THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE DESIGN OF POSITION AND VELOCITY SERVOS FOR MULTIPLYING AND FUNCTION GENERATION Edward 0. Gilbert March, 1959 IP-362

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TABLE OF CONTENTS Page I. INTRODUCTION............................................. 1 II. COMPONENTS.......... 1.......... 1 Potentiometers...................... 2 Gear Train.............................. 2 Tachometer.................................... 3 Amplifier.................................. 3 Two-Phase Servomotor................... 6 III. 1HE POSITION SERVO................................ 8 Block Diagram............................. 8 Nonlinear Performance Requirements...................... 10 Selection of the Gear Reduction Ratio.............. 12 Compensation of the Linear System..................... 14 IV. THE VELOCITY SERVO................................ 25 Block Diagram.......................... 25 System Design............... e.,....0...0......** 25 FOOTNOTES..... o.............. 32 ii

LIST OF FIGURES Figure Page 1 Typical ac Amplifier Gain Characteristic........ 5 2 Two-Phase Servomotor Speed-Torque Curves.....a...a 7 3 Position Servo Block Diagram..,......... 9 4 Open-Loop Gain for Position System with Error Rate Compensation When 25/Tm < 1/Ta........eoo..0000 16 5 Open-Loop Gain for Position System with Error Rate Compensation When 1/Tm < l/Ta < 5/Tm.oo.... 20 6 Open and Closed-Loop Gain for Position System With Tachometer Compensation...........,,,,,,..0..o...0 23 7 Velocity Servo Block Diagram..............o...... 26 8 Equivalent Block Diagram for Velocity Servoo..... 27 iii

I. INTRODUCTION The use of servo positioned potentiometers for multiplication and function generation in analog computation and simulation is wide-spread. Typically, the potentiometers are driven by a two-phase servo-motor through a gear train. The motor in turn is driven by an ac amplifier that receives a compensated error signal in modulated form from a synchronous chopper or other modulated waveform source. The performance of such a system is specified by such factors as bandwidth, static resolution, and velocity and acceleration limits. The design problem is to select suitable components, such as motor, gear train, and potentiometers, so that performance specifications may be realized or optimized. The particular difficulty of design is that these performance requirements are interrelated, and it is not generally possible to specify arbitrarily all the performance characteristics. It is the purpose of this paper to express the interrelationships so that compatible requirements may be understood and a successful design accomplished. II. COMPONENTS There are a number of component characteristics that are significant in design. Many of these characteristics are primarily determined by factors such as accuracy, reliability, availability, economy, and the -1

-2state of the art. For example, a high resolution, high linearity requirement would specify multiple turn, wire-wound potentiometers. This in turn would set quite definite values for the potentiometer inertia and frictional torque. In other cases} less stringent requirements would allow a more flexible selection of components, and hence, a wider range of component characteristics. Individual components will now be discussed. Potentiometers Important characteristics for servo design are: Tp = total potentiometer frictional torque. Ip = total potentiometer inertia including potentiometer coupling. ~]= resolution of the reference potentiometer. = number of potentiometer wires in full scale. np = number of shaft revolutions in full scale. For film potentiometers,' is essentially infinite. Desirable characteristics are low Tp and high r, which have rather definite limits set by potentiometer sizes resistance, accuracy and np. As will be seen, Ip is not usually significant in design. Gear Train Important gear train characteristics are: ng = gear reduction ratio between motor shaft and reference potentiometer. Ig = gear inertia referred to the motor shaft. Tg = maximum gear train frictional torque referred to the motor shaft. = gear backlash at the poteniometer shaft expressed as a fraction of full scale potentiometer travel

-3The gear reduction ratio is an important parameter fixed by design. The inertia Ig is a function of ng but may be considered independent of ng over a limited range which is sufficient for design considerations. The gear train precision is specified by Tg and ~. Gear friction is a rather erratic function of gear position, and Tg should be small relative to potentiometer and tachometer friction referred to the motor shaft in order to insure jerk-free operation. The backlash G should be considerably less than 6, the static resolution error expressed as a fraction of full scale. 6 is discussed more fully at a later points When some of the potentiometers are driven through gearing from other potentiometers, the reference potentiometer should be the one most directly geared to the motor, minimizing backlash in the closed-loop. The gear backlash between all potentiometers should be considerably less than 1/ so that wire-to-wire oscillation of the reference potentiometer is not possible without the frictional torque load.of the other potentiometers. Tachometer A tachometer may be used for damping or as the velocity reference in a velocity servo. The tachometer is coupled directly to the motor shaft to eliminate gear backlash in the velocity feedback loop. Important characteristics are: Tt = tachometer frictional torque. It = tachometer inertia including coupling. Kt = voltage constant in v./rpm. Amplifier Important characteristics are: K = linear region static gain in v./full scale error.

