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CA.S.R.E.TECHNICAL NOTE

GX- 55-1

AN ELECTRONIC SERVO SIMULATOR

by

R. W. RENSON

Ii

This document is the Property ct Her Majesty's Government and Crowncopyright Is reserved. Requests for permission to disclose Its contentsoutsde official cirrl( should be addressed to the !ssuing Authority.

1"733792

ADMIRALTY SIGNAL AND RADAR ESTABLISHMENTPORTSDOWN, COSHAM,PORTSMOUTH, HANTS.

AINh AITAGHE. LONu ,,,.

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(mhE VIEA'S EXPRESSED HEREIN DO 14OT NECESSARILY REPRESENTTHE CONSIDERED OPINION OF THE ESTABLIMENT)

A.S.R.E. TECHNICAL NOTE GX-55-1DATE: 14TH APRIL, 1955

AN ELECTRONIC SERVO SIMULATOR

by

R. VTI. REITSCN

Apprcvpd by V11. D. MallinsonHead of Division

SUWMLfIRY;

The construction of a simple electronic servo simulatoris described and some of the possible applications are indi-cated.

Admiralty Signal and Radar EstablishmentPortsdown, Cosham,Portsmouth, Hants.

P.T. . for Initial Distribution and References to Illustratione

Page 2

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Page S

A.S.R.E. TECHNICAL NOTE GX-65-1

LIST OF CONTENTSPage

1 . INTRODUCTION ... ..... . . .. ............. 5

2. THE EQUIPMENT 5

2.1 General ......................... .. 5

2.2 The Standard Block 6

2.3 The Input Circuits ....... ...... 6

2.4 Limiting Circuit 7

2.5 Backlash Circuit ................... 7

2.6 Low Speed Oscilloscope 7

2.7 R.iMi.S. Computer ...................... 8

2.8 Noise Filter 8

3. OPERATICN OF THE SIMULATOR .................. 8

3.1 General 8

3.2 To Set Up a given Transfer Function ,.. 8

3.3 Introduction of Delays and Simple Non-Linearities into the Resetting Loop 10

3.4 Synthesis of a given Electro-mechanicalSystem ............................... 11

4. APPLICATIONS 12

4.1 General ..6............................ 12

4.2 Use of Linear Transfer Functions 12

4.3 Simple Non-Linearities ............... 13

4.4 Simulation of Electro-mechanical Systems 14

5. CONCLUSIONS .................... ............ 16

6. ACKNOWLEDGMENTS 16

7. REFERENCES .......................... ...... 16

APPENDIX Simple Operations using Feed-back Amplifiers 17

Page 5

A.S.R.E. TECHNICAL NOTE GX-55-1

AN ELECTRONIC SERVO SIMULATOR

1. INTRODUCTION

A modified version of an electronic serve simulator,first described in a paper by Williams and Ritson, (Ref. 1). wasbuilt in the laboratory to investigate the servo parameters ofan auto-tracking radar. It proved such a useful instrument thata better engineered version has been built. This note willdescribe the equipment and indicate some of the possible uses.

The simulator is basically a special purpose analoguecomputer containing a number of variable elements. These repro-duce electronically the characteristics of the system to beinvestigated. It is emphasised that the simulator now describedhas been designed for the investigation of closed-loop problems.It would not be accurate enough for the different techniques usedin open-loop computation.

The basic element is a high-gain D.C. amplifier whichhas been carefully designed to give a stable performance. By theuse of selected passive networks in the input and feedback cir-cuits of this amplifier, basic operations can be performed. Thesimple theory of these circuits is included as an Appendix tothis note.

There are twenty-four standard blocks in the simulatorwith which quite complicated loops can be built up. Intercon-nection and feedback component selection is simple and relativelyquick.

To simulate practical problems, facilities are availablefor introducing limits to the velocity and acceleration of thesimulated output member. Backlash and resilience effects can bestudied.

