INSTRUMENTATION AND CONTROL TUTORIAL 1 CREATING MODELS OF ENGINEERING SYSTEMS

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1 INSRUMENAION AND CONROL UORIAL CREAING MODELS OF ENGINEERING SYSEMS hs tutral s f nterest t any stuent stuyng cntrl systems an n partcular the EC mule D7 Cntrl System Engneerng. he purpse f ths tutral s t ntruce stuents t the basc elements f engneerng systems an hw t create a transfer functn fr them. he tutral s manly nfrmatve an cnssts f examples shwng the ervatn f mels fr real harware systems. he self assessment materal s base n basc general engneerng knwlege. On cmpletn f ths tutral, yu shul be able t the fllwng. Derve the mathematcal mels f basc mechancal systems. Derve the mathematcal mels f basc flu pwer systems. Derve the mathematcal mels f basc thermal systems. Derve the mathematcal mels f basc electrcal systems. Recgnse the smlarty between mels f fferent systems. Explan the stanar frst an secn rer transfer functns. Explan the lnk between pen an clse lp transfer functns. If yu are nt famlar wth nstrumentatn use n cntrl engneerng, yu shul cmplete the tutrals n Instrumentatn Systems. In rer t cmplete ths tutral, yu must be famlar wth basc mechancal an electrcal scence. Yu shul als be famlar wth the Laplace transfrm an a tutral n ths may be fun n the maths sectn. Yu can als fn tutrals n flu pwer n ths ste. utral n ths seres gves a etale accunt f electrc mtr mels an yu may wsh t stuy ths frst. D.J.DUNN

2 . INRODUCION Engneerng systems s a very bra area rangng frm cntrl f a pwer statn t cntrl f a mtr s spee. he stuent nees t have a bra base knwlege f engneerng scence n rer t unerstan the varus elements an see hw many f them are mathematcally the same (analgues f each ther). Dfferent kns f engneerng systems ften cnfrm t smlar laws an there are clear analges between electrcal, mechancal, thermal an flu systems. he basc laws whch we use mst ften cncern Resstance R, Capactance C, Inuctance L an cnservatn laws. Yu nt nee t stuy all these n etal an the apprprate law wll be explane as requre. Here s a table shwng the man analgue cmpnents. It s useful t nte that capactance s a zer rer fferental equatn, resstance s a frst rer fferental equatn an Inuctance/nerta/nertance s a secn rer fferental equatn. MECHANICAL ELECRICAL HERMAL FLUID Sprng Electrcal Capactr hermal capactr Flu Capactr x C F (/k) F Q C V Q C M C x p Damper Frce k x velcty F k x/t rque k x Ang.vel Ohm's Law V R I V R Q/t Heat ransfer Laws R Φ R Q/t Flu frctn Laws nt cnfrm t ths pattern. Newtn's n Law f mtn Frce Mass x acceleratn F M x/t Law f Inuctrs V L q/t N equvalence Flu nertance p L v/t D'Alembert's Prncples Frce 0 Mment 0 Krchff's Laws current 0 Law f Cnservatn f Energy Energy cnstant Law f Cnservatn f Mass Mass cnstant Let s lk at the smlarty f the varus quanttes use n these systems. D.J.DUNN

3 . SIMILARIY OF ELEMENS CAPACIANCE he symbl C wll be use fr electrcal, thermal an flu capactance. Mechancal capactance s equvalent t /k fr mechancal systems where k s the sprng stffness. RESISANCE he symbl R wll be use fr electrcal an thermal resstance. Mechancal/hyraulc resstance s calle the ampng ceffcent an has varus symbls. INDUCANCE / INERIA / INERANCE he symbl L wll be use fr electrcal nuctance an flu nertance. In mechancal systems, mass M s the equvalent prperty fr lnear mtn an mment f nerta I fr angular mtn. OHER EQUIVALEN PROPERIES Q s the symbl fr electrc charge an quantty f heat. hs s equvalent t splacement n mechancal systems, these beng stance (usually x) r angle (usually ). V s the symbl fr electrc vltage (ptental fference r e.m.f) an s equvalent t temperature fr thermal systems, Frce F fr mechancal systems an pressure p fr flu systems. v r u s the symbl fr velcty n mechancal systems an ths s equvalent t electrc current (I r ) an heat flw rate Φ. 3 LAPLACE RANSFORM an RANSFER FUNCIONS Laplace s cvere n etal n later tutrals an n the maths sectn. he purpse f ths transfrm s t allw fferental equatns t be cnverte nt a nrmal algebrac equatn n whch the quantty s s just a nrmal algebrac quantty. In ths tutral we shul smply regar t as a shrthan meth f wrtng fferental ceffcents such that: n n becmes s becmes s n becmes s t t t 4 RANSFER FUNCIONS he mels f systems are ften wrtten n the frm f a rat f Output/Input. If the mels are turne nt a functn f s t s calle a transfer functn an ths s usually ente as G(s). Output G(s) he utput an nput are functns f s. Input Nw let s examne the mathematcal mels f sme mechancal systems. D.J.DUNN 3