-4es = saturation voltage of amplifier. Ya(S) = equivalent transfer function of amplifier in linear range, where s is the Laplace transform variable. The required amplifier gain K will be determined by design. The amplifier should have sufficient power capability so that the saturation voltage es approximately equals the maximum motor voltage emax. The existence and determination of the transfer function Ya(s) is open to question. This is apparent from the ac amplifier gain IYa(ja)j shown in Fig. 1. Here wo is the carrier frequency and aot + amn and Mo - m are the side band frequencies corresponding to a sinusoidal modulation frequency of can. For Ya to exist, Ya(+ji% + jo) Y(+ jW). This is certainly not true for C > 1l/Ta due to the nonsymmetrical gain characteristic about the frequency cbo. The exact effect of the nonsymmetric gain, including the demodulation characteristics of the motor, would be a worthy subject of analysis. For the design problem considered here) the equivalent transfer function Y(s) K (1) Yas+l has proven adequate. l/Ta is taken as the lower break frequency of the ac amplifier. Obviously, l/Ta <coo, and as 1/Ta approaches co, the transfer function given by (1) becomes less valid, and additional time lags should be considered. In some servos, a magnetic amplifier provides modulation and amplification. Typically, additional gain is provided by a dc amplifier. The above transfer function is still valid where Ta is now the time

o STWCq tD TDf +~y - tmTdw _ t6t II / 1 L Nt *l'1 I I. l I I'~!,.. u!oB JBa!J!ldwD 3D

-6constant of the control winding. Magnetic amplifiers commonly suffer two disadvantages: 1. the bandwidth l/Ta is considerably less thanoo, and 2. the dynamic characteristics are poor in the saturated state. The bandwidth may frequently be improved using current feedback in the control winding. The poor saturated amplifier dynamics are characterized by long recovery times from the saturated states which means that the full linear bandwidth is not usable. Such poor dynamics are minimized by good amplifier design or by introducing saturation at a point before the magnetic amplifier. Two-Phase Servomotor The important servomotor characteristics are obtained from the speed-torque curves for the motor. A typical family of such curves is shown in Fig. 2.in dimensionless form, where T is the motor torque, Gm is the motor angular velocity in rad.y sec., and e is the rms voltage applied to the variable phase. -.m max and Tmax are the maximum velocity and torque when e =emax, the maximum motor voltage. The family defines a torque function T = f(Gm e). Neglecting frictional torques IGm = f(6m e), (2) where I = I+ g + It + I p/ng2 Im is the motor inertia. This nonlinear differential equation may be linearized when Gml Om maxi which is frequently the case when the servo amplifier operates in the linear range, i.e.! when |el < es < e max * Then (2) becomes;m max emax

7-O?... 0 ~ ~~~~~~~~~~~~~o l~ Ibm

where 7, generally less than one in magnitude, is the slope parameter of the speed-torque curves defined in Fig. 2. Equation (3) defines the motor transfer function = Km (4) where the constants Km and Tm are expressed in terms of the parameters of the speed-torque curve for e = emax by Km 1 Qm mxm ax Nm ax d Y emax 607 emax' vsec. Tm 1 Q max I _ 2 Nm sec., (5) Tma 60 y Tmax Km = K - Tmax rad.. Tm I emaX v.sec, Nmax is the maximum motor velocity in rpm, and- K is the motor gain constant for frequencies above the break frequency l/Tm. Expression of the transfer function parameters in the above form is important since it allows design equations to be expressed in terms of readily available speed-torque curve parameters. III. THE POSITION SERVO Block Diagram Fig. 3 shows the block diagram for the position servo. The dimensionless input x and output y are selected so that the full scale range on x and y is unity. The input compensation Yi(s) and the feedback compensation Yf(s) are defined to have unit magnitude at zero frequeney, i.e., Yi(o) = Yf(o) = 1. For series compensation, Yi(s) = Yf(s).