The most useful input has been found to be a positionstep, but other step-function inputs can be used, e.g. a step-function of velocity. Alternative inputs are an S.H.M. of fre-quency between 0.05 c.p.s. and 100 c.p.s. and an input consistingof "noise", with an approximately flat spectrum from 0.1 c.p.a.to 10 c.p.s.

A low-speed oscilloscope is included as part of theequipment, although a pen recorder is often used to producepermanent records for comparison with the "real" system at alater date.

2. THE EQUIPMENT

2.1 General

The simulator is a self-contained unit and comprises:-

(a) The twenty-four standard blocks

(b) The input circuits

(c) Limiting and backlash circuits

(d) Low-speed oscilloscope

(e) R.M.S. computer and noise filter

(f) Stabilised power supplies

(g) Valve voltmeter for setting up circuits.

Page 6

.i equipment is mounted in two standard five foot PostIffice racks; the layout is shown in Fig. 1. Supplies of I ampof 24 volt D.C. and 4.5 amps of 180 volt 500 c.p.s. are required.

2.2 The Standard Block

A circuit diagram is shown in Pig. 2. The main ampli-fier, consisting of VI and V2, has a nominal open-loop gain of550. The variation in gain of twenty-four similar circuits didnot exceed + 10%. The saturation level is + 45 volts output andthe frequency response is flat within 3 dB To 7 kc/s. The res-ponse time is 1.4 milliseconds. Relay A is included for ease ofsetting up, i.e. any capacity in Z4 is discharged whilst the gridof VI is earthed.

The phase lag network (R, C) is necessary for stabilisa-tion. Since the natural frequency of the amplifier varies fromchassis to chassis, the value of R must be selected.

Amplifier V3 is a unity-gain stage by means of whichthe polarity of the output can be reversed when required. Whena capacity is used in the feedback position, it must be isolatedfrom the following circuit, unless this has a resistive input, byV3 or a standard block coupled as a unity gain stage.

There is provision for three summing components and onefeedback component. A range of the more usual values of components,selected to within + 1% of nominal value, are kept available forthe simulator.

2.3 The Input Circuits

(a) Step-functions

A circuit is shown in Fig. 3. A D.C. voltage step canbQ fed via the cathode-follower VI to the step-function relay onthe Low Speed Oscilloscope (see subsection 2.6). It is then fedto the simulator circuits under examination. The step-functionrelay can be switch operated if the oscilloscope is not beingused. (The output of the Sine Wave Generator and the Noise Sourceare also fed via this relay.)

Other forms of step-functions, e.g. exponentials, stepsof constant velocity, etc., can be obtained by feeding the D.C.voltage step to a standard block connected to give the requiredtime-function output.

(b) Sine-Wave Generator

This is a conventional phase-shift oscillator using afour-section R.C. network. The circuit is shown in Fig. 5.

The frequency range is 0.05 c.p.s. to 100 c.p.a. Theoutput amplitude can be varied from zero to 125 volts (peak topeak) by means of RV2. For one particular setting of RV2 theoutput amplitude will not vary by more than 5% over the fre-'quency range. The output zero is adjusted by means of RV1 andSWA.

(c) Noise Source

The circuit diagram is shown in Fig. 6. Noise is gene-rated in the thyratron V1 and heterodyned with a 3.3 kc/a squarewave in V3. The output is smoothed by a low-pass filter cf timeconstant 0.01 seconds and is amplified in V4. RV2 controls theoutput magnitude, which has a maximum value of 10 volts, andRVI controls the level. When a sharp cut-off frequency isrequired, the output is fed to the noise filter (see subsection2.8).

$Page 7

2.4 Limiting Ci.cai

Limit voltcs of up to 30 volts can be set into asimulated loop b; th "irciit 3hown in Fig. 4. The output of thecircuit is taken to t:L high il:q: dance limit point on the stan-dard block (See Fig. 2).

When setting up a limit voltage, the valve voltmetermust be connected between a monitor point on the limit chassisand the limit input on the standard block.

