{IHHHHI l I I HI I 1H1“ .THS_ ”lés‘éal‘EN“ fiHALYSis CA?- A {5. E 3ER”-fC:-QEMG§€$?R.A_T§®N 'UHE'E‘ 71652325 €332? #5210 Emma af M‘ S. MLCP‘iEGéAE‘J STATE {29% GE ',' " " 3-3 1 ‘: a" 1 g flaward ;: mews wedaugn ma V THESIS This is to certify that the thesis entitled mum. Analysis of A 0.3. Servo-Damnation Unit presented by Board Edmond: Ger-laugh has been accepted towards fulfillment of the requirements for _.!_08_0____degree in__§¢_!°..___ a Major professoa Date “bag: 27 ’48— M496 TRANSIENT ANALYSIS OF A G. E. SERVO-DEMONSTRATION UNIT By HOWARD EDMONDS GERLAUGH hut-u.- A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 1948 .1 f} I. Introduction Since the development of large sources of power, man has become increasingly interested in its control so that it may be made to perform more useful and exacting work. In particular it is the starting, stopping, and governing of these forces that are included in the field of "control? Systems which regulate the flow of energy may be class- ed into four broad classes: open cycle, closed cycle, dis- continuous, and continuous. In all of these different types the amount of control energy bears no relation to the power released or governed. If an electric motor is connected to a source of power and set in motion or stopped by closing or opening a switch, we have a simple discontinuous control. The system is dis- continuous in that the current to the motor is completely on or off and there are no intermediate positions possible. If we replaced the switch with a rheostat, the system would become continuous because now the power supplied to the mo- tor has a range values from fully on to off. Fc A 3 J A 7'4 Mo‘l’o r Figure 1 Howard E. Gerlaugh - 1 - 1:33;.- .r I} a). g x 5'..." .. 3's"..- ' l 1 ‘ u u / _ .- . r ‘, ‘ r I f 1 I . l 5‘ ’ K I 7 ~ . ‘ K | I . .‘ , . ' ' . t .' 1’ (_ - . H r f ‘ ' . a K h A ‘ ‘ . , r. l , . I l I .> ‘ '. . r . ~ '- n ‘ . ' ‘ A .\ ._ . ‘ .‘ . . ' ‘ r - 4 t : ‘ . ' ' ' .p .- I A I I‘ '_ f‘ . ‘ u . ' ‘ . Q g - ( . \ f ‘ V ‘ A \ - )- K ' I . u ‘ . v I A , r: a _ i \ l ‘— l . ‘ r ‘ r ‘ l I ,, , ‘ 0 ' ‘ I V . ‘ ‘ ' 1 . . I I - i 7 ~ , ‘ ‘ ' V ' ' ' ‘ u . ~ . i | ' J i I . ‘ \ i ’ ' v , I ‘ . t ‘ v : — A I ~ Kr . v ' ‘ w ' ' ' I‘ - ' . t ‘ d v v - .‘ . ~ . v . _- . '\ A I \ . l ’ I . | , ‘ .4 . . ~ . ’ ‘- . h V I . - ‘ l I ' . ,t ( ‘ - . v.7 . - ' ‘ A c . x I ‘ ' . . i l | ‘ l \ ' ‘ ‘ ' l . ~ - > u . I , _ , l <._ I l . A “I, I ‘ ' ‘ 7 ‘ , ‘ I ' ‘ ‘ Y: ' ' 9 a . ' . -- I l ‘ l r. I I . > x . - t t ‘ , , I ‘ r‘ ‘ I t l I . I I J ' . ), ‘ r . ' ‘lg I . l‘ I ‘ ' "LJ . .V r ' ~ . —-_ . - -.. . ..~ ,4” - .- ‘-0. nl— ‘4~"“‘-v <'_ 45m.-.._-_ -.~~‘. T. “”- , h . ‘ A ‘ l I - f ., , ~ -- ~r ’ . s an . ’ L t v 1 t 3 I} ~ \ . - 'f f I . l ‘ ' . l . . . s ) ' J '-i -. I . l . ‘ I I '" ~ . _ ~ g ‘ “‘ . l l ’, ""~‘V ~o- u.’ 3‘ A? g I” ‘ , A J ~ ‘ l I .. . _‘ ‘ ‘ J ‘ l l ‘ ,. a . n l . ta. ~ . . l a ,,v..‘hl-. ...‘ u a ~ ‘ c. . ‘J u I O. . v, ' I L . ....-., ,. ‘1 u .. . ,t, - . . . . i _ . - , - vi «.4 - v. .-A o—ua . h— ‘r ., ,~ . a In the system shown in Figure l the switch controlling the direction of rotation of the motor is operated by the difference in position between points A & B and some refer- ence. In this system the point C is made to follow the point A by letting the error or deviation control the rotation of the motor; thus a complete loop or cycle is present. For this reason the control is a closed cycle system and at the same time discontinuous since the switch either turns the motor completely on or off. If a slide wire rheostat, with the sliding arm connected to the differential, were to replace the switch in the above control, it would become closed cycle and continuous. A servo mechanism consists of an input member, an out- put member, a differential device which compares the in- stantaneous positions of the input and output, a controller consisting of an amplifier and output drive motor, and a stabilizing device. These components are so arranged that if the position of the input were x(t) and the output y(t), x(t) would equal ky(t). It shall be the purpose of this paper to describe a typical servo system, develop a mathematical expression for its performance, determine the constants of the elements, and to compare the calculated characteristics with those determined experimentally. II. Description of System Analyzed The analysis which follows was made on a closed cycle, continuous position control servo system. The system was - 2 - Howard E. Gerlaugh manufactured by General Electric and is designated as the "Amplidyne Servo Demonstrator". T The block diagram of the system is shown in Figure 2. at Diffcm‘lml e 00"!” J0 Mofor ‘fl r‘l_‘:" emphagnne Figure 2. The input shaft whose angular position is 91 is connected to a selsyn generator is converted to three single phase 60 cycle output voltages whose amplitudes are proportional to the cosine of the angle between the magnetic axis of the rotor and stator. The angular position of the output motor shaft 90 is connected through a gear train with a gear ratio of 24:1 to a selsyn transformer. The three out- put voltages of the selsyn generator is connected to the stator of the transformer and the output of the rotor is a 60 cycle voltage whose amplitude is proportional to the sins of the angle between the generator rotor and transformer rotor. The combination of the selsyn generator and trans- former are indicated in Figure 2 as a differential whose output signal is proportional to the error e, where 9 is the difference between 91 and ififl—. The error voltage is fed into an amplifier of gain ul which in turn supplies the field current of the amplidyne. - 3 - Howard E. Gerlaugh . ‘ V I ‘ I t | O 1 F . l, v I I ° Onv~ ' .. - . L\ , . f . l x ‘ a. it, ,_ .. . i - 2 I . ‘ s .s n ‘ l ' ‘. .-,_ __, . u - Y .- n .... In“. Q « - . ; .‘ \ h ’I q : ' ‘ ~ ‘. . . I ‘_ I. , L l . 4 . . )1; ~ ‘ . . , .. v... ’. _ I -. ) . ‘Q‘ G - cl, 2 5" ' ’5' 'l" " ”A" a . j, ' A ‘ t - ‘v ’1 3 n.‘ i g I “J' l g "F ‘ 4' t ' . 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' 4* ””""“""“‘ “’ ' ‘ iV/H/No AJ'IHuA/‘A/V ‘r J ;’ Mama 5 m . «- ; E 317 3 » d) + 02: NEPAL x: if; 7px;, 5%” t..} mar-F I 590M 45M [ {F9 ; *“Z‘vm - L-v—wwwwfibr- -¢-< ‘v-er’v-wh—ww- wwvm ' ‘ F i F. L‘““— "'.*'= ‘ O ‘- I L ‘ l L. .. l C L? ...n.,. .1 i e I a r ,7 _l .N J 1 ‘1 .1 1 F T ,. —. w fi M ‘F J 4 s 3' ~ F 4f ‘ . f +_ B d) , I 0 -* 1;) ~- A ~ 'AHI AH‘ Tl 74¢I'JCIHO.H In. ‘k¢4 5OJ j IISVAL P é/f/ [\l L ONT/POL UIV/r m i A 1 A , f ' ‘ .r * * ‘ ., i ‘ ‘ IL L— l HANDWHEEL F< .. .J -’ "" ’ _ ._ ._. l -,.._. _.- _ ‘. w. r——— ~~ -—+————“‘-t--— H-a—J I i _ ,, F J L...____._._.-_ % ,__, . . _, A, , - r f —F ’Eg. {IVE/5? PEGUM 70/? (AV/7" ! “A F , i 1"“ ’1 " F 7 (3/4)" /-'..L F 1 +5 l+_Q,_ F 4%)- f. 7 j LJM/v PAL - E U 5)’ I? ‘ NC 0 km& A.” A), O 1 :4 GENERATOR? ....,H-,_w_wm_ L‘O F ,, W VLJNI'R n 53.1. F FJFx ~ i4}- -- ~— 5 p, - 4 ow *1 F ’4'5/ t o. ' ‘5‘”le o' _ d .“‘I‘ Q I“! 5:... A: ‘ “ Uflx‘tg'q f./L‘-fi D .S YNC/Po C ONTPOL TRANS Fo RMEE’ _ ._. .gm—u fir: A’er/LIL/YIV...: t t L s. I"! J 11% MO raw 95A] I 'V J-‘orh‘f‘ -‘-d .‘. Fl .51. U og"\ o 1-3“ /""1 J“ 1;, b57- -4-m"~—v~ -v HG. 3 L.--” - . 4 ll 1' ,t , > 4 I ‘ .fi - . .3 A § .l ' V ,. V '~ F . F v- 1. .' V v" 1 o‘ 7 . . . . ‘.:" v ,‘ .F‘ . .. . . . ‘7' '7". I" . F ‘¢¢0. - o, ' . . . . 