AMPLIFIER AND COMPENSATION GEAR r- -- - ---- -' MOTOR TRAIN POTENTIOMETER I I 1 4 rx I K i Km 1 i I T:S+ I S(TS+ I) 8 2 I I 1 1 1 Itg I S L_ - Figure 3. APosition Servo Blcmk Diagram

-10The nonlinear characteristics of the amplifier, motors gear train, and potentiometer are not shown. Nonlinear Performance Requirements A number of the performance requirements are nonlinear in nature and are particularly important in the selection of a motor and gear train. In this section, these requirements will be defined in terms of the component parameters. Later$ their full significance in design will be considered. Two requirements are the velocity and acceleration limits Ymax and Ymax. From the blo6k diagram and definition of motor parameters Y Nmax (6) Ymax = 60 ng np (6) and Tmax Ymax = 2 n np I (7) In (6) and (7), the frictional torque is assumed small compared with Tmax Another parameter that may be important is yo, the first overshoot magnitude that results from a large step input. Exact determination of yo is difficult, but yo may be estimated as follows. Assume that servo amplifier inputs not causing saturation are small compared to yo, that the torque reverses to full negative value when c = x - y — 0, that the deceleration to zero velocity is constant at Ymax, and that velocity when C = 0 is Ymax. Then 2 Ymax Yo Ymax. For an actual system, deceleration (proportional to torque in the negative velocity range in Flg.2 )

-11may be somewhat greater than Ymax, which occurs for T = Tmax also' for typical damping characteristics and linear range, torque reversal is earlier. Thus a somewhat smaller yo than predicted resuit s and.2 ( )2 Ymax _( yo < m()I (8) 2 Ymax ng np Tmax Determination of the magnitude and number of successive overshoots for a large step input is difficult unless extreme simplifications are made. However, if yo is less than several times the amplifier input causing saturation, few excursions result with reasonable system damping. For a given motors yo is decreased by increasing ng or np. Considering I to be independent of ng (a reasonable approximation, typically), this means lower Ymax and Ymax Since frictional torque exists, there is a static error corresponding to motor torques less than the frictional torque referred to the motor shaft. To reduce this error to allowable values, a minimum gain is required. Thus, if b is the allowed static resolution error, Tmax Tp emax K TT + ng where TT = Tg + Tt, the frictional torques acting at the motor shaft. Solving as an equality yields the minimum gain for a specified static resolution T Kmin T + emax ( 6 Tmax

-12When tachometer damping is not used, TT is small and the approximation XIin IL. emax TT << (11) Kmin ng Tmax n is valid. It will be seen that K is limited by amplifier bandwidth, and therefore increasing ng is the only way of meeting a high resolution requirement for given potentiometers and motor. Selection of the Gear Reduction Ratio Evaluation of the above performance figures requires the gear reduction ratio. This ratio should be selected to optimize the performance in some sense.1 Here, some of the different factors will be considered and a minimum gear reduction ratio will be determined. Frequently, the gear ratio is chosen to maximize the acceleration Ymax subject to the torque limitation Tmax. If the frictional torques are neglected, this leads to the familiar result ng=p, (12) im + It + Ig Often, this yields a small ng. In fact, ng may be such that the motor torque Tma is less than TT + Tp/ng2, the frictional torque, so that the assumption of negligible frictional torques is certainly not valid. Neglecting TT but including Tp gives maximum Ymax when n- =P + m / )2 + It (13)

-13In many cases the inertia ratio in (13) is small compared with the squared torque ratio and ng9 2 Tp/Tmax. Thus the maximum motor torque is approximately twice the frictional torque at the motor shaft. For smooth tracking of a slowly varying input, the frictional torque must be small compared to the available motor torque. Since smooth tracking is important in accurate analog computation, the gear ratio should be selected accordingly. Experience has shown that maximum motor torque should exceed the frictional torque by a factor of at least five. Thus Tmax > 5 (TT + ) (14) ng or ng > Tmax (15) - TT When tachometer damping is used, TT is small and (15) becomes T T ng > 5 T TT <<. (16) max ng Inequality (15) or (16) frequently sets a minimum gear ratio that is considerably greater than the gear ratio for maximum acceleration, in which case the potentiometer inertia Ip is not significant in design. If Ymax Ymax,, and good tracking characteristics were the only performance figures, (12), (13), (15) or (16) would set the minimum