4 5. BASIC MODELS OF MECHANICAL SYSEMS 5. GENERAL PROCEDURE he general prceure fr mechancal systems s as fllws.. Apt a sutable c-rnate system wth an apprprate sgn cnventn. Fr lnear mtn, up s pstve an left t rght s pstve. Fr rtatn antclckwse s pstve an clckwse s negatve. hese may be gnre when cnvenent.. Ientfy any sturbng frces actng n the system (nputs t the system).. Ientfy splacements an/r velctes (utputs frm the system). v. Draw a free by agram fr each mass shwng all the frces an mments actng n t. v. Apply Newtn's n Law t each free by agram (F Mass x Acceleratn). v. Rearrange the equatn(s) nt a sutable frm fr slutn by a cnvenent meth. Nte that unless therwse specfe, gnre gravtatnal effects. Let s nw examne mechancal elements n etal. 5. LINEAR MECHANICAL SYSEMS. 5.. SPRING he basc law f a mechancal sprng s Frce change n length. he agram shws the mel wth mechancal symbls an as a blck agram. Fgure he relatnshp has n ervatves n t may be wrtten as a functn f t r s wth n transfrm nvlve. As a functn f tme we wrte F(t) k x(t) where k s the sprng stffness. As a functn f s we wrte F(s) kx(s) hs can be arrange as a transfer functn such that x F C s the recprcal f stffness an t s calle mechancal capactance. he use f k s usually preferre n mechancs but C s use n systems as t s rectly analgus t electrcal capactance. 5.. DAMPER r DASHPO (s) /k C Fgure D.J.DUNN 4

5 A amper may be ealse as a lsely fttng pstn mvng n a vscus flu such that the frce s rectly prprtnal t velcty. F v. Velcty v s the frst ervatve f stance s F x/t x he basc law f a ashpt s: F(t) k k s the ampng ceffcent. t Change nt Laplace frm. F k s x x Rearrange nt a transfer functn (s) F ks k s the ampng ceffcent wth unts f Frce/Velcty r N s/m. he agram shws the mel wth mechancal symbls an the cntrl blck MASS When a mass s accelerate, the nerta has t be vercme an the nerta frce s gven by Newtn s Secn Law f Mtn Frce Mass x Acceleratn. Acceleratn s the secn ervatve f x wth tme. x Basc Law F(t) M t Change nt Laplace frm. F Ms x x Rearrange nt a transfer functn (s) F Ms 5..4 MASS - SPRING SYSEM Fgure 3 Fr ths sprng - mass system, mtn nly ccurs n ne rectn s the system has a sngle egree f freem. It s nrmal fr the rectn f mtn t be expresse as the x rectn regarless f the actual rectn. he free by agram s as shwn. he nput s a sturbng frce F whch s a functn f tme F(t). hs cul, fr example, be a snusal frce. he utput s a mtn x whch s a functn f tme x(t). Let x be pstve upwars. Fgure 4 D.J.DUNN 5

