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Efficiency-optimal operating conditions for single-phase induction motors Cummings, Gerald Floyd 1995

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EFFICIENCY-OPTIMAL OPERATING CONDITIONS FOR SINGLE-PHASE INDUCTION MOTORS by GERALD FLOYD CUMMINGS B.A.Sc, The University of British Columbia, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF APPLIED SCIENCE in  T H E FACULTY OF GRADUATE STUDIES (Electrical Engineering)  We accept this thesis as conforming to the/vreauired/standard I  T H E UNIVERSITY OF BRITISH COLUMBIA April, 1995 © Gerald Cummings, 1995  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 available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by his or  her  representatives.  It  is understood that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  £u_E<rrg.i cA L. E*sk,i »o  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  Abstract The double-revolving field model is used to determine theoretically efficiencyoptimal operating conditions for single-phase induction motors equipped with an auxiliary winding. The algorithm developed determines the optimal values of main winding current, auxiliary winding current and stator frequency for any desired combination of torque and rotor speed within the motor's ratings. Computer simulations are presented to validate theoretical results and relate optimal terminal voltages to computed winding currents. Experimental measurements made on a 2 pole, 1 hp, 230V submersible motor are compared with computer simulations. These measurements reveal that while it is most important to control core losses by matching the V/Hz ratio to the load, it is possible to make further efficiency improvements by exciting the auxiliary winding.  ii  Contents Abstract  ii  List of Tables  v  List of Figures  vi  Acknowledgments  vii  1  Introduction  1  2  Theoretical Analysis  6  2.1 2.2 2.2.1 2.2.2 2.2.3 2.3 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.5  Equivalent Circuit Performance Calculations Impedance Matrix Input Power Output Power and Torque Efficiency Optimization of Running Performance Auxiliary Winding Current Phase Auxiliary Winding Current Magnitude Main Winding Current Magnitude Stator Frequency Optimization of Starting Performance  6 11 11 11 12 13 15 16 18 19 19 20  3.1 3.2 3.3  Simulation Results Validation of Theoretical Results Optimal Winding Currents Optimal Winding Voltages  22 22 25 29  4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.2 4.2.1 4.2.2  Experimental Results Apparatus Motor Load Supply Instrumentation Sources of Error Modelling Error Measurement Error  33 33 33 33 34 34 35 35 37  3  4  iii  .  4.3  Efficiency Measurements  38  5  Conclusion  44  Appendix A  Equivalent Circuit Parameters  49  Appendix B  Proposed Implementation  56  Appendix C  Waveform Analyzer Program  62  References  63  iv  List of Tables Table 1 Table 2  Equivalent Circuit Symbols Locked-Rotor Measurements  9 49  Table 3 Table 4  No-load Measurements Equivalent Circuit Parameters  50 51  V  List of Figures Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  Revolving Field Equivalent Circuit Efficiency vs. Auxiliary Current Phase Angle Efficiency vs. Auxiliary Current Magnitude . Efficiency vs. Slip Optimal Auxiliary Current Magnitude vs. Rotor Speed . . . . Optimal Slip vs. Rotor Speed Optimal Stator Frequency vs. Rotor Speed Maximum Efficiency vs. Rotor Speed Optimal Main Voltage Magnitude vs. Rotor Speed Optimal Main V/Hz Ratio vs. Rotor Speed Optimal Auxiliary Voltage Magnitude vs. Rotor Speed . . . . Optimal Auxiliary Voltage Phase Angle vs. Rotor Speed . . . Main & Auxiliary Voltage Main & Auxiliary Current Dual Winding Efficiency vs. Auxiliary Voltage Phase Angle . Dual Winding Efficiency vs. Auxiliary Voltage Magnitude . . Dual Winding Efficiency vs. V/Hz Ratio Single Winding Efficiency vs. V/Hz Ratio Efficiency vs. Mechanical Power Friction and Windage Losses vs. Rotor Speed Measured Motor Characteristics Simulated Motor Characteristics Voltage Utilization of Three Phase Inverter Auxiliary Winding Compensation Link Voltage vs. Run Capacitance  vi  7 23 24 25 26 27 27 28 29 30 