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Modeling, analogue and tests of an electric machine voltage control system Dawson, Graham E. 1966

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MODELING, ANALOGUE AND TESTS OF AN ELECTRIC MACHINE VOLTAGE CONTROL SYSTEM 'by GRAHAM E.' DAWSON ( j B.A.Sc, University of British. Columbia, 1963. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS, FOR THE DEGREE OF MASTER OF APPLIED SCIENCE. in the Department of Electrical Engineering We accept this thesis as conforming to the required standard Members of the Committee Head of the Department .»»«... Members of the Department of Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA September, 1966 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y . a v a i 1 a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r ex-t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n -c i a l g a i n s h a l l not be a l l o w e d w i t h o u t m y . w r i t t e n p e r m i s s i o n . Department o f E l e c t r i c a l Engineering The U n i v e r s i t y o f B r i t i s h C o l umbia Vancou ve r.,8, Canada Date 7. MM ABSTRACT This thesis is concerned with the modeling, analogue and tests of an interconnected four electric machine voltage control system. Many analogue studies of electric machines have been done but most are concerned with the development of analogue techniques and only a few give substantiation of the validity of the analogue models through comparison of results from analogue studies and from real machine tests. Chapter 2 describes the procedure and the system under study. Chapter 3 describes the methods used for the determination of the electrical and mechanical system parameters. The analogue model of the interconnected system is developed by studying the induction motor-amplidyne subsystem in Chapter 4 and the synchronous motor-dc generator subsystem in Chapter 5 . Both transient and steady state tests are carried out on the analogue and on real machines to substantiate the validity of the analogue models. The subsystems are then interconnected and the interconnected system analogue is further verified from transient and steady state tests on the analogue and real machines. Although the system under study is specific, the methods of modeling, analogue and testing are general, there-fore, they can be applied to the transient and steady state studies of other electric machine systems. i i TABLE OP CONTENTS Page ABSTRACT i i TABLE OF'CONTENTS i i i LIST OF ILLUSTRATIONS v LIST OF TABLES v i i ACKNOWLEDGEMENT ix 1. INTRODUCTION 1 2. PROCEDURE AND SYSTEM UNDER STUDY 2-1 Procedure 3 2- 2 System Under Study 5 3. PARAMETER DETERMINATION 3- 1 Self Inductance 7 3-2 Mutual Inductance 9 3-3 Speed Voltage Coefficient 11 3-4 Resistance H 3-5 Friction Coefficient, . 12 3-6 Moment of Inertia , 15 3-7 Induction Motor Torque 16 3- 8 Synchronous Motor Torque 17 4. STUDY OF INDUCTION M0T0R-AMP1IDYNE SET 4- 1 Voltage and Torque Equations 19 4-2 Parameters 22 4-3 Analogue Setup 22 4- 4 Comparison of Analogue Study and Real Machine Tests 25 5. STUDY OF SYNCHRONOUS MOTOR-DC GENERATOR SET 5- 1 Voltage and Torque Equations 31 5-2 Parameters ; • 33 5-3 Analogue Setup 33 5-4 Comparison of Analogue Study and Real Machine .. Tests 34 i i i Page 6. STUDY OP THE INTERCONNECTED POUR MACHINE VOLTAGE CONTROL SYSTEM 6-1 System Equations 39 6-2 Parameters 41 6-3 Analogue Setup 42 6-4 Comparison of Analogue Study and Real Machine Tests 44 6-5 Effect of Anti-Hunting Potentiometer Setting on System Stability 47 I 7. CONCLUSION 62 APPENDIX Machine Ratings 64 REFERENCES 65 iv LIST OF ILLUSTRATIONS Page Figure 2-1 Procedure for Modeling, Analogue and Tests 4 2- 2 Interconnected Voltage Control System • 5 3- 1 Circuit for Self Inductance Measurement 7 3-2 Amplidyne Field Winding Current Response 8 3-3 Circuit for Mutual Inductance Measurement 9 3-4 Circuit for Speed Voltage Coefficient Measurement 11 3-5 Plot to Obtain Speed Voltage Coefficient 12 3-6 Effect of Iron Losses on Friction Calculations .. 14 3- 7 linear Approximation of the Induction Motor Torque Curve 17 4- 1 Circuit of the Amplidyne 20 4-2 Amplidyne No load Test 27 4-3 Amplidyne load Test 27 4-4 Amplidyne Output Voltage Transients 28 4-5 Amplidyne load Current Transients 29 4- 6 Amplidyne Anti-Hunting Current Transients ....... 30 5- 1 Circuit of the DC Generator 31 5-2 DC Generator Open Circuit Characteristic 36 5-3 DC Generator Load Test 36 5-4 DC Generator Ouput Voltage Transients 37 5-5 DC Generator load Current Transients 37 5- 6 Synchronous Motor Speed Transients 38 6- 1 Circuit of the Interconnected Voltage Control System 40 v Figure Page 6-2 Thevenin Equivalent of DC Generator Armature and Feedback Circuit 41 6-3 Analogue Setup of the Interconnected Voltage Control System 43 6-4 Interconnected System at No Load 45 6-5 Interconnected System with Load 46 6-6 Amplidyne Field Current Transients Interconnected System, a = 1.0 48 6-7 Amplidyne Field Current Transients Interconnected System , a = 0.40 49 6-8 Amplidyne d-Axis Current Transients Interconnected System, a = 1.0 50 6-9 Amplidyne d-Axis Current Transients Interconnected System, a = 0.40,. 51 6-10 Amplidyne'Anti-Hunting Current Transients Interconnected System, a = 1.0 52 6-11 Amplidyne Anti-Hunting Current Transients Interconnected System, a = 0.40 53 6-12 DC Generator Field Voltage Transients Interconnected System, a = 1.0 54 6-13 DC Generator Field Voltage Transients Interconnected System, a = 0.40 55 6-14 DC Generator Output Voltage Transients Interconnected System, a = 1.0 56 6-15 DC Generator Output Voltage Transients Interconnected System, a = 0.40 57 6-16 Synchronous Motor Speed Transients Interconnected System, a = 1.0 58 6-17 Synchronous Motor Speed Transients Interconnected System, a = 0.40 59 6-18 Amplidyne Field Current Oscillations Interconnected System 60 6-19 Amplidyne Anti-Hunting Current Oscillations Interconnected System 60 v i Figure Page 6-20 Amplidyne d-Axis Current Oscillations Interconnected System 61 6-21 DC Generator Output Voltage Oscillations Interconnected System 61 6-22 Induction Motor Speed Oscillations Interconnected System 61 v i i LIST OF TABLES Table Page 3- 1 Measurements and Calculations of Mutual Inductance 10 4- 1 Parameters of