Modeling, Analogue and Tests of an Electric Machine
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 presenting this thesis in partial fulfilment of the
requirements for an advanced degree at the University of British
Columbia, I agree that the Library shall make it freely.avai1able for reference and study. I further agree that permission for ex• tensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.. It is understood that copying or publication of this thesis for finan• cial gain shall not be allowed without my.written permission.
Department of Electrical Engineering
The University of British Columbia 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.
ii TABLE OP CONTENTS Page
ABSTRACT ii
TABLE OF'CONTENTS iii
LIST OF ILLUSTRATIONS v
LIST OF TABLES vii
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
iii 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 vi 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
vii 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
viii ]
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 fin• 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 final 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 all the previous studies no comparison was made between the results from the analogue computer and those from the actual tests. 2
To carry out transient analysis of 2-phase and 3- phase induction motors on the analogue computer, Hughes and
Aldred (5) used measured parameters. They verified their model hy a comparison of results from steady state studies on the analogue with those from real tests. With this model they studied starting transients of a 2-phase induction motor and unbalanced and single phase operation of a 3-phase induction motor.
Riaz (6) in his voltage regulation studies gave a detailed analogue presentation for a synchronous generator including saliency, amortisseur winding and saturation. With approximations, he studied a zero power factor load. His computer results of no load and load steady state voltages and transient response times.compared favourably with test results.
Although the techniques of analogue computer studies of electric machine systems have been very well developed, a complete comparison of results from analogue studies with those from actual machine tests is still lacking. In this thesis, an interconnected four machine system, consisting of a synchronous motor-dc generator set and an induction motor-amplidyne set, is used to carry out the study. Both steady state and transient performances from analogue studies and real machine tests will be compared for the individual sets and for the interconnected system. A Pace 231R analogue computer is used to carry out the study. 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 STEADY STATE COMPARISON STUDY
t ELECTRIC PHYSICAL MACHINE MODEL EQUATIONS—& ANALOGUE VERIFICATION SYSTEM OF ANALOGUE i DETERMINATION TRANSIENT OF 4= STUDY COMPARISON?--* PARAMETERS
STEADY STATE STUDY MACHINE PERFORMANCE
TRANSIENT STUDY
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 field of the dc generator is excited by the amplidyne which in turn derives its 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 it 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 field 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, friction coefficients, moments of inertia, energy conversion torque expressions of induction and synchronous motors will 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 difficulty 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 is obtained
from a voltage regulated power supply (Sorenson Type QR 18-
1.5A). The current response is recorded on a storage oscil•
loscope (Tektronix Type 564) by measuring the voltage drop across
a known resistor R^ in series with the winding. The result is
plotted on semilog paper with ln (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 1 = (R+Ri?(t2"t1 )
A typical example of plotting from measurements at
different step voltages for the amplidyne field winding is
given in Fig. 3-2.
8-0
4-0
2-0 -
0-8 - 19f5>V..STEP' — >c 0-4 D 15V STEP -#—* 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"uo U'
L ^3 for the 15 V step ^ = (88.8) (200x10^)= ,-, na rr f ln(7.87)-ln(l.56) i:L-08 H> and for the 10 V step ^ = (88.8)(200xl0""3)= -i-i AC TT f ln(7.40)-ln(l.48) i:L'05 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 +^ R STORAGE VOLTAGE vf ) OSCILLOSCOPE EXCITATION + WINDING 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,
v2(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 v2(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) v2(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 = .980 udf 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 vQC versus
the field winding excitation current i when the armature is
driven at a constant speed to. The speed voltage coefficient
colaf = = (¥) 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 ool^ is the slope
of the linear portion of the q-axis open circuit voltage versus
the main field 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 friction coefficient measured is due to brushes,
bearings and windage. The determination of the friction co•
efficient of the complete synchronous motor-dc generator and
the induction motor-amplidyne sets is accomplished by operating 13
the dc machines as motors. The torque equation of the com•
plete sets is written as
J Te = tf + f w + TI M where. T = energy conversion torque of the dc motor,
J = moment of inertia of the complete set,
f = friction coefficient of the complete set,
to = speed,
T^ = load torque.