6 he nput frce s ppse by the sprng frce an the nerta frce (whch always ppses changes n the mtn as state n Newtn s thr law f mtn). Sprng frce k x Inerta frce M x/t D'Alembert's Prncple s that all the frces an mments n the by must a up t zer. In ths case t means F(t) - kx(t) - M x/t(t) 0 r F(t) M x/t(t) kx (t) Changng t a functn f s we have F(s) Ms x kx x [Ms k] F x(s) Ms k x(s) hs may be shwn as a transfer functn. G(s) F(s) s he blck agram fr use n systems s as shwn. ( ) F /M s k/m /M k/m Fgure SPRING DAMPER Frce balance as a functn f tme. Frce balance as a functn f s Rearrange nt a transfer functn. F(t) k x k x/t F(s) k x k s x x /k (s) F k /k s ( ) he unts f k/k are secns an ths s the tme cnstant fr the system k /k x /k (s).hs s the stanar frst rer equatn whch we shall stuy many tmes n these tutrals. F s Fgure 6 D.J.DUNN 6

7 5..6 MASS -SPRING - DAMPER SYSEM he nput s the frce F an the utput s the mvement x, bth beng functns f tme. Sprng frce Fs kx Dampng frce F k x/t Inerta frce F Mx/t he three frces ppse mtn s f the ttal frce n the system s zer then F F F Fs x x x /k F(t) M k kx F(s) Ms x k sx kx G(s) (s) t t F s (M/k) s(k /k) If we examne the unts f (M/k)/ we fn t s secns an ths s the secn rer tme cnstant als wth the symbl. he transfer functn may be wrtten as x /k G(s) (s) F s δ s δ s the ampng rat efne as δ k/cc an Cc s the crtcal ampng rat efne as (4Mk)½. he k k M/k k M k term δ s hence an s the frgng s crrect. C 4Mk M k k k c Fgure 7 hs s the stanar n rer transfer functn whch wll be analyse n etal later. WORKED EXAMPLE N. A mass sprng system has the fllwng parameters. Stffness K 800 N/m Mass M 3 kg Dampng Ceffcent k 0 Ns/m. Calculate the tme cnstant, crtcal ampng ceffcent an the ampng rat.. Derve the equatn fr the frce requre when the pstn s acceleratng.. Use the equatn t evaluate the statc eflectn when F N. v. Use the equatn t evaluate the frce neee t make the mass accelerate at 4 m/s at the mment when the velcty s 0.5 m/s. D.J.DUNN 7

8 SOLUION. (M/k) (3/800) 0.06 secns c c 4MK (4 x 3 x 800) Ns/m δ k / c c 0/ Fr a cnstant acceleratn s x a (acceleratn) an sx v (velcty) x /k (s) F kx( s δ s ) F s δ s F ( 0.06 s x 0.04 x 0.06s ) ( 3s 0 s 800) s x 0 sx 800 x 800x F x F 3 a 0 v 800 x Fr a cnstant frce an a statc pstn there wll be nether velcty nr acceleratn s the s an s terms are zer. F 800 x 0.05 m r 5mm x 800 v. Fr velcty 0.5 m/s an a 4 m/s F 3 a 0 v 800 x 0 800x 800 x he eflectn x wul nee t be evaluate frm ther meths x v /a 0.03 m F 46.8 N SELF ASSESSMEN EXERCISE N.. A mass sprng system has the fllwng parameters. Stffness K 00 N/m Mass M 5 kg Dampng Ceffcent k 0 Ns/m. Calculate the tme cnstant, crtcal ampng ceffcent an the ampng rat. (0. s, 68.3 Ns/m an 0.447).. If a cnstant frce f N s apple, what wll be the statc pstn f the mass? (8 mm) Calculate the frce neee t make the mass mve wth a cnstant acceleratn f m/s at the pnt where the velcty s. m/s. (396 N) D.J.DUNN 8

9 5.3 ROARY MECHANICAL SYSEMS he fllwng s the rtary equvalent f the prevus wrk ORSION BAR hs s the equvalent f a mass an sprng. A metal r clampe at ne en an twste at the ther en pruces a trque ppsng the twstng rectly prprtnal t the angle f twst. he rat / s the trsn stffness f the trsn sprng an s ente wth a k. s trque ( N m) s the angle f twst (raan) k s the trsnal stffness ( N m/ra ) Balancng the trques we have Change t Laplace frm. Wrte as a transfer functn. (t) k (t) (s) k (s) (s) k Fgure ORSION DAMPER A trsn amper may be ealse as vanes rtatng n a vscus flu s that the trque requre t rtate t s rectly prprtnal t the angular velcty. k s the trsn ampng ceffcent n N m s/raan (t) k t (s) k s G(s) (s) k s Fgure 9 D.J.DUNN 9