Induction Motor-Amplidyne Set 23 5- 1 Parameters of Synchronous Motor-DC Generator Set. 33 v i i i ] ACKNOWLEDGEMENTS I want to acknowledge with gratitude the guidance and encouragement given to me by Dr. Y.N. Yu during this thesis study. Acknowledgement is given to the University of British Columbia and the National Research Council of Canada for f i n -ancial support of the research. I would like to thank Mr. A. MacKenzie for drafting the illustrations and Miss H. Klassen for typing the thesis in fi n a l form. I am also indebted to Beverley, my wife, for her encouragement and for preparing the original manuscript. ix 1. INTRODUCTION Many analogue studies of electric machines have been done. Physical models are obtained, equations are written and analogue studies are carried out. However, these studies are mostly concerned with the development of analogue techniques. Verification of system behaviour from analogue studies by comparison with machine tests is seldom given,especially for interconnected machine systems. Krause and Thomas ( l ) , applying the d-q :coordinate transformations, studied on the analogue computer transient phenomena of a 3-phase induction motor with unbalanced stator voltages and unequal rotor resistors. Krause (2) also carried out transient studies of 2-phase and various types of single-phase induction motors. Dineley and Glover (3) modeled a synchronous generator on an analogue computer to study voltage effects of capacitive loading. The effect of saturation of the main axis flux was simulated by a function generator. O'Plaherty and Aldred (4), using the analogue computer, studied a synchronous-machine and transmission system with symmetrical and unsymmetrical faults. Stabilities of various faults at different locations along the transmission line were determined. Symmetrical components and Park's equations were used to set up the system on the analogue computer. In a l l the previous studies no comparison was made between the results from the analogue computer and those from the actual tests. 2 To c a r r y o u t t r a n s i e n t a n a l y s i s o f 2 - p h a s e a n d 3-p h a s e i n d u c t i o n m o t o r s o n t h e a n a l o g u e c o m p u t e r , H u g h e s a n d A l d r e d (5) u s e d m e a s u r e d p a r a m e t e r s . T h e y v e r i f i e d t h e i r m o d e l h y a c o m p a r i s o n o f r e s u l t s f r o m s t e a d y s t a t e s t u d i e s o n t h e a n a l o g u e w i t h t h o s e f r o m r e a l t e s t s . W i t h t h i s m o d e l t h e y s t u d i e d s t a r t i n g t r a n s i e n t s o f a 2 - p h a s e i n d u c t i o n m o t o r a n d u n b a l a n c e d a n d s i n g l e p h a s e o p e r a t i o n o f a 3 - p h a s e i n d u c t i o n m o t o r . R i a z (6) i n h i s v o l t a g e r e g u l a t i o n s t u d i e s g a v e a d e t a i l e d a n a l o g u e p r e s e n t a t i o n f o r a s y n c h r o n o u s g e n e r a t o r i n c l u d i n g s a l i e n c y , a m o r t i s s e u r w i n d i n g a n d s a t u r a t i o n . W i t h a p p r o x i m a t i o n s , h e s t u d i e d a z e r o p o w e r f a c t o r l o a d . H i s c o m p u t e r r e s u l t s o f no l o a d a n d l o a d s t e a d y s t a t e v o l t a g e s a n d t r a n s i e n t r e s p o n s e t i m e s . c o m p a r e d f a v o u r a b l y w i t h t e s t r e s u l t s . A l t h o u g h t h e t e c h n i q u e s o f a n a l o g u e c o m p u t e r s t u d i e s o f e l e c t r i c m a c h i n e s y s t e m s h a v e b e e n v e r y w e l l d e v e l o p e d , a c o m p l e t e c o m p a r i s o n o f r e s u l t s f r o m a n a l o g u e s t u d i e s w i t h t h o s e f r o m a c t u a l m a c h i n e t e s t s i s s t i l l l a c k i n g . I n t h i s t h e s i s , a n i n t e r c o n n e c t e d f o u r m a c h i n e s y s t e m , c o n s i s t i n g o f a s y n c h r o n o u s m o t o r - d c g e n e r a t o r s e t a n d a n i n d u c t i o n m o t o r - a m p l i d y n e s e t , i s u s e d t o c a r r y o u t t h e s t u d y . B o t h s t e a d y s t a t e a n d t r a n s i e n t p e r f o r m a n c e s f r o m a n a l o g u e s t u d i e s a n d r e a l m a c h i n e t e s t s w i l l be c o m p a r e d f o r t h e i n d i v i d u a l s e t s a n d f o r t h e i n t e r c o n n e c t e d s y s t e m . A P a c e 231R a n a l o g u e c o m p u t e r i s u s e d t o c a r r y o u t t h e s t u d y . 3 2 . PROCEDURE AND SYSTEM UNDER STUDY 2 - 1 Procedure The development of a complete analogue of an electric machine system for both the steady state and transient studies involves modeling, analogue, and comparison of analogue results with those from performance tests on real machines.. The pro-cedure is outlined in the block diagram of Pig. 2 - 1 . In order to develop the physical models, assumptions are made for the electric machines; they have linear parameters and there is no effect of saturation. The system equations are derived according to Yu's work (7 ,8 ,9) . They are non-linear differential equations because of the product of currents and of current and speed. Comprehensive tests are performed on the electric machines to determine the machine parameters for the analogue setup. The analogue is set up in such a way that the synchronous motor-dc generator set and the induction motor-amplidyne set can be studied separately or as an interconnected system. However, because of the number of analogue components available only the amplidyne and dc generator are represented in detail. As for the induction and synchronous motors, only their torque equations are included.. Details are presented in the succeeding chapters. Comprehensive steady state and transient studies are carried out on the analogue for, the interconnected system as well as the separate subsystems. The same studies are performed ELECTRIC MACHINE SYSTEM PHYSICAL MODEL EQUATIONS DETERMINATION OF PARAMETERS MACHINE PERFORMANCE —& ANALOGUE STEADY STATE STUDY TRANSIENT STUDY STEADY STATE STUDY 4= TRANSIENT STUDY COMPARISON t VERIFICATION OF ANALOGUE i COMPARISON?--* Fig. 2-1 Procedure for Modeling, Analogue and Tests 5 on the actual electric machine systems. The results from the two studies are then compared. 