1 \ F . Iv. . I ‘v‘ ,. I 4;. 'Il ,. . I I ‘- u“ . J | A i .1 . . —. u . r. o .v' ‘ I . 'V The complete wiring diagram is given in Figure 3. The amplifier is shown to consist of a pair of 6L6 tubes operating as half wave grid controlled rectifiers, the grid potentials being controlled by the error signal. An error signal of a given A.C. polarity increases the grid poten- tial and output of one tube and decreases the output of the other tube by a like amount. An A.C. voltage of oppo- site polarity reverses the tube system. The plate voltage on the tubes is supplied from the same 60 cycle source as the selsyns since it serves as a reference to determine the A.C. polarity of the error signal. The neon lamps across the grid circuits of the tube serve to provide overload protection. The feedback circuit of the ampli- fier returns a portion of the voltage across the armature of the output motor through a differentiating circuit. This provides feedback for the amplifier with a gain of u2. The output of the amplidyne is connected to the arma- ture of the motor. The voltage isia function of the error and also of the rate of change in error. When the rate of change of error is maximum the degenerative feedback voltage is largest, decreasing the armature voltage of the motor and_preventing the output shaft from overshooting the zero error point as far as it normally would with out feed- back. The output angle 90 is measured from the motor shaft and is connected through a step down gear train to the sel- syn transformer. The moment of inertia of the output is designated. - 4 - Howard E. Gerlaugh s a... . ' 4 ‘ . 4' I v . C O - g _ i . - ‘_ I K 1 o I » £ . / .' I , I i \ , . ‘ l ' . v V . . — . ' , I F ‘ 0 V I - I F . _ . ‘ . r . I O ' I ‘ . ‘ . . ,- . . F . _ . ' ' , I I a ‘ ‘ I O " V . x F ' ‘ V ' ~ I‘. I _ . . v ‘ , . r » . I , . ' I ‘ , ’ ‘ t , l . ' 0 . c ‘ L I - ' v r ' I L (a I F e I l . ‘ l I F' F ‘ .x . . . I F I | I , , . , ‘ \ ‘1 O a... . . . , . . , I I I j l " . r x -" . h - I , I 0‘ . N O O I l I ‘\ .. . , _ o . _ . ' F I V . I . a - ‘ |, I . .- - . . . ..... . . . F , ’ . J0 and includes that of the motor and associated gear train. The resistor across the terminals of the output motor armat- ure permits adjustment of the feedback voltage. The ratio h represents the ratio of the feedback voltage to the total armature voltage. In operation the output shaft, where the angle is _§2, is made to follow the input shaft with an angle of 91. The accuracy of correspondence during transient and steady state conditions is dependent upon the constants of the system. A quantitative analysis of the system follows. III. Derivation of Equations. If the input shaft displacement is made a sinusoidal function of time, the output shaft displacement and the difference between the two or error will also be a sinus- oidal function of time with the same frequency but of dif- ferent amplitudes and phase angles with one another. These functions are shown vectorially in Figure #. . d, \ . 0 1ft» 9 / / Gk F9. Figure 4 In Figure 4 we see that <9= 9.: e. (1) and that the output vector 90 lags behind the input vector 91 by the phase angler'7. «and flare the angles between the error 9 and input 91 vectors and the error e and output ' 5 - Howard E. Gerlaugh . D, " t ; g .1 ‘ \~ ' K \ ’ " '\ ‘l \ § " I it." .I‘ ,.. . o- ‘, w! . r 7 ~ I s" k _ . ‘ ' I " . . . . I ~ I I y r I . _ ‘ x ,v ,. a. .I - ' l v . l‘ v \ ‘, t I ‘ J \ . ‘. a n. \ '. '. 1‘ .‘ § ' ' a "A { . . 1.. ‘- ‘ . *‘ x If .5 rc- ' ‘ \ 4-. .p ‘ . . l .w. ' . . l , > V ~ . . I _ n . ' ’x \ ‘i up .‘. ‘Q ' ~ ' __‘ a I l ' . GO vectors respectively. From these geometric considera- tions, it is seen that the relation between the output dis- placement and the input displacement can be expressed by the equation e M -3-:=14<= 6‘ ‘ (2) Similarly the relation between the output and error is 29-: B:"3 (3) and the error and input 2 = J“ (4) 9‘ Ce . Using the above relations Be’fis QB a M = 94 .. £9 £3 £3 | (5) from which at J5 6 l —-=I+B€ —-.—.= = M (6 9 6. ”Bed" Ce. J Thus knowing .89. , it becomes possible to calculate 9/91 , and follows that e = m: a. .621: Be”. '5‘: Ae 6 x6. |+Be:—:-“a (7) '13 The term 88" is given the name of transfer function of a servo system since it relates the ratio between 90 and 91. In the derivation which follows the transfer function will be developed for the G. E. Servo Demonstrator. In the first case the feedback ratio h of the system will be made zero. The relation between the error 9 and the total amplidyne field current I, is I.= iii—:57 (8) where ul is the gain of the amplifier in ma./radian taken with respect to the output motor shaft, p is the Heaviside operator equal to the first derivative with respect to time, - 6 - Howard E. Gerlaugh . Inv- . and T1 is the time constant of the amplifier field Ll/Rl with the units_of seconds. The product of ul and 9 give the resultant field current in milliamperes. When the dif- ferential equation is solved it has the form I.=4-9(|‘é-%). (8a) In equation 8 we assume that the amplification of the ampli- fier ul is constant arer range of error found in the system; in the tests which follow this is found to be true. The torque of a D.C. shunt motor, such as is shown in Figure 3 to be employed in this system, is proportional to the field strength and armature current. 'In this servo system the field is held constant so that the torque is equal to a constant times the armature current. The output current I3 of the amplidyne in terms of the field current I1 may be given by the relation = K.I. (9) Ia I+Piz where K1 has the units of amperes/ ma. and T2 is the time constant of the armature of the amplidyne LQ/Rg. The ex- pression for the torque of the output motor may be immedi- ately written T3 K: I" P2561; (10) where K2 is a constant with the units of ft.-lbs./amp. Let KaK. Kg, (11) then Ta- -——l,7-KI a K39 . H’l’ t .(Hp’flflwfl’fi (12) - 7 - Howard E. Gerlaugh « up ”a ‘ ., . w . also I.» S r ‘\ 1 . ‘ q u I I . w . v . - 4 ‘n . I a n I I 1 .‘ o v t . . f r o. I :~ p J , . - V»'O"" ' the torque T having the units of ft.-lbs. Also the torque of the motor is related to the moment of inertia and friction of the output system by the equation T= J’one- +FP 5° (13) J0 having the units of slug-ft.2 and the friction coeffi- cient F, the units of ft.-lbs./radian/sec. Equating 12 and 13 K e , .. (H-P . Hp z , (lop +FP)6° (13a) and solving for the transfer function 89 we have 90 )3... KAI EN :5: - NP(|+PT¢)(|+PT7(JQP+F) (14) By dividing 14 by the gear ratio N, measurements of the output displacement may be taken at the selsyn transformer. For equation 13 to predict accurately the performance of the servo the output current of the amplidyne must be directly proportional to the field current and the torque be a linear function of the armature current. These assump- tions are true until saturation takes place in the former, I r I in the field of the amplidyne, and in the latter, in the armature of the motor. In the case where h has some finite value giving the amplifier degenerative feedback, the equation for the trans- fer function becomes much more complicated. As the first step, if the voltage across the armature of the motor is taken as E0: as shown in_Figure 5, the voltage at the out- put terminals of the amplifier E due to the feedback hEo at its input is given by the expression hEoflgKH'PRC) 52" \+ pRC 'uqthR—E - 8 - Howard E. Gerlaugh (15) ‘V‘ .