Report U M M - 24 A New S ervomechanismns 7Tchnique by D. M-Donald A Stuple Experimental Method for Improving the Response of Servomechanisms Aeronautical Research Center Engineering Research Institute University of Michigan June 1, 1948 WiXllow Run Airport Ypsilanti Michigan

Lithoprinted from copy supplied by author by Edwards Brothers, Inc. Ann Arbor, Michigan, U.S.A. 1948

AERONAUTICAL. RE~SEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM-24. The development of new technmques for the improvement of the response of servomechanisms has made it possible to apply servomechanisms to an ever increasing number of military and industrial uses. These techniques are based upon: (1) improved methods of analysis which show the factors that limit servomechanism response, and (2) the design, or redesign, of components which improve the response. This article describes a new and simple method for designing stabilization circuits which may..be used to improve servomechanism response, and offers a new approach to the design of stabilization circuits in general. The design of these stabilization circuits is based upon the following concepts: a circuit, having a transfer function which is the inverse of the transfer function of the servomechanism, can be used as a stabilization network; and this circuit can be approximated by a feedback amplifier having in its feedback path a network with a transfer function proportional to that of the servomechanism. Consequently the first step in obtaining such a stabilization circuit is to design a circuit having a transfer function proportional to the transfer function of the unimproved servomechanism. This latter circuit will be referred to as the Equivalent Servomechanism Circuit or the Equivalent, Circuit. Equivalent Circuit The first step in designing the Equivalent Circuit is to set up the,general form of the equivalent network.' The second step is then to determine the parameters of this network. These two steps;are illustrated by the examples of Figures 1, 2, and 3. In figure 1-A is shown the schematic of a simple: servomechanism, using a motor generator set as the power control component of the servomechanism. Figure 1-B is the electro-mechanical equivalent of 1-A. The transfer functions of the system are expressed by the equations of Figure 1. Figure 2 shows the general form of an electrical network having a transfer function of the same type as the velocity transfer function of the physical servomechanism. Comparison of the analytical expressions for the transfer function of the Equivalent Circuit of Figure 2, and for the physical servomechanism, demonstrates the equivalence of the network.,,,,,. ",', Page 1

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM-24 Figure 3 is a schematic of the set-up used in determining the values of the parameters of the Equivalent Circuit. This set-up consists of the unimproved open-cycle servomechanism, the Equivalent Circuit, a variablefrequency-sinusoidal-voltage source, a d.c. tachometer, and an oscilloscope having d.c. amplifiers to drive its deflection plates. The following paragraphs desc-ribe the operation and theory of the setup of Figure 3: With Switch S-2 open, and selector Switches S-3 and S-4 in Position 1, the voltage E1 of the Equivalent Circuit and the open-circuit output-voltage Eg -of the generator are placed across the deflection plates of the oscilloscope. By adjusting Resistor R1, it is possible to reduce the elliptical figure on the oscilloscope screen to a straight line. Then the first factors of the velocity transfer functions of the actual servomechanism and of the Equivalent Circuit are the same. In essence, when the ellipse becomes a straight line, the phase shift of the first stage of the Equivalent Circuit has been made equal to the phase shift of the first stage of the actual servomechanism. Therefore, the first factors of the velocity transfer functions of the actual servomechanism and of the Equivalent Circuit are the same because the servomechanism contains only minimum phase networks.2 The frequency of the variable-frequency-sinusoidal-voltage source should be varied over a large frequency range to insure that the first factors of the velocity transfer functions of'the actual servomechanism and of the Equivalent Circuit are the same for all frequencies. Non-linearities in the servomechanism may make it impossible to obtain one value of R1, which reduces the elliptical figure to a straight line for all frequencies. If this occurs, then that value of R1 which minimizes the width of the ellipse over the frequency range of interest should be chosen. With Switch S-2 closed, selector Switches S-3 and S-4 in Position 2, and with the rotor of the servo motor locked, the voltage E2 of the Equivalent Circuit and a voltage proportional to the current ia of Figure 1 are placed across the deflection plates of the oscilloscope. By adjusting resister R2 it is possible to reduce the elliptical figure to a straight line, This set-up, in a simplified form, was first developed by the author. in 1945 while a member of the M.I.T. Servomechanisms Laboratory. 2 Feedback Amplifier Design, by H.W. Bode; Bell Telephone System Technical Publications, Monograph B-1239 Page 2

