THE ANALYSIS AND SYNTHESIS OF CONTACTOR SERVOMECHANISMS ARMAND PIERRE PARIS A THESIS SUBMITTED IN PARTIAL FULFILMENT OF

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1 THE ANALYSIS AND SYNTHESIS OF CONTACTOR SERVOMECHANISMS by ARMAND PIERRE PARIS A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE n the Department of ELECTRICAL ENGINEERING We accept ths thess as conformng to the standard requred from canddates for the degree of MASTER OF APPLIED SCIENCE Member of the Department of Mechancal Engneerng Member of the Department of Electrcal Engneerng THE UNIVERSITY OF BRITISH COLUMBIA August, 1954

2 Abstract Ths nvestgaton s concerned wth the analyss and synthess of contactor servomechansms. The technques employed are based on Kochenburger*s quas-lnear representaton of the contactor descrbng functon for snusodal nput sgnals to the contactor. The frequency-response method of analyss and synthess, whch has been found practcal for treatng lnear servomechansms has been appled by Kochenburger to the contactor servomechansm and s explaned here. By ths method t s possble to determne whether the system possesses absolute stablty. The root-locus method of synthess whch has been appled to lnear servomechansms s appled to the contactor servomechansm. The root-locus descrbes the roots of the closed~loop system for all values of the control sgnal ampltude. The rootlocus method s valuable when consderng the problem of relatve stablty. For a smple contactor wth no hysteress effect, Kochenburger*s vector form of the contactor descrbng functon can be used drectly to obtan the root-locus. The contactor appears as a varable gan element for the varous control sgnal ampltudes. The contactor has no effect on the open-loop roots but the varatons n the contactor gan cause the roots of the closed-loop to travel along the root-locus obtaned from the open-loop roots of the system. The root-locus can also be obtaned when the contactor possesses hysteress. Kochenburger»s vector form s modfed

3 to the Laplace transform form of the contactor descrbng functon. Ths form of the descrbng functon shows that not only are the postons of the roots varyng for the closed-loop but also for the open-loop. A model was constructed to check some of the theory. The assumed over-all open-loop transfer functons approxmated the actual. Even for the assumptons made, the expermental work has verfed qualtatvely and to some degree quanttatvely the predcton of the model performance.

4 Table of Contents 1. Introducton 1 2. General comments on contactor servomecnansms 8 3. Mathematcal representaton of lnear components The frequency-response of contactor servomechansms The quas-lnear representaton of the contactor 23 descrbng functon. 6. Stablty crtera for contactor servomechansms Hoot-locus method of synthess Analyss of the model by the frequency-response and root-locus methods page a) Kahn's method for obtanng transent responses 37 b) Analyss of the model 38 c) Applcaton of a smple phase-lead type 44 compensatng network 9. Root-locus method when contactor has hysteress The experment a) The crcut and ts operaton 51 b) Calbraton of the model 53 c) Tests and results Summary and conclusons References Acknowledgements 63

5 Lst of Illustratons page Fgure 1. Block dagram of sngle-loop servomechansm 2 Fgure 2. Fgure 3. Controller Characterstcs Block dagramof a contactor servomechansm 2 8 Fgure 4. Typcal contactor characterstcs 9 Fgure 5. Representatve phase-lead network 13 Table 1. Symbols and unts adopted followng 17 Fgure 6. Relaton of control sgnal and correcton sgnal for a contactor wth hysteress 19 Fgure 7. Fgure 8. Graph of ampltude of harmonc components followng 20 of correcton sgnal aganst ampltude of control sgnal Plot of the fundamental harmonc contactor followng 24 descrbng functon, Gr-^ Fgure 9. Block dagram of lnear sngle-loop servomechansm 26 Fgure 10. Frequency polar-locus plot 27 Fgure 11. Super-posed frequency and ampltude loc 28 Fgure 12. Examples of super-posed loc 30 Fgure 13. Roots n the p-plane 33 Fgure 14. Root-locus of sngle-loop servomechansm 34 Fgure 15. Sngle-loop contactor servomechansm 37 Fgure 16. Graphs for Kahn's sem-grapncal method followng 37 Fgure 17. Block dagram representaton of the model 39 Fgure 18. Super-posed frequency ampltude loc for the representaton of the uncompensated model 39 Fgure 19. Fgure 20. Graph of the control sgnal ampltudes followng 40 whch gve ponts of equlbrum for the representaton of. the uncompensated model Root-locus for the representaton of the followng 41 uncompensated model

6 Lst of llustratons - cont'd. Fgure 21. Plots of transent responses to reference step nputs. page followng 43 Fgure 22. Frequency locus for.the of the compensated model representaton followng 44 Fgure 23. Fgure 24. Fgure 25. Fgure 2.6. Fgure 2.7. Fgure 28. Fgure 29. Fgure 30. Fgure 31. Graph of the control sgnal ampltudes whch gve ponts of equlbrum for the representaton of the compensated model Root-locus for the representaton of the compensated model Plots comparng transent responses for the. uncompensated and compensated model Plots necessary to obtan the fundamental harmonc contactor descrbng functon, gjyj_ Open-loop zero-pole confguraton wth contactor hysteress Root-loc wth contactor hysteress Crcut dagram of the model Schematc dagram of the model Change n rheostat voltage drop aganst tme followng 45 followng 45 followng 45 followng 48 followng 48 followng 49 followng 51 followng Fgure 32. Block dagram of the model followng 55 Fgure 33. Table 2. Transent responses to a step dsturbance Ampltudes and frequences of self-sustaaed oscllatons wthout compensaton followng 57 followng 58 Table 3. Ampltudes and frequences of self-sustan- followng 58 ed oscllatons wth compensaton

7 I* Introducton In the last two decades consderable progress- has been made n the scence of servomechansms and feedback control systems. Most of the lterature on these automatc controls deal wth contnuous control and very lttle deals wth dscon^ tnuous control. Ths s very sgnfcant when t s consdered that Hany of the frst control systems were of the dscontnuous type. The development of any exact scence requres that relatonshps be expressed mathematcally. It s then under** standable that the contnuous type of control was more amenable to theoretcal nvestgaton because relatonshps could be expressed as contnuous functons. The dscontnuous controls on the other hand have not been conducve to mathematcal nvestgatons especally when consderng synthess technques, hence the preponderance of theory on contnuous controls over dscontnuous. However, many dscontnuous types of controls are used and possbly more would be used f ther performance were more clearly understood. Therefore the nvestgaton of the analyss and especally the synthess technques of dscontnuous control systems has defnte practcal mportance. In ths paper, the type of dscontnuous control system to be studed s that referred to as on-off, relay or contactor type servomechansm*

8 Fgure 1 s a block dagram representng a smple feedback control system. + -M Error Control Sgnet/ Cortrol/ad ) ' /a.r»<lnts D Systam 0 Fgure 1. Block dagram of sngle~loop servomechansm Ths fgure represents a farly general sngle-loop servomechansm consstng of a reference nput, control elements, controlled system and a drect feedback comparng the controlled varable wth the reference nput. For contnuous control of the control elements the relatonshp of the correcton sgnal, D, to the steady-state error E s, may be represented as n fgure 2a. ; [ (or«tttvn S/yna! 0.Correcton 5'<tn«l 1 D J Jtajy-state trt-or JtloJ _a) Contnuous SgrvorxLcfan'S/n -$>) Contactor SzrVomzctctmsm Fgure 2:. Controller characterstcs Ths relatonshp s drawn as a straght lne passng through the orgn llustratng that not only s the correcton for the error contnuous but also lnear. Ths condton s the one for whch most of the theory has been developed. If the control elements of fgure 1 are substtuted by a contactor servomechansm the controller characterstc would be as represented

9 3 n fgure ab. It may be observed from ths graph that no correcton takes place wthn a certan zone* Ths zone s known as the nactve or dead zone. When the error s outsde ths zone the controller ntroduces a constant value of correcton sgnalvto the controlled system. It s an all or nothng type of control and the correcton sgnal s a dscontnuous functon of the error. Ths type of control s wdely used because of ts smplcty. It frequently requres much less equpment than contnuous controllers, hence ts many savngs, n weght and cost. The controller could consst of an electromagnetc relay, hence the term relay servomechansm, or some sort of contact-makng devce actvated, for example, pneumatcally, hydraulcally or electrcally, hence the term contactor servomechansm. The manner of operaton of the control suggests the other name pn-off servomechansm. For purposes of unformty the term contactor servomechansm wll be adopted n ths paper and used exclusvely henceforth; The early nvestgators approached the analyss of contactor servomechansms wth two sgnfcant ponts n mnd. Frst, mathematcal relatonshps could be expressed for the several condtons of control that s for postve, negatve and zero correcton. Ths means that although the control system s non-lnear when consdered over ts entre range of operaton t may be consdered lnear and dfferental equatons can be set up for each condton of ts operaton. The equaton chosen would depend on the condton of operaton The boundary condtons for the controlled varable would have to be matched for each contactor swtchng operaton. Secondly, experence had Indcated that contactor servomechansms were oscllatory by nature and tended to mantan sustaned oscl-

10 latons. Ths fact s seen to be reasonable by consderng the 4 manner by whch the contactor servomechansm corrects for error. Once the error s outsde of the nactve zone a step nput of correcton sgnal s ntroduced to the controlled system and ths same value of the sgnal perssts untl the error agan, comes nto the nactve zone. Wth suffcently hgh gan n the forward transfer functon of the open-loop system the controlled varable s caused to overshoot ts stable poston, that s, tne error travels outsde the nactve zone and correcton takes place as before n the opposte drecton* Because of the type of correcton sgnal a tendency exsts for the swtchng process of correcton to persst hence causng a generaton of sustaned oscllatons of the controlled varable. The early nvestgators were armed wth these two facts. The two ntal papers wrtten on the subject were one by Ivanoff 1 and the other by Hazen 2 } both, n Both, nvestgators approached the problem assumng a condton of sustaned oscllatons In the steady-state. Ivanoff, who was prmarly nterested n temperature regulaton assumed a symmetrcal rectangular correcton sgnal wave beng ntroduced to the controlled elements, whch, n hs case was a heat plant. He analysed the rectangular wave nto ts Fourer seres and assumed a transference characterstc for hs heat plant. He was able to demonstrate the relatonshp of the steady-state ampltude and frequency to the nactve zone wdth. By employng the Fourer analyss of the rectangular wave he also showed that the predomnant effect of the rectangular correcton wave was due to ts fundamental component, Hazen assumed a snusodal response for the controlled varable and then used dfferental equatons drectly. Ths method requres that dfferental equatons be