31 32 36 37 38 39 40 41 43 53 54 55 57 58 60  Acknowledgments The author would like to thank Dr. W . G . Dunford of the University of British Columbia Department of Electrical Engineering and Dr. J.B. Neilson of POWERTECH Labs for their financial support, patience, and valued advice. A portion of this work was completed at POWERTECH Labs and the use of facilities there was greatly appreciated. I am honored to express special thanks to my father who did an excellent job of designing and constructing the testfixtureused for experimental measurements. Others who have been very helpful include Mr. Lloyd Welder, Mr. Alan Prince and Mr. Dave Fletcher at the University of British Columbia Department of Electrical Engineering. While at POWERTECH Labs, Mr. Brian Ferguson, Mr. Jerry Albrecht, Mr. Walter Zarecki and Mr. Peter Wuntschek were valuable sources of advice. Mr. Kenneth George and Mr. Joe Weber of Franklin Electric Corporation were also very helpful in providing motor performance characteristics.  vii  1 Introduction Single-phase induction motors are commonly used in many applications. A typical home utilizes many fractional horsepower single-phase motors in appliances such as refrigerators, washers and dryers to name only a few. Small singlephase motors are often less expensive to purchase and install than similarly sized three-phase motors and so are often used even where three-phase power is readily available. In rural areas where three-phase power is not usually available, single-phase motors of up to 10 horsepower are in common use. In general the efficiencies of small single-phase motors are quite low. In the past, motors have been connected directly to the utility line and so have been restricted to operation at fixed voltage and frequency. In the past two decades, considerable attention has been given to efficiency optimization of line operated single-phase induction motors with the most common solution being the placement of a capacitor in series with the auxiliary winding^ ' 5  A n alternative arrange-  ment, described by Huang^, is known as the Wanlass connection and places a capacitor in series with the main winding. Substantial efficiency improvements are reported for both connections and these improvements are primarily attributed to equalization of the current densities in the main and auxiliary windings'^. Both solutions are limited by the fact that the capacitance required for optimum efficiency is strongly load dependent^ . In addition, as will be shown, efficiency is 1  1  affected by both the magnitude and phase of the auxiliary winding current and so a simple capacitor can not provide the maximum possible efficiency dictated by motor parameters. As the cost of power electronics continues to decrease, variable voltage variable frequency drives are finding widespread application. As energy costs increase, at some point it may become economical to retrofit existing singlephase motors with such power converters.  Although the greatest increase in  efficiency would result from applying this technology to both main and auxiliary windings, some improvement could be realized by leaving the main winding directly connected to the utility line and powering the auxiliary winding from a variable voltage power converter. Such a system would operate in much the same way as a capacitor run motor with the advantage that both the magnitude and the phase of the auxiliary winding current could be independently adjusted depending on the load. This approach has been used by Collins^ to effect a variable speed drive which is claimed to improve efficiency only indirectly. A variation of the approach is proposed by Lettenmaier^ and although reference is made to possible efficiency improvements, this author's objective is the minimization of pulsating torque. If variable frequency drives are ever to be applied to single-phase induction motors it will be important to know the optimal operating conditions for the motor. Although there are several papers which report efficiency improvements 2  by optimization of the run capacitor^ ' ^ and operation at reduced V/Hz ratios^, 3 5  the author was unable to find any analysis which