AERONAUTICAL RESEARCH'CENTER -'UNIVERSITY OF MICHIGAN Report No UMM-24 whereupon the second factors of the velocity transfer functions of the actual servomechanism and of.the.Equivalent Circuit are the same.. In a simslar fashion, with Switch S-2 closed and selector Switches S-3 and S-4 in Position 3, the voltage E3 of the Equivalent Circuit and a voltage proportional to pAo of the servo motor are placed across the deflection plates of the oscilloscope.'By proper: adjustment of the resistor R it is again possible to minimize the width of the ellipse and make the third factors of the velocity transfer functions of the actual servomechanism and of the Equivalent Circuit the same. Upon conclusion of these three steps, the transfer function of the Equivalent Circuit is proportional to the velocity transfer function of the unimproved servqmechanism. As mentioned before, non-linearities in the servomechanism may make it necessary to choose values of R1, R2,and R3 which tend to minimize the width of the ellipse over the frequency range of interest. From simple network considerations of the Equivalent Circuit, it is possible to determine what factors in the unimproved servomechanism limit its response. (Reference 1 and 2) If these factors cannot' be removed by redesign, then a stabilization network must be used. Application of the Equivalent Circuit as a Stabilization Network The schematic diagram of Figure 4 shows how the Equivalent Circuit- can be incorporated into a feedback amplifier and thus create a stabilization network. The resulting feedback amplifier will be referred to as the Stabilization Feedback Amplifier. In Figure 4 the unimproved open-cycle servomechanism,' represented by the transfer function KG1 of Figure 1, is cascaded with the stabilization Feedback Amplifier having a transfer function K4G4. The Stabilization Feedback Amplifier has a constant gain amplifier K5 in its forward branchaand the Equivalent Circuit of Figure 2 in its feedback branch. The expressions of Figure 4 demonstrate that if K3 is large enough, K4G4 approaches l/K2G2. It is further shown that the overall transfer function K5G5 reduces to l/bp and Ao/Ai to l/(l+bp). Thus by combining the Equivalent Circuit of the velocity transfer function of a servomechanism with an amplifier, it is possible to obtain Page 3

AE RONAUTICAL RESECARCH: CENTER-. -UNIVERSITY OF MICHIG(AN Report No:UMM-24l a stabilization- network.which -tends to reduce: both — the transfer.functionl of the compensated servomechanism to l/bp. and the output-to-input ratio to 1/(1 + bp). If the output-to-input ratio Ao/Ai of'-the-compensated servomechanism of Figure 4 is considered to be a transfer function K6G6? it.i possible..to design an equivalent network: for- this transfer function. This.:equivalent.: network may be obtained by: comparing K6G6:to a: singleRC stage.in;a.manner similar to that demonstrated'in:.Figure.. This single stage equivalent RC network can be used as the feedback branch of a Stabilization Feedback Amplifier and:the:resulting feedback amplifier cascaded with K6G6, as: shown in Figure 5.::If this is:, done,.:.the expressions of Figure 5 show that- K 7,:;the transfer function:of this:.tabilization Feedback Amplifier, approaches 1- + bp, and the.oyerall' ransfer function KOGO of. the twice-compensated: servomechanism approaches:.: Therefore, in the limit, Ao becomes directly proportional to Ai In the ex'amples of Figures'4 and:5, -it wasasssumed:that the.transfer function K6-6 approached 1/(l:+ bp); and:that, therefore,- theR C, product'of the equivalent circuit of K6G6 should equa: b.: In:general,it:miay not:be,: possible to reduce K G5 to exactly l/bp, and therefore''6G6wil::be sligtly. more complex than l/(l + bp). Consequently,when comparing the equivalent BRC network to K6G6, a value of R should be,chosen which tends "to minimize:.the width of the ellipse over the frequency range of interest. Figures 4 and 5 show that it: is'possible to::enhance- the stabilityofa servomechanism by:first cascading the open-cycle: servomechanism with a- Stabilization Feedback Amplifier containing the.equivadlent cit..' of its tran sfer function, and then by closing the servo feedback loop around these two cascaded components,. This procedure may be repeated successively until.'the desired improvement in-the response of the servomechanism'has been'achi:eved.;: Some General Considerations In any servomechanism it may not be necessary to compensate for every time delay or its equivalent. Consequently, the Equivalent Circuit used in the Stabilization Feedback Amplifier need not.always contain all of the. terms of the complete velocity transfer function. In fact, it may be more. practical not only to reduce the number of terms in the equivalent transfer function, but also to approximate these terms. Page. 4