For steady state operation and without load, TL = 0, equation
(3-4) becomes
Te = f co (3-5)
The energy conversion torque of the dc motor can be expressed as
L Te =(|_) af *a if (3-6) where P = number of poles,
la£ = 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 friction coefficient
f _/TA I..* i_ i (D^V^ • (3-7)
The substitution of I f found in section 3-3 results in
f =fl\ Voc(if)^a (3-8)
2J cos co 14
where v (ip) = open circuit voltage, a function of the oc i field 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 friction for the induction motor-amplidyne set.
"572-0X/0"5
EXCITING CURRENT ;> 10-OxW3 Lf* "14 A —«—»-
^ 8>0x10~3 if" -35A UJ O .j £ 6-0x10 U. UJ 8 4-0x10'3
O -3 £ 2-0x10 o
U, n 500 1000 1500 2000 SPEED (r/min)
Fig. 3-6 Effect of Iron losses on Friction Calculations
As seen from Fig. 3-6, the friction coefficient varies over the speed range. A mathematical expression fitting the test results is
(oo) f = c1 + 1 (3-9) c2oo + 1 15
For the induction motor-amplidyne set
f(o>) = 4.50x10^+ 1 84 2 f (co) = 1.20xl0~ + _i_ 52SwsU7 (3-ll) 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 field and armature circuits are opened simultaneously. The speed variation is sensed by a tachometer with its output signal displayed on a storage oscilloscope. Since Tg = 0 and 1-^ = 0 from equation (3-4), one has T dco — + f oo = 0 (3-12) Equation (3-9) is substituted into equation (3-12) and the result is integrated as follows, to i?2 > 4- J dt = I .dco = \ "2c2co+^l dco d \ ' J cof(co) J Cj^co +co(c1+l) oo^ oon J. -L resuiting in • t2"tl = i fL_ in aU)l+b + Cl in ^2 - lnfV^ J c1 | c^+1 aco2+b c1+l co1 aco2+b 16 aw +b =Ju- f in ^ - :JL_ ln l ^ (3-13) +D +1 aco0 §ince c^l as observed from equation (3-10) and (3-11), equation (3-13) reduces to t -t- ''9~*1 _1_ in +l3 J '" c-j_ \aa)2 y and the moment of inertia J = °1 (V*!? n ,-co.,+b/aco^+bN\ (3-14) ln \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 TJM = -d co + e (3-16) where d and e are constants and TIM = 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 STRAIGHT LINE APPROXIMATION OF TORQUE \ 5 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.SS = 1 EV sinS+ V2 (xd ~ Xq) sinzSl (3-17) SS W 2x X S g Pd d 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, Xq = 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 first 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 field winding axis and the q-axis in quadrature. The first stage consists of the field winding "f" and the q-axis commu• tator winding "q". The "qn 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 field 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 is 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 indi• cated in Fig. 4-1, the voltage equation of the field winding due to self and mutual induction in the d-axis can he written vf = (Rf+ plf) if+ p(Mfc-Mfd)id-pMfhih (4_1} The voltage equation of the "d" and "c" windings v = w(Ldq+lds)Vp(Mdf-Mof 5if + P^oh^dh^ (4-2) 21 The voltage equation of the "q" and "s" windings 0 = v(R +R )i +p(v L +1 +M +M . )i _i. q s' q * q s qs sq' q -coqlf f (4-3) w(L L )i +W i - qC- qd d V h The voltage equation of the anti-hunting winding loop The voltage equation of the anti-hunting potentiometer v = R(id-ig-<*ih) (4-5) For a resistive load, the terminal voltage of the "d" and "c" windings v = RTi ' (4-6) . l g In this thesis only the speed and1 torque variations of the induction motor are of primary concern, not its steady state and transient currents and voltages. Therefore, only the torque equation of the induction motor is used, i.e., L +I -dco+e = Jpco + fco + poles idi ( aa as^ 2 (4-7) iq(Vif+(Lqc_Lqd)id-Lqhih) The left hand side represents the linearized portion of the torque of the induction motor near synchronous speed, equation (3-3). On the right hand side of equation (4-7) the first two terms represent the acceleration and friction 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 For the analogue setup, equations (4-1) through (4-7) 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^, id, i^, i^, v, ig and co. The field current of the amplidyne from equation (4-1) 500i„=l 4.52 (I0vf) -7.70(500if) 090(lOid)+1.0l(250ih) (4-8) The d-axis current from equation (4-2) 107(l0id) - .526 v (4-9) Self Mutual Speed Voltage Resistance Induction Friction Moment of Inductance Inductance Coefficient Motor Coefficient Inertia9 . (H) (H) (P-) (A) Torque at Synchronous (N.m/rad/s^) (N.m) Speed (N.m/rad/s) Mfh. = 5'58 Lf = 11.1 Mfc = 1.00 Rf = 85.1 M^ = 0.98 Lc = .184 Mdc = -135 col. = 35.4 RH =. 