‘ v I 1". i x, é;V.‘ ‘ ‘ R U | -l . . where ug is the amplification of the feedback voltage in the amplifier with the units of volts/volt and R and C are the fundamental elements in the differentiating circuit. : :9nwbhffler 4, W flank/lay”: Error #01127: A“ , lac/J F‘ /?h 1 ‘ L 256 ' r' Figure 5 As before the component of the field current due to the error voltage is I -.:1L§LT,. . \"\'P.' (8) The component of the amplidyne field currents due to the feedback voltage is I ._ E 3 I150,“ |+ RC) . (16) "RJH’P’E R.(I+pT. I-I-p "A: The component of the output voltage of the amplidyne due to I, is as in equation 9 Es" ‘rfglfi (9) and the component due to I2 is :- KII I53: -r:i§%; Using the superposition theorem the amplidyne's total (1?) generated voltage Egt may be expressed in the relation Egt =Egt‘PESa (18) - 9 - Howard E. Gerlaugh ”- \f u!) . c . . a 1 LI.) 1.. .J u If. u . o . 4 .1; tt The voltage at the output terminals is seen to be 593 59*‘ItRz (19) where 13 is the output current. E0 may also be expressed in terms of the parameters of the motor by E.= K.¢Nm +13 Rs (20) Let ' K3¢=C| (21) and Nm=C3peou (22) so that C1_will have the units of volts/rad./sec. and 02 will have no units if the speed of the motor Nm is in radians/second. Rewriting 19 we find that EQ=C§C3P60 +I3 R“ (23) The motor torque T may be expressed as T: c, I,= 3'.p‘6o + Free (21+) or by equations 18 and 22, solving for I3 and substituting in 22, the motor torque is T" C3 53* - C.€z C; p 69 (25) - izeu*‘zz If now the right hand side of 23 is solved for I3 A and substituted in 22 and the result used to replace E0 in (16), the component of field current due to feedback is h [ma.c, p6,, +TRa]4;( \+p RC) 1" 3 R.(H- ptfichwscu-«MT (26) Combining 8, 9, 26, and 17 the total generated voltage of the amplidyne is - 10 - Howard E. Gerlaugh t... . . f. ’ I c “ 0" P (A. ““ Q ., 4': . L" ‘ r ' ' f .. 3' .J " V : ‘ w 7 1.. r. " h. -‘ .~ Q ~ l 7" 0' Li A d c ' b ,5 NC I -‘.I - I, I t “f :5? \J t . ' o n '.__ l‘ ..‘ v ’ ‘4' «“3 + " 1 . ' " I . " g "' . x- ' ‘ ' . . ii. I ‘u ’ n 1- . “'1 1' ’ ‘ ‘ '- ,n ' . if M K. a Eat: 59: +5913 (‘*PT?(‘*W— + Kl .1441. (C|C1C5 Pea +TRQ)('+PRC) R.C,(I+p'r.)(|+p"l’z§l H-p RC (~th or simplifying Kid-Lela; |+PRC("‘4%|\) E3" R.C3(I+Pfr.)| “+PT17f-1'PRC| -47_h)T (28) + K0 ’14; (COCL9390+TR¢)(I+ERC) R. c3(|+ pT)(I+PTz)EI+ PRCU-quhn substituting the value of T shown on the right side of 24 (27) in 28 and collecting coefficients of 9 and a {Kuhn KC: 0‘" PRC(I‘M1"\)J}O® {52. C3 (H PTNHPTT” PRC (""iz-hnj‘Qea) e. {Kw-u. (n+pRC)Ec. c2 c3p+(J.p'-e.+Fp)R.]}@ {R c,(H p’fian’I’z) [HPRQO a... HT}? Egta The circled numbers beside the brackets will be used in future expressions to refer to the bracketed quantity on its left. Combining 24, 25, and 29 we have an expression for the transfer function in terms of the system parameters 39.3.. g, 1 Ca {9} A . he) N N(:r.p +FpS-c, {@f+c.c,c,p{®} Equation 30 defines the operation of the system under transient and steady state conditions. After the constants of the above expressions have been determined experimentally, the equation will be solved for a sinusoidal input by sub- stituting P34” (31) - and then the transfer function plotted for values of w. J‘ By use of equations 6 and 7, CE and HEM may be plotted - ll - Howard E. Gerlaugh v-vu ... Q ._ Lu . .t) Z . .. ‘ a. It. ' ‘9 . n I 5 L.’ I... . I. ." V's. n}. J i! . .l .. ’1 I. .. ‘6 I , I. o} I ll’ b~ a . n ‘.l . . m 0. Q t ... 1' . 7 x .. 37.! ‘ x ... . ”V .5 . .0. .c I ‘Qi I b In . . I i. 'L ‘... f? . . I 1 - . . . ) t . - . K . a! r v v 'I J . .- '5‘ I. I u‘ A _ (n . ‘IC I O. .. >8. nip I I.- V la . ‘g 1. r8! . \luu 1‘ . K 5 § . a: .. . . A a .1. CL.» giving the complete representation of the performance of the system. IV. Experimental Determination of System Constants In the tests enumerated below the various constants of the system are determined. The tests were made over the same range of values as used in the system for a normal input. Using the resulting curves our assumptions may be proven to hold or be in error, as the case may be, and com- pensating corrections made. Test No. l. The output Inertia of the Motor and Gear Train. The output inertia of the system is contributed to by the motor armature and two large gears connecting the motor shaft to the output selsyn transflormer. The total inertia Jo will be computed with respect to the motor shaft. To accurately measure the moment of inertia of the irregular armature, a fly wheel whose inertia could be com- puted was suspended from a piece of long piano wire. It was set to oscillating as a torsional pendulum and its per- iod recorded. The fly wheel was then replaced by the motor armature and its period of oscillation recorded. Neglecting any damping due to the wire or air resist- ance, the period of a torsional pendulum is given in seconds by T‘ 2",? . (32) I being the moment of inertia of the system in slug-feet2 and K the spring constant of the wire. Solving 32 for I and setting up a ratio for two masses using the same wire we find that -12 - Howard E. Gerlaugh .22.- .. Io. " €1F2‘" ‘1. (33? and ‘ .133 ' Ti“ (33a) f’ 3 2.25:11. (1: 0L75rn, ~ ’- d w: 3.4lfb, Figure 6 The moment of inertia of the flywheel depicted in Figure 6 is -§Mnr1=osx.3__'4. x $.35): “neezuo aslug- f1“. "(34) I IG‘ "'2' 32. 2 The data taken in determining Tf of flywheel is : Test No. of oscillations Time Period #1 10 123 sec. 12.3sec. #3 10 124 12.4 Tf average - 12.3 sec. The data for T3 of motor armature is: Test No. of oscillations time Period #1 lb 83.0 sec. 8.3 sec. #2 10 82.5 8.25 #3 ' 10 83.0 8.3 Ta average : 8.3 sec. Usins equation 34 13 is found to be I43L862‘MO-3fi-‘9-é-3‘1: 848’H0 eslufl‘f“ (35) The two large gears were initially thought to contri- bute a larger amount of inertia than is actually the case. Their moment of inertia was calculated since they are of ‘ 13 ' Howard E. Gerlaugh O‘ a l o .‘V: .‘.¢ L .4. A. . .O .I‘. u at y I! . Q -l x. __ r. K}. a I . a\c - . a! 2. e . l' , ‘4. .. o. It § . , ~ I .. . r s a s t» 2.5! symmetrical shape and difficult to remove from their assem- bly. The calculations are given below: Gear A r : 2.05 in. density of steel a = 0.284 1b/1n3 d : 0.1875 in. , .‘(fig m: 0.02:6 elu M“ a g 35 (36) 3 I... = éMr": 0.5xo.02t6x(3:.%_§) .2 0.3.3xuo"slu.3-f*" (37) 3.03 in. 6. 1875 1110 Gear B. r d M, 3J4! 3.03:0.‘815 “$321332 = 0.0415 slugs (3s) (39) 16.3 O.SX0.0‘158(-l'§)‘ g Ls”x.o'35‘ug_ f-‘p‘ The moment of inertia of the gear as seen from the motor shaft is 'Ien I I 6" ‘ "M1 (39.) with a similar expression for I'GB. The gear ratio of the train is shown above each pair of us . _. [.4 N324- . "EEEEI|L=$ /bebw'5h6¢géf \ a gears and the number inside each gear represents the number of teeth on each gear. The total gear ratio N is the pro- duct of the two or 24. The total inertia referred to the motor shaft is - 14 - Howard E. Gerlaugh a“ A O! ‘O I..’ ‘4 6» + octane" Lanna“ Leawxno' 4‘ + a4: (39 b) I“: (848 +2z)no'° a 870 x Io‘“ slug- «ff? Test No. 2. The Time Constants of the Amplidyne. In order to determine the time constant of the ampli- dyne field a Westinghouse oscillograph was used to record the transient current when the circuit was closed. The set- up employed is shown in Figure 7. From the curve of the tran- sient, the length of time for the current to reach 62.3% of its final value was determined. The final current was 11mi~ ted by the resistor to 20 ma. The resistance of one half the field R, was found to be 830 ohms and that of the resis- tor Rh to be 300 ohms. Using the familiar expression for anRLcircuit E (._e-Et) t:— R the inductance of the one half of the field was determined. (40) -' i 13- ~+ ”flaw-9”: ; Em = no vol-h dc. lgszavnq Rhsaoan. Tm‘mfi \— osc :I/ogra-f’” 2.0:” Wave Element ‘ Figure 7 - A series of three oscillographs were taken, one of which is shown in Figure 8. The average of the three time constants was found to be Ti 3 0.0262 seconds since - 15 - Howard Gerlaugh in .7”, ‘1': . _u ,_I, .5 o , 5'" D I s " ‘ ' - r ..'. . . ; ‘ b' ‘ ; _ ‘ 1 . A ‘ r. ‘ ‘ ' " . 4' '. " r‘ .2“ - ‘mJ " I .1 . _ O D . . .-— ‘ ‘ - . ' ‘ _ . I Q 1 v . I — . ‘ ' ‘ V. ' D I' . ‘ . v - ’ ‘ I ‘ I J . o 5 w ‘ v m . ‘l ‘ ‘ h . .' . l - .. £ , . u ‘1 f '0“ U ‘ ‘ ‘ - o - I- : * l - 0' I v -’ M o u— . o: M \- - . a. a o i . a .H ..\ A \ ‘t ‘ (f u. - ‘ —- ~‘ . v. e v - v ' . v . . ‘ . x . _ ‘b ’. ,,. x -»- ’i .. ' I, \ 3 ' .1- l " N I a n .a . - J‘r‘ _. \ . " ’ » ' f. ‘A (‘l 9 a , « “I. f fl: .5‘ r \ '1 . l h—« I . A" 5.54 i . ' r 7- ‘ _ |‘ o «.l.‘ '. .1 ', . ‘ .‘ . ‘ - ~ . a . , . a". ._. , ‘1 . - ~ ‘ o ' U . ‘- . 2 I . A - O- n' . at '. ‘ e v. . ”c . .n' ‘ . a 'o , ‘ t ‘ ’ . ’ I‘ ‘ . i . 3‘ f e “r 4‘ ‘ “‘ ' 5“ . 7 u . ‘ “ ‘ x ‘ R ‘ u " Q C ‘ . , r ,‘ L m h... V , r' u' 1. ‘3 . O 3 ' f; . ‘ F a‘ u. t ‘ .4 ‘4 I . ’v I d‘ L t. .i ‘ ‘ . ‘ I .. .-. .v ‘ 7 c .’ ‘3: , . .1 ' . I C .1 ' , I u .. l . V f i ' ' . t ‘ f I '.I y ‘ .3. s 'a I . n ‘ r . . . . , p. . .4 -~- s c t .. ‘ '.I" “ Er, . ‘3‘ a (' V ;.'~.o' I t r \, ‘ . . T ' I \ ‘ . L,’ ".I l‘ . ' v’ -' ' .',i ‘ '\ ' . ’ - . . . , , . - - _ .5. 'IA ’ - ‘ > i v I 7 V I . . , I. H ' ' I v f‘ . ~ to‘ I ti‘. Vt. D..an Uthofinxxk‘ -Uvo fl JNO. u ...__.. _ Lo _ ‘Lt R1- ' 830 1- 300 L. = 29.6 henracs (408) Thettime constant of the field alone is found by ._L - Fh' 1: .. __.... ._ H30 40b 1:..- - T. #1.. -m%0.026230-0357 53C.( ) it In Figure 9 is shown the circuit diagram followed in obtaining the oscillograms of the amplidyne armature current when the field is suddenly applied. 1 ConTN/Ecld I thfbr' Em. 1/0 “/fs <16 Tfivnmny ' . Mac 05:; //og rap/r f O '" tflkznend‘ ' fif’ Figure 9 The resistors thand RE were set to give an 11 of 20 ma. and I3 of l ampere respectively. When measured Rh was found to be 300 ohms and R5. 260 ohms. The resistance of the armature R2 measured by the volt-ammeter method while the motor was running had the value of 54 ohms. A series of six oscillographs were made, two of which were disgarded because the motor voltage Em varied from 110 volts. The time to rise to a value of armature current e- qual to 0.634 amperes was scaled from each one and found to have a value of 0.0333 seconds. A typical oscillograph is shown in Figure 10. - - l6 - Howard E. Gerlaugh I ... U ‘ ’ a. ‘ h i. I - - . r ~ h. o ’W' ' . . .0 _ ' A u ’ - ~ . ‘1’ ~ .4 I" K“ H ‘ l . , v‘o‘ ;' v ‘ "' ’ . ’ H‘ 'u' . _. y . a ll 0 I .x-' > . v . ‘ . ‘ 7 k .1 . a - - “‘ . r - * t 4 V A . \ - 1" i ' ’ " ’I ‘ " l . .gu _- e. ‘y . 77- V .« 3' A4 1 v A I ‘ ‘ D - "“ ”w. -- -- a . ‘ ~ ‘ I a i ' " i r I 1 I, . . A ‘ ' . . .I l . ‘ ‘ ' ‘ . . ‘ g . I '8 ‘ _.‘, o- n... .. A . .-~ 0 on \~ as - .- mr v- .- I a - ' i' M ' “ “ ‘ C ‘- \ . \ IV A! w . . 4 l w I I — ' w I > ' - ‘ I' ‘ ’7 .< I . 'v 4 ‘ ‘ ' on . u ’ . ' ( ’ ' n . I. u. v ’ . ‘ ' \ ‘ I . . P ‘ ‘ ‘ . : _' " r \ l ' .k f . I I , -- _ ‘-..C- ‘C- \l ‘ “ . .. f. a- n V ‘ v 1 . ‘ul .- - ‘ . t 1 . ‘. u' k i : ‘.'.\'b, 5 ~ ‘ In A. P . 'I . -‘ s" ' ~" w l . t " at ' \ u "' \ 5 . - 3 .\ . x v «4 ' o I I ~ h ‘ i ‘1’- V I I' -" ‘ ‘ l- . L O »m~‘- 9 e “ ~ I ‘ . . -" ‘ wt: O '. . ' . ~ ; ‘ur I I", 1-: 0‘ I I u' ‘ ‘ ¢ - . "n ,3 ‘ . w . \ b u‘ ‘ .< a a. . r. 5 f 1 7 . ‘ I u ‘ I» i ’ I ,u v ‘- l I ‘ ‘ . . « . t ‘ ‘ .- . , a . \ ’ . ~ r . ' 3 ' n , . e ' H ' I . t c ‘ - f . .‘ ' ‘ ' 4 ‘. u 1 ,.. ,0 k , ..,.. . . !’ ‘ ‘ r . ‘ .' .. C " " ‘ ‘ K V ., \ \ . n \ " \. ‘-. .- . ‘ \, . . 9 . \' 0" a i 1 V I . . . . l I '1 I v ’ . . a . V w \. . | ' ‘ ‘ . .- ‘ o . o ’ I ‘ ' :' a L w l ‘D . . a O . ‘ A ‘ I , . o . . - .— ' l . ‘ _ s r “ A ~ L I ' ‘ . . _ a‘ \ _ . .. . . . . | ’ 7 . ‘ ’ ~ I I I ' 1‘ ‘ . . . \ 7 .. ’ , c ' ' I .- c . ' 1 . I v , w , ~ . x o ‘ . . . '. ‘ I ‘ 70 I _ ._ ' . " 1 " ‘ . I . r’ r ' ~ 0 v o ’ ' ‘ . 9 ' . . 2 o . i ' , 1 The current I} follows the equation -%t *‘l 13: £5... \+ ___€_‘-__.+ (41) When the values taken from the oscillograph and the above circuit constants are substituted in 41, the resulting trans- cendental equation may be solved by trial and error for L2. The value found for L2 is 1.9 henries and the equation 41 appears as 0333 . W... __ §|fl%.°33§ I3: ‘ ‘L e z L9 1 J— / "9 _') +(026283.4 .4) (42) kozezx 3M. LB * With Hg in the circuit the time constant is Téa—Lé—z- =-;-;—2-= 0.00605 sec. (43) 2 and of the armature alone Tz.= $73.. = 0.0552 sec. (44) Test No. 3. Speed-Torque Curve of Output Motor with Input From Lab Supply In this test the voltage across the motor armature was obtained from the lab outlet as shown in Figure ll. The motor was coupled mechanically to a small one-eighth horse- power generator with adjustable shunt field which served as a dynamometer. The motor was loaded by increasing the gen- erator field while its output was dissipated in resistors. The torque of the motor was measured by a spring balance connected to the generator and the speed of the motor by a stroboscope. The resistance of the armature Hg is 23 ohms. - l7 - Howard E. Gerlaugh r. ‘ ‘ f ‘ 7‘ _‘ CI *. -.- - - " v c l a ‘ . n? e. . .' r . '3 . 1 J o ‘_ ’ ‘- ‘J. ‘ I . Kat, . ‘ . ‘ u; 1‘»! 1“ :w ,‘v .‘ U n ' h . - ,‘ ‘ ‘ A4. I w — - , ‘ _ o o u . r I ’ 9 x. fl I: IVOTBrflZDaflflt “ca/2544C. (zigzafiavdkn (/4/1/7 -/a. l8b0r7017. (‘00. 5"": be scope Dyna mam: ”A? Figure 11 The data taken in the test is as follows: Part A Eb : Va : 250 v. Vf : 109v. N-rpm F "Oz. T- ftolbo 1960 0.5 '0.0156 1940 4.0 0.125 1900 8.0 0.250 1870 12.0 0.375. 1840 16.0 0.500 1800 20.0 0.625 1780 24.0 0.750 Part! E E0 = V8. = 180V. Vf : 1®VO 1445 0.5 0.0156 1420 4.0 0.125 1385 8.0 0.250 1542 12.0 0.375 1320 16.0 0.500 1290 20.0 0.625 Part C E0 2 Va 2 120v Vf : 109v 910 0.5 0.0156 870 4.0 0.125 840 8.0 0.250 820 11.0 0.344 and is plotted in Figure 12. Of interest is the observation that the speed torque relationship is linear since curves are straight lines. The curves are parallel and their dis- tance from one another are proportional to the armature - 18 - Howard E. Gerlaugh .. .. n... a. .. . ,‘ A: ’v I. v... on .I in -v V ‘l 2 .J u at}, . 11“ . .9 .. .- u o \- . h. . .o m. x 4 [..II I . s0 . w m V .. . I 1‘ u .1“ D . . .. y - I a. o . U x - 7 . . (m . L. . . § . 5 . . L. ., . - a. . . .. . .7 m . .o A V . a 1 . .C . 4 . a 7 ., n .2 . u I .- ... . V ; .0 If. . . .xn‘! .\ . w , v .. M... u. I Q a. . . I .\ 1b . Os. . . \ . I. 1 ...I\. m .15 1 1. a . b .s . .r . u A; \ if .lw ‘. . . . - I ._ . V I ~ | \ I A I. q ‘ '§ . n .. .... u.