A EINATICAL RESEEARCH CENTER UNIVERSITY OF -MICHIGAN Report; No UMM-24Following -this philosophy, an. Equivalent.Circuit can be designed which is composed of only one or two RC networks or an L C R network. With the.setup of "Figure 3 ~.these.RC networks can be made equivalent for only those. frequencies where the- maximum phase shift and resonance in the servomecnanism.response:has -been observed.. Then an approximate Stabilization Feedback Amplifier. can -be constructed,and. the procedure of.Figure 4 followed. The:..remaining.,Uncompensated factors) which limit the response of the Servo-.mechanism.-.c.an:either be neglected. or removed,by another approximate Evuivalent Circuit and Stabilization Feedback Amplifier, as was done in Figure 5. The.nu.mber.. of.;cathode.followers used in the Equivalent Circuit may be recuced if:the.mpedance. of: one:RC network is several times greater than the impedance of the, preceding network, thus making it unnecessary to isolate these -.tw.o networks. In::'.some cases.the inertia of the d.c. tachometer of. Figure 3, might be c omparable,to the iinertia ofl.. the servo motor which would change -the effec - tive characteristics: of the servo motor. If this condition arises, the d.c. tachometer':s:hould- be..coupled to the motor through a gear reduction. This gearing.down, uualy. can be accomplished by coupling the d.c.. tachometer into the: output:gear.train at -some.convenient gear mesh. It is a worthwhile precaution, when combining the Equivalent Circuit with an amplifier, to form a Stabilization:Feedback Amplifier to use no more than two time constants, or their equivalent, in one feedback branch. Otherwise the possibility exists that the Stabilization Feedback Amplifier may oscillate,..If:there are:.more than two time constants, or their equivalent, in an:;Euivalent~ Circuit,..:then: several Stabilization Feedback Amplifiers- should' be.'us.ed.: —:Good results-: can be obtained. from al Stabilization Feedback'Amplifier with a;value of K3 of: Figure 4, of: only 7 to 10 for each time constant. The overall gain K4 of the Stabilization Feedback Amplifier can be made any -value desired by proper choice of the value of K2 of. the Equivalent Circuit. It is worth noting that the amplifier K3 does not have to be an additional amplifier, but may be a stage of K1.. Page 5

'AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No tUiM-24 Further Equivalent Circuits and Stabilization Feedback Amplifiers In many applications the output motion of the servo motor is restrained by a spring,-as in the case of a servo torque. motor actuating the pilot valve of a hydraulic system. Figure 6-A shows the electro-mechanical circuit of a servo motor restrained by a spring of stiffness K, and Figure 6-B shows the Equivalent Circuit of Figure 6-A. In this example the servomechanism amplifier stages are not shown, and the servo motor is assumed to operate on d.c. The output voltage E0 of the Equivalent Circuit may be made proportional to the output voltage E0 of the physical system in a manner similar to that demonstrated in Figure 3. In this case it is not possible to subdivide the Equivalent Circuit or the electro-mechanical circuit so that only one variable may be adjusted, or compared, at a time. This difficulty may be overcome by noting that at high frequencies the voltage Eo is primarily a function of the LC product, and that at low frequencies it is primarily a function of the RC product. Therefore, with the frequency of the variable frequency sinusoidal voltage source high, C should be adjusted until the width of the ellipse has been minimized. Then the frequency should be reduced until the width of the ellipse is quite large. At this frequency the value of R should be adjusted until the width of the ellipse has again been minimized. Some estimate of the approximate value of L and C may be obtained by visual observation of the natural frequency of the unexcited servo motor and spring. So far,the servomechanisms described have been d.c. servomechanisms. Figure 7 shows part of a 2-phase induction motor servemechanism and its electro-mechanical equivalent. The amplifier components have not been considered because, in general, straightforward amplification stages act upon the side bands of the modulated suppressed carrier signals and therefore produce only very small time delays. On the other hand, the servo motor essentially demodulates the modulated suppressed carrier signal and acts only upon the envelope or modulating signalsand,thus may produce large' time delays. Figure 8 is a schematic of the Equivalent Circuit of the 2-phase induction servo motor of Figure 7. As will be noted from the analytical expressions of Figure 8, the equivalence of this Circuit holds only over a limited region of the modulating signal frequencies. This situation is generally true of any a.c. network containing only linear elements. Page 6