2-2 System Under Study The system under study is depicted in Fig. 2-2. The machine ratings are given in the Appendix. Fig. 2-2 Interconnected Voltage Control System The dc generator is driven by a synchronous motor. The f i e l d of the dc generator is excited by the amplidyne which in turn derives i t s excitation from a reference voltage and a negative feedback from the output of the dc generator. Although the component machines in the system under study are laboratory size, the system is a typical machine system because i t includes a generator, a motor which corresponds 6 to the prime mover of a large system, an amplidyne motor-gener-ator exciter and a negative feedback voltage in the exciter fi e l d circuit. The system also includes an anti-hunting winding in the amplidyne circuit to prevent instability of the complete system. 7 3. PARAMETER DETERMINATION In this chapter, methods of determining self and mutual inductances, speed voltage coefficients, resistances, f r i c t i o n coefficients, moments of inertia, energy conversion torque expressions of induction and synchronous motors w i l l he developed. 3 - 1 Self Inductance ,..... Based on Saunder's (10) and Thaler's and Stein's (11) works, the self inductances of electric machines are determined hy the transient method. The d i f f i c u l t y in recording transient currents has been overcome through the use of a storage oscilloscope. The circuit for self inductance measurement appears in Fig. 3 - 1 . STORAGE OSCILLOSCOPE STEP VOLTAGE Fig. 3 - 1 Circuit for Self Inductance Measurement The rcachine winding has a resistance R and an inductance 8 I, which are assumed linear. A step voltage V i s obtained from a voltage regulated power supply (Sorenson Type QR 18-1.5A). The current response is recorded on a storage o s c i l -loscope (Tektronix Type 564) by measuring the voltage drop across a known resistor R^  in series with the winding. The result i s plotted on semilog paper with l n ( i _ _ _ - i ( t ) ) as the ordinate and time as the abscissa. The slope equals - l / T where T=L/R+R . The self inductance is A typical example of plotting from measurements at different step voltages for the amplidyne f i e l d winding i s given in Fig. 3-2. 1 = (R+Ri?(t2"t 1) 8-0 4-0 2-0 -0-8 -19f5>V..STEP' — >c 0-4 15V STEP -#—* D 10V STEP -o &• 0 50 100 150 200 TIME(ms) Fig. 3-2 Amplidyne Field Winding Current Response The results from Fig. 3-2 combined with equation (3-1) are for the 19.5 V step L» = (88.8) (200x10"^) = -I no -a f ln(7.87)-In(1.56 ) - u " u o U ' ^3 for the 15 V step and for the 10 V step L^ = (88.8) (200x10^)= ,-, n a rr f ln(7.87)-ln(l.56) i : L - 0 8 H> ^ = (88.8)(200xl0""3)= - i - i A C TT f ln(7.40)-ln(l.48) i : L ' 0 5 H ' The average value of self inductance is 11.1 henries. 3-2 Mutual Inductance The circuit for the mutual inductance measurement appears in Fig. 3-3. -2a*. STEP +^ VOLTAGE vf ) R EXCITATION WINDING STORAGE OSCILLOSCOPE + RESPONSE WINDING V2(t) o Fig. 3-3 Circuit for Mutual Inductance Measurement The mutual inductance is derived by the assumption of linear parameters. The mutual inductance-10 where V = magnitude of the step voltage, 1 = self inductance of the excitation winding, v 2 ( t ) = voltage response, R = resistance of excitation winding, T = L/R, time constant of excitation winding. The storage oscilloscope is used to record the transient response v 2 ( t ) . The results of a typical measurement and computation of a mutual inductance are tabulated in Table 3-1. Mutual Inductance Between Amplidyne Field Winding and d-Axis Winding Step Voltage V = 6.58V Ld= .105 H Td= .023 s Storage Oscilloscope Calculations i : (s) v 2 ( t ) (V) e ' d Mdf(H) 4 x 10~5 1.54 .968 1.00 20 x 10"3 1.39 .852 1.03 37 x IO'5 1.20 .743 1.02 58 x 10"3 1.00 .628 1.00 80 x 10"3 .82 .526 .981 100 x 10~3 .68 .449 .955 120 x 10~3 , ,:54 .382 .880 Average M u d f = .980 Table 3-1 Measurements and Calculations of Mutual Inductance 11 3-3 Speed Voltage Coefficient The speed voltage coefficient is defined as the linear portion of the plot of armature open circuit voltage v Q C versus the f i e l d winding excitation current i when the armature is driven at a constant speed to. The speed voltage coefficient col a f = = ( ¥ ) linear portion (3-3) The circuit for the speed voltage coefficient measurement appears in Fig. 3-4. FIELD WINDING 4 Fig. 3 - 4 Circuit For Speed Voltage Coefficient Measurement A typical example is given in Fig. 3-5 where o o l ^ is the slope of the linear portion of the q-axis open circuit voltage versus the main f i e l d excitation current. 3-4 Resistance The determination of armature resistance of the 12 O o Pig. 3-5 Plot to Obtain Speed "Voltage Coefficient amplidyne and dc generator, including the brush effect of the commutator winding, is achieved by loading the armature circuit. For a fixed excitation, a linear plot of armature terminal voltage versus load current is obtained. Prom the armature voltage drop, i.e. the internal induced voltage minus the terminal voltage, and the load current the armature resistance is determined. Other resistances are measured with a type 1650A General Radio Impedance Bridge. 3-5 Friction Coefficient The f r i c t i o n coefficient measured is due to brushes, bearings and windage. The determination of the f r i c t i o n co-efficient of the complete synchronous motor-dc generator and the induction motor-amplidyne sets i s accomplished by operating 13 the dc machines as motors. The torque equation of the com-plete sets i s written as Te = J t f + f w + T I M where. T = energy conversion torque of the dc motor, J = moment of inertia of the complete set, f = f r i c t i o n coefficient of the complete set, to = speed, T^ = load torque. For steady state operation and without load, T L = 0, equation (3-4) becomes T e = f co (3-5) The energy conversion torque of the dc motor can be expressed as (3-6) T e =(|_) L a f * a i f where P = number of poles, l a£ = a coefficient; tol^being the speed voltage coefficient, i = armature current of the motor, i = exciting current of the motor. The combination of equations (3-6) and (3-7) and the elimination of T , gives the f r i c t i o n coefficient f _/TA I..