11 5 solved for each, correcton nterval ntated by tle contactor and that boundary condtons be matched for each swtchng operaton* Ths method becomes very awkward even for relatvely smple cases. Hazen, however, was able to demonstrate many of the pecular characterstcs of contactor servomeonansms such as the effects due to the nactve-zone and back-lash wth hs drect dfferental equaton approach. Methods of analyss based ndrectly on the transent response have been developed. Wess 3 and MacColl 4 appled the graphcal phase-plane method, whch had been used n non-lnear systems, to the analyss of contactor servomecnansms. The dfferental equatons are put n such a form that the soluton s obtaned n terms of the frst dervatve wth respect to tme of the controlled varable and the controlled varable. If the controlled varable be dsplacement then the frst dervatve wth respect to tme s velocty. The plane formed by havng the frst dervatve of the the controlled varable wth respect to tme as the ordnate and controlled varable as abscssa usually expressed n terms of velocty versus dsplacement s known as the phase-plane; Trajectores n the phase-plane are plotted n the dfferent regons and from the web of the trajectores the stablty of the control system can be determned. The transent response of the system can also be obtaned graphcally from the phase-plane plot. Recently Flgge-Lotz 5 has gven extensve treatment of ths method wth specfc applcaton to guded mssles. Kahn 6 has developed a method whch Is qute analogous to the phase-plane. Hs graphcal method s based on plots of the frst dervatve of the controlled varable versus tme. Both these methods of analyss suffer serous lmtatons n that they are satsfactory only for systems whch can

12 6 be descrbed by a second order dfferental equaton. Kahn&has further developed a graphcal method of plottng the transent response for arbtrary dsturbances* Prac-? tcally, the soluton of only one dfferental equaton s necessary and ths s the open*loop response of the contactor control sgnal to the step correcton sgnal. Ths method wll be elaborated upon later as t wll be used to obtan transent responses for certan portons of the nvestgaton to follow. The dsadvantage of all the methods descrbed s that they are ncapable of drect use for synthess. Ths, of course, s a serous dsadvantage to the desgner of the contactor servomechansms. Kochenburger 7 has ntroduced a novel method of analyss whch s conducve to synthess based on the frequency-response of the system. Ths method s based on approxmatons whch are vald for most systems encountered n practce. Ths approxmaton enables the contactor to be represented n terms of a quas-lnear descrbng functon. The chef advantage of ths approxmate method, whch sets t apart from other methods, s that t allows lnear technques to be appled to non-lnear mechansms, so that other technques already developed for lnear control systems may also be used. The wrter suggests n ths paper that the rootlocus method developed by Evans for lnear oontrol system synthess may be used as an added technque for the synthess of contactor servomechansms n conjuncton wth the Kochenburger descrbng functon and ts modfcatons. The object of ths paper therefore s to outlne

13 7 Koehenburger's method of analyss and synthess of contactor servomechansms and to show how the root locus method of synthess may be used, wth the concept of the descrbng functon of the contactor, to determne the relatve stablty of a contactor servomechan sm. A model was constructed for the purpose of checkng some of the theory developed. Unfortunately not all the theory developed could be checked expermentally because the hysteress effect n a contactor could not be reproduced n the contactor of the model The expermental results obtaned are based on the assumptons that postve and negatve correctons are equal and all components of the servomechansm other than the contactor are lnear. These assumptons are necessary but only approxmately true. Therefore two types of error are to be expected n the results; the frst source of error due to the approxmate method of analyss used and the second due to the assumpton that components other than the contactor are lnear whch was not exactly the case for the model. In the chapters to follow, Kochenburger'a frequency-response method and the root-locus method wth the concept of the quas-lnear representaton of the contactor descrbng functon wll be explaned. These two methods of analyss and synthess, wll be appled,n. anal*» yzng the model frst wthout a compensatng network, then wth a smple phase-lead compensatng network. The mathematcal analyss wll be followed by a descrpton of the model, and a report on the expermental tests performed and the results obtaned.

14 2» General Comments on Contactor Servomechansms 8 servomechansm. Fgure 3 represents a typcal sngle-loop contactor Control EI<L/nertts Hahrtna J Input -\~ f \ terror Compensaton > Contactor means >rr<. <:tar\ Control/ol El<unants Cont rolu< 0 Fgure 3* Block dagram of a contactor servomechansm The block dagram for the contactor servomechansm s qute smlar n form to the lnear contnuous type. The power amplfer whch s essental to the control elements of the contnuous servomechansm s replaced by the contactor means whch acts as a dscontnuous power amplfer. The compensatng network s very often ncluded n the control elements to ad the characterstcs of the system. Ths compensatng network s whenever possble placed on the low power sde of the contactor for purposes of economy. In fgure 3 the output s compared drectly to the reference nput. Drect feedback s not essental for contactor servomechansms and more complcated transfer functons may be ncluded n the feedback loop. Wth a compensatng network the error sgnal s altered before beng appled to the contactor. Wthout a compensatng network the error s, of course, the same as the control sgnal to the contactor. Fgure 4 represents typcal characterstcs of contactor

15 means. In these representatons symmetrcal operaton of the 9 contactor s assumed, whch means that the negatve and postve correctons are equal n magntude. It s convenent to express the correcton sgnal as havng a unt dlmensonless ampltude so that the postve correcton wll be plus one whle the negatve wll be mnus one. + A Correcton S/aoal D -H Corracton S/qnal y * 0 -I Control S/qtcf/ 3 c Cotltrvl 5/amxl c a) Wth mactva zotf<z, Oy b)wth ndctvd Zona.,Ccl anjhyste^st Zon<L t Fgure 4; Typcal contactor characterstcs Fgure 4a represents the characterstcs of a contactor wth nactve zone -only, the wdth of the nactve zone beng Da» No correcton wll take place unless the control sgnal s greater than Ca/2' and less than -0^/2. The nactve zone wdth represents the range of permssble error wthn whch no correcton takes place and t can be no larger than desgn specfcatons. Makng the nactve zone smaller than necessary complcates the problem of stablty. As the nactve zone wdth approaches zero the system wll at best mantan sustaned oscllatons about some equlbrum pont. Fgure 4b represents the characterstc of a contactor wth nactve and hysteress zones. The physcal meanng of

16 hysteress n a contactor as may be observed from ths fgure s 10 that the control sgnal requred to cause correcton s greater than the control sgnal requred to cease correcton. Ths pheno^ menom may be observed, for example, n electromagnetc relays where the col current necessary to close a relay s greater than that necessary to open the relay. Of course the assumed characterstc of the contactor to account for hysteress s dealstc* Nevertheless the effect of ths dealstc characterstc may be shown and t can be reasonably assumed that an actual relay wth hysteress wll have much the same effect. It wll be shown that hysteress n a contactor has an adverse effect on the stablty of a system. Bascally the performance crtera of contactor and lnear contnuous servomechansms are the same. The control system attempts to mantan the controlled varable equal to some desred value of reference nput. Statc accuracy n contactor servomechansms, whch s the possble range of error under steady-state condtons, s of prme mportance. Ths accuracy s determned by the nactve zone. Dynamc accuracy s the measure of error when the system s respondng to a dsturbance. Ths accuracy may be mproved by ncreasng the rate of response of correcton or effectvely as n contnuous servomechanms ncreasng the amplcaton of the con* troller. Ths Improvement n accuracy tends to cause oscllatons n the dynamc response whch ntroduces the problem of stablty; For purposes of dynamc accuracy t s mportant that the frequences, assocated wth the oscllatons, be as hgh as possble* Stablty requrements demand that oscllatons be suffcently damped. The measure of the dampng of these oscllatons s referred to as the

17 11 relatve stablty of the system, A dstncton must be made between stablty requrements for contactor and lnear servcnmechansms. For contactor servomechansms self-sustaned oscllatons of fnte ampltude are qute possble and at tmes permssble whle for lnear servomechanms self**sustaned oscllatons of necessty ncrease and are destructve* Wth respect to the problem of stablty t s nterestng to note that for contactor servomechansms, the stablty of the system s dependent on the type of dsturbance ntroduced nto the system, A partcular system could, for example, be absolutely stable, that s ncapable of mantanng self-sustaned oscllatons, for a step nput of dsturbance whle be n a state of sustaned oscllaton for a "velocty" nput of dsturbance. Ths s unlke lnear servomechansms n whch the frequency and dampng of the transent oscllatons are ndependent of the dsturbance and dependent solely on the natural modes of oscllaton of the servo* mechansms determned from ther system transfer functons. For* tunately, however, t s rarely necessary to desgn contactor servos mechansms for contnuously varable control of output* Ths greatly smplfes the analyss of contactor servomechansms as well as the equpment necessary for ther sutable operaton.

18 121 3# Mathematcal Representaton of Lnear Components In usng Kochenburger r s frequency-response method or the root-locus method, wth the contactor descrbng functon, the assumpton made s that all elements other than the contactor are lnear. These elements are represented mathematcally n terms of ther transfer functons or ratos of outputs to nputs. Referrng to fgure 3 the transfer functon G s (s) of the controlled system s defned as G s (s = Qjs) (1) where 0(s) and D(s) are the Laplace transform of the controlled varable 0(t) and the correcton sgnal D(t) respectvely, subject to zero ntal condtons* If, for example, the controlled system conssted of a servomotor, the transform functon could after beng factored have the followng form G s(s) = Rs (2:) s(ts+ 1) T s a tme constant of the motor; R s s the gan of the transfer functon and n ths case gves the slope of the response of the motor to a unt step Input after a suffcently long tme. For ths reason R$ s referred to as the runaway velocty of the motor. All other lnear components may be expressed n a smlar manner. Consder the phase-lead compensatng network of fgure 5.