10 5.3.3 MOMEN OF INERIA Rtatng masses ppse changes t the mtn an Newtn's n law fr rtatng masses s I /t I s the mment f nerta n kg m. (t) I (s) I s G(s) (s). Nte many text bks als use J fr mment f nerta. t Is Fgure 0 SELF ASSESSMEN EXERCISE N. Derve the transfer functn fr a mass n a trsn bar ftte wth a amper an shw t s anther example f the secn rer transfer functn. s trque an J s mment f nerta. G(s) (s) (J/k)s /k (Jk /k)s s /k δ s D.J.DUNN 0

11 5.3.4 GEARED SYSEMS When a mass s rtate thrugh a gear system, the affect f the nerta s ramatcally altere. Cnser a mtr cuple t a la thrugh a spee changng evce such as a gear bx. here s ampng (vscus frctn) n the tw bearngs. Fgure m s the mtr rtatn an the utput rtatn. he gear rat s Gr / m Snce ths s a fxe number an s nt a functn f tme, the spee an acceleratn are als n the same rat. m /t ω m /t ω Gr ω /ω m ω s the angular velcty m /t α m /t α Gr α /α m α s the angular acceleratn. he pwer transmtte by a shaft s gven by Pwer ω. If there s n pwer lst, the utput an nput pwer must be equal s t fllws that ω m m ω hence m ω /ω m Gr (In realty frctn sgnfcantly affects the trque) Cnser the nerta trque ue the nerta n the utput shaft I. I α I α m x Gr m x Gr I α m x Gr Nw cnser the ampng trque n the utput shaft. k ω k ω m Gr m x Gr k ω m Gr Nw cnser that there s an nerta an ampng trque n the mtr shaft an n the utput shaft. he ttal trque pruce n the mtr shaft s m I m α m k m ω m Gr I α k ω m I m α m k m ω m Gr { I α k ω } m I m α m k m ω m Gr I α m Gr k ω m m α m (I m Gr I ) ω m (k m Gr k ) (I m Gr I ) s the effectve mment f nerta Ie an (k m Gr k ) s the effectve ampng ceffcent ke. he equatn may be wrtten as In calculus frm ths becmes m α m (Ie) ω m (ke) m /t (Ie) /t (ke) D.J.DUNN

12 Changng ths nt a functn f s we have m (s) s(ie) s (ke) s{sie ke} he utput s the mtr angle an the nput s the mtr trque s the geare system may be presente as a transfer functn thus. (s)/m (s) (/Ie)/s{s Ke/Ie} Fgure WORKED EXAMPLE N. A DC Serv mtr has a mment f nerta f 0.5 kg m. It s cuple t an aeral rtatr thrugh a gear reuctn rat f 0. he rven mass has a mment f nerta f. kg m. he ampng n the mtr s 0. N m s/ra an n the rtatr bearngs t s 0.05 N m s/ra. Wrte wn the transfer functn / m n the smplest frm. Calculate the trque requre frm the mtr t. urn the aeral at a cnstant rate f 0.0 ra/s.. Accelerate the rtatr at ra/s at the start when ω 0 SOLUION. I e (I m Gr I ) (0.5 0 x.) 0.5 kg m. Ke (k m Gr k ) (0. 0 x 0.05) 5. N m s/ra. (s)/m (s) (/Ie)/s{s Ke/Ie} /Ie (s) s s k /I m m m I e α 0.5α ( ) K e e ω 5.ω e If the rtatr s mvng at cnstant spee α (acceleratn) s zer. Hence: m 5.ω 5. x Nm. When acceleratng at ra/s the mtr acceleratn s 0 tmes larger at 0.05 ra/s. m 0.5α 5.ω 60.5 Nm when ω 0 D.J.DUNN

13 SELF ASSESSMEN EXERCISE N.3 A DC Serv mtr has a mment f nerta f kg m. It s cuple t an aeral rtatr thrugh a gear reuctn rat f 4. he rven mass has a mment f nerta f 5 kg m. he ampng n the mtr s 0. N m s/ra an n the rtatr bearngs t s 0.4 N m s/ra. Calculate the trque requre frm the mtr t. urn the aeral at a cnstant rate f 0.5 ra/s. (3.3 N). Accelerate the rtatr at 0.0 ra/s at the start when ω 0 (0.6 Nm) D.J.DUNN 3