optimizes efficiency over all degrees of freedom. The theory developed here assumes a linear model and is used to determine the optimal choice of frequency, main and auxiliary winding currents and auxiliary winding phase for any desired operating point. A secondary result is the optimization of starting torque which will be shown to be a special case of efficiency optimization. Variable frequency drives are in some instances already used in conjunction with single-phase induction motors in remote areas where utility power is not available. Here, one of the most common power sources is solar energy provided by photo-voltaic arrays. As the cost of these solar panels is typically $5 per peak Watt, considerable savings could be realized by optimizing the efficiency of the motors they power and so reducing the size of the array required. Starting performance is also a consideration in these applications and so a system which could also optimize starting torque could provide additional value. One example of such an installation is the aeration of remote northern lakes. During the winter when these lakes are frozen over they become low in oxygen and as a result some species of fish do not survive.  To increase the oxygen  levels it is necessary to provide continuous aeration. To this end, a number of installations utilize Air-O-Lator fan type impellers coupled to submersible singlephase induction motors which are powered by photo-voltaic arrays. One motor 3  commonly used in these installations is a Franklin Electric model 2143084416 which is rated for 1 HP at 3450 RPM. The same motor is used here for experimental work so that the results derived in the remainder of this work might have immediate practical value. What follows is a theoretical analysis of a single-phase induction motor. The results of this analysis determine operating conditions which theoretically yield the highest possible efficiency. The majority of this analysis is done in closed form but, due to the complexity of the problem, the final step of the optimization is done numerically. To complete the numerical optimization, a knowledge of equivalent circuit parameters is required and these are determined in Appendix A . Computer simulations are presented in chapter three to demonstrate that the results of the theoretical analysis are correct. Both the theoretical analysis and the corresponding computer simulations assume that the magnitude and phase of the winding currents can be independently and directly controlled. In practice a voltage source is the most cost effective way of supplying power to the windings. A second set of computer simulations is therefore presented to illustrate the effects of changing the magnitude and phase of the winding voltages.  4  In chapter four experimental results are compared with those predicted by theory and in general there is good agreement. Where there are differences, these can for the most part be attributed to inadequacies in the model used for the analysis. Despite these differences the higher efficiencies predicted by theory are shown to be practically realizable.  5  2 Theoretical Analysis  0  2.1 Equivalent Circuit Two of the most common models used for steady-state analysis of singlephase induction motors are the Cross-Field^ Field  10,  1 3 , 9  '  1 2  '  1 4 ]  model.  2  '  1 ]  model and the  Revolving-  The Cross-Field model is based on a D-Q axis  representation of the machine while the Revolving-Field model assumes two sinusoidal flux waves rotating in opposite directions. Although both models can easily be extended to include an auxiliary winding, the Revolving-Field model is somewhat more popular and is therefore used as the basis for this work. A paper presented by W.J. Morrill  [ 1 0 ]  in 1929 is one of the earliest and most  complete derivations of the theory. His analysis has since been followed by many authors^ ' ' ' 1. 