19 R, n ohms I Input vo/taj 0 Output vo/tay Fgure 5. Representatve phase-lead network. The transfer functon of ths network G c (s) s 60(a) = R c (m 2 s-r- 1) Tgs+ 1 (3) where Rc = Rg, m 2 - RjC, T 2 - B3R3C1 Rl+^a R1+R2 Tg and mg are tme constants of the transfer functon. Transfer functons can be expressed as products the gan and the functon of s, g(s), defned Gr(s) = Rg(s) (4) so for the transfer functon gven by equaton (2) G s (s) = ( R s ) = R s g s (s) (5) TsTf][sTTT where gs(s) s 1 S(TTS + 1) Smlarly for the transfer functon of the phase-lead compensatng network gven n equaton (g) where G c (s) = R c (mls-- 1) = H c g c (s) (T2.8+ 1) g c (s) s m s -» 1 (6)

20 Snce the transfer functons of the controlled system and the compensatng network are lnear they may be combned as the open-loop transfer functon from contactor output or correcton sgnal to the contactor nput or control sgnal. For the system as represented n fgure 3, consstng of the controlled elements, a compensatng network and the contactor, the^open-loop transfer functon s then G(s) - C(s) DTI7 E = constant «0 (7) - G s (s)g c (s) (8) = R s S c g s (s)g c (s) = Rg(s) (9) Where R = RsRc, g(s) - gs(s)gc(s), G s (s) and G c (s) are the transfer functons of the controlled system and compensatng network respectvely* The transfer functons Rg(s) as already descrbed n factored form wll be referred to as the Nyqust form of the transfer functon. In ths form the complex varable n a factor s multpled! by the tme constant of the factor so that the numerator has factors of the form (ms+1) and the denomnator has factors of the form (Ts+ 1). The advantage of ths form s obvous for use wth the modfed form of the Nyqust crteron of stablty where attenuaton n decbels and phase-margn are plotted versus angular velocty. However ths form of the transfer functon s not convenent for use wth the root-locus method of synthess.

21 Consder the open-loop transfer functon n Nyqust form G(s). Rg(s) = R (ms-h) (10) al'ls^dl'fas + l) The root-locus form of the transfer functon gven by equaton 10 Is F(s) s Rmx (s+ 1 ) (11) ' m TxT 2 s(s + l )U^1 ) Tx T a and rewrtng F(s) as the product of the gan and the functon of s n an analogous manner to G(s) - Eg(s) for the Nyqust form, f.(s) = A f(s) (12) for the root-locus form. Therefore for equaton 11 A = Rm "TlTa (13a) f(s) = (s + 1 ) V s(s+ 1 )(s+ 1 ) T± T 2 (13b) For the transfer functon factored n root-locus form the complex varable, s, s multpled by unty. The zeros and poles of F(s), the root-locus form of the transfer functon, may be obtaned drectly from the factored form of ths transfer functon. Zeros and poles are those values of s whch make the transfer functon zero and nfnte respectvely. For equaton (13b) a zero occurs at - 1, and poles at 0, - 1, and - 1 m x T T on the complex s-plane. It wll be found most convenent to represent both the Nyqust and root-locus forms of the transfer functon n

22 dmensonless tme. Ths s accomplshed by substtutng the complex varable s whch has the dmensons seconds-1 by the dmensonless complex varable p where P = t b 8 (14) and t D s the tme base selected. The tme base selected s usually one of the tme constants of the transfer functon G(s). Transent solutons wll therefore be functons of dmensonless tme 0 where - t (15) ^b t beng the elapsed tme n seconds. To obtan transfer functons, the dmensonless-tme Nyqust form of consder agan the case of the open-loop transfer functon G(a) gven by equaton 10. The tme base selected wll be % r 1\ G(s) - R (mts+1) = RT! (m-l^s- 1) s(ts-r l)(t 2 s + 1) T Txs (T x s + 1 j (TgTxS + 1) For p s Ts (16) S(p) r RTUP + l) (17) m plp~+ttiyp +1) waar* cf, = rp- and y 4 = A Separatng the transfer functon G(p) nto a product of the gan and a functon of p G(p) r Kg'(p) (18)

23 for equaton 17 K =. RTT_ (19a) and g(p) = (qp+1) (19b) plpttjty 2P + 1) Smlarly for the dmensonless-tme root-locus form of the transfer functon F(p) s Bf(p) (20) and for the oase under consderaton F(p) = RTd! (p+1 A = Bf(p) (21) <U y 8 p(p+l)(p+l ) 7Z where B r R^qj. (22a) and f(p) - p+_ (22:b) *1 P(P + l)(p^_l_) y'a- For the transfer functons n dmensonless-tme form, the gan K of the Nyqust form and the gan B of the root-locus form have the same dmensons. Ths s convenent for the study to follow because the transfer functons are frequently changed from one form to the other. Table 1 gves a synopss of the adopted notaton for the system propertes n ther varous forms and the sgnals throughout the system, along wth ther varous dmensons.

24 to follow page 17 Table 1 Symbols and Unts Adopted Quantty Symbol Unts Tme base Elapsed tme Complex varable of Laplace Transform H Elapsed tme S3 P> <D Ej Complex varable of EH Laplace Transform tb t Ss/H-jW P=sttl-cr+-ju Seconds Seconds Seconds" 1 Dmensonless Dmensonless Controlled Varable c or Output ' H cju Ref erence Input «3 Error td dra Control Sgnal S u o Correcton Sgnal H 'CO I B.I-0 C D Output-unts Output-unts Output-unts Output-unts Dmensonless Inactve zone Hysteress zone Transfer Functon Propertes: a) Nyqust Form Of controlled elements Of compensatng network. O 0) H Over-all open-loop o Tme constant n numerator o) Tme constant n denomnator >> CO V bj Root-locus form of Controlled elements Of compensatng network Cd G s (s) = R s g a (s)ro(s) D(s) a c (s)=r c g c (s} = C(s) G(s) 5 G s (s)g c (s) m T E-slsJsAsfgtsJsOCs) D(s) g c (s)-acf c (s) sc s Output-unts Output-unts Output-unts Dmensonless Output-unts Seconds Seconds Output-unts Dmensonle s s

25 to follow page 17 Table 1 (contnued) Quantty Symbol Unts Over-all open-loop F(s) = F s (s}f c (s) Output-un ts c) Dmensonless-tme Nyqust form Overall open-loop G(p)=G a (p)g c (P) Output-unts ertes K g(p) =Kg(p) Output-unts Dmensonless ;em Prop f 01 >> co Tme constant n numerator Tme constant n denomnator t b Dmensonless Dmensonless d) Dmensonless-tme root-locus form Overall open-loop F(p)=F s (p)f c «Bf (p) B f(p) (P) Output-unts Output-unts Dmensonless o

26 18 4. The frequency-response of oontaotor servomechansms. The frequency-response method of analyss and synthess has been found valuable for lnear servomechansms^ Ths method, based on certan.approxmatons.can also be used for study of contactor servomechansms. The over-all open-loop transfer functon G-(p) may be descrbed n terms of ts steady-state response to snusodal nputs of varous real frequences by substtutng ju for p where u s the dmensonless angular velocty. Then G (ju) = Kg(ju) (23) g(ju) vares wth the appled frequency only hence g(ju) s a frequency varant porton of the system. The rato of output to nput of the contactor for snusodal nputs cannot be represented by a transfer functon because of ts non-lnear characterstcs. Kochenburger has expressed ths rato by what s called the descrbng functon of the contactor. It wll be found that the descrbng functon s dependent on the ampltude but ndependent of the frequency of the appled nput sgnal. The contactor descrbng functon s then an ampltude varant porton of the system. Kochenburger*s frequency-response method bascally s the combnaton of the frequency varant porton of the system and the ampltude varant porton of the system nto one scheme subject to nterpretaton by the frequency-response method used n lnear servomechansms. Consder a contactor wth the characterstcs of fgure 4b

27 whch has both an nactve and hysteress zone. Let the control 19 sgnal C be represented by C = C m cosu<t> (24) The contactor wll ntate a postve correcton sgnal when C = G d /2'+ C^/2 and cease correcton when C = C d /2 - Cfc/2. It wll ntate a negatve correcton sgnal when C =-0^/2 - C^/2 and cease negatve correcton when C =-0^/2+0^/2. These nstants of tme are desgnated by the angles u< ) equals a - b, a + b,tr + a - b, and r+a * b respectvely. These relatonshps are shown n fgure 6a. a)assontad snuto/<ja/ s/)apt of control j/jna/ ' 1 Correcton Sgnal, D rvnj<mc*taf component f 0,. b) Rt3»ft<*" form of correcton scjnql and ts fundamental component" Fgure 6. Relaton of control sgnal and correcton sgnal for a contactor wth hysteress. The pulse wdth of the rectangular wave s gven by the angle 2b as shown n fgure. 6b,. The fundamental harmonc component of the rectangular wave D]_ s supermposed on the rectangular wave n fgure 6b and the angle, a, represents the phase lag of the fundamental component of the correcton sgnal behnd the control sgnal,

28 If the correcton sgnal conssted solely of the 20 fundamental harmonc component D, then only snusodally varyng sgnals would exst n the system. The contactor would then appear, as a so-called quas-lnear transfer devce n that t would operate as a lnear amplfer for any gven constant ampltude of control sgnal. The contactor would not operate as a truly lnear devce because of ts nonlnear relatonshp between correcton sgnal and control sgnal ampltude. Consderng the contactor as such a quaslnear devce the frequency-response method may be used for any partcular sgnal ampltude. Ths approxmaton of the control sgnal, whch neglects the hgher order harmoncs of ths rectangular wave, s essental to the concept of the quas-lnear descrbng functon. The justfcaton for ths approxmaton wll now be consdered. An analyss of the rectangular wave of the correcton sgnal wll show the relatve mportance of the harmonc components. The relatve ampltudes of the harmonc components are a functon of the pulse-wdth whch n turn s a functon of the ampltude of the control sgnal and the nactve and hysteress zones. Fgure 7 llustrates the relatve mportance of the thrd and ffth harmoncs wth respect to the fundamental for a contactor wth nactve zone only. The ampltudes are plotted aganst the rato of the ampltude of the control sgnal and one-half the nactve zone. The graph of the ampltude of the fundamental s taken as beng assymptotc to unty. From the graphs t may be observed except for very small values of the abscssa that the ampltude of the fundamental s greater than those ampltudes.of ts harmoncs. But the ampltudes of the harmoncs can no way be consdered as beng neglgble as far as the correcton sgnal only s concerned. However^ the transfer functons of the controlled elements usually act to suppress the

29

30 21 hgher harmoncs. Consder for example the controlled element to be a servomotor whose transfer functon s of the form K$ P(y]P+ 1) The transfer functon descrbed n terms of ts steady-state response to snusodal nputs of real frequences s obtaned by substtutng for p, ju, j3u, j5u for the fundamental, thrd harmonc and ffth harmonc respectvely. It s obvous from makng these substtutons that the hgher harmonc components wll be greatly suppressed by the controlled element so that although the hgher harmonc components appear n the correcton sgnal ther effect on the output sgnal can be consdered to be neglgble. If sgnals other than the fundamental are neglgble n the output then the error and the control sgnal can be sad to contan only the fundamental component. Ths s the bass for consderng the contactor characterstc n terms of a quas-lnear descrbng functon gj>-j_ where the subscrpt 1 ndcates that the descrbng correcton functon consders only the fundamental component of the sgnal. Ths descrbng functon neglects the hgher harmoncs wth the followng justfcaton as set forth by Kochenburger. 1, The normal frequency spectrum of a rectangular wave nvolves progressvely smaller ampltudes for ncreasng orders of the harmonc components. 2. Most servomotors (Kochenburger here consdered a partcular type of controlled element) serve as effectve low-pass flters and mnmze the mportance of the hgher-harmonc components. Kochenburger, by more exact analytc methods and by test, has compared results wth those predcted by hs approxmate method and found the comparsons to be qute good. Ths frequency-response

31 method when applcable s apparently satsfactory engneerng approxmaton.