* i _ i (D^V^ • (3-7) The substitution of I f found in section 3-3 results in f =fl\ V o c ( i f ) ^ a (3-8) 2J cos co 14 where v (ip) = open circuit voltage, a function of the oc i fi e l d excitation i ~ and the speed oo . i s Equation ( 3 - 8 ) is based on the assumption that there are no iron losses in the machine. This can be approximated through the use of small exciting current. Fig. 3 - 6 demonstrates the effect of iron losses on the calculation of f r i c t i o n for the induction motor-amplidyne set. "572-0X/0" 5;> 10-OxW3 ^ 8>0x10~3 UJ O .j £ 6-0x10 U. UJ 8 4-0x10'3 O -3 £ 2-0x10 o U, n 500 EXCITING CURRENT Lf* "14 A —«—»-if" -35A 1000 1500 2000 SPEED (r/min) Fig. 3 - 6 Effect of Iron losses on Friction Calculations As seen from Fig. 3 - 6 , the f r i c t i o n coefficient varies over the speed range. A mathematical expression f i t t i n g the test results i s ( 3 - 9 ) f (oo) = c 1 + 1 c2oo + 1 15 For the induction motor-amplidyne set f(o>) = 4.50x10^+ 1 84<co<210 (3-10) 2.44co+l and for the synchronous motor-dc generator set f (co) = 1.20xl0~ 2 + _ i _ 5 2 S w s U 7 ( 3 - l l ) These results are used to calculate the moments of inertia of the two sets. 3-6 Moment of Inertia The moment of inertia is obtained from a retardation test. The test is performed at no load and the f i e l d and armature circuits are opened simultaneously. The speed variation is sensed by a tachometer with i t s output signal displayed on a storage oscilloscope. Since T g = 0 and 1-^ = 0 from equation (3-4), one has T dco — + f oo = 0 (3-12) Equation (3-9) i s substituted into equation (3-12) and the result is integrated as follows, to i?2 > 4- J dt = I .dco = \ " 2 ^ dco d \ ' J cof(co) J c2co+l Cj^co +co(c1+l) J . - L resuiting in oo^  oon • t 2 " t l = i f L _ i n a U ) l + b + C l i n ^ 2 - l n f V ^ J c 1 | c^ +1 aco2+b c 1 +l co1 aco2+b = J u - f i n ^  - : J L _ l n a w l + b ^ +1 aco 0 + D 16 (3-13) §ince c ^ l as observed from equation (3-10) and (3-11), equation (3-13) reduces to t -t-''9~*1 _1_ i n J '" c-j_ \aa) 2 + l 3y and the moment of inertia J = °1 ( V * ! ? n ,-co.,+b\ (3-14) ln /a +^bN\a,oop+b/ where a = c-^ c^  and b = c +1 = 1. (3-15) These are the equations used to calculate the moment of inertia. 3-7 Induction Motor Torque Since the induction motor operates near the synchronous speed, a linear approximation of the torque curve shown in Fig. 3-7 is used in this thesis study. The approximation is expressed in the following form T J M = -d co + e (3-16) where d and e are constants and T I M = 0 at synchronous speed. The torque curve over the complete range of speed is obtained from a circle diagram constructed from no load and load tests and stator resistance measurements. 17 5 STRAIGHT LINE APPROXIMATION OF TORQUE \ SYNCHRONUOS SPEED «t SPEED OJ (rad/s) Pig. 3-7 linear Approximation of the Induction Motor Torque Curve 3 - 8 Synchronous Motor Torque The steady state energy conversion torque of a salient pole synchronous motor is P. S S SS = 1 EV sinS+ V2 ( xd ~ Xq) s i n z S l (3-17) S W g P d 2 x d X q where P = synchronous motor steady state mechanical power s s output, to = synchronous motor speed, E = excitation voltage, V = terminal voltage, S = torque angle, angle between E and V, x • direct-axis synchronous reactance, X q = quadrature-axis synchronous reactance. 18 For the transient study, the torque expression is approximated t by using a transient reactance x^ instead of the synchronous i i reactance x, since x = x . The transient reactance x-, can d q q d he determined from a 3-phase short circuit test. 19 4. STUDY OP INDUCTION MOTOR-AMPLIDYNE SET The modeling, analogue and tests of the electric machine voltage control system are carried out in three steps; the study of the induction motor-amplidyne set, the study of the synchronous motor-dc generator set and the study of the inter-connected four machine system. The f i r s t step, namely, the induction motor-amplidyne study is included in this chapter. 4-1 Voltage and Torque Equations The amplidyne has four windings on the stator and two commutator windings on the rotor. The d-q coordinates are applied to describe the machine with the d-axis on the fie l d winding axis and the q-axis in quadrature. The f i r s t stage consists of the f i e l d winding " f " and the q-axis commu-tator winding "q". The "q n winding is short-circuited through a series compounding winding "s" to provide the excitation in the q-axis. The relative motion with respect to this f i e l d generates a speed voltage in the d-axis commutator winding "d". A compensating winding "c" is supplied on the stator of the d-axis to compensate for the magnetomotive force of the "d" winding. To stabilize the system to which the amplidyne i s connected, an anti-hunting winding "h" is connected across the output terminal of the "d" and "c" windings through a capaeitor C. The degree of stabilization can be adjusted by an anti-hunting potentiometer R. The details of the amplidyne circuit are given in Fig. 4 - 1 . 20 Es- MMF POLARITY Fig. 4-1 Circuit of the Amplidyne According to the MMF polarity of the windings in d i -cated in Fig. 4-1, the voltage equation of the f i e l d winding due to self and mutual induction in the d-axis can he written v f = (R f + p l f ) i f + p(M f c-M f d)i d-pM f hi h ( 4 _ 1 } The voltage equation of the "d" and "c" windings v = w ( Ldq + lds ) V p ( Mdf- Mof 5 i f + P ^ o h ^ d h ^ (4-2) 21 The voltage equation of the "q" and "s" windings 0 = (R +R ) i +p(L +1 +M +M . ) i -col _ i . v q s' q * v q s qs sq' q qf f (4-3) - w ( L q C - L q d ) i d + W V i h The voltage equation of the anti-hunting winding loop The voltage equation of the anti-hunting potentiometer v = R(i d-ig-<*i h) (4-5) For a resistive load, the terminal voltage of the "d" and "c" windings v = R T i ' (4-6) . l g In this thesis only the speed and1 torque variations of the induction motor are of primary concern, not i t s steady state and transient currents and voltages. Therefore, only the torque equation of the induction motor is used, i.e., -dco+e = Jpco + fco + poles i d i ( L a a + Ias^ 2 (4-7) i q ( V i f + ( L q c _ L q d ) i d - L q h i h ) The l e f t hand side represents the linearized portion of the torque o f the induction motor near synchronous speed, equation ( 3 - 3 ) . On the right hand side of equation (4-7) the f i r s t two terms represent the acceleration and f r i c t i o n torques respectively and the remaining terms, the energy conversion torque of the amplidyne. 