3  9  12  14  The operation of all rotary induction motors depends on the presence of a rotating mmf in the air gap. If there is any differential in speed (slip) between the rotor and this flux wave, the rotor bars cut the lines of flux and emf s are induced in the rotor. The induced emf s generate currents which flow through the rotor impedance. Torque is produced when the flux due to these induced rotor currents interacts with the flux due to the stator windings. In multi-phase motors the necessary rotating mmf is produced by spatial displacement of the stator windings and temporal displacement of the stator currents.  Figure 1: Revolving Field Equivalent Circuit^ ! 13  The mmf produced by the main winding of a single-phase induction motor is stationary and varies sinusoidally with time. Central to the Revolving-Field theory is the fact that such a stationary flux can be resolved into two constant amplitude counter-rotating fluxes, each with half the amplitude of the stationary flux.  Provided the rotor is stationary, the currents induced in the rotor bars  by these fluxes are equal and opposite and consequently no torque is produced. However, if the rotor turns in either direction, the induced emf s are of different frequencies. As a result the rotor impedances are different as are the amplitudes of the induced currents. 7  The model shown in figure 1 is derived by Cyril G. Veinott in his book Theory and Design of Small Induction Motors  and except for differences in  [13]  nomenclature is the same as Morrill's model. This model describes a singlephase induction motor with a main and auxiliary winding assumed to be in space quadrature. The parameters used in the model and throughout the rest of this work are defined in table 1. All rotor quantities shown are referred to the main winding. The apparent rotor impedances to forward and backward rotating mmf s, as listed in table 1, are defined by Veinott^ ] as: 13  _ x  R  x [r /s]  m  2  _ X  x  m  H/s] + 2  m  x  m  [x  m  +  Xf X  h / ( 2 - s)]  2 [ /( - ,)] + [  b  +  2  r 2  2  _ x [r2/(2-  s)]  m  b  + Xi]  m  h/s] +  m  + Xlf  m  2  x  x  [x  + X2[x  fo/s}  2  2  2  ~  2  X m  -rx [x 2  [ / ( - s)} + [x 2  m  }  2  + x]  m  2  r 2  X l  1  +  }  2  Xl  The left half of the circuit of figure 1 represents the main winding and the right half represents the auxiliary winding. The blocks in the horizontal branches represent stator impedances including an optional run or start capacitor in series with the auxiliary winding.  Blocks in the vertical branches model the rotor  impedance to forward and backward mmf s. These rotor impedances are referred to the main winding and so terms which appear in the auxiliary winding branch are scaled by a to account for the turns ratio. 2  a  Auxiliary/Main Turns Ratio  p.u.  Backward Auxiliary E M F  V  Backward Main E M F  V  Forward Auxiliary E M F  V  Forward Main E M F  V  Auxiliary Current  A  Main Current  A  ri  Main Stator Resistance  ft  Ho  Auxiliary Stator Resistance  n  ^2  Rotor Resistance  ft  Rotor Slip Frequency  p.u.  Auxiliary Voltage  V  Main Voltage  V  X!  Main Stator Leakage Reactance  ft  &la  Auxiliary Stator Leakage Reactance  ft  ^2  Rotor Leakage Reactance  ft  •Km  Magnetizing Reactance  ft  Main Stator Impedance  ft  Auxiliary Rotor Impedance  ft  Apparent Backward Rotor Impedance  ft  Z = R + jX  Run/Start Capacitor Impedance  ft  Z=  Apparent Forward Rotor Impedance  ft  Auxiliary Relative Phase  rad  Rotor Speed  rad/s  Synchronous Frequency  rad/s  Ej = a ZjI 2  a  E  =a Z I 2  / a  /  E/ =Z/I m  I  a  a  m  a  v v  a  Z& =  Rb+jXb  c  c  f  <f>= L \  c  Rf+jX  f  a  -  ZI  m  Table 1: Equivalent Circuit Symbols Voltage sources shown in the vertical branches model transformer action  between the main and auxiliary windings.  To understand the nature of this  coupling it is important to remember that the main and auxiliary windings independently produce both forward and backward rotating mmf s all of which have a constant amplitude.  To simplify the discussion, consider the forward  rotating mmf produced by the main winding current. This mmf produces a back emf which appears in series with the main winding. Because this back emf is proportional to the main winding current it is most conveniently represented as an impedance Z / = E / / I . m  m  The same mmf also produces a back emf which appears in series with the auxiliary winding and is given by +jaEf . m  As in the case of the main winding,  peak voltages are induced when the rotating mmf is aligned with the axis of the winding.  