1.36 Mch = .505 col** = 14.4 R 67 -ld = .105 c= « Mdh = .503 Mcd = Mdc Mcf = Mfc Mdf = Mfd Msq = .045 wLqf = 173 1 = .087 Rq = 2.43 M M if^ 7.4 mA q. qs = sq R 54 1 = .026 col f = 290 s = ' s if>7.4 mA col d = 38.7 colqC = 38.7 colQh = 163 \ = 2<83 %f =-Mfh Mhc = Mch 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„ = 10(.460(500if) - .866(250ih)) _.925(25i )l p 100 V/ (4-10) for ifs7.4mA and 25i„ = 14^8 •/*>(-770(500^) -866(2501^)) .923(25IJ 100 ,4-11) for 1^7.4mA Two values of Lf for the ranges if<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) 2501 h - l&L (250ih) + \ 135(lOi,-10iJ -5.40(2501 g 4(4-12 ) - 2p^p 2 (250i ) T+ .050(101,.V^^V^ ) + .985(5001.) The output voltage from equation (4-5) v = 30.6(10id-10i ) - 1.22a(250ih) (4-13) The load current from equation (4-6) lOi = 10 v (4-14) The speed from equation (4-7) to - 1_ f-41.4 a + 259o - .977(lOid)(25ia) 3 P L 100"^ -.148(251 )(.460(500if) -.866(2501^,)) (4-15) 100 for if<7.4 mA} 25 and 3 = i~ JJ-41.4 w + 2590 - .977(10id)(25iq) 3 100 ^ (4-16) - .148(251 )(..77O(50Oif) -.866 (250ihi) P 100 for if> 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 field 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 • • REAL TEST — *- *- —ANALOGUE 30 40 FIELD CURRENT if(mA) Fig. 4-2 Amplidyne No Load Test 70 60 50 40 Ui 30 REAL TEST o •ANALOGUE 20 - .ft. 70- o 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 •i —f — r- •2 '3 -4 -5 -4 -5 TIME (S) TIME(S) 0 Load Rejection, i^ (c) Load Rejection, i^0fa=1.0 REAL TESTS ANALOGUE X 4 5 4 5 TIME (S) 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 TIME (S) UJ Qc S o REAL TESTS - ANALOGUE l 3: L •25 (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 field winding "g"- and the commutator armature winding "a", as shown in Fig. 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 Fig. 5-1. For the field winding v = (R +pL ) i (5-1) For the armature winding co (5-2) v_a = gag L ig - (^a^a'R +pL ) ai w / For a resistive load, the terminal voltage of the armature winding "a" E \ = L ia (5-3) The torque equation of the synchronous motor-dc generator set is approximated by EV. sinS + V2(x'-x ) sin 2S> + (-d'co +e') o 1 i 8 W ST"^-- J (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 first term of the left 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 first and second terms of the right hand side of equation (5-4) represent the acceleration and friction 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 will affect the torque angle value, it will not affect the speed. As seen from equation (5-5), 3co^ will 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 Speed Voltage Resistance Friction'at Moment of Inductance Coefficient Synchronous Inertia 0 (H) Speed (N.m/rad/s^) (N^m/rad/s) 1 = 32.1 R = 84.4 g g 3 oo 1 = 136 L = 5.47xl0" g ag -R'a •= 0.675 ci -2 4.4x10 .70 Synchronous Motor Torque Tsm = _1_(33070 sin$_i470 sin2S) (N.m) Wg 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 (5-12). The synchronous motor-dc generator voltage and torque equations are solved for the following variables: the field current from equation (5-l) 10 i = g 312 v - 2.63(l0i ) (5-6) 34 the armature current from equation (5-2) 2ia = Ml f 1>22 + 17>6 (.75o)g)(lQig) _ .4H(2i )) P L 100 J(5-7) the open circuit armature voltage from equation (5-2) . Vq = 14.5 (.75co )(10i ) . 