22 These seven equations provide the basis for the induction motor-amplidyne study. 4-2 Parameters The parameters in equations (4-1) through (4-7)» determined by methods described in Chapter 3* are listed in Table 4-1. 4-3 Analogue Setup are written in the form of equations (4-8) through (4-16). Parameter values of section 4-2 are substituted into these equations. The computer voltage signals which represent the variables of the actual system must be amplitude scaled between two voltage bounds. The upper hound is the maximum operating voltage and the lower bound is a voltage large enough to suppress the noise voltage. For this study time scaling is not needed. The induction motor-amplidyne voltage and torque equations are solved for the following variables; i ^ , i d , i ^ , i ^ , v, i g and co. The f i e l d current of the amplidyne from equation (4-1) For the analogue setup, equations (4-1) through (4-7) 500i„=l 4.52 (I0v f) -7.70(500if) 090(lOi d)+1.0l(250i h) (4-8) The d-axis current from equation (4-2) 107(l0i d) - .526 v (4-9) Self Inductance . (H) Mutual Inductance (H) Speed Voltage Coefficient (P-) Resistance (A) Induction Motor Torque (N.m) Friction Coefficient at Synchronous Speed (N.m/rad/s) Moment of Inertia 9 (N.m/rad/s^) L f = 11 . 1 Mfh. = 5 ' 5 8 M f c = 1.00 M^ = 0.98 Rf = 85.1 L c = .184 - l d = .105 Mdc = - 1 3 5 Mch = .505 Mdh = .503 Mcd = Mdc Mcf = Mfc Mdf = Mfd col. = 3 5.4 col** = 14.4 RH =. 1 . 3 6 R c = « 6 7 1 = .087 q. 1 = .026 s M s q = .045 Mqs = Msq wLqf = 173 i f ^ 7.4 mA col f = 290 if>7.4 mA col d = 38.7 colqC = 38.7 col Q h = 163 Rq = 2.43 R s = ' 5 4 \ = 2 < 8 3 % f =-Mfh M h c = M c h Mhd = % h Rh = 57.0 -1.49co+280 2.63x10-5 .036 Potentiometer resistance R = 306-Q. Anti-hunting winding circuit capacitance C- = 450p.F Table 4-1 Parameters of Induction Motor-Amplidyne Set 24--The q-axis current from equation (4-3) 25i„ = and p 10(.460(500if) - .866(250ih)) _ . 9 2 5 ( 2 5 i )l 100 V / for ifs7.4mA (4-10) 25i„ = 14^8 •/*>(-770(500^) -866(2501^)) . 9 2 3 ( 2 5 I J 100 for 1^7.4mA Two values of L f for the ranges i f<7.4 mA and i^>7.4 mA are inserted, one in equation (4-10) and one in (4-11). i The anti-hunting loop current from equation (4-4) ,4-11) 2501 h - l&L (250i h) + \ 135(lOi,-10iJ -5.40(2501 g - 2p^2 ( 2 5 0 i ) + .050(101,) + .985(5001.) p T . V ^ ^ V ^ The output voltage from equation (4-5) 4 (4-12) v = 30.6(10i d-10i ) - 1.22a(250ih) The load current from equation (4-6) (4-13) lOi = 10 v The speed from equation (4-7) to -3 1_ f-41.4 a + 259o - .977(lOi d)(25i a) P L 100"^  -.148(251 )(.460(500if) -.866(2501^,)) 100 } for i f<7.4 mA (4-14) (4-15) 25 and 3 = i ~ JJ-41.4 w + 2590 - .977(10i d)(25i q) 3 1 0 0 ^ (4-16) - .148(251 )(..77O(50Oif) -.866 (250ihi) P 100 for i f > 7.4 mA These equations are the hasis of the induction motor-amplidyne subsystem computer setup which is part of the interconnected voltage control system setup in Fig. 6-3. 4-4 Comparison of Analogue Study and Real Machine Tests In this section the results from steady state and transient studies of the induction motor-amplidyne set on the analogue and from real machine tests are comparedi Fig. 4-2 shows the no load output voltage characteristic versus the f i e l d current of the amplidyne for steady state operation. There is no load except a small current (0 to 0.35A) in the anti-hunting potentiometer. The results of the analogue are indicated by the dashed line and show good correlation with those from real machine tests indicated by the solid line. Fig. 4-3 compares the results of a load test. A slight discrepancy between the analogue results and the real machine tests is noticed. The results from transient studios are presented in Fig. 4-4 through 4-6. Two transient phenomena are observed, the sudden application of a resistive load and sudden load rejection. Studies are carried out with and without the anti-hunting winding. Fig. 4-4 indicates the results of transient 26 studies of the amplidyne output voltage v, Fig. 4-5 indicates that of the d-axis current i ^ , and Fig. 4-6 indicates that of the anti-hunting current i ^ . It is noticed that a high frequency oscillation occurs in the analogue setup of this study hut disappears in the interconnected system study. The induction motor speed was also observed but no change was noted for steady state or transient operation. 27 70 60 50 • • REAL TEST — *- *- —ANALOGUE 30 40 FIELD CURRENT if(mA) Fig. 4-2 Amplidyne No Load Test Ui o .ft. o 40 30 20 -70-REAL TEST •ANALOGUE 6 7 8 LOAD CURRENT lg (A ) Fig. 4-3 Amplidyne Load Test 28 •4 • *> u -1 u 'J ** TIME (S) TIME (S) (a) Load Rejection, i , = 0 (c) Load Rejection,i , ?^0,a=1.0 •Pig. 4-4 Amplidyne Output Voltage Transients 29 sr 5h 5 o •i •2 '3 -4 -5 TIME (S) Load Rejection, i ^ 0 i —f — r--4 -5 TIME(S) (c) Load Rejection, i^0 fa=1.0 REAL TESTS ANALOGUE 4 5 TIME (S) X 4 5 TIME(S) Load Application, 1^ =0 (d) Load Application, i ^ 0,a= 1 . 0 Pig. 4-5 Amplidyne. Load Current Transients 30 (a) Load Application, a=1.0 •75 r UJ Qc S o l 3: •25 L TIME (S) REAL TESTS - ANALOGUE (b) Load Rejection, a=1.0 Fig. 4 - 6 Amplidyne Anti-Hunting Current Transients 31 5. STUDY OF SYNCHRONOUS MOTOR-DC GENERATOR SET In this Chapter, the study of the synchronous motor-dc generator set is presented. This is the second step in the modeling, analogue and tests Of the four electric machine voltage control system. 5-1 Voltage and Torque Equations The dc generator is separately excited and has two windings, the f ie ld winding "g"- and the commutator armature winding "a", as shown in F ig . 5-1. The voltage equations are =*- MMF POLARITY Fig . 5-1 Circuit of the DC Generator written according to the MMF polarity of the windings indicated in F ig . 