AERON AUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM-24 The parameters of this Equivalent Circuit may be determined by the procedure demonstrated in Figure 3. As there are two variables, L and Rs, to be adjusted -together, tuning the LO components so that they resonate at the carrier frequency leaves only R to be adjusted. As mentioned before, the a.c. servo motor demodulates the modulated suppressed carrier signal, and,therefore,~it is difficult with linear elements to synthesize a network which is the equivalent of the a.c. servo motor. This factor makes it quite difficult to obtain as much stabilization from linear a.c. networks as from linear d.c. networks. It is,therefore, suggested that a small 2-phase motor, directly coupled to a small a.c. rate generator, may'be used as the Equivalent Circuit for any a.c. servomechanism. This small 2-phase motor and generator combination can then be used as the feedback branch of a Stabilization Feedback Amplifier, and the resulting amplifier will serve as a stabilization network for 2-phase servomechanisms. Figure 9 demonstrates the method by which this may be accomplished. It may be noted that if an additional inertia is coupled to the combination shaft of the small servo motor and rate generator, the time constant of the feedback path of the Stabilization Feedback Amplifier may be made large enough to compensate for more than one a.c. time delay or servo motor. The A.C. Amplifier No. 2 of Figure 9 does not need to be a separate amplifier, but may be a stage from either K3 or A.C. Amplifier No. 3. In review, the response of a servomechanism can be improved with the aid of an approximate Stabilization Feedback Amplifier as shoiwrn in Figure 4. Furthermore if the response is not sufficiently improved after the use of a single stabilization network, additional networks may be added. The designr of these additional networks usually is simplified by the use of a minor feedback loop around the first stabilization network and unimproved servomechanism in cascade. This is demonstrated 1in Figure 5. -_, I~~~~~ Page 7

AERONAUTICAL. RESEAR'CH CENTER -UNIVERSITY OF MICH-IGAN Report No..UMM-24.REERENCES (1) "Application of Circuit Theory to Design of Servomechanisms" By A.C. Hall, Journal of Franklin Institute. T-242, P-279. (2) "Electrical Analogy Method Applied to Servomechanism Problems"' By G.D. McConn, S.W. Herwald, H.S. Kirschbaum (Translation ATEE V-65, P-91, 1946).Pae 8

AE-RONAUTICAL RESEECARCH CENTER - UNIVERSITY: OF MICHIGAN Report No. TM — 24 SERVO MOTOR A. ~ GENERATOR PA FIG. Io GENERATOR FIG, lb IFIERd if R. Jrrm MOMENT OF INERTIA OF MOTOR I | E ki T mEQUIV. DAMPING COEFF.OF MOTOR. - 4 R = RESISTANCE OF GENERATOR AND Ra MOTOR ARMATURES, a 1+ La p L = INDUCTANCE OF GENERATOR AND p a R MOTOR ARMATURES. Lf INDUCTANCE OF GENERATOR Tm = k ia FIELD. Tm~ = Jm p)^AMPLIFIER AND FIELD. mk,,k,,&k CONSTANTS OF PROPORTIONA LITY. _, 2c 13. K& K; = REAL GAIN FACTORS OF pAo - RT R; TRANSFER FUN<T1ONS AND _ _L__ L J ARE NOT A FUNCTION OF IE fi- 1 1-pIG, FREQUENCY E FREQUENCY iK -G 1 p- E = TRANSFER FUNCTIAOON M O TORAND K +Ra = R ESINTANCE OF GENERATOR. -.AND K, G,. -p K; G;, FIG. I.BG~Nsze1 L.2 EMPa' a~ w R'a-gO MOTOR ARMATURES. Lf'INDUCTANCE OF' GENERATOR P,,'e 9 TIONALITY. k,.,.