1=0 a g_ y a g_ '(5-8) 100 the load voltage from equation (5-3) \ = \ 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) -6.81(.75w )+645-.063(.75w ) - 11.8 (10 g g _ 100 J and the torque angle from equation (5-5) S= 10^0 |100(.2l6) - ,229(.75wg)| (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 Fig. 5-2 and 5-3. Because of the linearized analogue model, a discrepancy is observed on the dc generator open circuit characteristic in the 35 saturated range. For the lower portion of the curve, the real machine test results are obtained after demagnetization. Fig. 5-4 and 5-5 compare the analogue study with the transient test results of the dc generator output voltage and load current. Both load application and rejection results are presented. The transient responses of the synchronous motor speed to load application and rejection are depicted in Fig. 5-6. Because of the number of analogue components available, the salient pole synchronous motor energy conversion torque is not set up in detail. Instead, a conventional torque ex- pression is used with x^ replaced by x^„ This had been pre• sented in equation (5-4). Consequently, it gives a fairly accurate result for speed deviation magnitude and frequency of oscillation. A speed deviation of approximately 20 r/min in 1200 r/min is observed. 36 140 120 100 80 60 o REAL TEST 40 - * *- - ANALOGUE 20 •2 -4 -6 -8 1-0 1-2 P4 FIELD CURRENT lg(A) Fig. 5-2 DC Generator Open Circuit Characteristic 100- 80 = •554 *60 = -58A Ui to = .51A S | 40 REAL TESTS ANALOGUE Q. 20 10 20 30 40 LOAD CURRENT iQ (A) Fig. 5-3 DC Generator Load Test 37 LOAD REJECTION ES. 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 39 6. STUDY OF THE INTERCONNECTED POUR MACHINE VOLTAGE CONTROL SYSTEM The comparison of analogue results and performance tests on the real machines in Chapter 4 and 5 indicates the validity of the models for steady state and transient study of the induction motor-amplidyne and the synchronous motor-dc generator subsystems. As a final step, the .study of the inter• connected four machine voltage control system is undertaken in this Chapter. 6-1 System Equations The circuit of the interconnected voltage control system is shown in Pig. 6-1. Equations (4-1) through (4-5) and (4-7) of the induction motor-amplidyne set and equations (5-1), (5-2), (5-4) and (5-5) of the synchronous motor-dc generator set remain unchanged. Two connection equations are introduced to complete the description of the system. First, the amplidyne field voltage is obtained from the difference of a reference voltage V and a feedback voltage from the out- ° s , put terminals of the dc generator. Secondly, the equation of the dc generator output voltage, equation .(5-3), includes-the effect of the feedback circuit in parallel with the load resis• tance R-jy These two equations are obtained from a Thevenin equivalent of the dc generator armature, the feedback, and the amplidyne field circuits as shown in Fig, 6-2, Fig 6-1 Circuit of the Interconnected Voltage Control System 41 R1 R2 RP Za R Z Rl + R-2 Rf+pLf p+ a rA/V + vt - - R 2 Pig. 6-2 Thevenin Equivalent of DC Armature and Feedback Circuit The amplidyne field voltage R^ TT R-, R^ (6-1) .= 2 V - 1 2 i - v 1 3 f a R1+R2 R1+R2 The dc generator output voltage R Z R n a L co 1 i 1^ + P g ag g (6-2) a R +Z f R 4Z p a p a where R = RfbRL and Z = R +pL . p T, +t> a a ^ a RfbtKL 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 Rfl3 = 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 field voltage 196(500if) (6-3) 10 v = 10 j{ •.906(.833V S) - va and from equation (6-2) the dc generator output voltage 54- 1 + 51 f v = Rl a : ' ^< 7-i-(500i^)-i'4'i^v5Q0iP)-14^4... - . » . 