5-1. For the f ie ld winding v = (R +pL ) i (5-1) For the armature winding v_ = co L i - (R +pL ) i (5-2) a g a g g ^ a ^ a ' a w / For a resistive load, the terminal voltage of the armature winding "a" \ = E L i a (5-3) The torque equation of the synchronous motor-dc generator set is approximated by o 1 i EV. s i n S + V2(x'-x ) s i n 2S> + (-d'co +e') W S T " ^ - - J 8 (5 -4) = J pco + f co + poles 1 i i g g g g 2 ag g a The derivative of the torque angle = co - 3co (5-5) s g The f i r s t term of the l e f t hand side of equation (5-4) represents the energy conversion torque of the synchronous motor and the second term the induction motor torque due to the amortisseur winding. The f i r s t and second terms of the right hand side of equation (5-4) represent the acceleration and f r i c t i o n torques and the last term the energy conversion torque of the dc generator. Equation (5-4) is used for transient studies to obtain the speed and the frequency of osoillation for a torque disturbance. The same equation is used for the steady state study for con-venience without changing x^ to x^. Although this w i l l affect the torque angle value, i t w i l l not affect the speed. As seen from equation (5-5) , 3co^ w i l l be forced equal to cog when£= 0. 33 These five equations provide the basis for the synchronous motor-dc generator study. 5-2 Parameters The parameters in equation (5-1) through (5-5)> determined by methods described in Chapter 3, are listed in Table 5 - 1 . Self Inductance (H) Speed Voltage Coefficient Resistance Friction'at Synchronous Speed (N^m/rad/s) Moment of Inertia 0 (N.m/rad/s^) 1 = 32.1 g R = 84.4 g L = 5.47xl0"3 ci oo 1 = 136 g ag -R' •= 0.675 a -2 4.4x10 .70 Synchronous Motor Torque T s m = _ 1 _ ( 3 3 0 7 0 s i n $ _ i 4 7 0 sin2S) (N.m) W g Table 5 - 1 Parameters of Synchronous Motor-DC Generator Set 5 - 3 Analogue Setup For the analogue setup, equations ( 5 - 1 ) through ( 5 - 5 ) are written in the form of equations ( 5 - 6 ) through ( 5 - 1 2 ) . The synchronous motor-dc generator voltage and torque equations are solved for the following variables: the f i e l d current from equation ( 5 - l ) 10 i = g 312 v - 2.63(l0i ) ( 5 - 6 ) 34 the armature current from equation (5-2) 2 i a = Ml f 1 > 2 2 + 1 7 > 6 (.75o)g)(lQig) _ . 4 H ( 2 i )) P L 100 J(5-7) the open c ircui t armature voltage from equation (5-2) . V q = 14.5 (.75co )(10i ) . 1 = 0 a g_ g_ y a 100 the load voltage from equation (5-3) '(5-8) \ = \ 10(21 ) (5-9) 20 a the speed from equation (5-4) .75w g : - \ 100 (2.66(100 sinS)-.240(100' sinScosS)') P L .75co « (5-10) 100 J -6.81(.75w )+645-.063(.75w ) - 11.8 (10 g g _ and the torque angle from equation (5-5) S= 10^0 | 1 0 0 ( . 2 l 6 ) - ,229(.75w g)| (5-11) These equations are the "basis of the synchronous motor-dc gen-erator subsystem computer setup which is part of the inter-connected voltage control system setup in Fig . 6-3. 5-4 Comparison of Analogue Study and Real Machine Tests The steady state results of a no load test and load test of the synchronous motor-dc generator set from the analogue study and real machine tests are compared in F ig . 5-2 and 5-3. Because of the linearized analogue model, a discrepancy is observed on the dc generator open c ircuit characteristic in the 35 s a t u r a t e d range. For the lower p o r t i o n of the curve, the r e a l machine t e s t r e s u l t s are obtained a f t e r demagnetization. F i g . 5-4 and 5-5 compare the analogue study w i t h the t r a n s i e n t t e s t r e s u l t s of the dc generator output v o l t a g e and lo a d c u r r e n t . Both l o a d a p p l i c a t i o n and r e j e c t i o n r e s u l t s are presented. The t r a n s i e n t responses of the synchronous motor speed to l o a d a p p l i c a t i o n and r e j e c t i o n are d e p i c t e d i n F i g . 5-6. Because of the number of analogue components a v a i l a b l e , the s a l i e n t pole synchronous motor energy c o n v e r s i o n torque i s not set up i n d e t a i l . Instead, a c o n v e n t i o n a l torque ex-p r e s s i o n i s used w i t h x^ r e p l a c e d by x^„ T h i s had been pre-sented i n equation (5-4). Consequently, i t g i v e s a f a i r l y accurate r e s u l t f o r speed d e v i a t i o n magnitude and frequency of o s c i l l a t i o n . A speed d e v i a t i o n of approximately 20 r/min i n 1200 r/min i s observed. 36 140 120 100 o 80 60 40 20 REAL TEST - * *- - ANALOGUE •2 -4 -6 -8 1-0 1-2 P4 FIELD CURRENT lg(A) F i g . 5-2 DC Generator Open C i r c u i t Characteristic 100-80 *60 Ui to S | 40 Q. 20 = •554 = -58A = .51A REAL TESTS ANALOGUE 10 20 30 40 F i g . 5-3 DC Generator Load Test LOAD CURRENT iQ (A) 37 LOAD REJECTION E S . LOAD APPLICATION -REAL TESTS ± — - ANALOGUE 0 1'0 2-0 TIME(S) Fig.- 5 - 4 DC Generator Output Voltage Transients h*—LOAD APPLICATION LOAD REJECTION REAL TESTS ANALOGUE JL 10 2-0 TIME (S) Fig. 5 - 5 DC Generator Load Current Transients 3 9 6. STUDY OF THE INTERCONNECTED POUR MACHINE VOLTAGE CONTROL SYSTEM The comparison o f analogue r e s u l t s and performance t e s t s on the r e a l machines i n Chapter 4 and 5 i n d i c a t e s the v a l i d i t y of t h e models f o r s t e a d y s t a t e and t r a n s i e n t s t u d y of the i n d u c t i o n motor-amplidyne and the synchronous motor-dc g e n e r a t o r subsystems. As a f i n a l s t e p , the .study o f the i n t e r -connected f o u r machine v o l t a g e c o n t r o l system i s u n d e r t a k e n i n t h i s C h apter. 6 - 1 System E q u a t i o n s The c i r c u i t o f the i n t e r c o n n e c t e d v o l t a g e c o n t r o l system i s shown i n P i g . 6 - 1 . E q u a t i o n s ( 4 - 1 ) t h r o u g h ( 4 - 5 ) and ( 4 - 7 ) o f the i n d u c t i o n motor-amplidyne s e t and e q u a t i o n s ( 5 - 1 ) , ( 5 - 2 ) , ( 5 - 4 ) and ( 5 - 5 ) o f the synchronous motor-dc g e n e r a t o r s e t remain unchanged. Two c o n n e c t i o n e q u a t i o n s a r e i n t r o d u c e d t o complete the d e s c r i p t i o n of the system. F i r s t , the amplidyne f i e l d v o l t a g e i s o b t a i n e d from t h e d i f f e r e n c e of a r e f e r e n c e v o l t a g e V and a feedback v o l t a g e from the o u t -° s , put t e r m i n a l s o f the dc g e n e r a t o r . Secondly, the e q u a t i o n o f the dc g e n e r a t o r output v o l t a g e , e q u a t i o n . ( 5 - 3 ) , i n c l u d e s - t h e e f f e c t of the feedback c i r c u i t i n p a r a l l e l w i t h the l o a d r e s i s -t a nce R-jy These two e q u a t i o n s a r e o b t a i n e d from a Thevenin e q u i v a l e n t o f the dc g e n e r a t o r armature, the feedback, and the amplidyne f i e l d c i r c u i t s as shown i n F i g , 6 - 2 , Fig 6-1 Circuit of the Interconnected Voltage Control System 41 R1 R2 Rl + R-2 rA/V Rf+pLf RP Za  Rp+Za R 2 + vt - -Pig. 6-2 Thevenin Equivalent of DC Armature and Feedback Circuit The amplidyne f i e l d voltage R^  TT R-, R^  .