Because the auxiliary winding is in electrical space-quadrature with  respect to the main winding, the peak voltages induced in the auxiliary winding precede those induced in the main winding by one quarter cycle. Consequently, this emf is modelled as a voltage source proportional to the back emf induced in the main winding but phase shifted by one quarter cycle. Because the auxiliary winding does not have the same number of turns as does the main winding, the magnitude of the emf induced in the auxiliary winding is scaled by the turns ratio.  10  2.2 Performance Calculations 2.2.1 Impedance Matrix An impedance matrix is easily derived by applying Kirchoff's voltage law to the circuit of figure 1. The equations defining this model can be written as:  V  m  = [  + Zf + Z ]l  Zl  h  J ^ + J ^  ~  m  a  J  V  a  = [Z -rZ c  +  ia  (2)  a a (Z -rZ )]l +jaE -jaE 2  f  b  a  frn  bT  The impedance matrix is obtained by substitution of the four back emf s defined in table 1 and is the same as derived by Matsch^.  "V  'z + Z / + Z 6 . ja(Z -Z ) Z -rZ  -ja(Z -Z ) + a (Z + Z )  1  f  b  c  f  (3)  b  2  ia  f  b  2.2.2 Input Power The input power is calculated as P = $t{V I e  m  + V I * } . After substitution  m  a  of the terminal voltages from the impedance matrix this equation can be written as: P =5R{ [(z + Z e  a  f  + Z )l b  - ja(Z  m  - Z )l ] J 4 ) +  f  fc  a  (4) K{ [ja(Z  - Z )l +(Z  f  Recognizing that ImJ  m  b  + a (Z + Z ))l„]i:}  1 B  f  6  a  + R ]l +  [R + r -ra (R  2  f  +  + z  C  and I Ia are both real, this equation simplifies to:  P = [ n +R e  m  2  b  2  m  c  la  f  K{ja{Z -Z )[I r -I r }} f  b  11  m  a  a  m  +  R )]l  2  b  (5)  Since I I * - I I ^ = I I * - ( I I * ) * = - j 2 7 / sin <f> and Rf and R m  a  m  m  m  a  b  are the real  parts of Zf and Z j , equation 5 can be reduced to: Pe = [ri +R  + Rb] I  + [Rc + r  2  f  + a (R 2  la  m  f  +  R )]l  2  b  a  (6) + 2aI I (Rf m  - Rb) sin</>  a  where <f> is the phase of the auxiliary winding current measured with respect to the main winding current.  2.2.3 Output Power and Torque The torque produced by the machine can be determined by examining the net power transferred across the air gap. The forward rotating mmf s contribute to a positive torque while the backward rotating mmf s contribute to a negative torque. The difference between the forward and backward emf s therefore determines the net power. P  and P  gm  ga  are defined here as the air gap power produced by the  main and auxiliary windings and can be written as:  P  gm  = m(E -jE /a) fm  -  fa  (E m+jE /a)]r } b  ba  m  (7) P  ga  = 9fc{ [ ( E  / a  - jaE )  - (E +jaE )]i:}  fm  6a  6a  Substituting the back emf equations from table 1 yields: Pg  =  Pgm  "T"  Pga (8)  = *{(Z/  - Z )(l  + ja(Z +Z )(I r  + a ll)  2  b  2  m  b  f  m  -  a  I TJ} a  After simplification the air gap power can be written as: P = (R g  - R ) (l  2  f  b  + a1) 2  m  2  12  + 2aI I m  a  (R  f  + R ) sin <j> b  (9)  This equation is derived in slightly different ways by other authors^ ' . The air 13 9 ]  gap power is given in synchronous  Watts^  and so must be multiplied by the  per unit rotor speed to compute the total mechanical power. Mechanical output power is given by the difference between total mechanical power and friction and windage losses and can therefore be written as:  = (1 - s)P - P  P  g  m  Average torque^ '  13 9 ]  rotor frequency of u>m  (10)  fw  is computed by dividing the mechanical power by the  = (1  —  S)LO . S  _ (1 ~ s)Pg — Pfw _ Pg ~ Pfw  rp _  oj  (1 - s)u>  m  u  s  s  2.3 Efficiency The efficiency, can be written as: Prn _ (1 ~ *)P  t  V  ~  P  e  ~  (l-s)P  g  ~ Pfu,  (  n  )  + P  d  where Pj represents the electrical losses within the machine. By inspection of equation 12 it is obvious that efficiency is affected by friction and windage losses as well as electrical losses. Because friction and windage losses depend only upon the operating point, assuming constant slip the only term which affects efficiency at a given operating point is electrical losses. It is therefore of