-14gear ratio for the constraint of a given motor and potentiometer. Howevery requirements on yo or static resolution may require an increased minimum gear ratio. Thus (8) bounding Yo may require higher ng to obtain an acceptable value of Yo. Similarly, specified Kmin and 6 may require higher ng to satisfy (11). Linear analysis will show that Kmin is indeed limited by the servo amplifier bandwidth. With the above limitations in mind. a minimum gear ratio may be determined for specified motor and potentiometers. If the requirements for velocity and acceleration limits are not met with this ng, a motor with greater maximum torque and/or velocity must be selected, and a new set of performance figures must be computed. Compensation of the Linear System To assure smooth tracking, the gear ratio has been selected so that motor torque exceeds frictional torque by a factor of at least five. This means that there is at least a range of five-to-one where motor torque can exceed frictional torque and be proportional to amplifier input. In this range, system design on a linear basis will yield reasonable results. In this section different compensation techniques to achieve specified linear characteristics such as bandwidth, damping, and steady-state errors will be considered. In certain instances, fundamental limitations on the characteristics exist and may influence the selection of components and the gear reduction ratio. These limitations will be apparent from expressions relating the component parameters and linear and nonlinear performance figures. First, consider series compensation where Yi = Yf is the series compensating function. The closed-loop transfer function Y(s) relating

-15the Laplace transform of y to that of x is then Y(s) =Yi Yo Y- - (17) Yf 1 + YO 1 + Yo where Yo(s)= Yf K Km (18) 2t ngnp s(Tms+l)(Tas+l) For simplicity, perfect error rate or lead compensation, Yf =Yi = 1 + Ces, (19) will be used. Imperfect and physically realizable compensation yields similar results. Three cases will be analyzed. The first and most common occurs when 25/Tm < 1/Ta. The second and third cases are defined by 5/Tm < 1/Ta < 25/Tm and 1/Tm < 1/Ta < 5/Tm. Analysis will be based on plots of db. magnitude of Yo/K vs. log XU with compensation selected so that Yo has a slope of approximately -9 db./octave when IYol = 1 This will give a phase margin of about 45~ and hence a reasonable resonant peak in the closed-loop transfer function magnitude. The peaking frequency Lp will be estimated as the frequency where IYol = 1. The accuracy of this procedure will be sufficient to determine the required interrelationships between system parameters. Fig. 4 shows the first case where Ce has been selected to maximize K. Here 1 1 1 Ce =5 Ta 5 Ta (2)!> 5 since > 25 Ce Tm Ta Tm

-16-6 db/oct \. 20 Log KI12 WC I B h~~~~og wa 20 Log8(COuec) -- — ~ --- I \ II~~~ j12 S | \8~~~~~~~~~~~~~~~~~~~ Consea8o mke~ \57 <0 1Q' 8. t 20Lg( ke)............ -. \ B'a Figsure 4. Open-Loop Gain. for PositilonSysteg.withP Error Rate Compenstionl When 25/Tm 6 1/r~

-17Since (p = 3/Ce, it is apparent from Fig. 4 that 1 -(1 (C)2 (21) where Km K, c= /2 it ng np m 2C ng np Substituting (22) and (2-0) into (21) gives K = 3 2i ng np I emax (23) K=235(23) 25 Ta Tmax For given motor, potentiometers amplifier, and gear train, this is the maximum amplifier gain. If the above K value equals or exceeds Kmin determined by (10) or (11), then the specified resolution O is obtained. If K does not exceed Kmin, the resolution requirement cannot be met unless certain system parameters are changed. The interrelationship of these parameters is most easily seen by substituting in K > Kmin from (23) and. (10)leading to the inequality TT + - 1 > /\ 25 2T (24) Ta 6x ng np IT or 1 > 1 / T T << (25) Ta ng /n 6 np I ng Tese - inequalities are useful in selecting components that are compatible with specified static resolution. For example, if potentiometers, motor,