32 23 5. The quas-lnear representaton of the contactor descrbng functon. In the precedng secton the form of the control sgnal C was assumed to be G = C m cosuc > whch mples the duraton of the postve and negatve correcton sgnal pulses to be the same. Ths s not the most general form of control sgnal but t s the case whch wll be consdered n ths paper. For unequal postve and negatve pulses of the correcton sgnal the control sgnal would be of the form C ~ G 0 + Cm cosu * w n re G 0 s a constant, descrptve of the average value of the control sgnal. The choce of the form of the control sgnal presupposes a knowledge of the form of self-sustaned oscllatons of the control sgnal. For an assumed control sgnal wth an average value of zero t s mpled that the nput ha.se zero average rate of change of nput and that all sgnals wthn the system have zero average value. For a control sgnal C = C^cosuQ the fundamental of the rectangular wave s as shown n fgure 6b. A Fourer analyss of the rectangular correctve wave yelds a fundamental component ^1 = D m cos(u(t) + /D ) (25) Here D m = 4 sanb (2.6a) fol = -a (2.6b) where b * 1 (cos-^ca-cn) + cos~ 1 (G C +-Ch) (2;7a) ^ (2Cm F (2Cm ) and a = l(cos- 1 (C d -C 1 ) ~ cos" 1 (0^+0^) (27b) 2 (20 m ) (20m ) The contactor descrbng functon may be expressed n a manner smlar to the lnear transfer functon as the rato

33 of tne output to the nput. The descrbng functon gjj_ s 24 defned as the rato of the Laplace transform of the correcton sgnal and the control sgnal subject to zero ntal condtonsnor gd]_ «L[p m ooa(u»+ /D)l (28) L[GaCOSU< )] Ths expresson wll be found useful when consderng, the relatve stablty of a system usng the root-locus method for a contactor wth both nactve and hysteress zones. For the tme beng equaton 28 wll be gnored because t does not contrbute drectly to the frequency-response method. The descrbng functon s found to be more useful for the frequency-response method when t s wrtten n vector form as obtaned from the rato of the output vector and the nput vector. The descrbng functon Gp^ n vector form s GDI = Dm /&1 * 4snb/-a (2:9) T T Cm Ths form of the descrbng functon of the contactor s the one developed by Kochenburger. It s a functon of nactve zone, hysteress zone and control sgnal ampltude and ndependent of the frequency. In fgure 8 s plotted the graph of the magntude and phase angle, of the. vector form of the contactor descrbng functon, aganst the ampltude of the control sgnal for varous ratos of nactve and hysteress zones. The plot of the magntude llustrates that for values of Cd/2 less than unty that the magntude Is gero or no correcton sgnal s present because the control sgnal s wthn the: nactve zone. Wth an ncrease n control sgnal the magntude of the descrbng functon ncreases then reaches a

34

35 25 maxmum, and decreases untl n the lmt as the ampltude of the control sgnal approaches nfnty the magntude of the descrbng functon approaches zero. When hysteress s present the phase angle s greatest for small values of control sgnal and ths phase angle decreases wth an ncrease n control sgnal. There s zero phase-shft when the hysteress effect s non-exstent.

36 86 6. Stablty Crtera for Contactor Servomeonansms. Consder a lnear sngle-loop lnear servomechansm wth unty feedback and an open-loop transfer functon equal to Kg(p), as represented n fgure 9. I +^ T J 0 < Fgure 9. Block dagram of lnear sngle-loop servomechansm. The rato of the output and nput for snusodal exctaton s gven by Ojjuj - Kg(u) (30) I 1+ Kg(ju) The nverse response rato je(ju) may be expressed n the 0 followng form 1 (ju)» g- 1 (ju)-vk (31) 0 K where g" 1 (ju) = 1 g( Ju) A smplfed verson of the Nyqust crteron for stablty whch apples for a sngle-loop system n whch drect feedback s employed and whch has a mnmum-phase forward transfer functon may be stated as follows: A system satsfyng the condtons and havng an nverse response rato as n equaton 31 wll be stable f, for a polar-locus plot of g~!(ju) drawn for values of u from -ooto*«o t the pont - K s not enclosed and such a system wll be unstable f, for the polar-locus plot of g _ 1 (ju) drawn for values of u from -oototoo, the pont - K s completely enclosed.

37 Tmoq. aks 27 Fgure 10, Frequency polar-locus plot Fgure 10 shows a typcal frequency polar-locus plot. For a transfer functon gan of K-j_, the system, accordng to the crteron s stable. However for a gan of K 2 "the system s unstable. Actually for 0< K<K D the system wll be stable and for K>Kb "the system wll be unstable. For the gan equal to Kb the system, at least theoretcally, wll be capable of selfsustaned oscllatons. Consder now a contactor servomechansm wth an over-all open-loop transfer functon equal to Kg(p). For a sngle constant control sgnal ampltude the contactor descrbng functon wll determne the partcular value of the gan and phase-shft due to the contactor. The descrbng functon GT^, may be consdered n the same manner as a conventonal transfer functon. Therefore the equvalent gan of the system wll be KGr^. Ths value of the gan however s not a constant and depends on the ampltude of the control sgnal. Applyng the smplfed Nyqust crteron for stablty that has already been stated t may be found that the system s stable for some ampltudes of the control sgnal and unstable for others. The smplfed Nyqust crteron restated for the contactor servomechansm s as follows: For an over-all openloop transfer functon Kg(p) and a contactor descrbng functon GJJ^,

38 the system wll he stable f, for a polar-locus plot of g* 1 (ju) 28 drawn for values of u from -o» to+o, the pont - KG-T^, s not enclosed and such a system wll be unstable f, for the polar-locus plot of g-l(ju) drawn for values of u from -«o to+«, the pont - KGrr^ s completely enclosed. In order to obtan the complete pcture of stablty all possble values of - KGDI must be consdered. Therefore a second locus s drawn jonng all the - KG.DT. vectors for a l l possble ampltudes of the control sgnal. Ths locus s approprately termed the ampltude locus and s super-posed on the graph of the frequency locus plotted for the functon g~ 1 (ju). Condtons n <x«tmpl<l plottax k Imoa. amy <Cc/ZK=.or Jftassacton u~3.l>c m lql2 = /.24- J8 Fgure 11. Super-posed frequency and ampltude loc Consder fgure 11 whch s an example of the super-posed loc. Only a porton of the frequency polar-locus plot for u postve s plotted for the functon g^tju) r ju(ju+ 1) (32.) The ampltude locus s obtaned from fgure 8 for condtons Ca/K.05 and C n /Ca =.2. The frequency and ampltude loc for u 2~ negatve wll be the complex conjugate of the loc for u postve.

39 For ths example n fgure 11 the frequency and ampltude loc ntersect at 0^0^/2 = The values of - KGj^ for 1.2< Oj ] y/c C 3:/2< 1.24 wll be completely enclosed by the frequency polarlocus plot. Therefore the system wll be unstable for 1.2< C^/CCL/2< The values of - KGr^ for values of Cm/ca/2> 1.24 are not enclosed by the frequency polar-locus plot. Therefore the system wll be stable for > Keepng n mnd that Ca/2: represents one-half the nactve zone wdth and Cm the ampltude of the snusodal control sgnal, for very small dsturbances the ampltude of oscllaton wll ncrease because the system s unstable for small ampltudes, up to Cny/Cc\/2'= For ampltudes of oscllaton greater than ths value the system s stable. Therefore the condton where the two loc ntersect represents a pont of equlbrum. Ths pont of equlbrum s descrbed as a pont of convergent equlbrum because wth reference to the ampltude at ths pont hgher ampltudes of oscllaton decrease and lower ampltudes norease. Therefore at ths pont of convergent equlbrum oscllatons wll tend to be mantaned. At the ntersecton of the loc the angular velocty u r 3.1. Therefore from the super-posed loc t s expected that the selfsustaned oscllatons wll have an ampltude rato Cm/ca/2: st 1.24 and a dmensonless angular velocty of 3.1. Kochenburger has checked ths partcular example expermentally and found the agreements concernng ampltude and frequency wthn 6 per cent and 3 per cent respectvely. Fgure 12 llustrates several confguratons of frequency and ampltude loc. From the crteron of stablty the system represented by fgure 12a s stable for all operatng condtons and s called an absolutely stable system whle that of fgure 12b s unstable for all operatng condtons.