The grid base of the output cathode follower of thestandard block is 0.6 volts and this is the.eforc the least valueof voltage which can be used or the limit c-ircuit.

2.5 Backlash Circuit

It will be seen from Fig. 7 that a positive-going inputhas to overcome the bias on V2A, selected by RV2, before a changeoccurs at the output. Similarly, for a negative-going input, thebias on V2B must be overcome.

After, say, a positive-going input has reached a maximumand is reducing, the output will not change until the condenser Chas discharged through V2B. This will therefore simulate theeffect of backlash between th, motor and the load with startingfriction, but no inertia, on the load. The result of passing asawtooth wavoforrm through this circuit is shown in Fig. 8.

It is necessary to isolate the input and output of thiscircuit by cathode followers. To maintain unity amplification inthe reset loop, an amplification of 1.15 is used following thebacklash circuit. Because of the input cathode follower, specialcare must be exercised in setting the bias of the diodes to cor-respond to the vrltage scale of the resetting loop. RV1 and RV4are used to cancel the grid bases of VIA and B.

2.6 Low Speed Oscilloscopu

The circuit diagram is shown in Fig. 9. The principlefeatures are:-

(a) Time base speeds selected in three ranges. Each rangehas preselected calibration pips which can be switched off.

Range (i) 5 seconds to 1 second, 1 second calibrationpips.

Range (ii) 1 second to 0.2 second, 0.1 second cali-bration pips.

Range (iii) 0.2 second to 0.04 second, 0.01 secondcalibration pips.

(b) A step-function relay operated at a constant distancefrom the start of the sweep. The polarity of the step-functioncan be seltcted by a switch.

(c) The time base is normally repetitive, the time delaybeing 5 seconds on range (i) and 1 second on ranges (ii) and(iii). Single-stroke or external triggered operation can beused.

(d) The X and Y plates can be fed from D.C. amplifiersgiving a sensitivity of up to 50 mm/volt. Facilities areavailable for feeding the plates direct.

2Page 8

(e) A reference voltage supply is included for "backing off"when the input is not about earth potential.

() It is worth recalling that the relative phase of twosinusoidal voltages can be measured with the oscilloscope. Thevoltages are fed to the X and Y plates and adjusted separatelyto give equal deflection. The direction of the major axis ofthe ellipse will give the relative phase shift. It is sometimesmore convenient to compute the phase angle as tan - ' Y2/Yi whereY2 and Yi are quantities shown in Fig. 9(a).

2.7 R.M.S. Computer

The circuit diagram is shown in Fig. 10. The inputvoltage is scaled, squared, and integrated by normal circuittechniques. The output appears as a meter reading. The circuitis normally used in conjunction with a noise input. Integrationtimes of 10, 25, 50, and 100 seconds can be used and are measuredwith a stop watch. The circuit scaling is adjusted by SVIA. Theintegrator is reset by SWB when not in use. The mean square volt-age output is that indicated by MI multiplied by a scaling factor(1, 0.5, or 0.2) selected by SWC. The accuracy of the circuitis 5%.

2.8 Noise Filter

The filter has been designed to give unit magnification,1 1 dB, to all frequencies less than w after which the attenuation

is 30 dB/octave.

The circuit is shown in Fig. 11. The input noise is fedthrough three simple R.C. low-pass filters followed by a secondorder filter with a transfer function (see subsection 3.2) of

w /(p2 %2a +W )

The damping factor, a, of the second order is made 0.1 and thenatural frequency, , is 50 rad/sec. The time constants of eachof the R.C. filters equal 1/w.

When the input has a frequency equal to w, the outputvoltage from the low pass filters is -8 dB and, for increase infrequency, falls at the rate of 18'dB/octave. The second-orderfilter has a sharp peak of +8 dB at frequency w after which theoutput falls at 12 dB/octave. The sum of these filters providesthe required characteristic.