14 6. HERMAL SYSEM MODELS 6. HEAING an COOLING Cnser a mass M kg at temperature. he mass s place n a ht envrnment at temperature an heat Q s transferre nt the mass causng ts temperature t rse. he system cul be fr example, a resstance thermmeter, an we want t knw hw lng t takes fr the sensr t warm up t the same temperature as the lqu. Fgure 3 he laws f heat transfer tell us that the temperature rse s rectly prprtnal t the heat ae s: Q Mc C c s the specfc heat capacty. C Mc s the thermal capactance n Jules/Kelvn. Dve bth ses by t an: Q Φ C t t he rate f heat transfer nt the mass s Φ C /t an the rate s gverne by the thermal resstance between the lqu an the mass. hs beys a law smlar t Ohm s Law s that: Φ ( - )/R R s the thermal resstance n Kelvn per Watt. Equatng fr Φ we have C t R t RC t RC RC In all systems, the pruct f the resstance an capactance s the tme cnstant s we have t Changng frm a functn f tme nt a functn f s we have s (s ) (s) (s ) Fgure 4 Nte that ths transfer functn s the same stanar frst rer equatns erve fr the sprng - amper system an thermal capactance C s equvalent t /k an resstance R s equvalent t k. D.J.DUNN 4

15 6. INDUSRIAL HEAING SYSEM he agram shws a schematc f an nustral prcess fr cntrllng the temperature f a tank f lqu. he pneumatc cntrller wll nt be explane here but t has an nput temperature set by ajustment f the cntrl. he temperature f the lqu s measure wth a sutable evce an turne nt a stanar sgnal n the range 0. bar. hs s cnnecte t the cntrller an pressure sensng evces pruce anther ar sgnal (0. bar) epenng n the errr. hs s sent t a valve that s pene pneumatcally wrkng n the stanar range. he verall result s that f the lqu s t cl, steam s allwe thrugh t heat the lqu. If the valve pene nstea f clsng an a clng flu was use nstea f steam, the system cntrl s by clng. he cntrl equpment cul just as lkely be all electrnc. Pneumatcs are use n angerus envrnments such as heatng up l tanks. Fgure 5 he mel fr the abve system wll nt be erve here but t wll be mre cmplcate than smply (s) because the cntrller has the faclty t mre than prprtnal cntrl. (hree term (s ) cntrl s cvere n later tutrals) WORKED EXAMPLE N.3 A smple thermal heatng system has a transfer functn (s) (s ) he temperature f the system at any tme s an ths s at 0 C when the set temperature s change frm 0 C t 00 C. he tme cnstant s 4 secns. Deuce the frmulae fr hw the system temperature changes wth tme an sketch the graph. D.J.DUNN 5

16 SOLUION s s s a cnstant (00 C) at all values f tme after t 0 (the start f the change). s t Let x Dfferentate an - Rearrange an t Integrate wthut lmts Substtute fr x x x x t t he equatn becmes x t ln(x) ln( - A ) A When t 0, startng temperature Hence t 0 ln( - ) A A - ln( - ) - change n temperature t ( - ) Substtute fr A an ln( - ) ln( ) ln t ( - ) ake ant lgs an e e t ( - e t ) Put n the values x t 00 80e -t/4 Evaluatng an plttng pruces the result belw. It s an expnental grwth. Fgure 6 D.J.DUNN 6

17 7. HYDRAULIC SYSEM MODELS he basc thery fr hyraulc an pneumatc cmpnents may be fun n the tutrals n flu pwer. 7. HYDRAULIC MOOR he fllwng s the ervatn f a mel fr use n cntrl thery. he frmula relatng flw rate Q an spee f rtatn ω s Q kq ω kq t k q s a cnstant knwn as the nmnal splacement wth unts f m 3 per raan. s the angle f rtatn n raan. Wrtten as a functn f s ths becmes Q k q s If we take the flw rate as the nput an the angle f rtatn as the utput the transfer functn s: G(s) Q k s he frmula that relates system pressure p t the utput trque s k q p q Fgure 7 If pressure s the nput an trque the utput then G(s) kq hs s a further efntn f the cnstant k q. p WORKED EXAMPLE N.4 A hyraulc mtr has a nmnal splacement f 8 cm 3 /raan. Calculate the trque pruce at a pressure f 90 bar. SOLUION p k q 90 x 0 5 (N/m ) x 8 x 0-6 (m 3 /ra) 7 Nm D.J.DUNN 7