- .. ..... - . 1 . . . f . .xu \ h I r. .. . . . . x t u ' I . k...“ x i k u . .. 1‘ ... v ( i .. .n. .l .. .. .«i u. . v .\‘r. s \ o e A . . . ., C. , n . A. n — voltage. Data could not be secured for higher values of torque because of the low rating of the generator. Test N0. 4. Speed- Torque Curve of Output Motor with‘ Input from Amplidyne This test is a duplication of Test N0. 3 except that the armature voltage is supplied from the amplidyne. The amplidyne field current was held constant in each part. The circuit used in the test is shown in Figunal}. The torque and speed of the motor were measured by the same method as used in Test No. 3. I. ‘ . Cosbfml v 1 ”afar FMId ' OZH’po‘I’ - J; A ———- {Ecru Phfibr 1n ' , ____J toss» ‘ ' 0" ‘ . 4.5 80 Figure 13 DJ’MMO‘M‘ r The data taken in the above test is as follows: Part A Vm : 109 v 11 = 20 ma. N - rpm F- 020 T‘ftolbo 2710 0.8 0.025 2580 4.0 0.125 2420 8.0 0.250 2220 12.0 0.375 2100 16.0' 0.500 <1960 20.0 0.625 1815 24.0 0.750 1670 28.0 0.875 0 2.05 - 19 - Howard E. Gerlaugh 1” MM“- - '.| I . l p O i V P u r a 'l - Del- - z i 3 a 1' r .7 ‘ ... u , I iv: 7 .. U. C a t r , e t. \- u. ... 9. I. It n.» In! It . .9 ._ . . x. I r . .c p u n - \ .- . . . ... . . . . . . . . .. ,. .. . . . . . . . . . . .H. . ... .H .1 . H.. .. H HgH . .. 4..-... . H . . H . H .. H . . . . ... . . . . ., ..i . . ,. . . . . . . . ,. .. . . 4 -1119-11... 1.1 . 19-11 1.1 . .1111 11 . 1.1 -1. :1 . .111.-. 11.4 1111 .1 .11..-.1111411- 14.7-1-1-11. . . -. - 1. - T. . .1 . 1 - .11. . . . - - .11- ...11 i -- -4 11-1 11 1 L1 1- 1.911111 2a.. . L 1 ,_ .. i . . . :7. ..... ,. . . i ., 1 . \, . . .H.. . . .v. P. H .. .H...H. . H .H. . . . .. H . . ..y -. .. . . H. A . - v .r .h . . . o . ... . . + . H .- . . H . . . x . . . . . . .. 4. .H . .H .H . .. a . . . . . H . . . H . - . - . - . , 4 . 1 1 1 K , .H.1 q- - .. . . w. . . . . H ..H. H . .H H . .‘ .4- 1.1 3?... . H- H... .. .1 a. l “..._g M H . H . .. ... i l. \: ...". F . ... . . _. . . m ... H . fl. _ . H . . . u . . .\ . ....«... . ... 0 W1 1 I11. .111! 1dro1. w-1bl1. H111 . 1|4.l1. 1 1H. . . 1. . 1. v .141 11.1.1- 10Hv-1ll 1 LT1W1IMIY'IIIH 11016 I ”It 1 ...... a W11 o I Hi 1 1 .911 1 - 1 I -110 v a 11 v11 1 s «V I t H 9 011.1011: . . i . ... H. . H _ .. . - . .. ,. i . . ”Wufi. H H. H . H l . .. . . . . . . . \l . . . .. H . . q . . . .. . . . . . .+ . 1 . 4 1, ”1 . .. l. K . ,. 1 _ . l . . . . . m. . H. . . . H i H ., . . H . H .H‘ . H . . 1 _ v . . . . e - . — . _ . . . H . . . . : i H l . : _ \ _ H : H . t .11 A 11 - H1 11 '1- 9 1 A 1 w . 1T o . 9 1.911 11* 11 111 . v «11 I11 11 1.11111 n 1 . F1 1 .11 111110 1 .1 1 1 . .. - 1 o 1 $1 - 1 1 9 q . .. H.111 ‘1 ' 11H. l 1 p .1 1 1 *11 11.. o D . u . H 1. 1LT» a 1111‘ 111. L . . . . .. ... ... . . .. . l . H . H . .U . o . .. . m \ . _ H .. . 4 . . _ . . H. l H .. . . . . . . . 4 H 4 1 .1 V . 114111111 11H 11 .1111 1 . 111-117-1171 . u f U . l .. H 1: w H . _ . a \_. . .. H U U _ H _ H a. . H . H . H . r . . l . . . . . . . . _ _ . ,1 , . i _ _ _ .. . . a _ \ _ . . . H - . 1 ... v .1111; -.- 11-.. 1.11-1.- 1.11 3+. . 1 . 1+1-.. 1 4 I - 1.1 . . .1 4111 - 1.111.114 211.. 11- . . 1 1117 .11- .1 01.11 - - 71v . 11 11 v . -1 A .-. . ..1 1 11.. - . 1 l 1 . 11 i 11. 1 01 1,» 111 4.141.111- - .11-. . . . . . . . _ . . ... .. . . . .. H . _ H . ..H . . H- . . H . H \. h H H H. . H . . . n . . . . . . . . . n H . \ . . . - 1 - N1 1 1 F .H .4 . . H . . H- .- 1._--_.1. 53H 1h. ,Ht . 1. 9 -~—--4._._‘. 1 1 l 1 1 o - o l.\ 1 -.HH- I 1 1... “17’“ 1 1 1.-H "“ -3. 11HHH -.--.h . 5 . ‘ \ . a: 1 1 7. 1 1 1 1 1 1 T ~ I 1 ......_-.- u u o 1 1 A Y , n 1 2-1.- I Q 1 ‘ h—{H‘~-.4 1 v .- 1 oH—Hyowau. .... . l l L . H H . n a 1 2.1- . .1: i . 1. - . l . .1111? 1.... iv . 111 I . .. . 1 . -1. . 0.1-1.1111 01 11. r . 11» o .o . - 1- .1 . o . H . _Y . a . . . H . . . .- \ . ... n . H . . . . . . .. H .0. . H . H . H l \ . H . W11 1 4 ... H . H. n 4 v A. . I _ 1h 1 H W a 1‘ o y . . v H . a . . . .. A . 1 . . ..... . . _ _ a . -1. . . ._ . . 2 . H l ., L H H . ...-l . ... . . - .. . . . . .. . .. .1 v 1 1.- ...-1. ..-+-...--..1 1. -- 1.1-1.1. .11. .1 7:11-17- L- 1 1---- . -.\1. -1. 141-1- 1.11. 11h N . h. H .H . . . . . 1 . U H \ . l H . ..H; - U _ i i i H H H l . H ..x M ._ H H g... - - 11- .1 La- 1 .1. u -- .- 1 . . .. H 4 . . . _ . v. . . . . . . . . . . . . . . . . 1 if” .7 H. m h H H n M .m ”1 a. H. . \H m .. m. .. 7H2- ...-f. ..H .. . . .. . . . . . . . . . . . 1. .. . 1 1 .1; .11 - - 1 1h1n -. 11 1 -1 .11HT.11 . .1 .11- .11 11.-..11- 7111.11.11 .1. .1 , . T. . 11p .1 1-11 H 11.? -1 11 at 11 - . - 1- -11\ - H1 1 v- - 11 H- 1 1H . . - .1. 1 . '11 . . . . .. . . H . 1 .. . . u . . .- 1. . H.. . H H. . ..H. M _ H H 1 . 1 . .. l \ . . a H . ..... 1 -.H H . 1. . . H . .h .. . H . 1 . H H . .. .-H . . . . . p .1 \ . . p . .. , .1 ; . 1. a w _ .1. . 2 . . H H . .. .H.- .H- H a 1 . . . . .. . , . . . c . H . . _. ._ . _ i .. l H ._ . ‘ .x u . . . e . ._ .-.-.19 ..Ho1.-1c 1 . 11H. - 1 11 1. 111 l -1 II? 1 1. . 141- 11 -1 11. . - - 111 - 11 v11 - -+.11. .lc w .1 1.11 - t 1- - -. - 1 u . 4 . 1 I 11; 1 . 1 up I - - o1 . w .. 0.1..» 10.. 11 . ... . . .. e _, _ H . . .. . . \ - a H . 1.0" .. H. . . . .. . . . . ... . ... . . . . . . . .. . h. i . . H .. _ . i L \. . . H . ... .... .. . H . .. . . H . . .H 1P -. . . _ . a . _ r. 1 1 1 1 .1. < 1 II1-11.1 1 > .. s . a . . . a. . l. .. . . . . \- . . .1 i . . . .. . ... . . . . ._ . . .. . i .- . . . .. . ..H H . . . . ,— . .. . . . . . . i 1 .. . . H14....--.1-_--.-..:....:-:1-1.1.4111l1---.-HW- 1-.-- . -..-.11 .. 1 . -1 .- . .. -. . ...... . . . . «I. . . > . v . .. ... .H . .. .. H. . m .H H .. .4.. H . . _ a . 1 .H. . w H. H - . i. . . . H .... f ..L.TH . .. . _. H . f H. ._ A l . . . . H .H . W . . 4 9 . .. . . . . . . . . . . _. - . . . ... . . -H .... . .. .. . . . . _ . - . . H i H T-.1+-..1...H11-Hv1!.1. 7-11....111-11 H .-v .-1- H111 . 1- 1. 1 . . ........ 111.- 1.1+.-11o-1 42 -11.. I . ., . . . -. - . l . . . .o. . Hi. f l . H u ... H H U. H ..h . 1 . . l . . 4 . 41111-11111- 11. w 1 + . . . . . . H. o . H. . . .H . .. .4 . . H.... . . p . . . . . 1 T 0 OIJIW 1011i :1 9 v1hlo|n 0 1f 6 11 u e - :1 o .1 q :1 - I It. 1 I I u 4 1 .1+11 1L1: 11.. L a 11 I 1111 .. . _ . . ... ., .. ....- H. ..... 4 . .... . a. . W. .w. H 1» .1 w p. 1.1.1.. . w . xi . H _ i H . h ... ....4 . . . . . 1 u . o v .- ... . . . . . t T q 1 1 A.H.I1u1o.|n 1 1. 111“”...1- -1 1 c 1.11 - - w . 1. - 1r 5 1 W1. c r ..41 ..... 1.1 111 .0111 1.11 t 111W 101111 .H .-.. . . H . l . . . . . . . . . H ... H :H. _ . . H , ; 4r » 5.111.711 w11 ... . v H . . u c a .. ..... 1H.- . M . .. H ... . ..-. .H-. « .... . H . . . v H .o. . .L-h-«11.wp1.-.-1-1...1H1.1.1.1 -- -1 w 1H 111 -7 - .11 H 1-1L-.1. . 4. . ... . .H. . H . l H . .., .. .. .. .... .. .. . ... . . .. .. ...... ..-.H.... ..r . . l m . .M . .. ... Part B Vm : 109 v 11 : lO ma. N- rpm F - oz T - ft. lb. 1620 0.4 0.0125 1500 4.0 0.125 1375 8.0 0.250 1245 12.0 0.375 1185 16.0 0.500 O 1.425 Figure 14 shows the curves plotting the above data. The points showing the blocked rotor torque are to the left of the values indicated by the extended dotted lines because of the saturation in the motor armature. There is a greater decrease in the blocked rotor torque at 20 ma. because of the saturation in the amplidyne field. From these curves the K of equation 11 was determined and is found to be “Lg—g.