AERONAUTICAL RESEOARCH CENTER - UNIVERSITY' OF MICHIGAN Report No. -UMM- 24 R, ) R2 ~,/R3 Era - 2 -- To E E aE, E aE2' I+RC,ip 1+ R2Cp E= I I+R3C3p Eo = I a a Kz G E I + RC, I + R2p I+ RC3p 2 HERE THE SOURCE IMPEDANCE OF THE CATHODE FOLLOWERS HAS BEEN ASSUMED TO BE ZERO. FIG. 2 Page 10

GENERATOR rSERVO MOTOR AMPLIFIER TACHOMETER iISOURCE Lq -o VARIAB ILE E " )I' 0 FIGI.~I II FIG. 3.

,-AERON AUTIGCAL, RES EACRCH -CENTER-UNIVER S ITY OF MI CHIGAN Report No. TMM- 24 K5 G5 E KK "'q4 K2G2 4 GK4 G+ K23KG2G2 IF' K Kz Gz >G>1 K G KG KK4 G K G5K -z lA I+K.G, +bp6 FIG. 4 FINALLY IF K Pae 12K, GI ~.~........=....~.aKGo

AERONAUTICAL" RESE.:ARCH CENTER - UNIVERSITY OF,MICHIGAN Report No. UM' - 24 K Go. A K FROM FIG. 4 K? G7 = K7 K6G I+ K6 G6 - -, 1+pR+C K8 IF l> I THEN, KoGo=KG K8 G66I+ pRC K - I+ pRO A- FI A I+ KoGo 2 AND IF RC =b K7 G7F I+sb p FIG. 5 Page 13

AERONAUTICAL RESE:ARCH CENTER -UNIVERSITY OF MICHIGAN Report No. h4M 24 FIG 6a Tm kmia = (JmPttmP K) Ao WHERE K Jm km Ao km Ao Jm ia J Pt fmP + K -a [m pIT, 2] + j I ia L R E, C Eo FIG. 6 b Eo I Eo C.)nt' - 3,,.)' + j E, LCp2+ RCp+ I E, (W )J WHERE C.) n LC FIG. 6.age 14

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No. UMM 24 LINE 0 2 PHASE INDUCTION SERVO MOTOR E FIG 7a, EQUIV, <a 15 E SERVO - MOTOR f r ~-.1 _. I-N E0 FIG. 7b Tm = K3 E ( I + P ) fmp Ao E = KG' WHERE' = dt E I ~~fm ~Jm = MOMENT OF INERTIA OF MOTOR. fm= EQUIV DAMPING COEFF. OF MOTOR. FIG. 7._.______ _i Page 15

AERONIAUTICAL RESEiARCH CENTER- UNIVERSITY OF MICHIGAN Report No. UMM 24 C L R2 E oR Eo Rs- SELF RESISTANCE OF INDUCTANCE L Eo a RCp ON A FREQUENCY BASIS: E LCp2+ RCp+ I.P E ja R:C O WHERE R a R - RTRZ E (- 2L)+ j RC IF LC = % C C (c,= CARRIER FREQUENCY E- e sin Cst sinCct - e' [cos(cC +Ws) t+cos(C, -)s t ] THEN CA)oc + (AS EO ja RCW Jo R~CC(CL -G)~ c O( - I, F. IF (As << ('~c E a (Ca)sa THIS APPROXIMATE RELATION HOLDS ONLY AS LONG AS FIG 8 Page 16

LINE | K G, K4G4 0f iHA. G. A' 0. AMPLIFIER TII AMPLIFIER,3 0 A 2 PHASE INDUCTION A.C. RATE GENERATOR SERVO MOTOR.-ADDITIONAL INERTIA K CD L 0 AMPLIFIER2 t G \ S. s | 2 PHASE INDUCTION SERVO MOTOR K2 G2 FIG 9 FIG. 9 0 ~'~l 2PHASE INDUOTIO I~~~~~~~~

UNIVERSITY OF MICHIGAN 3 9015 03125 9008