40 /1 Tma< at/ /1 Tma< Ral dxis Fgure 12. Examples of super-posed loc. The system represented by fgure 12c has two ponts of equlbrum* Pont P represents a convergent pont of equlbrum already descrbed. Pont Q represents a dvergent pont of equlbrum so called because for ampltudes slghtly less than that ampltude at pont Q the system s stable therefore effectng a decrease n ampltude whle for ampltudes slghtly greater the system s unstable therefore effectng an ncrease n ampltude. Ths pont of ntersecton of the loc then does not correspond to a pont of sustaned oscllaton but s rather a boundary condton and one for whch ampltudes wll tend to shft away. For ths system represented by fgure 12c for small dsturbances the system wll oscllate at the ampltude and frequency determned by pont P. If a suffcently large ds«- turbance were to cause an ampltude of oscllaton greater than that determned by pont Q, then the ampltude would grow and the system would be unstable. The problem so far consdered has been that of stablty only. It s very often not suffcent to consder just the problem

41 31 of stablty* Relatve stablty or the amount of dampng present for oscllatons after a dsturbance must very often be consdered, Kochenburger approaches ths problem from hs ampltude and frequency loc usng the well known concept of the M crteron used n lnear servomechansms M s defned as the j 0 (ju) and n lnear servomeonansms the maxmum value of M, Mp, s often chosen by rule of thumb. Lnear systems are often consdered to have suffcent dampng f Mp<1.3. Kochenburger suggests, from results obtaned from< expermental tests, that an Mp of 2.0 near the cut-off pont and of 1.3 for hgher control-sgnal ampltudes, provdes a satsfactory degree of relatve stablty for most applcatons. It s suggested n the work to follow that the rootlocus method used n the synthess of lnear systems can be used to great advantage for contactor servomechansms especally wth respect to.the relatve stablty of a system. From the rootlocus the statc servomechansm gan can be determned drectly for a specfed amount of dampng.

42 32 7. Root-locus method of synthess. 8 The root-locus method was developed by Evans. Ths method uses the roots of the open-loop transfer functon to fnd the roots of the closed-loop transfer functon. For the lnear sngle-loop servomechansm represented n fgure $ the open-loop transfer functon was Kg(p) n Nyqust form, and ths transfer functon can be expressed as Bf(p) n rootlocus form. The rato of output to nput s gven by the expresson, 0 (p) = Bf (p) (33) 1 1+ Bf (p ) The problem of fndng the roots of ths closed-loop transfer functons appears n the form of fndng those values of p whch make the denomnator of equaton 33 equal to zero. Ths condton s satsfed f the open-loop transfer functon. Bf(p) = - 1 (34) or f f(p)" 1 = - B (35) f(p)~^ may be wrtten n vector form f(p)" 1 = N& 0 (36) For the value of the p to be a root of the closed-loop system N/ - - B (37) Therefore N = B ' (38a) T - 18oU+2n) ' (38b) where n s any nteger. To fnd the root-locus the followng procedure s adopted. Consder the open-loop transfer functon. Bf(p) ~ B 1 (39) L (P+ Jd (P+ ) v l?2

43 where the dmensonless tme constants q.^, Y^> y 2 '» a r e rea^- numbers. The zeros and poles of the open-loop transfer functon are plotted on the complex p-plane as n fgure 13. A zero s located at -1_, and a pole at - 1, and -1. The two requrements q.1 y 7Q mposed regardng the phase angle and the ampltude of the transfer functon are consdered separately. Fgure 13* Roots n the p-plane. Frst a locus s obtaned whch makes the phase-shft angle of f (p)*" 1 equal to 180 (1 + 2n). Ths s done by choosng an exploratory p pont and jonng the postons of the zeros and poles to the p pont to form vectors whch represent all the complex factored terms nvolved n the open-loop transfer functon. The angle of each vector s measured wth respect to a lne parallel to the postve real axs. Ths operaton has been performed n fgure 13, If the exploratory p pont s on the root locus then the sum of the angles must equal 180 (H-2n) or from fgure ]+*! s 180 (1+ 2n) (40) Ths procedure s carred out untl the complete "180 phase locus'* or root-locus s obtaned on the complex p-plane. The next step s to fnd the ampltude of the nverse transfer functon on the root-locus, ths ampltude beng equal

44 to tae statc gan B of the open-loop transfer functon. The statc gan B for any pont on the root-locus for ths example s gven by B = Pr+ 1 I n ya Pr + *1 *.on where p r s/the root-locus and the factors are the absolute values of the vectors jonng the poles and zeros to pr» Therefore, gven the poles and zeros of the open-loop transfer functon and the statc gan B, the roots of the closed** loop system can be obtaned drectly from the root-locus. Fgure 14. Root locus of sngle-loop servomechansm. Fgure 14 llustrates the root-locus for a sngle-loop servomechansm wth open-loop poles at 0,»I, and - The root formed y 72 from the pole -, travels n an ncreasngly negatve drecton y2 along the negatve real axs. The roots formed from the poles at 0 and - approach one another, at frst, then form a complex cony jugate par wth a further ncrease n gan. From ths root-locus t may be seen that the troublesome roots are those whch have formed nto the complex conjugate par. For a suffcently hgh gan they may appear n the rght half p-plane so that the complex conjugate

45 55 par wll have postve real parts resultng n an unstable system. However the gan can be set so that the complex conjugate roots appear n the left half p-plane wth a suffcently large value of the negatve real part to ensure suffcent dampng for the natural oscllatons whch would result from a dsturbance to the system. Consder a sngle-loop contactor servomechansm wth poles of the overall open-loop transfer functon as those n fgure 14. The contactor n ths dscusson s assumed to have no hysteress zone. For zero hysteress, the vector form of the contactor descrbng functon has zero phase-shft for all ampltudes of the control sgnal. Therefore for all snusodal control sgnal ampltudes the value of the descrbng functon shows the contactor to appear as a varable gan element, the gan beng dependent on the ampltude of the control sgnal. It s mportant to observe that the contactor characterstc expressed n terms of ts descrbng functon has no effect on the open-loop roots of the system. The open-loop zeros and poles are obtaned from the lnear porton of the crcut and reman fxed n the p-plane. However, the closed loop roots are not fxed and are dependent on the effectve open-loop gan, BGj^, whch s a product of the statc lnear gan and the contactor descrbng functon. For a partcular value of control sgnal ampltude the contactor gan s gven by the descrbng functon and for a known value of the lnear gan the closed-loop roots of the system may be obtaned from the root-locus. Therefore as the ampltude of the control sgnal s vared, the roots of the closed-loop system vary n a manner determned by the root-locus. The root-locus n conjuncton wth the descrbng functon of the contactor can be used as a synthess

46 36 technque for contactor servomechansms and becomes qute effectve when consderng the problem of relatve stablty. Ths descrpton on the use of the root-locus n syntheszng contactor servomechansms has been general. However n the followng secton,whch consders a partcular example, the use and the sgnfcance of the root-locus method wll be treated n specfc detal. o

47 37 8. Analyss of the model by the frequency-response and root-locus methods* a) Kahn's method for obtanng transent responses. The transent response of a control system to a dsturbance s of major sgnfcance. Ths statement s true for both contactor and lnear servomechansms. Transent responses wll be examned n the nvestgaton to follow. Such responses are mportant when correlatng relatve stablty from the rootlocus n an analogous fashon to lnear servomechansms. The transent responses are all obtaned by applcaton of Satn 1 s semgraphcal method. The followng s a summary of ths method. Consder the sngle-loop contactor servomechansm as represented n fgure Fgure 15. Sngle-loopj contactor servomechansm For a step reference nput the transent response of the controlled varable could have the form as shown n fgure 16a and the correcton sgnal would then be as represented n fgure 16b. The relatonshp between the correcton sgnal and the controlled varable far the quanttes expressed n ther Laplace transforms ls D(s) - Iff} - ^(s) (42)

48 to follow page 37 FtQur G-raphs hr Kahn*5 sant-yrafatca/ tdthod

49 38 where l a) s a term contanng ntal condtons. For ntal condtons equal to zero (43) For D(t) as shown n fgure 16b D(s) r f [l - e~ t l S - e- t 2 S + e'^s + B'^S - e^ 5 S -...] (44) To obtan the transent response graphcally t s necessary to solve one equaton only, and that s the nverse Laplace transform L - l [Gs] = 0 R (t) (45) The physcal sgnfcance of ths nverse transform s that t s the output of the controlled varable whch results from a steady applcaton of the correcton sgnal +1 when the controlled varable s ntally at rest. Therefore to obtan the complete transent o response the "runaway responses"' can be added graphcally snce 0(t) s 0 R (t) - OR(t-t) - OR(t-tg) + 0R(t-t3)+... t>0 t>t t>t 2 t>t3 (46) and ths has been shown n fgure 16c. In all the transent responses whch follow, zero ntal condtons have been assumed for smplcty. b) Analyss of the Model The contactor of the model contaned neglgble hysteress effect. Therefore ths analyss s based on zero hysteress zone for the contactor. Further the over-all open-loop transfer functon of the model can be approxmated by the transfer functon G-(p) * K_ (47) PTP^TI ) D

50 39 Fgure 17 s a block dagram representng ths system. For a snusodal nput sgnal the 1 + Contactor mo.clns o f>(p*) z 0 Fgure 17. Block dagram representaton of the model to nput rato of output s gven by A 0(j U ) - &D G-U u ) (48) where GUu) = Kg(ju) s K JuUu+ I) 2 The nverse frequency locus g _ 1 (ju) = ju(ju+l) 2 s plotted n fgure 18. The ampltude locus -KGr^ wll travel along the negatve real axs snce no phase-shft exsts for GQ^ when the contactor has no hysteress zone. For ratos of Cm/ba/2 equal to unty and nfnty, Gr^ equals zero. Therefore the ampltude locus has ts orgn at the orgn of the complex plane for 0^0^/^ r 1.0 then proceeds along the negatve real axs to a mnmum pont then reverses and fnally as the control sgnal ampltude approaches nfnty t agan approaches the orgn. From the smplfed verson of the Nyqust crteron the system wll be stable for values of K G T J 1 < 2;.0 and wll be absolutely stable, that s stable for all control sgnal ampltudes, f the maxmum value of KGj^^KGT^max.), s less than 2.0. Ponts