18 7. HYDRAULIC CYLINDER Fgure 8 he flw rate an mvement are relate by the law Q A x/t. Expresse as a transfer functn wth x x beng the utput an Q the nput we have: G(s) Q As Frce an pressure are relate by the law F pa. he transfer functn wth p as the nput an F as the F utput s: G(s) A p 7.3 MODEL FOR A FLOW MEERING VALVE AND ACUAOR Fgure 9 he nput t the system s the mvement f the valve x. hs allws a flw f l nt the cylner f Q m3/s whch makes the cylner mve a stance x. D.J.DUNN 8

19 Makng a bg assumptn that fr a cnstant supply pressure the flw rate s rectly prprtnal t the valve pstn we may say Q k v x k v s the valve cnstant an examnng ts unts we fn they are m/s he area f the pstn s A m. he velcty f the actuatr s v x /t an ths s relate t the flw an the pstn area by the law f x cntnuty such that Q k v x A t Changng t a functn f s ths becmes k v x Asx x Expresse as a transfer functn we have G(s) (s) x (A/k )s he unts f A/k v are secns an we euce ths s yet anther tme cnstant. x G(s) (s) x s Nte that ths s nt qute the stanar frst rer equatn /{s} an the fference s that the utput wll keep changng fr a gven nput, unlke the prevus examples where a lmt s mpse n the utput. If the actuatr s a mtr nstea f a cylner the equatn s smlar but the utput s angle nstea f lnear mtn. v WORKED EXAMPLE N.5 A hyraulc cylner has bre f 90 mm an s cntrlle wth a valve wth a cnstant k v 0. m /s Calculate the tme cnstant. Gven that x an x are zer when t 0, calculate the velcty f the pstn an the utput pstn after 0. secns when the nput s change suenly t 5 mm. SOLUION A πd / x 0-3 m A/k v 6.36 x 0-3 / secns G(s) sx x t x x x x (s) x t s x velcty m 0.03 s 0.56 m/s Velcty stance /tme velcty s cnstant. stance x v t 0.56 x r 5.6 mm assumng the D.J.DUNN 9

20 SELF ASSESSMEN EXERCISE N.4. A hyraulc mtr has a nmnal splacement f 5 cm 3 /raan. Calculate the trque pruce at a pressure f 0 bar. (60 N m). A hyraulc cylner has bre f 50 mm an s cntrlle wth a valve wth a cnstant k v 0.05 m /s Calculate the tme cnstant. Gven that x an x are zer when t 0, calculate the velcty f the pstn an the utput pstn after 0. secns when the nput s change suenly t 4 mm. (0.039 s, 0.0 m/s an 0 mm) D.J.DUNN 0

21 7.4 ADVANCED HYDRAULIC MODEL Cnser the same system but ths tme let the actuatr mve a mass M kg an have t vercme a ampng frce. Further suppse that the valve nw meters the pressure an nt the flw rate such that the pressure apple t the cylner s p kvx. Cnser the free by agram f the actuatr. Fgure 0 he apple frce s ue t pressure F an ths s etermne by the pressure actng n the area A such that: Fp pa. he apple frce s ppse by the nerta frce F an the ampng frce F. F M x/t an F k x/t. x x Balancng frces gves pa M k t t x x Substtutng p kvx. we have k v x A M k t t In Laplace frm we have k x A Ms x k sx Rearrangng t nt a transfer functn. v G(s) x x (s) (M/Ak If we examne the unts we fn M/Akv where s a tme cnstant. he crtcal ampng ceffcent s Cc (4 M A kv) an the ampng rat s δ k/cc x he transfer functn becmes: G(s) (s) x s δ s v )s (k /Ak v )s Fgure Nte the smlarty wth the stanar n rer equatn /{s δs }. he fference s agan ue t there beng n lmtatn n the utput. If the actuatr s a mtr nstea f a cylner, the transfer functn s smlar but the utput s angle an angular quanttes are use nstea f lnear quanttes. D.J.DUNN