- 0,-sef+.1b./ma1. (45) In equation 13 and 24 the motor torque is equal to the force due to inertia and the frictional force which is proportional to the speed of the motor shaft. The torque necessary to overcome the friction is given by T= FPeo - (216) 01“ 1- F= ‘55. 1471 This is the inverse of the slope of the motor's speed- torque curve; the friction coefficient becoming - 1-56 .. -3 th. ‘ F- ”130,111.28 .. 9.151110 {5 find/sec. ‘ (48) 6o - 2O - Howard E. Gerlaugh ..M I 1‘. Test No. 5 Saturation Curve of Amplidyne A saturation curve of the amplidyne was run in order to determine if there was a linear relationship between the field current and terminal voltage and to determine the constant K1 of equation 9. The circuit is shown in Figure 15 and the data taken in the test follows. ‘fluwbl ‘e Figure 15 11 ' ma Egt - volts 0 5 2 32 4 69 5 109 8 151 12 229 14 266 15 298 18 326 20 348 The data is plotted in Figure 16 and K1 is determined to be K.'= ig—i— =|9.22uo|+s/ma;. (49) ‘ o The voltage is shown to be linear up to about 17 ma. - 21 - Howard E. Gerlaugh : 1 . ZML- :7 i I ”i , E ' . ” "7 1 I - I” f I I 7 M I I I..‘._ ‘ I I ’ ' i‘ I I f : . V I I, g I ‘ * M 7 A - $§*t , ‘ - I I“ Q 1 _' . 5 w m— , ~ j 13 I ‘_ i :.-..I-: Q Q I, a-“ Q _, i. 3 I k3 3 '~ , If u. r f I (rm , - , I K” ‘ 1 Lag“; I ,_ i ii 3 , 1* * ‘f i I ~-~-—(20» » . j i Q leaf- , ’ 3 1 I a? I '3 Ir , I :9 ' If: I" *J“;-Im I w l : II 7*, ' I I I . : “II -' If I, 3.4 71/494, Tia/V Cat/Rm ; “g "I "I ‘ . I I“. AMPLIDVNE aidikyo: ’5' “ ’ = , " I Déuoavsrzmnay JIM/7f 1:? I I ~ i * ‘ I I ‘ I I ‘ “1‘ WI I I TWA. I I . I ' I " . I ' I . If ; . I II a I O- IL’MLAIYI ”I? I I 9 II fl ___4 ‘ I . ‘ b 4..-. «a; g I4? i, ILL/5 ‘ Wz'o ; ' ' ‘ 1[Wanna Cale r-m , I I; I I ' Test No.6. Amplidyne Output Voltage vs Current In studying further the characteristics of the ampli- dyne a current vs. voltage curve was taken. The circuit diagram is shown in Figure 17 followed by the test data. 1-—-.’-.-— —-———_—-—— —--—--’ r lflofbr GD Ewn - 80: «1c Confr‘a/ field flm/b/Idyne Figure 17 Part A Em : 109V’ 11 : 10 ma. E0 - volts i2 - amps 216 0 191 0.33 175 0.50 154, 0.75 137 1.00 118 1.25 E0 - volts 12 - amps 362 310 0.52 286 0.75 260 1.00 236 1.25 In the curves plotted in Figure 18, we see that there is not quite a linear function between the field current and armature current because of the saturation in the field at 20 ma. The curves are not exactly parallel; whether this is due to the fact that the machine was cold during the 10 ma run or to plotting is not known. - 22-0 Howard E. Gerlaugh I | - . ' ' . ‘ u ' ' ‘ v . ' I , ‘ '7 u v - ‘ w v ‘ a D I " , . ' v \ . I u ‘ " . u not -- ..a .- ~ ' -‘ ‘ J 9" -l‘ an. - (A. h' n- .7 I- - 4' a, ... .x , _, - ..- u ‘-uw-¢.n.v-'-¢-- r, r V. I - L o s ‘& . l | l ‘ - .v'. \ ~ '\ ~ ”-1" N n" ' . . ' " ‘ - . . ' - n‘ ." ‘ * - . f . \-‘ u . .- P ’ - A ‘ -' g . .I 7 so . — - .-. ‘ - ' I .-.o' 'L r n. ‘ ' O | f n : \ 0 n . f . i v "‘. .-.. - -. ~ - ~ -- - - .2. ~ ' \ ~ I. -_ ..t '. -"\-v...,v, k‘. I I l‘ ‘ f“ ‘ -‘ R. , . ‘ ‘ v ‘0 ' .. . 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V .V . . . I V V 7 * V V V n V w . V . V I V V V V V . . V V V V V V V V . V V V V. I . II) V I III¢.. VIIIIII IVVV III: a . IVV .o I I I .VVII V I I V ~ II V + I VI III.— I. III I V VIII. III III V I; I|II+IV H I V I V . V V V V. V. . V. V V V V . V. V V I V _ _ V v H V V. v V V V V H V V V V V V V . V V V. V . . V V . V . V . V . V V V V M V _ V .V 4 < I 4 1 V V . . V V . . . .I d V V V I V V V V . V V V. . .V V . V .V V V. V V n V . VV ._ V . r. ‘ a V . + V, V V . . . . . :~ . ..V V . I .V I V V . . I I. ILIIII I 4. V I I I:V?II . I . I I I I o I V V +VI V Y I . I .I I III V. II II. o . .IIIIII. I I I IV 1 . I VIII V .14 IVIII I I V I . V V III I V .VII. I. II. IIIIJIIII V V V V . V V V V V . .V V _ V V V . V, V . V V V V V V V . . . . I V V . V V V V . V V V V V. V V. V . V . V I H V . fl V I V I . . V I " V V . + . V V .V V VI I III III V IIII [ITII .IV I I.IV9II.IIIII+VIVIIII. I IIIIVI I, II IVIIIIIIIIIIIIVTIIII V I II VI IIIIIIII Ara . a. .V . a . V . V . . I . . V V . . . V. V V V V V. V «I V _ V V + V V . V . V V . m V . V T .. V V I V e . V . no. V . . .V I . . V . V . . V V . V V”. V . h V V V I . v I III‘ I III *VIII . III I I III IVYII I V H . I I I. II V I .IfI II It V II ..III A .III. II v V I I III F I IIIVI Vol I III III . I W V— » . V V _ . . I . V . V . V V V V .. V I I V V V V I C I Ii . V V V V —. V V V . . V. V . , . . V V — I V. V V V I I . 1:. V V V I . I V M V . V . . V V V V _ V . . H u . I _ V V. I IV . V I V I I. . V I V . . V V V . V . V V . V . V H V I V VI V I V V . .I IV V I IFIIIII I I I + w I I I I V. V . . V _ H . V V V V V V V V H . V. V . . V V. V .. V . V VV. V V V V V V V V V .V V V V V V V .V . ..VII V . V V . V . V ... V V r V V V . V . . V V V . . V V . I V V I V V . V V V V . V V. V a V . V V . o . V V . V V . I V V V V _ V V . V H II I I V V V VPV I V I II. V- I I I V 4: IIIV . I I I I I I . I I I». I . III I I I I III . I III I + I V I, 0 I I III. I V v IIII V I I I III?! II I VI III I I I; .V I I I Y I I I V V II I III . I I IV? IVIII . . V . III VIIIII . o IIIIII vII I I II I III I I II I I o II III I a II 11 .V. . VI. . V V. H . V V . V V . V . V; V V V I .V I I . V V . .V I V V . V V . V . .V I .V a V . V V * V V . . V . I V“. _ V V V. .V V V V . . . .V I V R . .V . V .V _ . . V V I I V - III.I.IIII4IIIIIIII + II V . I. III. I II I Lv IIIIVVIIIIIIII. II V I I V ¢ . V V VV V V V V V .V V _ V V V 7 V . . . ¢ I . . .fi . _ . V c I - V o . V V * V V _ V . . V V V V I . . V V . V V . V V V V V V . V, . . . V V V . V V V I I . . I IIVV . .IV I. I IIII II I II . IIV I I . V . V*III 4 V II II V I I V OI V I II VI. .VII. .. III. I V ..V. . V I I oI I I .sVI I . ... II I I I . V .V V V V w _ . V V . . . V .. V V V _ V H V V V V V V. V V . . V V ~ . I V I . V v . V V _ . V V . .V o _ V — . I . ‘V V . V V V V _ V V _ V I I I V V V . I V VI _ 1 I 4 V V . . V V V . V . V V V V V V. V_. V . V V. V V V V . . V V . V V _ V _ V V V . V V V V . V V V . . V V V ~ — V V V V . V-.I....V..I ...I- .. .VI..L ....I IV IIII- .I.VV.I V .I. V. VI. VVVVV V . VIII VIII V V . . N _ _ V _ - V . . V . V .V V . V _ . V _ . V . k _ . . V V . V V V V V. .V V _ V V V V V V V V. , V V V V . H V V .V V . . V V V V. VII. IVII; m VIIIII . I III oVII I .VVIIII .VIIII I . IIIIVVrIIIV p III. V Ifi I I I. I I I QIIII+I I II I. IIIII V VIII IV III. I I II . II I I IoVV I IV «III; II + IL II I III! IVIIIIIIIII ?ILI¢IIIIIVIIVIIIIII IIL . . V . . V . . V V V V . . V V p ... V . + V . V V H . V V V V . V V . V . . V V V . V V .. . V I .V . V . V V V . V H . . .m V . V V . V n . . V V . V V V V V V . . I H a .IV V . Vfi . V V . V V V . V V V V V V V V V V . . V . . II V. V V. IIVIIH V I II +I I .