dt dt n n 0 u Derental equatn s lnear cecents are cnstants r unctns nly tme t. Lnear tme-nvarant system: cecents are cnstants. Lnear tme-varyng system: cecents are unctns tme.
Mechancal Systems Fundamental Law Mechancal Systems Translatnal Systems Newtn s Law x D Alembert s Prncple m mx ma mx r s appled rce, n mx 0 m s mass n g x s dsplacement n m. 3
Mechancal Systems Trsnal Systems T J J T s appled trque, n-m J s mment nerta n g-m s dsplacement n radans s the angular speed n rad/s T J J r T J 0 4
Mechancal Systems - sprngs Translatnal: x x s tensle rce n sprng, n s sprng cnstant, n/m Imprtant: Nte drectns and sgns ( x x ) Rtatnal: T are external trques appled n the trsnal sprng, n-m G s trsnal sprng cnstant, n-m/rad T G( ) 5
Mechancal Systems dampers r dashpts Translatnal:. x. x s tensle rce n dashpt, n b b s cecent dampng, n-s/m b( x x ) Rtatnal: T s trque n trsnal damper, n-m b s cecent trsnal dampng, n-m-s/rad T b ( ) 6
Usng superpstn r lnear systems Due t x : x x x Due t x : x Due t bth x and x : ( x x) Due t : T G Due t : T G Due t bth and : T G( ) 7
Usng superpstn r lnear systems Translatnal damper. x. x x Due t : Due t : bx bx x b x Due t bth and : x b( x ) x Rtatnal damper: Due t : T b Due t : T b Due t bth and : T b ( ) 8
Example Derve the derental equatn relatng the utput dsplacement y t the nput dsplacement x. b A y x Free-bdy dagram at pnt A, d A s Nte: Drectn s and d shwn assumes they are tensle. Snce m = 0, ma gves s d 0 d Snce and by ( x y) s Thus Or ( x y) by 0 by y x 9
The Transer Functn The transer unctn a lnear tme nvarant system s dened as the rat the Laplace transrm the utput (respnse) t the Laplace transrm the nput (actuatng sgnal), under the assumptn that all ntal cndtns are zer. Prevus Example by y x Assumng zer cndtns and takng Laplace transrms bth sdes we have bsy ( s) Y ( s) X ( s) Transer Functn G ( s) Y ( s) X ( s) bs Ths s a rst-rder system. 0
Example Fr the sprng-mass-damper system shwn n the rght, derve the transer unctn between the utput x and the nput x. m x Free-Bdy dagram m x Nte: s and d assumed t be tensle. b s d ma gves s d mx x Thus ( x x ) b( x x ) mx Or m x bx x bx x And ms X ( s) bsx ( s) X ( s) bsx ( s) X ( s) G( s) X X ( s) ( s) ms bs Transer Functn. Ths s a secnd-rder system. bs
Electrcal Elements Capactance e C Or E Cmplex mpedance I sc q C e q Ce dq dt C I C(sE) IX c de dt X c /( sc) Resstance e R Unts R: hms ( ) Inductance e L Unts L: Henrys (H) e e R e R E IR L L d dt t 0 e dt Or E IX L I(sL)
Electrcal Crcuts- rchh s Laws Current Law: The sum currents enterng a nde s equal t that leavng t. 0 Vltage Law: The sum algebrac sum vltage drps arund a clsed lp s zer. e 0 3
Electrcal Crcuts- Examples RC crcut: Derve the transer unctn r the crcut shwn, E IR IX c R and E IX c e C e gvng E E X c R X c RCs /( sc) R /( sc) Ths s a rst-rder transer unctn. 4
Electrcal Crcuts- Examples RLC crcut: R L and E IR IX E IX c L IX c e C e gvng E E R LCs X X L c X c RCs /( sc) R sl /( sc) Ths s a secnd-rder transer unctn. 5
Operatnal Ampler Prpertes an deal Op Amp v A( v ) v Gan A s nrmally very large s that cmpared wth ther values, ( v ) s assumed small, equal t zer. v The nput mpedance the Op Amp s usually very hgh (assumed nnty) s that the currents and are very small, assumed zer. Tw basc equatn gvernng the peratn the Op Amp ( v v v v ) 0 r and, 0 0 6
Operatnal Ampler Example Z v =0 Z S - + v Fr the Op Amp, assume =0 and v s =v + =0. 0 0 Then r V Z V Z Therere V Z R V V V ( s) ( s) Z Z 7
Operatnal Ampler Example Z v =0 Z S - + v Fr the llwng V Z Z V Z R sc V Z R V R R RCs p s 8
Permanent Magnet DC Mtr Drvng a Lad R a L a e e T J b Fr the dc mtr, the back em s prprtnal t speed and s gven by e where e s the vltage cnstant. The trque prduced s prprtnal t armature current and s gven by where s the trque cnstant. Relevant equatns: T t e T R a t L a d dt T J e t d b dt Nte: By cnsderng pwer n = pwer ut, can shw that e = t 9
End 0