3. OPERATION OF THE SIMULATOR

3.1 General

The simulator can be set up to prcduce a particulartransfer function or can be used to synthesise a given systemfrom a knowledge of its components. Once the system has been setup, measurements are carried out to determine the transientresponse, frequency response, etc., using the equipment describedin section 2 above.

3.2 To Set Up a given Transfer Function

The transfer function is the differential equationrelating the output of the servo to the input. In low-powerservos operating in a linear regime, a simple transfer functiongenerally gives a very close approximation to the behaviour ofthe servo.

Page 9

Let 2 be the operator d/dt, € be the input quantity and0 the output. The three systems most commonly encountered havetransfer functions given by:-

8 bi

p 2ab+b§. _ ..a. .. (2

Sp2 +2abp+b

2

2 2 3E 2abP +cb p+b

Sp 3 +2abp2 +cb2 p+b3 ........

These represent servo systems having finite velocitylag, zero velocity lag and zero acceleration lag respectively.

One method of producing each of these transfer functionsis shown in Fig. 12.

The simplest system is that producing the zero velocity-lag transfer function, equation (2). From Fig. 12(b), using thesimple expressions for the blocks developed in the Appendix,:-

Let e =-(0+0)

(R 3 1 1

and e + -

.(pR. 3 /RCR 2 .)+(1RCR2 2 (4)

S p2 +(pR3 /RCR, )+(1/RCPC 2 )

Hence, if R/RiO2R 2 2ab

1/RICIR2 C2 =b 2

equations (2) and (4) are identical.

The system producing the finite velocity-lag transferfunction, equation (1), is shown in Fig. 12(a). It will be seenthat

e =-(€ + a + 2C2 pC)

and -1x x eRO pR C

~R CR, R pR C

Therefore = " . R 1 0, 2 02

p +(pR/I RRC, )+(i/R C,1 lC 2)

Page IC

For ... ith ucuation (1):-

R/RRC = 2ab

1/RcRo = b2

Figure 12(c) shows a method of producing a zero accelera-tion-lag transfer function, equation (3). Proceeding as before,

_ 1 1 R4 1 1 + R x71-[_ + x + (0+9)

and for identity with equation (3):-

R/R 5 R3C 3 = 2nb

R 4 /R R C R 3 C 3 = cb2

1/RICRZC 2 R3 C3 = b3

Two practical points should be noted at this stage:-

(a) When integrators are used in cascade, their time con-stants should be approximately eq'jal. If this is not possible,the shortest time constant should occur at the end of the chain.Otherwise the short time constant integrator will reach itsmaximum output voltage before the resetting loop has had timeto reduce the input step-function, i.e. the integrator willsaturate.

As an example, a practical system was set up to simulateequation (2) with a = I and b = 10. Integrators of time con-stant cne second followed by 1/100 second gave correct performanceto a 10 volt step. Integrators of time constant f/20 secondfollowed by 1/5 second gave substantial errors. When the timeconstants were equal at 1/10 second, the system was again satis-factory.

(b) Great care must be exercised to ensure that the sign ofthe output of each block is correct. As stated in subsection2.2 the output of each block can be selected positive or nega-tive. Fig. 12 shows tho polarity of the output required fromeach block in these particular examples.

3.3 Introoauction of Delays and Simple Non-Linearitiesinto the Resetting Loop

It will bp apparent from Fig. 12(d) that delays andsimple non-linearities existing in the resetting loop of a "real"system can be introduced directly into the simulated system. Astandard block is used to provide the transfer function i/(l+pT)wNhere a simple exponential delay (of magnitude T seconds) occursbetween the output and the resetting signal. The circuit simulat-ing backlash was described in subsection 2.5 and can be used toA -mulate backlash between motor and resetter. If the output hasa maximum velocity, this can be simulated by restricting the levelA' the voltage fed to the last integrating stage.

Delays present in the error signal, due to smoothingetc., can be introduced by placing the requisite block in the pathof the error signal. This position is shown dotted in Fig. 12(d).