-18and amplifier are given, (24) determines a minimum ng for specified 6. This ng may very well exceed that given in (13) or (15). The advantage of high amplifier bandwidth and low-frictional torques is clear. It is also clear that nonlinear characteristics cannot be divorced from the linear system design. The above discussion concerns the limitations placed on gain and static resolution by the compensation restrictions imposed by amplifier bandwidth; the converse is sometimes true. For example, suppose 2 6 equals the resolution range 1/q of the reference potentiometer; certainly, 2 6 cannot be less than the wire-to-wire jump 1/ri. In this case, Kmin also becomes a maximum K = Kmax since larger K could result in wire-to-wire hunting, causing excessive potentiometer wear., That is to say, an -error corresponding to one-half of a potentiometer wire can produce torque exceeding frictional torque when K is greater than Kmin = Kmax. When Kmax is less than that given by (23), the compensation of Fig. 4 must be changed. By increasing Ce over the value of (20), K' can be decreased while ctp is maintained near 1/Ta. In the above work, it was assumed that 25/Tm < 1/Ta. For relatively low speed, high torque to inertia motors with TY 1, Tm given by (5) is small and the inequality may not be valid requiring a modification of the results. For 5/Tm < 1/Ta < 25/Tm, Fig. 4 is still correct in form, but lp and K may be increased somewhat because of the additional phase lead contributed by the motor transfer function. Suppose, for example, that 5/Tm~ 1/Ta; then K may be increased so that cp 1/Ta and the three is replaced by five in (21) and (23). In (24) and (25), the factor 25/6 is replaced by 5/2.

-19When 1/Tm < 1/Ta < 5/Tm, a different compensation must be used to maximize K and (p. This is shown in Fig. 5 where 1 1 Ce Tm (26) 1= Ta and Ta 1 60 e (27) K ng np ex (27) Km Ta Nmax max 2i ng np Again using K > Kmin} it is found that T 1, 1 Nmax TT 1 (28) Ta ng np 60y Tmax 6 or ~' ~ 1 Nmax Tp 1 1 1 Nmax T TT << - (29) Ta n np 60y Tmax 6 ng An important figure of merit in linear system design is the steady state error for ramp inputs. In determining this error, it is useful to take the following more general, physically realizable forms for Yi and Yf: Ces Yi = 1 + alCes+l (30) Yf = 1 +.Cd.;o12Cds+l For series compensation, Cd = Ce and OC2 = c1 ~ Using the final value theorem gives the linear-system steady-state error

- 2020 Log | YO| | 1 6 dWOCO. 20 Log K'~ T-~ ~ _ 2 noW n\ I 5t12 Figure 5, Open-Loop Gain for Position System with Error Rate Compensation when 1/Tm < 1/Ta < 5/Tm

-212 [ ng np + C( 2) Ce( ) V c= ~...... (31) K 1Km for the input x = Vt. For series compensation the last two terms cancel. In certain input circuity, Yi and Yf can be adjusted separately and Ce and al may be selected to make ce = 0. This does not change the pre, vious results on series compensation since Yi and Yf will differ little for Ey =0 Zero steady-state error for the ramp input is equivalent to zero slope of phase vs. frequency at zero frequency in the closed-loop system. This characteristic is particularly desirable in servos used for analog computation. However$ in many cases the first term of (31) is small and cancellation is not warranted. In addition to the linear-system error, there is the nonlinear steady-state error, v n |V|~~~~~ ~~(32) = - I for the input x = Vt. Thus an error equal to 5 must exist before motor torque is available to exceed frictional torque. Since this error is independent of IVI, it cannot be cancelled out for all V. Derivative control through tachometer feedback is another commonly used form of compensation. In this case Yi = 1 (33) TS Yf = 1+ Cd S where the high pass filter Ts/(ts +1) in the tachometer path is used to