500 f h+C. 675)183 P , \1+|1y^l+( .675)111, i(6-4) (.75wa)(lOig) 100 These fifteen equations are the basis of the computer setup of the interconnected four machine voltage control system. The analogue setup is 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 is 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 is set at 100$ for convenience. From steady state no load tests, the amplidyne field voltage and d-axis current and dc generator field 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 circuit. The full load armature current is 40A. The results of load tests are pre• sented in Fig. 6-5. The amplidyne field voltage and d-axis current and the dc generator field 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 field, d-axis, and anti-hunting currents, dc generator field and output voltages and synchronous motor speed deviations are presented 45 Uj 3 Q UJ 20 40 60 80 100 20 40 60 80 100 DC GENERATOR OUTPUT VOLTAGE Va(V)DC GENERATOR OUTPUT VOLTAGE Va(V) (a) Amplidyne Field Voltage (b) Amplidyne d-Axis Current 60r REAL TESTS 0 0 20 40 60 80 100 DC GENERATOR OUTPUT VOLTAGE VQDC(V) GENERATOR OUTPUT VOLTAGE VQ (V) (c ) DC Generator Field Voltage (d) DC Generator Current Fig. 6-4 Interconnected System at No Load 46 2-0 CD 1-0 s UJ k. JL 70 20 30 40 50 DC GENERATOR LOAD CURRENT ia(A)DC GENERATOR LOAD CURRENT LQ(A) (a) Amplidyne Field Voltage (b) Amplidyne d-Axis Current 100 r 700- ki O 50 50 S 2 •REAL TESTS —ANALOGUE 2: o tr 0 JL J_ J J_ 0 10 20 30 40 50 0 10 20 30 40 50 DC GENERATOR LOAD CURRENT DC GENERATOR LOAD CURRENT LQ(A) (c) DC Generator Field Voltage (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 it cannot he detected. For all cases, the analogue is ahle to predict the transients up to 0.4 seconds fairly 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 first instant are clearly' ob• served in the amplidyne field 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 is so small that it cannot be detected. Good correlation of results is also observed from this study. 48 REAL TESTS - ANALOGUE / (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 r 2: Uj 0 oc QC i> o -50 Q ul -700 REAL TESTS U. ANALOGUE (b) Load Rejection Fig. 6-7 Amplidyne Field Current Transients Interconnected System, a=0.40 50 V2r hO 0-8 ^ 0-6 ki REAL TESTS Q: 3 0-4 - ANALOGUE o co •8 VOj TIME (S) (b) Load Rejection Fig. 6-8 Amplidyne d-Axis Current Transients Interconnected System, a=1.0 51 REAL TESTS ANALOGUE ? -4 '6 '8 7-0 (bj Load Rejection Fig. 6-9 Amplidyne d-Axis Current Transients Interconnected System, oc=0.40 52 ki 8 TIME (S) O CD REAL TESTS § ANALOGUE a: i (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 REAL TESTS 20 — ANALOGUE J I •4 -6 •8 1-0 TIME(S) (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 -8 1-0 TIME (S) (a) Load Application v5» ki co 5 Q REAL TESTS kj —ANALOGUE k, •8 hO (b) Load Rejection TIME(S) Pig. 6-13 DC Generator Field Voltage Transients Interconnected System, a=0.40 56 140 Ui CO 60 s 40 — REAL TESTS — ANALOGUE 20 o ± j 1 -2 •4 '5 TIME(S) (b) Load Rejection Pig. 6-14 DC Generator Output Voltage Transients Interconnected System, a=1.0 •2 -4 -6 •8 1-0 TIME (S) (a) Load Application 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 TIME(S) REAL TESTS Ui ANALOGUE CO '20 (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 60 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 •2 '4 '6 6 TIME (S)1'0 ?±g. 6-21 DC Generator Output Voltage Oscillations Interconnected System 12- 72 - Fi£. 6-22 Induction Motor Speed Oscillation; 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 fill 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 first 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 is 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 friction coefficient is not constant. The moment of inertia is 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 utilized. 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 if 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 is 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, it can also be used for design studies. APPENDIX'S MACHINE RATINGS General Electric Synchronous Motor 3 phase Excitation 110/220V 125 V 26.2/13.1A 3.5A maximum- 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 65 REFERENCES 1. Krause, P.C. and Thomas, CH,, Simulation of Symmetrical Induction Machinery. IEEE Paper No. 31 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. Ill, 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. Ill, 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, 1951, 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.