+. I V. «IV V V.V. VI V I I I V . I.II .. I V VII. V . IIII . II. IV I II. III .VV II vI V 11! I . v V II. II .V III. I . VII..HVIII. IvIIII VIIIVIIII w; 1. VIII I I H . TVI V V. V V . V V . . V . . V .V . V . V V V . V V V V V V . V V V . V V V . V V V. H V I . , V . V V V _ . H V V V . V V .V . V . V I V V I . V V. . ..V . V V .. . H . n V V V V V V V I V V V . . V . V a V V L I I h . V V ... V V V V V k V . V V . V V V _ V . V + . V VI V V V I V V . I _ V I V I I V . I k V r I V I V 4 V V V V V V . .. V . V V V V . . V . . . . V V .V I . _ V V V .. V V V V . H . V V V . . V V V. V V V v V V V V V V V V V V V V _ V V . V. VI V . V .V . V . V I. V .4. V . V V . V V . . . V V V V V V . . . V. V _ .VL I . . II . . I y _ V .V V V V V .V V V V . V V V V . V V V V . V .. I. V V V I V -V . V . V a I .V. II. Iv...IIV.V.III .VIV. I. . .VIII~V I I .V I v . V VVI . .I. . II VV.L. ILI . I.I II; IIIIVIII VVI.III I+III..?VI III III .oIIVIIIrVVIVIVIIVIIIIVVVII . II .HIII V V . o w . I V I V fl — I . V .4. .. V . . V a .V .. V I V V. . . . V . . V V V . I . a V . V V _ V V . . V V . V V . V V V V V. . V. V j V . .. V V V V V . V V V V V . o . . . V V V V a I . V V I , L I I V V H VV V IL I I I I I I _ 1 4 V . V I I4_I I H V . V V V _ V V V I V. . . . V . . I . . V . V V . . V- V V V V V . . V V V . V V V . V _ . V V V . . V V V . V 4 V V V V. .V . V V V V . . . V V V V . V. V V . b V . V . V V V . V V V, . . V V V. . V V V . V . . m V V. V V V . IV I I V I I IILYI I V . .II I I I + VII I I I I I I v I I I I I Y I I I I I I V I m I I . I VIII. I . t. I II I I V I II I H I I I M II I I I I III I +IV I IVYI . . I I I III III III . III I I V A III I II..- I V I I I IolIII .11.?le It. I I V4 III I 5 I I . A . V . V V V V V . V V V . . . . V I .V . . V . . . V V . a. . V _ . V V“ . V V . V V V I V I. I V . V V I .V. V .V V V V V . V . . V V V V .V V V V V V I . .V V V . V . .o V V I V . . I . V . . V V . V V. V . . p I I . ._ _ V V V V V VI V V V V V D I IV V V I ‘I 4 .- I o . V V. . V m . I V I V . V. I“ V V . V. V I .V . A . . . VI. .V V.. fiIVVV V. V. .V V. V.. . V V V.. V .V V . . ,. V V V I I I I . W. V . V . I . I V V I . I V V V * , I . . 0 I V I . II . V .~I V V I V .V V . VI V I V. V . V . . V . V r I V .. V V . V . V. . V V V V I V V . _ . , . . IV. V V V + .II I .V V . V +I I V V V V V ~ V V . I V V _ V . I V .V 1 V *. VI V o — V. V . . I I V . [III IIWVVIII a I. I III. I o. I I I IIIIIIIIIVIV I I . L 4 4IIII VI. ‘ IIII!I.4. I II.I«II I I tI.IIII «ILIIVIIII. IIIIIVIII+IVIVVI:I IJI.I IIIILIVIVIIIIIIIIIII I I [LIIIIIIVV I.IIIIVIIIIIIIIIIIII.IIIII9II.II1II+VI IIIIIIVYIIII IIIVI. IIIFIIII I III I V «. LVV .V V. V I. .V. V IIVV ... V V V” V. v V . . . .V .V . V .V V .V V V .V . V V V. . V. V V . V V I . . V .V V . . . V . V V V . V V.. V. m . V V V V I . ..V V V .V V . . V V v w, . _ . V. a V. . . .V o .— V . V V . . . . . .V .. . V V. V V . V V. . III I III IIII I IIIIIIVII.II IOVIIII VII IIII V II. VIIIIIII IIIIIIIIIII Iiwlo ..4IIIIIIII IIII+I .III I »1 1 I I I II L V V. V V . V a . . V V V . o . .7 V. n V I . H V V V V. . V V V V . V . . V . V . .V. - . .. V V . . . V . . V . V V V . . V I V o I .V V . — V V V V V . I V w . V H V V . V I. V v v _. V .V . . V . V V . . V . IV I I I .. . V . V . . V I I . I.I I I.V~.VI I I I H I III I vI I I I I oi Io I I . . I I V . I . IF. II . I . I . IL IIIIOI . V. II I v . h V H. _ V . .HI . . . V V V . I IV I u I II VII . A IIIIIIVw I I 7. III. III I III I IlilIlololv II III I I I III III. I I I III I IIII . _ V . V . . V A . . I V V V . I V V V V 4 V Icl V v I . V — V . . . I V . V V _ V . V. . .V V V V _ V . r k V V V .V V V. V . V V V V . V V V V . V . V V V V V . . V V . H . I a w . . . V V n V . I V . . I . m V v V I . c . V V . V V V I I V V V V. V V V V V In I Test No. 7. Curvé of Armature Current vs. Torque In order to determine the constant 03 in equation 24 a curve of armature current vs. torque of the output motor was made. Also a check as to linearity of the torque and armature current can be made. The same equipment was employ- ed as in Test No 3 with the addition of an ammeter in the! motor circuit to measure the armature current. .taken in the test is plotted in Figure 19. The data The armature vol- tage was taken from the lab supply and held constant at the values indicated. Part A Em = 108v Ef = 108 v 13 - amps F- oz T-ft. lbs. 0.14 0.6 0.0188 0.20 2.2 0.0688 0.27 4.0 0.125 0.36 6.0 0.1875 0.43 8.0 0.250 0.50 10.0 0.312 0.52 11.2 0.350 Part B Em : 220v Ef i3 - amps F - oz T- ft. lb. 0.14 0.5 0.0156 0.21 2.0 0.0625 "0.28 4.0 0.125 0.38 6.0 0.1875 0.44 8.0 0.250 0.52 10.0 0.312 0.58 12.0 0.375 0.75 16.0 0.500 0.91 20.0 0.625 1.06 24.0 0.750 1.20 28.0 0.875 The constant 03 which is the inverse of the slope of the curve is found to be 3 2.9.9. 3 0,334. {-11 Ib./4-m P. C: 0572 - 23 - Howard E. Gerlaugh l 2" ifmk7 , 1 mm : 7' 7 ! t '3 1'7 :. 3 I I 7 -. . _-__._1. ....._-. .. 7 7 7 7 x J J MAME: 7:1 7 I 1 5r- F t. r i. {P u use ' ‘ [-920 a i 0740* 7; had ~_'J_o7ad7 1.294 P 1.20 - »> - » WWW“ 7—3 - ——— ; r 7 17 . 7 ' we *7- . ; V ‘ ' ARM 741R; Cafimz j ‘. * ‘ -A , w , ‘ k5 089M F:QR .55.?[0 ' -7 . ' g 7 I 7 7 I '{ 0a7’Rarl‘707‘on7 Test No. 8 . Gain of Amplifier Without Feedback 7 In this test we will determine the relation between the angular error and the resultant field current at the output of the amplifier. The rotor of the selsyn trans- former on the output shaft was blocked, making the angular displacement of the selsyn gererator on the input shaft the error. Figure 20 shows the arrangement of the equipment in making the test. The feedback voltage was zero. Selsyn er or?» , L mnsf . fim/b/m’y’nc 6 ;,.¢:: 0 ”WP/deer Field 0 5:13;,» Gen: \ 80v ’ 7 in 60~ F 0 a; igme2 G — teeth V1 -v01ts,ac il-ma 1 -ma 9-degree (11-12)-ma -10 17 0 31.0 -19° . -41.0 -8 1308 O #100 ‘1502 -41.0 '6 10.6 1.0 40.5 '11.4 -39.5 ’4 701 204 3900 -706 -36.6 -3 5.2 5.4 ‘37-} "5.7 —31.9 -2 3.4 10.7 .33.4 ~3o8 -22.7 -1 1.7 16.8 27.7 -1.9 -10.9 ~% 0.9 20.0 24.8 -0.95 -4,8 0 0.08 23.9 21.3 0.0 2,6 % 0.23 26.1 19.2 0.95 6.9 1 1.04 29.8 16.0 1.9 13.8 2 2.8 36.4 9.2 3.8 27.2 3 4.4 41.0 4.6 5.7 36.4 4 6.0 43.5 2.4 7.6 41.1 6 9.5 45.0 1.0 11.4 44.0 8 13.2 45.0 0.5 15.2 44.5 10 16.3 45.0 0.0 19.0 45.0 Equilibrium position: V1 3 0032‘, ; 11 = 2402 ma; 12 = 23 ma. 190 teeth in gear. - 24 - Howard E. Gerlaugh . ... . - . a . .\.\ . u \. . 7. . .... ..-7...... . . . 2 u c o a I o n O m ‘1‘. n‘ .l\\ ‘ . ., _ . o I O O c o I I I O u . . . . _ w _ -: .... 1 . 2 1 . u A . . .. c O o o o a g D o o _ v . u . .... a. .. ..\ r... .7. . . -' .\ .. . J .- . 4 . . . . . . . p s . * a 4 ., I O u a o c c v 9 a - M - I. .s't- "U v 0 it&’5.. t- into». '.|' .IL 05'! 7 7 L fi.‘ 7 7 RFth 1 1 7 P ‘ l - fez/a 1-1 ‘ 7 l + 7 l - .~ ""."Il'-l . . 7 ......ll . ‘ 7 ‘ g .. l 1 ---.l. 7-.- 2 .7 -' . 77 1 7 - (”g/R19 yq ' u; 5.9;; 7 ._ 7 i 1 619461196 ,- Gifugu‘, .17 an}? . I . . . l ... . . ..V m “7. 7 . _. ,— H7, fl . . «7 . - .7- N 77477 . .. 7 . - 7 .. 7 . 7... . _ 77....- 777. 7 7L1 .... . 77.7 7 77 77 77 775.77 . .71 I . 7.7 . M .. 7 - m 67_7:- .. 77.7, 2.7;--,.o--.7- 2.777 . --7 a, _ m M. . . -m, ‘ .H_...7..7 ; 0.: \ .umxnk§fix<\7fi.m\7w¢ . _ . _ . 7 .. .. . . . . . : . .. .7 . . . 1.4.. r. .11 4 . 7 ‘7 7 .. 7‘ 7 . . I, n 4 The curve in Figure 21 shows that the resultant field current is nearly a linear function of the error up to about 5 degrees. The humps on either side of zero are due to the error voltage flattening out. The gain of the amplifier “1 is -ALhLEzJ. .. §_§_Z. +2.3, . A" A e " a:.1+3.~u=."