22 WORKED EXAMPLE N.6 A hyraulc cylner has bre f 90 mm an mves a mass f 80 kg. It s cntrlle wth a valve wth a cnstant k v 0000 Pa/m. he ampng ceffcent s 80 Ns/m. Calculate the tme cnstant, Cc an δ. Gven that x an x are zer when t 0, calculate the ntal acceleratn f the mass when the nput s change suenly t 5 mm. Calculate the acceleratn when the velcty reaches mm/s. Calculate the velcty when the acceleratn s zer. SOLUION A πd / x 0-3 m (M/Ak v ) {80/(0000 x 6.36 x 0-3 ) secns Cc (4 M A kv) 0.78 Ns/m δ k/cc 0.89 x G(s) (s) r n terms f tme x s δ s x ( x acceleratn) (δ x velcty) he ntal velcty s zer a 0 a 7.95 x 0-3 m/s When v a ( x 0.89 x x 0.00) a { ( x 0.89 x x 0.00)}/ x 0-3 m/s he system ntally accelerates an wll eventually settle wn t a cnstant velcty wth n acceleratn. Put a ( x 0.89 x 0.793) x velcty velcty m/s r 3.53 mm/s. SELF ASSESSMEN EXERCISE N.5 A hyraulc cylner has bre f 50 mm an mves a mass f 0 kg. It s cntrlle wth a valve wth a cnstant k v 80 Pa/m. he ampng ceffcent s Ns/m. Calculate the tme cnstant, Cc an δ. (7.98 s,.5.7 Ns/m an 0.798) Gven that x an x are zer when t 0, calculate the ntal acceleratn f the mass when the nput s change suenly t 0 mm. (0.57 mm/s ) Calculate the acceleratn when the velcty reaches 0. mm/s. (0.37 mm/s ) Calculate the velcty when the acceleratn s zer. (0.785 mm/s) D.J.DUNN

23 8. ELECRIC SYSEM ELEMENS MODELS 8. RESISANCE Fgure Applyng Ohm's Law we have V I R V/I R hs may be a functn f tme r f s. he equatn may be expresse n terms f charge Q. V Snce I Q/t I(s) sq hence G(s) (s) Q hs s smlar t the mel fr the amper. sr 8. CAPACIANCE he law f a capactr s Q C V V/Q /C hs s smlar t the mel fr sprng. Dfferentatng wth respect t tme we have Q/t C V/t Q/t s current I s the equatn may be expresse as I C V/t As a functn f s ths becmes I (s) C sv V he transfer functn s G(s) (s) I sc Fgure INDUCANCE I Q Faraay's Law gves us V L L t t hs s smlar t the mel fr a mass an can be ether a frst r n rer equatn as requre. Expresse as a functn f s we have V (s) L si r LsQ V G(s) (s) sl r s Q I Fgure POENIOMEER Fgure 5 f the supply vltage s cnstant an the current s neglgble, the utput vltage V s rectly prprtnal t the pstn x r angle s a smple transfer functn s btane. G(s) V (s) x cnstant k p (lnear) G(s) V (s) cnstant k p (angular) D.J.DUNN 3

24 8.5 R -C SERIES CIRCUI Fgure 6 he transfer functn s then G(s) V V (s) he nput vltage V s the sum f the vltage ver the resstr an the capactr s V I R I /Cs V I (R /Cs) he utput s the vltage ver the capactr s V I/Cs I/Cs I(R /Cs) RCs he unts f RC are secns an ths s anther electrcal tme cnstant. he transfer functn may be wrtten as V G(s) (s) V s hs s the stanar frst rer equatn an s the same as bth the sprng an amper an the thermal example. 8.6 L - C - R n SERIES Fgure 7 he utput vltage s V I/sC he transfer functn s then V G(s) (s) V hs s 3 sub-systems n seres. In ths case we wll take the utput as the vltage n the capactr an the nput as the vltage acrss the seres crcut. he nput vltage s the sum f all three vltages an s fun by ang them up. V I R I sl I/sC G(s) V V I(R I/Cs sl /Cs) RCs (s) s CL src If we examne the unts f CL we fn t s secns an we have yet anther tme cnstant efne as CL an the equatn may be rewrtten as: V G(s) (s) V s δ s R L he ampng rat δ s efne as δ an 4 s calle the crtcal ampng value. L C 4 C Nte ths s the stanar n rer equatn entcal t the mass-sprng-amper system. CLs D.J.DUNN 4