Examples of these systems will be described in sub-section 4.3.

!Page ii

3.4 Synthesis of u given Electro-mechanical System

(a) Introduction

This alternative method of setting up the simulatoris used when a transfer function is not known or when it ismore complex than those discussed in subsection 3.2. It isgenerally necessary to use this method when high-power servosare being considered, the motor characteristics, sta.rting

* friction .nd othcr noh-lin aritius being important.

The method is described in Ref. 1. Briefly, the completedifferential equation of the mechanical system is set up on thesimulator. The resetting loop is closed and the servo isstabilised using the method intended for the "real" system.This simulation is straightforward but attention must be givento the scaling of voltage and mechanical motion. Care mustalso be exercised in choosing constants that do not saturatethe electronic units.

(b) Motor and Load

The differential equation is

T = Ip2 0 + fpO

where T is the output torque of the motor( is the angular displacement of the motorI is the total inertia at the motor shaft

and f is the motor dynamic breaking coefficient

In a real system, torque is often produced by passing

current through the motor field. Hence

T = Gke

where e is the input misalignment signal' is the gain of the amplifier

and k is the conversion factor; milliamps in thefield of the motor to oz in. torque.

This system is shown simulated in Fig. 13(a) where

RIC 1 f/- R/R 3 1 and C2 R3/R2 = I/f

The input voltage to this circuit represents torque and hencecan be limited to simulate the maximum torque available from thereal system.

When the output torque is not a linear function of inputvoltage, the desired law may be cbtained by a curve-fitting cir-cuit such as those described in Refs. 2 and 3.

(c) Friction and Windage Torque

In a practical system the motor has to do work other thanaccelerating the load, e.g. friction must be overcome. Since thesign of friction torque is always such as to oppose the appliedturque, the system shown in Fig. 13(b) is used. For small move-ments of the input or output an equal and opposite voltage appearsin the simulator at the point equivalent to torque. The outputtherefore remains stationary. As the input increases, the limitcircuit optrates, a signal appears at e', and the output changesto null this signal.

The value of friction torque varies with speed in a mannerthat is not precisely known. Its value when motion just commencesis considerably lower than the value at standstill and this is amajor cause of jerky motion at low speed. It can be simulated byreducing the bias on the diodes by a relay operated when the out-put of the first integrator of Fig. 13(a) moves from zero.

Page

Another source of torque whicL. must be overcome by themotor is that due to windage. This will be appreciable on alarge aerial and can easily be simulated. It must be scaledback to the motor, converted into voltage units and subtractedfrom the signal e of Fig. 13(a). This torque will probablyvary as the output position or velocity changes. These func-tions occur as voltages in the simulator and can therefore beused to modify the simulated windage torque.

(d) Resilience and an "External Load"

nwing to the resilience cf shafting in high-power servos,the instantaneous position of the load can be appreciably dif-ferent from that cf the resetting shaft. The load is then saidto be "external" to the servo loop. A pro er treatment of thissubject is outside the scope of this Note (see Ref. 4), but themethod of solution will be indicated.

To simulate an external load, two effects of resiliencemust be considered. These are the displacement between theresetting shaft (0) and the load position (eL), and the reflec-

ted torque (TL ) at the resetting shaft. It can be shown that

the equation relating the load position to the resetting shaftis:-

0L/%o = KL/(ILP +fLP+kL)

,;;here IL, fL and KL are respectively the inertia, viscous damping,

and stiffness of the load. The equation will be recognised asthe same form as equation (1) of subsection 3.2. The reflectedtorque is given by:-

TL = ()o -'L)K L

A diagram showing the simulated system is shown in Fig. 13(c).

4. APPLICATIONS

4.1 General

The details of some applications of the simulator arebriefly described.

It is difficult to describe these experiments withoutconsidering the servo problems involved, a task outside the scopeof this note. The examples have therefore been selected to demon-strate the use and accuracy of the simulator.