‘57’3a 389majrad. (50) A graph of individual field current components vs. error voltage is shown in Figure 22. Test No. 9. Gain of Feedback Circuit of Amplifier The feedback voltage is applied to an RC differ- entiating circuit which feeds a signal to the grids of the tubes which is proportional to the rate of change of the error. This circuit effectively increases the damping of the system without increasing the steady state error to any appreciable extent. In the test the rotor of the selsyn transformer was blocked and the input shaft set at a fixed error for each part of the test while the feedback voltage was varied. The circuit is shown in Figure 20, The values of R and C used in equation 15 are: R: 1.2 megohms and C: 2 microfarads. The data taken in the test is as follows: Part A e = O hEG- volts 11 - ma i2-ma (11'12)' ma. 0 25.1 2205 166 502 2102 2761 “5.9 10 18.1- 2908 “1107 15 15.0 32.4 ~17.4 20 12.0 4.2 -22.2 24.9 10.0 5.9 ~25.9 30 709 3702 -2903 - 25 - Howard E. Cerlaugh Part B 9 n 2.50 hEo ~volts 1 -ma 1 -ma 1 -i -ma 0 141.0 25.0 ( 36 2) 5 40.1 6.1 34 10 59.5 7.8 31.7 15 38.0 10.0 28.0 20 36.8 12.2 24.6 25 34.9 14.8 20.1 30 31.9 18.2 13.7 Part C 9 a -2.5° ' hEO-volts 1 -ma. ig-ma. (i -i )~ma. O 1862 3508 1'2 .6 5 6.2 3606 -3004 10 409 3702 -3253 15 356 3801 -3405 20 207 3805 "3508 25 2.0 38.9 ~36.9 30 1.5 39-2 '37-7 The curves in Figure 23 are not parallel because of the fact that the tubes draw appreciable grid current and the input transformer becomes saturated at higher values of feedback voltage. In the determination of the gain u2 in equation 15, an average value was determined to be 4.: ffilfl-Rp 12%;? = 0.462 uoH’s p.71; (51) . ‘ . IV. Calculations and Performance Curves. In the last section the constants of the servo system with and without feedback were determined. As the next step we will make the substitutions and draw a polar plot or Nyquist diagram of the transfer function. A sinusoidal input will be assumed and it follows that the error and output will also be sinusoidal functions of the same fre- quency but differing in phase and amplitude. Since this is to be a sinusoidal analysis, the differential equations will be solved by making - 26 - Howard E. Gerlaugh P‘J‘” (52) Equation 14 represents the transfer function of the servo without feedback; when the constants have been substituted it becomes ' 3/3 69'- 2.300.000 Be 3 ask p4+ee.17?5+7eee“§’-+aasop '53) Letting p become jw we have 2-52 N - - . (54) {w [17.578 + 0.001092w‘]+) [-915 -o.o 708 571]} to" In table A are given the computations for the polar plot 565133 of 8 and the resulting curve is shown in Figure 24.“ When w = 0, the vector B is at minus infinity and as the fre- quency is increased swings into the third quadrant. Since the plot crosses the real axis to the left of the point (-1.0) the system is unstable.- This was confirmed when the servo was operated without any feedback signal. Taking the case with feedback §oltage and letting h: 0.33 we shall plot the Nyquistdiagram of the transfer function. Upon substitution of the constants, equation 30 becomes .- ... 7 87235: 979.6 + aeitgho" . . " (55) 24p (l.715p3+Ioo.9 p‘+ 7453 75-;- 689) _ and when p: jw - 1 'il ' '1 . +7364- 703' (I 9 8 no) (56) 319 < . * ' Be " 24C§7nez7u3~|4539+i(1'°°-e“"*689)] The computations are shown in table B and the curve in Figure 25. The two curves shown are from the same data - 27 - Howard E. Gerlaugh .. w- .. 7Q.l. I I? . . u .r i . is . c r . . . 4 h a ...! I . C 7. ..r‘ 5‘ ...... w ... . . O 7o 7. 9A . .I COO . o . . . 75D 5 J n . . I. , . I. .x . u . 0 v o . . . 7 a 5 . 7. .. o . . t . I 5, | 7 . . ..— OI. .. .7 .~ ‘5‘ . n a V n. f ‘ . I C .l n a , . 7 .. 90 . . . ~v . . . . . .7 I O. I. 7 u . r. . an 5 . I.- . . - .. . . {7. 4 .- . r .- .. . .- . ‘7 I I . 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" {I - \ \ x \ '\ ‘7 ‘x \ K ‘ ‘ .7 J ,; V ' :v‘ "‘ ”v 6 , " N . ._ w... ‘ 1;“ ~‘ ‘- :3. w .' ~§ ‘. Lt. -. ”‘1‘"-.‘vv O“ V § 3 M . ‘ i :2 .. ,- and J ‘u, 3 K “I ‘ . ' \ a ‘ v ‘v ‘ I u '- X \ x ,. ‘ '1 '- J} i. "i ... ‘ , U H I ‘ l. . l‘. "J.’ ‘ Q. Q Q I.. \ 5‘ l_‘ x u | “ i \ ~‘ ’ “ v‘ ... h ‘ ’ ‘ fo‘. ‘. '- ‘ a! &\ b ... a ‘ ‘ 0" “ .“ ,1 \I \ ‘. \ \ \ \ \ .“‘ ‘ . . ‘ -.g. ~ 0 a, ' \ .aa ‘3‘ \ "V ‘ :‘n .‘3 k'. .I ,‘ L" vvl\ _ - I "' \.‘.‘,'_1\ ~‘\ 2 I l‘ '/ V/ ‘5.- \§ .4 a“. K ‘3, .“ \fl I “ \ \ l '1 \ ‘ I \ \ ‘, \ ‘I 7%; 1 - _.- ‘LALA A__._ but are drawn to different scales in order to bring out its general shape and to see its characteristics when crossing the real axis. It crosses at the point (-0.196,0) and the servo is therefore stable. The radius of the circle with center at the point (-1,0) and tangent to the transfer locus is a measure of the sta- bility of the system and its dampening constant. It is interesting to note that no matter what type of damping the system employs, all systems with the same damping coef- ficient will have the stability circle tangent at the same point. With a radius of 0.615, the damping coeffi- cient is 0.45. Some period of time was spent in trying to obtain the curve in Figure 25 experimentally. The loop was opened by blocking the rotor of the selsyn transformer on the out- put shaft and disconnecting it mechanically from the out- put motor. Selsyn generators were connected to the input and output shafts so as to obtain an A.C. voltage proportional to their respective angular displacement. The input shaft was then driven back and forth about the zero reference point sinusoidally. This was done by mounting a long rod eccentrically on the shaft of a drive motor and connecting the other end of the rod to the outer rim of the flywheel geared to the input shaft . As the motor revolved the input was moved with a sinusoidal motion of about 2.5 degrees amplitude and with the same number of cycles per second as the motor made revolutions per second. - 28'- Howard E. Gerlaugh My The speed of the motor was varied for each test and an oscillograph used to photograph the voltages from the input and output selsyns. The amplitude of the error was constant and the am- plitude and phase of the output could be then measured from the oscillographs. Using this data the polar plot of the transfer function could be made. The difficulty encountered in the above procedure was due to the fact that the output shaft of the servo did not oscillate about one point but slowly "crept". This was caused by not having the input vibrating about the exact zero error point of the system. This point is very critical and changes with small fluctuations in line voltage and operating temperature. Further investigations into experimentally obtaining this polar plot of the transfer function might use the output voltage of the amplidyne and convert it into the output displacement by certain derived expressions. - 29 - Howard E. Gerlaugh -.\ I. 4. 5. BIBLIOGRAPJY Mac Coll, Le Roy A: "Fundamental Theory of Servomech- anisms," D. Van Lostrand 00., Inc., Eew York, 1945. 9. James, Hubert 127.; Richols, Rathaniel R.; Phillips, Ralph 8.: "Theory of Servomechanisms," Kc Graw - Hill Book Co., Inc., New York, I947. Lauer, Henri; Lesnick, Robert; hatson, Leslie E.; "Servomechanisms Fundamentals," MC Graw— Hill Book Co., Inc., New York, I947. General Electric Bulletin, "Description Arplidyne Servo Demonstrator," General Electn.c Company, Schenectady, N.Y. , I943. "The Amplidyne Generator," Reprint from General Electric Review, march, I940. '30- Howard E. Gerlaugh (1 I!