-22eliminate the steady-state error for ramp inputs [the second term in (31)]. For T > Cd (T ~ Cd usually gives acceptable damping) the approximation (34) Y (jow) 1 + Cdja,L > Cd is useful since Y(jw) then becomes Yi Yo 1 Yo 1 Y(jD) =y 1 + Y 1 + Cd> 1 (35+ Y C The factor 1/(t + CdjC) gives increasing attenuation for Lc > —/Cd so that the IY(jc) l may not exhibit great peaking even though IYol/1l + YoI does become large. This is apparent from Fig, 6 where the plots of db. gain vs. log 0 for Yo /K and Y are shown for a value of K much greater than that given by (23). The phase margin for Yo is now very small resulting in a large peaking in IYol/ll + Yo at Xc = 0p. However, IYI does not greatly exceed one at cu = ap, because of the factor 1/(1 + Cdjo). Evidently, the derivative control allows greater gain than series compensation. The useful bandwidth is 1/Cd, The gain advantage of derivative control would indicate that static resolution was not limited by amplifier bandwidth for this form of compensation. Unfortunately, large increase of gain is not usually possible because of additional time lags neglected in the analysis. Such lags could reduce the phase margin to a negative value resulting in an unstable system. The maximum increase in K through derivative control can be obtained by a more complete analysis considering the additional time lags and using an inverse Nyquist plot. It is difficult to generalize on the possible gain increase due to the wide variation of

-23-6 d b/oct. 20 Log I r -- X - 20 Log - _ __ Log w 20 Log Y' Figure 6. Open and Closed-Loop Gain for Position System With Tachometer Compensation

-24neglected time lags, but a factor of two to ten above the gain for series compensation is not uncommon. The increase in K does not necessarily mean a corresponding decrease in the static resolution error. This is apparent from (10) since the added frictional torque from the tachometer requires increased K to maintain a given 6. In addition, the added tachometer inertia reduces the acceleration limit. The lower cost and great reliability of series RC compensation may also outweigh any advantage obtained with tachometer damping. Another means of increasing K above the value given by (23) is to use a series lead-lag compensating function Ci =lCi 1 + 1 + a2cd The second factor approximates the error rate control; the first factor approximates integral control and permits an increase in K by a factor of about l/a1 for correctly chosen Ci. Such compensation has two main disadvantages. First, the compensating network is more complicated and may require excessively large capacitor values for realization. Second, lag action introduces additional open-loop negative phase shift that may cause a nonlinear instability or long recovery time for inputs causing amplifier saturation. For these reasons, such compensation has limited use.

-25IV. THE VELOCITY SERVO Block Diagram Fig. 7 shows the block diagram for the velocity servo. A tachometer voltage proportional to the motor shaft velocity is the feedback so that shaft velocity is ideally proportional to input voltage. The tachometer is mounted on the motor shaft to eliminate gear backlash in the feedback. The dimensionless output y is again selected so that the full scale range on y is unity. The input and feedback compensation are defined to have unit gain at zero frequency, i.e., Yi(o) = Yf(o) = 1. Ki is chosen so that the output velocity y is proportional to the input voltage ei by the desired factor. For design purposes, it will be useful to consider the equivalent block diagram of Fig. 8 where m is a dimensionless input voltage and Ky (37) 60 ng np Kt System Design Much of the previous work applies for the velocity system. Thus (6) and (7) still give correct values for Ymax and Ymax. Yo no longer has significance. The same approach for gear reduction ratio determination is valid and (12), (13), or (15) gives the minimum value for ng. The velocity system has a velocity resolution error instead of a position error. That is, the velocity error e = u -ymust exceed a

GEAR TRAIN AMPLIFIER AND COMPENSATION AND r-....___A -_ —_ — MOTOR POTENTIOMETER: "1 i' 1 ei 1 ~s~i t ~:e~S~inS+]:...:. 2 ng np gy I 0 K7*yeast 1 1 K L god m=. —. - -', —-- -.aB, ~ igure 7: Velocity Bervo Block Diagram

-27E.. II I~~~~~~~~~ 1#1. X +,'''._._. B M a0, I E~~~~~~~~~~~~~~~~~~~~~~6 t i,~~~~~~~~~~~i 1~~~~~~~~~~~~~~~~~~~