A unity time scale is normally employed. This impliesthat servo natural frequencies, time delays, velocities, etc.,have the same time scale as the "real" system. This is not essen-

_ial, th(! time scale used can be arranged to suit the problem,providtd that consistency is maintained in all the parameters.

4.2 Use of Linear Transfer Functions

The transfer functions (1) and (2) of subsection 3.2were set up on the simulator using the method of Fig. 12(a) and(b). In one particular experiment the transfer functions were:-

9/(p2 +3p+9) and (4p+4)/(p2+4p+4) respectively.

Page 13

?hc components used were:-

R = 1 Megohm R = i Megohxm

R3 = I Megohm RA = R2 = 0.5 Megohm

C = C2 = 1 microfd C = C2 = 1 microfd

R, = R2 = 0.33 Megohm R= 1 Megohm

When tested with a 10 volt input step, the outputresponse of each loop did not deviate by more than 2% from thetheoretical value.

In a specific problem these two servos, operating incascade, had two possible inputs; a step-function of constantvelocity or a st p-function of position having an exponentialrise (i.e. 1-e 2). It was required to know the magnitude ofthe error between input and the final output and the manner inwhich this varied for changes in the paraneters of the servosystem. To have calculated the error would have been verylaborious. The actual figures arc of no general interest butthe parameters selected for the final electromechanical systemgave results within ± 3% of those forecast by the simulator.

4.3 Simple Non-Linearities

(a) Delays

It can be shown that if a servc, described by equation(2), be connected with two simple expcnential delays in theresetting loop, the syste. will oscillate if

4T TT +T +, , +T'

where T and T, are the values of the two delays and a = b in

equation (2).

The frequency of oscillation will be

W 2b

TI +T2

As a check on the accuracy of the simulator, delayswere introduced in the loop and their value increased untiloscillations were just maintained. The results obtained arecompared with calculated figures:-

Measured Calculated Error f Measured Calculated Error

Ti 0.55 0.546 0.7% 0.14 0.14 -

T 0 0 - 0.145 0.14 3.5%

W 3.59 3.66 1.7% 4.87 4.77 2.1%

_ _ _ _ _,__,__,__,__ _ _ ...

Page 14

As an example nearer a practical case, Fig. 14(a) showsthe effect on transient behaviour of delays of 0.1 sec. and0.2 sec. in the resetting loop of a servo where a = b = 2.Figure 14(b) shows the effect when the delay occurs in the pathof the error signal. These results are traced from pen-recordings.

(b) Backlash

Figure 14(c) shows the effect of backlash between motorand resetter. The servo simulated had a transient responsegiven by equation (i) where a = 0.5 and b = 2. The input wasa 10 volt step and the bac'lash was set up for 1 volt. Thissimulates the effect of backlash between motor and resetter, ofthe order of one degree, for a velocity feed-back stabilisedservo. It shows the "backlash oscillation" effect when motorfriction damping is negligible.

(c) Velocity Limiting

An investigation was made into the additional smoothingachieved from a servo whose output velocity was deliberatelylimited. A signal from the sine-wave generator was fed to thetransfer function of equation (2) with a = b = 2. The inputamplitude was 5 volts and the frequency varied from 0.05 c.p.s.to 2.0 c.p.s. The output amplitude was measured with no limit-ing and then with limiting set at 5.3 volts/sec. and 3.0 volts/sec. (i.e. the voltage limit to the grid of the last integratorwas 2.65 volts and 1.5 volts). A cross-check on the accuracyof the limit voltage can be obtained by actual measurement ofthe maximum output velocity from pen recordings.

The theoretical curves are shown in Fig. 15, with themeasured values shown as X. The maximum error is 5% and theaverage error is 2%.

An input from the noise source was later used in this

experiment.