= 2 V - 1 2 i - v 1 R1+R2 3 R1+R2 f a (6-1) The dc generator output voltage Lf a R Z n a R +Z p a R 1^ + P R 4 Z p a co 1 i g ag g (6-2) where R = R f b R L and Z = R +pL . p T, +t> a a ^  a R f b t K L These twelve equations, equations (4-1) through (4-5) and (4-7) of Chapter 4, equations (5-1), (5-2), (5-4) and (5-5) of Chapter 5, and equations (6-1) and (6-2), provide the basis for the interconnected four-machine voltage-control system study. 6-2 Parameters The same parameter values used in studies in Chapter 4 and 5 are used for the interconnected system study. In 42 equations (6-1) and (6-2) the feedback resistor R f l 3 = 54a and the potential divider resistances Rx = and R2 = 40£1 for Vj^ = 91 volts. 6-3 Analogue Setup For the analogue setup, equations (4-8) through (4-13), (4-15) and (4-16) of Chapter 4 and equations (5-6) through (5-8), (5-10) and (5-11) of Chapter 5 and equations (6-3) and (6-4) are used. From -equation (6-1), the fi e l d voltage {• and from equation (6-2) the dc generator output voltage 10 v = 10 j .906(.833VS) - v a 196(500if) (6-3) 54-1 + 51 v = Rl \1+|1y^l+( .675)111, f < -i-(500i^)-i'4'i. a : ' ^ 7^v5Q0 P)-14^4.. - . » . 500 f h+ C . 675)183 P , i (6-4) (.75w a)(lOi g) 100 These fifteen equations are the basis of the computer setup of the interconnected four machine voltage control system. The analogue setup i s shown in Fig, 6-3e Equations (4-14) and (5-9), the load equations of the subsystems, also are incor-porated in the computer setup so that with simple switching AMPLIDYNE FIG. 6-3 ANALOGUE SETUP OF THE INTERCONNECTED VOLTAGE CONTROL SYSTEM 44 the same computer setup is used for the study of either the interconnected system or the subsystems. 6-4 Comparison.of Analogue Study and Real Machine Tests The validity of the analogue model of the interconnect-ed four machine voltage control system in steady state and transient operation i s substantiated by comparing input and output currents and voltages of the subsystems from the ana-logue study and system tests. Throughout the following tests, the potentiometer R^k in the feedback circuit i s set at 100$ for convenience. From steady state no load tests, the amplidyne f i e l d voltage and d-axis current and dc generator f i e l d voltage and armature current versus the dc generator voltage performance curves are obtained. Fig. 6-4 indicates good correlation of the results from the analogue study and the system tests. However, a small current, 0 to 2A, flows through the dc gen-erator armature due to the feedback cir c u i t . The f u l l load armature current is 40A. The results of load tests are pre-sented in Fig. 6 - 5 . The amplidyne f i e l d voltage and d-axis current and the dc generator f i e l d and output voltages are depicted in Fig. 6-5 a, b, c and d respectively. Transient .load application and rejection studies are carried out for the anti-hunting potentiometer set at a = 1.0 and a = 0.40 The comparison of results for the amplidyne f i e l d , d-axis, and anti-hunting currents, dc generator f i e l d and output voltages and synchronous motor speed deviations are presented 45 Uj 3 Q UJ 20 40 60 80 100 DC GENERATOR OUTPUT VOLTAGE Va(V) (a) Amplidyne Field Voltage 20 40 60 80 100 DC GENERATOR OUTPUT VOLTAGE Va(V) (b) Amplidyne d-Axis Current 60r 0 DC (c GENERATOR OUTPUT VOLTAGE VQ(V) ) DC Generator Field Voltage Fig. 6-4 Interconnected System at No Load REAL TESTS 0 20 40 60 80 100 DC GENERATOR OUTPUT VOLTAGE VQ (V) (d) DC Generator Current 46 2-0 CD s UJ k. 1-0 JL 70 20 30 40 50 DC GENERATOR LOAD CURRENT ia(A) (a) Amplidyne Field Voltage DC GENERATOR LOAD CURRENT LQ(A) (b) Amplidyne d-Axis Current 100 r 50 tr 0 JL J_ J 50 0 10 20 30 40 DC GENERATOR LOAD CURRENT (c) DC Generator Field Voltage 700-ki O S 2 2: o 50 •REAL TESTS —ANALOGUE J_ 0 10 20 30 40 50 DC GENERATOR LOAD CURRENT LQ(A) (d) DC Generator Output Voltage Fig. 6-5 Interconnected System With load 47 in Fig. 6-6 through 6-17. The results from the analogue tests are shown with dashed lines and those from real tests with solid lines. The induction motor speed change is so small that i t cannot he detected. For a l l cases, the analogue i s ahle to predict the transients up to 0.4 seconds f a i r l y accurately. For time greater than 0.4 seconds, the analogue produces larger transients than those in the real machines although the time for the transients to decay to zero is approximately the same. The switching transients at the f i r s t instant are clearly' ob-served in the amplidyne f i e l d and anti-hunting currents and the dc generator output voltages from both the analogue studies and real machine tests* 6-5 Effect of Anti-Hunting Potentiometer Setting on System  Stability The anti-hunting potentiometer setting affects the stability of the voltage control system. The system is stable for a>-0.30, Fig. 6-1. For a setting of the anti-hunting potentiometer a = 0.30, the system currents, voltages and speed oscillatei They are observed from both system tests and ana-logue studies. The results are compared in Fig. 6-18 through 6-22. The speed deviation of the synchronous motor i s so small that i t cannot be detected. Good correlation of results is also observed from this study. REAL TESTS - ANALOGUE 48 / (a) Load Application 3 N REAL TESTS — ANALOGUE (b) Load Rejection Fig. 6-6 Amplidyne Field Current Transients Interconnected System a=1.0 49 750 — s 700 s 50 2: Uj 0 oc QC i> o -50 Q ul U. -700 r REAL TESTS ANALOGUE (b) Load Rejection Fig. 6-7 Amplidyne Field Current Transients Interconnected System, a=0.40 50 V2r hO 0-8 ^ 0-6 k i Q: 3 0-4 o co REAL TESTS - ANALOGUE (b) Load Rejection Fig. 6-8 Amplidyne d-Axis Current Transients Interconnected System, a=1.0 j •8 VO TIME (S) 51 REAL TESTS ANALOGUE Fig. 6-9 ? -4 '6 '8 ( b j Load Rejection Amplidyne d-Axis Current Transients Interconnected System, oc=0.40 7-0 52 ki 8 O CD § a: i TIME (S) REAL TESTS ANALOGUE (b) Load Rejection .Fig. 6-10 Amplidyne Anti-Hunting Current Transients Interconnected System, a=1.0 53 (a) Load Application. \ 1-0 ^ _ y