51 to follow

52 of equlbrum occur for KGJJ^ =r 2.0. ForCKG^max.^ 2..0, two ponts of equlbrum, both for u = 1.0, occur on the super-posed loc. The pont of equlbrum for the smaller control sgnal ampltude s a dvergent pont of equlbrum whle that for the larger control sgnal ampltude s a convergent pont of equlbrum. For example, the ampltude locus drawn n fgure 18 s for Cfl/K r.25. From fgure 8 thereforeckg^max.)» 2.44 and KGr^ equals for Cm/cd/2 equal to 1.11 and 2;.33. Therefore the dvergent pont of equlbrum occurs for Cm/ca/2. = 1.11 and the convergent pont of equlbrum for Cm/ca/2: r A plot of the ampltudes of the control sgnal whch gve ponts of equlbrum versus the rato Ca/K s shown n fgure 19, The ponts of convergent equlbrum are those of specal nterest because these ponts represent the possble self-sustaned oscllatons of the system. It s dffcult to ascertan the meanng of the ponts of, dvergent equlbrum for small values of the ampltude rato because Kochenburger's justfcatons for the use of the frequency response method do not apply. From fgure 7 for C m /C a < 1.1 the frequency spectrum of a rectangular wave does not nvolve progressvely smaller ampltudes for ncreasng orders of the harmonc components. However ths has no effect on the analyss because of such small ampltudes nvolved. The graph of fgure 19 shows the system to be absolutely stable for values of Ca/K > _ l. For Ca/K^, 1 the system s capable of self-sustaned oscllatons and the ampltudes of the selfsustaned oscllatons ncrease wth a decrease,n Ca/K. The 2~ frequency of these self-sustaned oscllatons, regardless of ampltude, remans constant at u =

53

54 41 The rato Cg/K s sgnfcant. For ths case, K, a represents the slope of the output response to a unt step correcton sgnal after a suffcently long tme. Therefore the greater the K the qucker the correcton response. However, the stablty of the system s not only dependent on ths gan factor but also on the nactve zone, Cg. Ths s a property pecular to contactor servomechansms. The graph of fgure 19 although separatng the system nto an absolutely stable reg/rn for Gd/Kv 1 and an oscllatory regon for Ca./K^ 1 gves no nformaton about the relatve stablty of the system n the absolutely stable regon. The conventonal M-crteron method employed n lnear servomechansms may be used as prevously mentoned. For ths example the root-locus method wll be employed to nvestgate the relatve stablty of the system. For ths case both the Nyqust and root-locus forms of the over-all open-loop transfer functon are the same* F(P) S JL G(P) = _K (49) P(p + D a P(P+D 2 where B = K and g(p) = f(p) s 1 T P(P + 1) f(p) has three poles on the complex p-plane, one at the orgn and a double pole at p» -1. These poles as well as the root-locus are plotted n fgure 20. For snusodal sgnals n the System the effectve open-loop gan.s BGjjjSnce K = B, then KGTJI s B G Dl* Values of Ca/K are also plotted on the root-locus. 2 The value of KGDT correspondng to the value of Cd/ K o n ^ e root-locus 2 s the maxmum value of KGy^ for that value of Ca/K. For example, 2~*

55 j to follow page Jj-l 1 j - - t 't j j j. ; '. - - ^ - l - t ' {, -,- ;- j- : C 1 ' :! ' : ; 1 /n\= A \ j 1 J ] 1. ' ; I : 1 [ I 1 I : '1! I ' : 1-1! - 1 t 1, u r ;-[ - j ' 1 ; ; 1 ;!! / / 1 * 1 A* 5 * j j f.-v. / j 4 1 1!! I \ 1/ 0 l 1 / J _ 1 1 ; 1!! I - r<3f t 1 f / I / 6 4 > 1 J! 1 t " 1 1! 1 I V 1 f J < cr >S A / 'S t Oc " r 7 ( - - I 1!. j V \ 5~. j 1 1 > j j - I 1 _ j 1 1 1! 1 1,. 1 f l t I I - 1 F/aure. 20. fact-lows for tho, refreentatfon of th. vtkdrtf>as>sctt<d /#dcj«./

56 42 the values KGQ 1 r.376 and d/ K = 1.69 correspond to the same pont on the root-locus of fgure 20. Therefore for a rato of Gd/K = the maxmum value of KGrj-^ s equal to.376 and ths pont on the root-locus determnes the mnmum dampng for oscllatons n the system. Smlar nformaton regardng stablty can be obtaned from the root-locus as from the super-posed ampltude frequency loc. The complex conjugate poles cut the magnary axs at u = 1.0 whch frequency corresponds to the frequency of convergent equlbrum n the frequency-response plot. The open-loop gan from the root-locus at u x 1.0 s KGTTJT^ r 2.0 whch also corresponds to the value already obtaned. From the root-locus the settng of the gan can be set for the desred mnmum dampng. For example, f the mnmum dampng rato <r/u s to be no less than 1//3* then from fgure 20, KGD- can be no greater than.376. The rato of Cd/K s therefore Gven the nactve zone C^, whch s determned from the requrements for statc accuracy, the gan K can be found. The root-locus gves some nformaton about transent responses at least qualtatvely and n many cases "sem-quanttatvely*. The contactor descrbng functon GTJ-^ s ndependent of frequency and dependent only on the ampltude of the control sgnal. However, the root-locus mples that for a partcular ampltude of control sgnal that only one frequency s possble. Ths s true because the root-locus determnes the natural mode of oscllaton of the system yet the gan for-a, partcular oscllatory condton

57 on the root-locus determnes from the descrbng functon the ampltude of the control sgnal for that natural mode of oscllaton. In fgure 21 are plotted transent responses of the controlled output to a step nput of the reference. The response n fgure 21a s for Ca/K =.4. 2~ From the root-locus the maxmum value of KG/jjj_ s The angular frequency and dampng f a c t o r correspondng to ths value of KGr^ on the root locus are u -.91 and T: -.04 respectvely. From the transent response plotted the predomnant frequency s u».796. For an ncrease n the value of the rato Ca/K t would 2~ be expected that the transent response for a smlar type of dsturbance would have a lower frequency and greater dampng. For Ca/K =»5 the maxmum value of KG^ s 1.2:7 and the frequency and "a dampng factor on the root-locus correspondng to ths value are u s.81 and<ts From the plot n fgure 21b the predomnant frequency s u :.689 and t s seen that the dampng s slg;htly greater n ths case than n the prevous one. It s qute reasonable that the predomnant frequency of the transent response s approxmately equal to that gven for the condton of mnmum dampng on the root-locus. especally n ths case because mnmum dampng also Ths s true corresponds to the maxmum possble value of the angular frequency. Further ths mnmum dampng occurs for a maxmum value of KGT^, for whch Cm/Cd/2'm 1.4 whch means that the control sgnal ampltude s relatvely small and the system s close to comng to rest. It s, of course, very dffcult to come to any defnte conclusons regardng the transent response from an nspecton of the root-locus because the transent responses are dependent on the form of the external dsturbance. Possbly wth suffcent

58 Favra, Zl. Plots of transent r<tsponsa.s to r«fs.ra.ncq. stap nputs

59 experence and practce a satsfactory rule of thumb crteron could be developed. However, the root-locus wth the contactor descrbng functon does yeld a practcal technque to help the desgner n choosng the gan of the system for satsfactory relatve stablty, c) Applcaton of a smple phase-lead type compensatng network. For the system just analyzed the rato C^/K had to be 2~ relatvely large to ensure stablty. The applcaton of a compensatng network allows ths rato to be made smaller. Ths means for a partcular value of nactve zone,,that K can be made larger whch results n mproved dynamc accuracy. The compensatng network added to the basc model s the type shown n fgure 5 whch has a transfer functon Therefore the over-all open-loop transfer functon s now F(p)» B(p-r-.65) (51) P(P-t-l) s (P + 2,.8) Placng the above transfer functon n ts Nyqust form G(p) = K(1.588p + 1) (52.) P(P+ l)k(.357p + l) ( 1 K =.65 B r *2S5B (53) The frequency-response of g~ 1 (ju) - ju(ju+l) 2 ' (,3573m-1) (54) U.588Ju-t-l) s shown n fgure 22, Ths plot cuts the real axs at for u = 2*0. Therefore the boundary condton for stablty s KOD-L S 3.69.

60

61 The system wll be absolutely stable for KG^ < 3,69 whch s an mprovement over the uncompensated case whch was absolutely stable for KG DL < 2.0. Fgure 23 s a smlar graph to that n fgure 19 for the uncompensated case. It conssts of a plot of the ampltudes of the control sgnal whch gve ponts of equlbrum versus the rato Oa/K. The forms of the two graphs are alke. In the regon 2: of possble self-sustaned oscllatons, the ampltudes of oscllaton ncrease wth a decrease n the rato C^/K. The frequency of the possble self-sustaned oscllatons wfl now be u = 2.0 n comparson to u for the uncompensated oase. The root-locus s shown n fgure 24. The effect of the compensatng network s to nclude a zero at p = -.63 and a pole at p n addton to the pole at p - 0 and the double pole at p = -1. In an analogous manner to the uncompensated case, the values of KGJJ^ as well as the values of Ca/K correspondng to the 2 maxmum value of KG^ are plotted on the root-locus. The effect of the addton of the compensatng network on the transent response s demonstrated n fgure 25. Ths fgure gves the transent responses of both the uncompensated and compensated systems to equal step nputs of the reference for equal values 6f the rato Ca/K equal to.40. The mproved dynamc response 2; of the compensated case s obvous. In fgure 25a whch s the transent response wth no compensaton, the contactor wll have to undergo many swtchng operatons before the output response fnally rests n the nactve zone. In contrast s the response of the compensated system shown n fgure 25b. In ths case merely two pulses of the correcton rest. sgnal are necessary to brng the system to

62

63 to follow page ^ yur~ 2J4. Koot'/ocu^ for th<l r<tpr<l)<nt C/on of fjfv. company (a4 r

64 .. ^ : to follow page *J-5 t ; : - T - ; - l - ""! " ' f :! > 1 1! > \& C = 1!, f :. 1 : A A A A \ 1 \ 1 / l \ 1 1 / 1 -T~Mj -:- \ M l : \ f- / \ I \ \ /' /1! ' 1 f 1 1 J 1 I f 1! 1 1 ' j ' 1 j I C '. I - I \ a - M r "3!. *N /.1 N j /! 1. j " r 1 - I 1 1 j a J\ ' I J ) > I.3 0 ;! -D! - : \ ' - ton hss fcrr)<l j ' 1 c L B * a s (Lntja t.on a. urc Off 1 t 1 I _ 7TS/C >. J f j 1! j I >. ; \C:, lt -. - > c $- 1 / - t - s o < t ' T h) tat dtt * tat dtt en_ > 1 _ U t I