4.4 Simulation of Electro-mechanical Systems

To clarify the arithmetic involved in the simulation ofan electro-mechanical system, consider the following example. Alarge serve-motor is driven by a D.C. amplifier from magslip errorsignals. The system is to be stabilised by a differentiated tacho-generator signal. The following parameters are known:-

(i) Motor torque - 40 oz in. maximum

(ii) Inertia at motor shaft - 10 oz in.

(iii) Viscous damping at motor shaft - 0.2 oz in./rad./sec.

(iv) Starting friction at motor shaft - 10 oz in.(Assumed constant with speed)

(v) Gear ratio between motor and load - 360:1

(vi) Scale of resetting signal - 2 volts/i0 misalignment.

(vii) Amplifier/motor constant (kG) such that full torqueis produced by the motor for a misalignment equal to0.b degrees.

(viii) Backlash between motor and load equal to 0.5 rev. of

the motor.

(ix) Tacho-generator produces 2 vots/r.p.s.

(x) Smoothing time constant of error circuit - 0.1 see.

Ii Page 15The system is shown in Fig. 16, and the procedure adopted

was as follows:-

t (a) Blocks (i) and (2) represent the error detecting cir-cuit of the system and include the smoothing time constant.The voltage at point B represents torque and this must belimited to 40 oz in. An arbitrary scale of I volt = 10 oz in.was chosen for this point and the limit block (10) set to4 volts.

(b) The scale of voltage representing 6o is given by (vi)above as 2 volts/lO.. Since there is 360:1 gear ratio betweenthe motor % , and 0 the scale of Om becomes 2 volts/rev. This

is rather low for the backlash simulation, but is used to avoidfurther scaling. Block (9) is therefore set up for a "backlash"of I vClt.

(c) The input to the simulated "motor", blocks (5) and (6),is given by (vii) above. Hence point (c) must have a scale1 volt = 40 oz in.

(d) Since the viscous damping, f, has a small value, themethod of 3.4(b) is not suitable. The equation of motion is

T = Ip 0 ~m

using the same nomenclature as in 3.4(b).

Hence = (1- n)/&p

Neglecting viscous damping we have:-

T-0 40 x2 2 = 490 volt/sec2

Assuming equal integrating time constants, for an input scaleof 1 volt = 40 oz in., time constants of 1/22 secs. will givethe performance specified.

The torque due to viscous damping is 0.2 oz in./rad./sec.At the output of block (5) this becomes 0.005 volt/rad./sec.With an integrating time constant of 1/22 sec., the voltage atthe output of block (5) is 2/(22xw) = 0.0145 volt/rad. /sec.Hence this voltage multiplied by /3 in block (4) is subtractedfrom the torque voltage and provides the correct viscous damping.

(e) The remainder of the simulation is straightforward. Itshould be noted that, for stability with the large smoothingdelay, it was found necessary to incorporate "phase advance"of the error signal. The circuit of Fig. 17(h) was used. Itwas assumed that, in the real system, this phase advance wouldbe incorporated with no loss of D.C. gain of the error signal.

The servo performance was adjusted to give a naturalfrequency of about 2 rad./sec. The response to a position-stepinput had an overshoot of 16%.

The scaling used in this simulation is not necessarilysuitable for all investigations; it was chosen to demonstratethe method involved. In particular, the scaling of the limitsand the backlash is as small as practicable.

Page 16

6. CONCLUSIONS

sThe simulator described is simple and is easy to con-

struct but has an inherent accuracy sufficient for the investi-gation of most closed-loop problems. It is a versatile instru-ment and the inclusion of a low frequency noise source has beena useful development for dealing with servos associated with aradar.

The simulator has been used by the author almost exclu-sively on combinations of various transfer functions; simplenon-linearities have been included. It has facilitated theoptimisation of these systems and reduced the labour in calculat-ing the performance of the "real" systerr. It has been useful indetermining the importance of various mechanical parameters as itwill indicate a deviation from the simple transfer functionascribed to the system. High orders of accuracy from the compo-nent units of the simulator are not necessary for this work. Anaccuracy of 3% is a reasonable limit.