-28certain value before the motor torque exceeds frictional torque. This velocity resolution error 5v is given by T TT + n 6v = (38) Tmax 60 emaxK K Kt iK must be large enough so that 5v is sufficiently small. It will be seen that there is no particular limit on K as there was previously. Since the velocity servo is a type zero system, it has a steadystate linear-system error for u = U, a constant. The error is given by U E 60 1 + K K K (39) U emax 60 Kt K Km Kt K Nmax UKt In addition there is a nonlinear resolution error U bv (40o) Typically, l enl >> IEl In fact when U = Ymax, En is usually larger than en. Thus design for specified 6v usually assures steadystate errors not greatly exceeding by. The compensation of the velocity servo is easier than the position system since the open-loop system no longer has a l/s term present. The main problem is to achieve sufficient gain to meet a requirement on 5v. Frequently this can be done with a simple series lead compensation network. The more complicated compensation given by (36)

may be necessary. In this case there is no stability problem for satu-. rating inputs because of the simpler open-loop characteristics. V. AN EXAMPLE To illustrate the above principles, a particular design will now be considered. The problem is to design a position servo with five multiplying potentiometers. Line frequency and the choice of a ten watt magnetic amplifier designate a 60 cps., two pole, 4 oz. in. servomotor. The potentiometer linearity requirement is 0.1% and the static resolution error is 0.05% of fu$l scale. For simplicity and reliability, a simple series RC lead compernsating network is to be used. Under these restrictions, the servo is to be designed for the maximum possible velocity and acceleration limits and bandwidth.. Examination of (6) and (7) shows that for a given motor, highest Ymax and Ymax are obtained for smallest np. This dictates single revolution potentiometers and an np = 1. Consistent with reliability, cost, and similar considerations, a six gang potentiometer is selected for minimum Tp and Q > 1000. Ganged 20,000 ohm, wirewound potentiometers with the following characteristics meet the specifications. Tp = 4 oz. in. = 2,08-10-2 lb. ft., p = 120 gm. cm.2 = 8.8.l0-4 slug ft.2, = 1800+ Ip includes the inertia of the potentiometer coupling. The high torque-to-inertia-ratio servomotor has Tmax = 4 oz. in. = 2.08.10'-2 lb. ft., Nmax = 3400 rpm.,

-30emax 1= 15 v.,5 =.8, Im= 5 gm. cm.2 = 3.66- 10-7 slug ft.2 The maximum power input per phase is 9 watts so the.magnetic amplifier has suffic.ient power capability. The gear train has Tg << 4 oz. in. and 5 << 1/T. For a fairly large range in ng, the gear train inertia referred to the motor shaft is Ig = 2 gm.cm.2 = 1.47.10'7 slug ft.2 Design for maximum acceleration, neglecting frictional torque, is given by (12) yielding ng = 4.1 Consideration of frictional torque using (13) gives ng = 5.3 The minimum Tmax to Tp/ng ratio of five requires ng > 5 Thus design for maximum acceleration considering frictional torque is possible, neglecting resolution limitations. To determine the effect of the- resolution requirement, it is first necessary to compare 1/Ta and 1/Tm. With current feedback, the magnetic amplifier has 1/Ta = 125 rad./sec. By (5) 1 = 91 1 17.1 TM 1 + 7 1ng and for ng > 5.3, l/m < lTa < 5/Tm Thus (29) is appropriate; substitution of the specified 6 = 5o10-4 and other parameters gives the inequality ng > 33.6

-31Hence the linear system design and resolution specification require a much larger ng than that for maximum acceleration. For ng < 33.6, it is not possible to achieve design resolution and acceptable stability. Setting ng = 35 gives 89.5 Ce = Tm = 8-'01 and by (27) K 7100 Computation of performance figures yields Ymax = 1.62 ymax = 182 Yo <.0072 For a ramp input x = Vt the steady state linear-system and nonlinear errors are given by (31) and (32)..e =.008 V V en=.0005 -Fv The linear-system error may be cancelled by designing the input circuity so Yi and Yf differ by an appropriate amount.

-32Footnote (1) The following authors have determined motor suitability and the gear reduction ratio considering ymax and Ymax requirements and frictional torque limitations but neglecting the other performance figures and their relation to linear system design. Harris, H.,. "A Comparison of Two Basic Servomechanism Types," A.I.E.E., Vol. 66,, Pt. II, pp. 83-93, 1947. Newton, G. C., Jr., "What Size Motor?," Machine Design, Vol. 22, N. 11, pp. 125-130, Noveriber.1950.

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