65 46 9, Root-locus method when contactor has hysteress For the contactor wth no hysteress zone the vector form of the contactor descrbng functon G^ has zero phase-shft for all ampltudes of the control sgnal. The contactor therefore appears^' as a varable gan element, the gan dependng on the ampltude of the snusodal control sgnal. The varatons n the gan of the contactor hawpno effect on the over-all open-loop roots and affected only the closed-loop roots of the system. For the contactor wth a hysteress zone, phase-shft exsts n the vector form of the descrbng functon G^. Therefore the argument that the contactor has no effect on the over-all openloop roots no longer apples. It wll be shown that the contactor wth hysteress now not only affects the closed-loop roots of the system but also the open-loop roots of the system. Ths of course complcates the root locus consderably. Referrng back to chapter 5 where the descrbng functon of the contactor was represented as a rato of the Laplace trans** forms of the correcton sgnal and control sgnal, and gven by equaton 28 SD s L [Dm cos(u<t> + /P].)] L [Gm cos u<t>] where from equatons 2:6a and 26b respectvely Expandng equaton 28 D lm z 4 sn b "* IT A>1 = -a s. L[DIHJICOS ud>) (cos a)4(sn u<t>) (sn a jj] L [C m cos ua» ] (55)

66 47 and placng equaton 55 nto ts Laplace transform t can be expressed n the form Dla Cm cos?.a f p T- u tan a 1 - h ( 5 6 ) The descrbng functon n ts Laplace transform ntroduces a pole at the orgn and a zero at p = - u tan a. Ths open-loop zero s not fxed on the p-plane and vares as a functon of u, the angular frequency, and the phase-shft angle, a. Therefore the descrbng functon of the contactor wth hysteress not only affects the roots of the closed-loop system but also those of the open-loop system. For zero hysteress, a s o for all ampltudes of the control sgnal, cos a then equals unty and tan a equals ^ zero, therefore the descrbng functon gj^ degenerates to SD * prn (57) whch for ths case s exactly the same as the vector form of the descrbng functon whch s to be expected for zero hysteress. Consderng the gan, D m - cos a, of the descrbng Cm functon; the rato Dm s equal to the absolute value of the Crn vector form of the descrbng functon GT^. T l e descrbng functon can then be rewrtten e D = ICDII c o s a p + u t a n a j P (58) In fgure 8, the product of the lnear gan, K, and IGTJJ, KlGj^l, as well as the phase-shft angle -a have been plotted aganst the ampltude of the control sgnal for varous constant ratos of

67 Ch/Cd«From these two graphs s plotted a new graph, KlGj^l cos a aganst the ampltude. It must be remembered that the gan K s that for the Nyqust form of the lnear open-loop transfer functon. The graph of K IGDJ cos a and the phase-shft angle '-a aganst the control sgnal ampltude for the rato Ch/Ca= 0 2: has been plotted n fgures 26a and 2:6b respectvely. The method of obtanng the root-locus for a partcular example wll now be consdered. The lnear open-loop transfer functon chosen s the same as that whch was assumed to represent the model that s the root-locus form s then where B s K G(p) = K (59) P(P + D 2 B (P) -, B. D (60) plp+ 1) 2 ; The example s worked out for the rato of hysteress zone to nactve zone r 0--12' The rato of output to nput s gven by. (P) r SDT. B(p) (61) l + SDB(p) and SDB(p) tt B'Gpxl cos a(p + u tan a) (62) p 2 2 (p+ l ) For a phase-shft angle, -a^,and for the exploratory p vector havng an magnary component u^ the pole and zero confguraton s as shown n fgure 27. The four roots of the closed-loop system for the open-

68

69 t ; : I.! t -1.:_ :M -1 ::; :::.( ::-.\ 1 ' t :. p ;:.: :..j ~PU --- t + - ';,!.....::. L:_. Of')-- - :: 'f. \:, ---Ish) ; com por to! ' : ' I ', fr co/7 "o':c tor = q.n't of \ u j -. r y p -rfh- J ' ::! t :.. - : _ : _ '.... ; ; ; ; ; - H (:.-: :....ll :..±.LL 4 - : : : - ^ *-t t.... ; f. t....p _ _.... follow page k& ;. 1 ' ' ' > 'A. A-:.:- - :-- \ rlt- : : '' J.: r. / J.: y -a [ / - H- ': HI''-:: /.L..TT ::.[..: /! :. y.r- n - :,;; : J.J : n rrrr - *!'.':. : " 1 :;::]T ; ;. ; ; : ; ; _:... '-' \.: I ; : : ' ' ' - ' '... l ' : ;' : : ;'': :: :, :' : ' ' : ^, 1 dt f- ;:: I ;.:. / ~ * " r /.:: j /. j t.{.--.~::- *. -: *::-.-, AJ M! :; :.;. -,';;[:; jv ; :; : :.:!' ^ 1.::. - ; 0 ; : ' ~ " J... : t. _ ;. _ t, : : j: :::: t. : :: j-: T :. :[. j- ' t : Tf a < -1 ;-.. : ; I:'' moi, :;::; j p - :--[-- ::: ;::: :;:: : :; : ;:; :::: ':.: J : : :: ; ' ' <. :::j!: :.... :::: ; :: :;:: : :!:t ::.: rr ~~- r r r ~ Tr- :;t;- ; [ ; * TT^TT UT r ';,(.. TT*T ; t ; "'. :: 1;!' ' 11 ] l!;! jhl: ; J : : ::.. :::: '.:!: : j: :. - -, -tr...! 1 1. :.....;. f. ITTT '- T; :... I:-'..::... I; t.. ::, r... ;... : :. :; :.:.:.:(. 4 :. I...:.1 ' I =L I ^>.y_1 j ::: );: -rrhf ' Pov!»/cj. ::! ; : ; ; : ; ; ;!'.: : : : ::.: :; ; ;!:: ; ;::.::: :::: :::* t - ;;; v.-- ;; : : : : : : I : : rrrf TH: * -', t:;!!;! ;!L tf: HI; -j r;n '. '.! ft H jt,!j'j,;-., : ll' f!.r! rf. TrTT ;::! 11 m. t l M ; " ' t ' t ::.:'::;:. 1 :/ :.' : '. : -. ; : _ at p =bl_.. : ; j.;: - :. 1: ' T" P7 : ::: 1 ' ,. 4 >:. ;. :: r. H. '. '.". \....., 4 :!:_.- :.: ::;:...!..:.: :.: ': ':' :: JL- ' ~ ' ' -j: 1 J, :::.j: ll: \\\rha - M : ll :; 1 "-;<',- t: '' 1 1 " 1!] ; I ll!:;!:;j - 14,1 * j! ' 1.!-H! :! ': " l!!? j >! '..-* :. ; '..; :. ' f-t 'T :! "; :!.:: r ::: -.: ^..:'' : ': r!'-.:'; :::: -. '..} 1H..j...,.. ll:! J::; ;.'.'.I r::. ;;;; : ;.;;j : ;r;r - : ' 11 : 1 ' 1 1 t-}! ;;;; ; 7.. _, TT r-l ' - :. : J f-- \.. I, ; j!.: : ; :. ;;;!-;;! ;;;;;;; ; :: :; -f : <d:: f/jvrd 27. 0/»w -loop tdro-p^lt conf/juraton tlth contactor bystates's * ::;: ;f

70 49 loop zero-pole confguraton of fgure 27 can be located approxmately by nspecton. Two of these closed-loop roots le on the negatve real axs; one of these les between the zero and the orgn whle the other les somewhere to the left of the pole at The other two roots, form a complex conjugate par. Root-loc for constant values of the angle, a, can be constructed. These constant phase-shft root-loc wll for u o have two roots on the negatve real axs:, as already descrbed, one between the zero and the orgn whle the other wll be to the left of the pole at The complex conjugate par wll break from the real axs at the pont where the complex conjugate roots for the root-locus of f(p) = 1 break the real axs. Ths s so p(p+ D 2 because for u = o, the zero and pole ntroduced by the descrbng functon of the contactor are supermposed at the orgn and cancel out the effect of one another. Therefore, the complex conjugate roots of all the constant phase-shft root-loc wll converge at the same pont on the real axs. The troublesome roots whch determne the natural mode of oscllaton of the system are those whch form the complex conjugate par. Only these roots are traced on the root-loc of fgure 2.7, Further, for convenence, only the half of the complex conjugate par root-loc for u postve s traced. The other half, for u negatve, can be located by nspecton because of the symmetry of these roots wth respect to the real axs. Referrng to fgure 28 the root-loc have been obtaned for several constant values of the phase-shft angle. The effectve gans whch equal BIGDJ cos a are plotted on the constant

71 to follow page

72 phase-shft angle root-loc and lnes of constant gan are constructed. Snce n ths case K = B these lnes are constant KIGQ^I COS a lnes. In the general case where K ^ B, the values of BIG/DTJ c o s a would have to he multpled by the approprate factor relatng K and B to obtan the constant KlG^I cos a lnes. Root-loc have been drawn for constant phase-shft angles of the contactor descrbng functon. Constant gan lnes have also been super-posed on these root-loc. The next step s to obtan the actual root-locus for varous values of the control sgnal ampltude. The procedure s as follows: Referrng to fgure 26, assume a rato Ca/K =» 1.0. For an ampltude rato Cm/Ca/2: r 1.2, a = 25 and ~~2 KlGj^lcos a =.44. From a knowledge of the phase-shft angle and the gan the root for ths condton can be obtaned on fgure 28. Ths procedure s repeated for ncreasng values of the control sgnal ampltude and hence a root-locus for all ampltudes of the control sgnal for a constant rato C^yK t 1.0 s obtaned. Ths: 2 procedure s then repeated for other ratos of Ca/K. A famly of ~2 root-loc for several values of the constant rato Ca/K has been plotted n fgure 28. The root-loc plotted n fgure 2,7 show that contactor hysteress has an adverse effect on the system. Ths adverse effect due to hysteress can of course be demonatrated also by the auper-posed frequency-ampltude loc and the Nyqust crteron. However the advantage of the root-locus method s that the values of the frequences and dampng factors can be obtaned drectly for all values of control sgnal ampltudes.