High power servos, including complex mechanical charac-teristics, can generally be simulated in some detail. Thedeleterious effect of backlash and delays in high-order high-natural-frequency systems can be demonstrated; and a useful guideto final performance can be obtained. Since the mechanical pro-perties of the "real" system are seldom known with a high degreeof precision, a more accurate simulator would be of doubtful advan-tage.

In future work on the simulator, it is hoped to investi-gate some ON-OFF control systems.

6. ACKNOWLEDGMENTS

The author of this technical note wishes to acknowledgethe work of Mr. P. F. C. Griffiths on the first simulator andMr. P. E. H. Pearce for the detailed circuit design of the simu-lator described in this note.

7. REFERENCES

1. WILLIAMS, PROF. F.C. "Electronic Servo Simulators"RITSON, F.C. J.I.E.E. Part IIA, Vol. 94,

1947, p. 112.

2. BURT, E. G. C. "Function Generators Based onLANG, 0. H. Linear Interpolation"

R.A.E. Technical Note G.W.2"4

3. BENJAMIN, R. "The Representation of Mathe-matical Functions of a SingleVariable by a Succession ofLinear Approximations"A.S.R.E. Technical Note NX/50/8.

4. BROWN, G.S. "Principles of Servomechanisms"CAMPBELL, D.P. John Wiley & Sons, 1948.

Page 17

APPENDIX

SIMPLE OPERATIONS USING FEEDBACK AMPLIFIERS

1. METHOD

In Fig. 17(a) Z, and Z2 are passive networks of

resistances and condensers and A is a high gain D.C. amplifier.Then:-

Vo-= -Ag

and, applying Kirchoff's law,

V0 AZZ 2

Hnev o AZ ........ M1Hence q= - (A+I)Z, + Zr

V o Z

V0 z2

and - - when A is very large ..(2)i i

Figure 17 sumnmarises the most usual functions.

2. EFFECT OF FINITE GAIN OF D.C. A1PLIFIER

(a) Introduction

As the gain of the D.C. amplifier used in this simulatojis finite, it is worth considering the error introduced byusing equation (2) instead of equation (i). Two units will beconsidered: the adder and the integrator.

(b) Addition

Applying the method used to derive equation (1) to theaddition circuit shovm in Fig. 17(b) we have,

A R (VI +V )

1+ -~ R1 +(R, j

The simplified performance is described by

R1

The error is therefore equal to

(A-x

+A 2I

5+

AUNCLASSIFIED

Page 18

If A =500

when R2 = 5, error = 2%

/R1 1, error 0.4%

R2 / 1 = 0.2, error = 0.1%

These are acceptable errors. However, when the adderis used in the resetting loop, V2 is negative and, in thesteady state condition, equals V, . The error in this conditionis zero.

(c) Integration

Applying equation (i) to the integrator circuit ofFig. 17(c) we have,

VO A

- (A+I)pT+I where T = R1 01

As a function of time,

VO = -ViA1 - exp{- TAY+l]

By the simplified theory

Vo = -V i

If V i is maintained constant after t seconds the error is equalto

[ AT exp( t X1C

The following table indicates the effect of amplifiergain G and the ratio t/T on this error.

A = 1000 A = 500 A =250 A = 100

tit 2.5 0 2.6 .0 2.6 60 2.5 50

% error 0.13 2.46 0.25 4.84 0.5 9.37 1.25 21.3

It is emphasised that these are percentage errors fora fixed input. Consideration of a simulated closed loop withstep-function inputs will produce much smaller errors. Thisis because the input to the integrators is, after a short time,reduced to zero. Consideration of the zero-velocity lag servodescribed in subsection 3.3 of this technical note shows thatthe maximum error due to the gain of the amplifiers being 500,and not infinity, is less than 0.4%. This is negligible com-pared with component tolerances and the initial assumptionsused in deriving the transfer function.

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Date of Search: 29 May 2008

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