TIME IS) REAL TESTS _ ANALOGUE (b) Load Rejection Fig. 6-11 Amplidyne Anti-Hunting Current Transients Interconnected System, a=0.,40 54 100 r 80 60 40 20 REAL TESTS — ANALOGUE J I •8 1-0 TIME(S) •4 -6 (b) load Rejection Fig. 6-12 DC Generator Field Voltage Transients Interconnected System, a=1.0 55 co o Q kl 0 2 -4 -6 (a) Load Application -8 1-0 TIME (S) v5» ki co 5 Q kj k, REAL TESTS —ANALOGUE •8 hO (b) Load Rejection TIME(S) Pig. 6-13 DC Generator Field Voltage Transients Interconnected System, a=0.40 56 140 Ui CO s o 60 40 20 — REAL TESTS — ANALOGUE ± j 1 -2 (b) Load R e j e c t i o n •4 '5 TIME(S) Pig. 6-14 DC Generator Output Voltage Transients Interconnected System, a=1.0 •2 -4 -6 (a) Load Application •8 1-0 TIME (S) REAL TESTS _ -ANALOGUE I 1 1 I I I 0 -2 '4 -6 -8 1-0 (b) load Rejection TIME (S) Fig. 6-15 DC Generator Output Voltage Transients Interconnected System, a=0.40 58 c I 2: o Q Ui CO '20 TIME(S) REAL TESTS ANALOGUE (b) load Rejection Fig. 6-16 Synchronous Motor Speed Transients Interconnected System, a=1.0 20r (b) Load Rejection Pig. 6-17 Synchronous Motor Speed Transients Interconnected System, a=0.40 6 0 REAL TESTS ANALOGUE Fig. 6-18 Amplidyne Field Current Oscillations Interconnected System Fig.- 6-19 Amplidyne Anti-Hunting Current Oscillations Interconnected System 61 7-2^ •2 -4 ' 6 • . * TIME(S) 1'° Pig. 6-20 Amplidyne d-Axis Current Oscillations Interconnected System 720 h 80 40 REAL TESTS ANALOGUE 12-•2 ' 4 ' 6 6 TIME (S)1'0 ?±g. 6-21 DC Generator Output Voltage Oscillations Interconnected System 72 -F i £ . 6 - 2 2 Induction Motor Speed Oscil lation; interconnects: .cjys;;sr;i 7. CONCLUSION Many analogue studies of electric machines have been done. Most of them are concerned with the development of analogue techniques and only a few of them give substantiation of the validity of the analogue models through comparison of results from analogue studies and from real machine tests. In order to f i l l this gap, an interconnected four electric machine voltage control system has been modeled, set on the analogue and the results from the analogue study are compared with those from real machine tests. The model for the four machine system is developed by f i r s t testing the induction motor-amplidyne and the synchronous motor-dc generator subsystems. Steady state load and no load and transient load application and rejection tests are performed on both the analogue and the real machines. After the analogue models of the subsystems have been verified through the comparison of analogue and test results, they are interconnected and tested proving the validity of the model of the interconnected system. It i s found that in order to develop a valid analogue model, accurate values of parameters must be used. The paramet-ers determined by methods described in Chapter 3 are found sat-isfactory. The self and mutual inductances are measured by a transient method. The speed voltage coefficient is obtained from an induced voltage test. The armature resistance, including brush effects, must be determined from a load test* The f r i c t i o n coefficient is not constant. The moment of inertia i s determined 63 from a retardation, test. The energy conversion torque of the induction motor is approximated hy a straight line and that of the synchronous motor hy a conventional torque expression with transient reactance x^ replacing the synchronous reactance V In the analogue setup of the interconnected system, eight multipliers, twelve integrators, thirty-four summers and inverters, and two function generators are used. Out of seventy-two amplifier units sixty-six are uti l i z e d . In addition, seven comparator relays and sixty potentiometers are used. Complete sets of induction and synchronous motor equations and the saturation effect of the dc generator, synchronous motor and amplidyne could he included in the analogue setup i f more components were available. The results of the study, however,show that the ana-logue can be used for the study of complex problems. Although the problem of this study i s specific, the methods developed are general for the study of other electric machine systems. Since the analogue setup is based on equations with system' parameters explicitly expressed, i t can also be used for design studies. APPENDIX'S MACHINE RATINGS General Electric Synchronous Motor 5kVA 4kW 1200r/min General Electric DC Generator 125/125V 40A 5kW 1200r/min General Electric Amplidyne Motor-Generat Induction Motor Input 3 phase 220/440V 7.2/3.6A 3hp 1725r/min DC Output 125V - 12A l,5kW 3 phase 110/220V 26.2/13.1A Excitation 125 V 3.5A maximum-65 REFERENCES 1. Krause, P.C. and Thomas, CH,, Simulation of Symmetrical Induction Machinery. IEEE Paper No. 3 1 TP 65-120. February 19, 1965. 2. Krause,. P.C., Simulation of Unsymmetrical Two-Phase Induction Machines. IEEE Paper No. 31 TP 65-121, February 19,' 1965. 3. Dineley, J . l . and Glover, K.J., Voltage Effects of Cap-acitive Load on the Synchronous Generator1, Pro-ceeding IEE, Vol. I l l , No. 4, April 1964, pp.789-795.' 4. O'Flaherty, T.M.M. and Aldred,- A*S.. Synchronous-Machine Stability Under Unsymmetrical Faults. IEE Paper No. 3996S, October 1962. 5. Hughes, F.M, and Aldred, A.S., Transient Characteristics and Simulation of Induotion Motors. Proceedings IEE, Vol. I l l , No. 12, December 1964., pp 2041-2050. 6. Riaz, M„Analogue Computer Representation of Synohronous Generators in Voltage Regulation Studies, _ AIEE Transactions, Vol. 75* Power, December 1956, pp 1178-1184. 7. Yu, Yao-nan, The Impedance Matrix.and Analysis of Commutator Machines. IEEE Paper No. 71 CP 65-61. 8. Yu, Yao-nan, The Torque Tensor of the General Machine. IEEE Transactions on Power Apparatus and Systems, February, 1963, pp. 623-629. 9» Yu, Yao-nan, The Impedance Tensor of the General Machine» AIEE Transactions, pt* I (Communication and Elec-tronics), Vol. 75» May, 1956, pp. 181-187* 10* Saunders, R»M,Measurement of D^-C Machine Parameters. AIEE Transactions, pt. I, Vol. 70, 1 9 5 1 , PP« 700-705. 11. Thaler, G„J. and Stein,- W.A., Transfer .Function and Para- meter Evaluation for D-C Servpmoters. AIEE Trans-actions, pt. II (Applications and Industry), Vol. 74, January, 1956, pp. 410-417. 

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