73 The experment. a) The crcut and ts operaton. The crcut dagram of the model tested appears n fgure 29 and a convenent schematc dagram of the model n fgure 30. Wth reference to these two fgures, the voltage drop, V 0, across the resstance R]_ of the potentometer, whch may or may not nclude the compensatng network, s energzed by the voltage drop across the motor-drven rheostat. A second voltage drop, V R, across R]_, s produced by the potentometer across the balancng voltage source. The voltages VQ and are opposte n polarty. The dfference of these two voltages, VR - Vo, s appled across the crtcally damped crcut galvanometer. If the lght reflected by the mrror of the crcut galvanometer and n turn reflected by the lght-splttng falls on the phototubes, the thyratrons are drven beyond mrrors grd cut-off and are non-conductng. Except for a narrow strp on the face, each phototube s completely masked by tape. The lght spot shnng on the phototubes s rectangular n shape. The edges of ths spot parallel to the axs of galvanometer rotaton, are parallel to the lght-acceptng slts of the phototubes. For a suffcently large deflecton of the crcut galvanometer the lght wll move off one of the phototubes causng the thyratron controlled by ths phototube to conduct. For a suffcently large deflecton of the galvanometer n the opposte drecton the lght spot wll move off the other phototube causng ts thyratron to conduct. For a certan range of galvanometer deflecton the lght remans

74 to follow page 51 -HOv -H Thyratron Photo multpler Tube 93/-A J,/2meqD. each y ^~ ' 1 \ )' Cr-ttcqlly damped* Record ma &al^onomet<tr frjure 1% Crtvtt dktgrem of the model

75 X, y Motor Cref/v. Q-dhanootethr' Defecton C, Contacto r /Mean % Armcttbre, ',urr&n"c gop"ttflg 7 IOK+ y x^-s 39.7 X r-aaaaaa-, LyvWSAA-j f a. O 5 Com pen s <*t na j^-f NO \ (xalvanometsjr\ 5 ± Deflecton, $ t2. F/gure 30. Schew'ot/c Ptaqram of the Model

76 on both phototubes and both thyratrons are non-conductng. Ths range determnes the nactve zone of the contactor. The drecton of motor rotaton depends on whch thjratron s conductng. Both the thyratron plate supply voltage and the voltage appled across the feld of the motor are alternatng. Consder that one of the thyratrons s conductng. Ths thyratron conducts over a half-cycle of the plate supply alternatng voltage If the other thyratron conducts, t wll do so over the other half cycle. However the current from ether thyratron whch s the current through the armature of the motor s undrectonal regardless over whch half-cycle thyratron conducton takes place. But the drecton of the feld current s reversed over each halfcycle. Therefore the drecton of motor rotaton depends on whch thyratron s conductng. In summary, control takes place In the followng manner. For a suffcently large dfference of the voltages, VR - VQ, the crcut galvanometer wll be deflected, so that lght wll move off one of the phototubes. The thyratron controlled by ths photo tube wll conduct. The motor wll run and turn the shaft of the rheostat n such a drecton as to correct for the unbalance. If the unbalance s n the reverse polarty then the other thyratron wll conduct. The motor wll run n the opposte drecton and turn the shaft of the rheostat n the opposte drecton n order to correct for the unbalance. From the crcut dagram t s seen that there are two galvanometers n the system, one referred to as the crcut galvanometer whle the other s referred to as the recordng galvan-

77 53 ometer. All the calculatons are based on the control sgnal whch s the nput sgnal to the contactor and whch for the model s the deflecton of the crcut galvanometer, Q]_. Snce t was mpossble to "break nto" the crcut to observe the deflecton of the crcut galvanometer, the recordng galvanometer was ntroduced n such a manner that the deflecton of the recordng galvanometer, C2> would be dentcal to the deflecton of the crcut galvanomet er. b) Calbraton of the model. Ballstc galvanometers of the D'Arsonval type were used for both the crcut and recordng galvanometers. Nether galvanometer possessed electromagnetc dampng on open crcut. The perods of the galvanometers on open crcut were 6.46 and 6.45 seconds for the crcut and recordng galvanometers respectvely. For convenence t was assumed that both galvanometers had an open crcut perod of 6.46 seconds. Suffcent resstance was added to each galvanometer so that both galvanometers were crtcally damped. For crtcal dampng, the tme constant of each galvanometer s therefore 6.46/2T = seconds. The senstvtes were found to be 370 radans/volt and 477 radans/volt for the crcut and recordng galvanometers respectvely. The transfer functons of the galvanometers neglectng ar dampng and nductance n the galvanometers for the condton of crtcal dampng galvanometer are therefore: for the crcut galvanometer: G ff (s) = and for the recordng galvanomter S (T f 025s + 1) radans/volt (63a) G g 2 (s) radans/volt (63b) (1,029s + l ) 2

78 54 The nactve zone was determned n the followng manner. A voltage just suffcent to fre one of the thyratrons was appled across the crtcally damped crcut galvanometer, then a voltage just suffcent to fre the other thyratron. The dfference of the two voltages, whch gves a measure of the nactve zone, was obtaned. Ths procedure was repeated several tmes and the average value of ths voltage was found to be 5.9 mcro-volts. Snce the output of the system s consdered to be the deflecton of the galvanometer, the nactve zone, C a, n terms of galvanometer deflecton s Cd x IO" 6 x 370 = 2.18 x 10~ 3 radans. The transfer'functon from the motor armature current to the voltage drop across the rheostat drven by the motor was obtaned n the followng manner. It was assumed, ven a thyratron conducted and therefore the armature of the motor carred current, that the motor mmedately attaned a certan speed and mantaned ths speed constant as long as t was runnng. Ths assumpton was observed to be approxmately true although some tme would have to elapse before the motor came up to speed. Ths perod s very small compared wtn the tme constant of the galvanometer and hence as an approxmaton ths perod can be neglected. Therefore wth ths assumpton the change n the voltage drop across the rheostat caused by the motor rotaton would be proportonal to t me. The voltage dfference between two settngs of the rheostat arm poston was obtaned together wth the tme requred for the rheostat arm to travel the dstance between these settngs. From ths, nformaton and the assumpton of constant motor speed,

79 55 the transfer functon of ths porton of the system can be defvad. For example, when the voltage output of the varable voltage supply, Vg, was equal to 445 volts, the dfference n voltage drop between two settngs of the rheostat arm poston equalled 1.57 volts/ For the case where the rheostat arm requred 8 seconds to traverse the dstance between these two settngs, the change n the voltage drop across the rheostat appears as shown n fgure 31. /57 " trrta. In S tconds Fgure 31 - Change n rheostat voltage drop aganst tme. Assumng, for convenence, the nput to be a dmensonless unt step functon, the transfer functon, G^U), of ths porton of the system s 1 G m (s) = 1 R m g m (s) * 1.57 ( 64) "s s 8^ Gm(s) = Rmg m ( s ) s -1965/s volts (65) where R m =.1965 volts/second and g m (s) = 1/s The value of the gan, R m, was changed by changng ether the value of the supply voltage, V s, or the speed of the motor, or both. The system s depcted by a block dagram n fgure 38. For the condton that the deflecton of the crcut galvanometer and the recordng galvanometer be the same, 370 Rp]_ must equal 477 R p2. R p s equal to 1^/23,100 and R p2 s equal to R 2 /22,400 for the uncompensated case. Rpj_ s equal to R^/51,200 and R p2; s equal to R 2 /51,720 for the compensated case. R]_ and R 2 are expressed

80 to follow page 55 cr V 4. partest/jjucr} MS T y \ ^

81 56 n ohms. The relatve values of R]_ and R g were always such that the condton for equal deflecton of the galvanometers was satsfed. Both of the values of these resstances were always small compared wth resstance of the galvanometer crcuts to ensure neglgble effect on the postons of the poles of the galvanometer transfer functons* from fgure 32, the over-all open-loop transfer functons, G(s), for the uncompensated and compensated cases respectvely are G(s) = R (66a) s(1.029s + l ) 2 G(s) = R (1.635s» 1) (66b) s(1.029s + 1J«(,367s + 1) Where R = 370 R m R pl. The over-all open-loop transfer functons n dmensonlesstme form for the tme base t^ = seconds for the uncompensated and compensated cases respectvely are G (P) = K (67a) p(p + D 2 and G(p) = K(1.558p ( 6 7 b ) where p = 1.029s p(p + l ) 2 (.357p + 1) and K = R(1.02$) output-unts, whch for ths case equals radans; The above over-all open-loop transfer functons are dentcal wth those of the two systems analyzed n chapter 8. c) Tests and Results. Transent responses for both the uncompensated and compensated cases were obtaned for equal step nput dsturbances. The dsturbance ntroduced appears as a step nput of the crcut

82 57 galvanometer dsplacement and was obtaned In the followng manner. The contactor was set so that t rested just on the edge of the nactve zone. The balancng voltage, V^, was then changed n all cases by an amount equal to 13.5 mcro-volts correspondng to a galvanometer deflecton of 5 x 10~ 3 radans. The feld swtch of the motor was left open so that correcton could not take place. After the crcut galvanometer had reached ts stable poston the swtch n the feld crcut of the motor was closed to allow the system to correct for the deflectons. Ths appears to the system as a step nput of 5 x ICP^ radans to the control sgnal. The transent response was observed on the recordng galvanometer. Graphs of transent responses for the model uncompensated and compensated are drawn n fgure 33. Galvanometer deflecton s plotted aganst dmensonless tme 0, where 0 = t/ From the graphs t may be observed that the dynamc accuracy for comparable values of C^/2K s mproved by the compensatng network. The effect of the compensatng network on the frequency of the transent response s evdent from the graphs. For a rato C d /2K =.588 for the Compensated system the predomnant frequency s nearly twce as great as the predomnant frequency for the uncompensated system wth a rato G^/2K =.583. From an nspecton of the root-loc for the two cases, ths ncrease n frequency for the compensated case s tobe expected. > Values of ampltudes and frequences of self-sustaned oscllatons were also obtaned. Table 2 s a tabulaton of the ampltudes and frequences of these oscllatons for several

83 to follow page 57 11: 1:-: I' 1 : Al1 LA,.; L ll 11 l t rll 11 fjur& Jj. TranslnC r<zspc}t,<l<> ta a st<p d/sturkant