25 WORKED EXAMPLE N.7 A Capactance f 00 µf s cnnecte n seres wth a resstr f 0 kω as shwn n fgure 6. he transfer functn s (s) (s ) he vltage acrss the resstr s suenly change frm 3V t 0V. Calculate the tme cnstant an erve frmulae fr hw the vltage acrss the capactr vares wth tme. Sketch the graph. SOLUION RC 0 x 0 3 x 00 x secns he ervatn s entcal t that n example 3 smply changng t V we get the result. V V V e t Put n the values 4 V V 0 V 0 7e -t/4 Evaluatng an plttng pruces the result belw. It s an expnental grwth. Fgure 8 SELF ASSESSMEN EXERCISE N.6. Calculate the tme cnstant fr an RC crcut wth a resstance f 0 Ω an capactance f 470 nf. (03 µs). Calculate the secn rer tme cnstant an the ampng rat fr a R-L-C crcut wth L 5 µh, C 60 µf an R6.8 Ω. (7.3 µs an.8 ) D.J.DUNN 5

26 9 ELECRIC MOORS hs s cvere n greater epth n tutral. Here are the basc mels. 9. FIELD CONROLLED MOOR, he man thery f electrc mtrs s cvere n anther tutral. It can be shwn that fr a.c. serv mtr wth fel cntrl kf f If the mtr rves an nertal la an has ampng the ynamc equatn becmes Fgure 9 G(s) Is I k t f k (s) s t Is (Is k f sk k hs mels the relatnshp between the shaft angle an the cntrl current. s) k f f 9. ARMAURE CONROLLED MOOR It can be shwn that the trque s relate t armature vltage k an resstance by the frmula Va k t R a he trque must vercmenerta an ampng as befre s Fgure 30 Wth rearrangement we fn Is k Equatng we get Is G(s) (Is k a s k s s) k s R a Is ( V ks) a k R a V (k/r hs mels the relatnshp between the angle f the shaft an the cntrl vltage. V (s) sk a a ) k k R a s/r a k R a V a k s R a Wrke examples an self assessment exercse fr ths sectn may be fun n tutral. D.J.DUNN 6

27 0. CLOSED LOOP SYSEMS RANSFER FUNCION WIH UNIARY FEED BACK. Cnser a smple system wth an nput an utput relate by the transfer functn G(s). If the system s t be a cntrlle system n whch we requre the utput t change an match the value f the nput (set value), we must make the nput the errr e nstea f the set value. he errr s btane by cmparng the utput value wth the nput value wth the sgnal summng evce. hs pruces the result e - an because s subtracte, ths ea s calle NEGAIVE FEED BACK. he blck agram shws that the sgnal passes arun a clse lp hence the name CLOSED LOOP SYSEM. G(s) G(s) G(s) e G(s) - - G(s) G(s) substtute ve the bttm lne by rearrange r e G(s) G(s) /G(s) nvert - - G(s) he transfer functn fr the clse system s hence G(s) s the transfer functn f the pen lp system. Fgure 3 /G(s) Let s revst the hyraulc pen lp transfer functns erve prevusly. When the hyraulc valve an actuatr s turne nt a clse lp system, the tw transfer functns becme: G(c.l) fr the frst rer versn an G(c.l) fr the secn rer versn. s s δ s hese mels are mathematcally entcal t the transfer functn f the mass-sprng- amper an the L-C-R crcuts. Nte that fr any system wth an pen lp transfer functn G(s) the clse lp transfer functn wth unt feeback s G cl /G(s) D.J.DUNN 7

28 WORKED EXAMPLE N.8 An pen lp system has a transfer functn G(s) /(s s ). Derve the clse lp functn when unt feeback s use. SOLUION G cl /G(s) s s s s s s 3 SELF ASSESSMEN EXERCISE N.7. An pen lp system has a transfer functn G(s) 5/(4s s ). Derve the clse lp functn when unt feeback s use. G cl 5/(4s s 7). An pen lp system has a transfer functn G(s) 0/(s 3 5s). Derve the clse lp functn when unt feeback s use. G cl 0/(s 3 5s 0) D.J.DUNN 8