interest to segregate the electrical losses which are distributed between 13  the rotor and stator. These losses are given by the difference between electrical power and total mechanical power:  P = P - { l - s)P d  e  =P -P  g  e  g  (13)  P  +  s  g  After the appropriate substitutions are made this equation can be written as: P =[r d  + sR  1  +  f  {2-s)R }l b  + [Rc + ru + a [sR  m  + (2 - s)R ]} I  2  (14)  2  f  b  sin (f>  + 2a [sRf - (2 - s)R ] I I b  m  a  a  A more intuitive understanding can be obtained if the equation is rewritten as: Pd=rill  + {Rc +  + s[(R -  Rb) (II + a 1 ) + 2aI I (R 2  f  2  + a 1 - 2aI I 2  m  + Rb) sin<f>]  2  m  + 2R [l b  ria)l2 2  m  a  a  f  (15)  sin <f>]  In this form, the first term represents stator losses. The second term, which is the product of slip and air-gap power, represents slip dependant rotor losses which are inherent to induction machines.  The third term represents additional rotor  losses which are due entirely to the asymmetry of the rotating mmf in the air gap. This term can be shown to be zero if and only if I  m  — al  a  and <f> = TT/2  in which case the air-gap mmf is symmetrical and the machine is said to be balanced. This condition also eliminates pulsating torque: a result derived by a number of authors^ ' ^. 7 4  14  2.4 Optimization of Running Performance Steady-state efficiency is optimized under the assumption that a particular operating point is desired. The operating point is defined by specifying both torque and rotor speed. The controllable variables which affect efficiency are phase of the auxiliary winding current relative to the main winding, the magnitudes of main and auxiliary winding currents, and the stator frequency. Because many terms in equations 6 and 10 are complicated functions of slip, a direct solution seems intractable. To simplify the problem, optimization is done in two stages: an analytical solution which determines main and auxiliary winding currents and their relative phase at an arbitrary stator frequency, followed by a numerical solution which yields the optimum stator frequency. Because the rotor speed is specified by the operating point, if the analytical solution is done assuming constant stator frequency, the slip is constant since s — (LO — u> )/u> . s  m  s  The problem described above is a constrained optimization and so is solved by means of Lagrange multipliers^. The objective is to minimize input power with a constraint of constant mechanical power. The solution which gives the optimal efficiency is therefore constrained to the surface defined by equation 10 which must be solved simultaneously with:  VP (7 ,7 e  m  (  <f>) =  \(l-s)VP {I ,I ,<j>) m  m  a  (16) A(l-s)V[P,(W 15  (  <l>) ~ Pfw]  Since Pf  w  is independent of all electrical quantities, and slip is assumed constant,  the relationship can be reduced to: V P ( / , I ,</>) = A V P ( / , I , <f>) e  where A = A(l — s).  m  a  3  m  a  (17)  In total there are four equations which must be solved  simultaneously. The variables in the optimization constitute the first three degrees of freedom and the fourth is the Lagrange multiplier A.  fr = ^ §r = # dl  (19)  < >  dl  m  m  dl  dl  a  (1 - s)P  g  0 2  a  = Tu  +P  m  (21)  fw  2.4.1 Auxiliary Winding Current Phase Equation 18, after substitution and simplification, can be written as: (7?/ - R )l I b  m  a  cos (j> - \(Rf  + Rb)l I m  a  cos <f>  (22)  One solution to this equation is given by: A  = frr  Two other solutions are possible when I  m  (23)  = 0 or I = 0. However, it is easily a  shown that if any of these three solutions are used, there are no valid solutions 16  for equations 19 and 20. The only valid solution for equation 22 is therefore given by cos  <f> = 0. This equation has roots at (f> =  which optimizes efficiency is given by <j> = ir/2.  ±TT/2  and the condition  This is justified by careful  examination of the air-gap power and power dissipation, as given by equations 9 and 14 respectively. By inspection, the air-gap power is maximized with respect to <f> when <f> = 7r/2.  The same condition minimizes power dissipation and so optimizes  efficiency. To prove minimum power dissipation, it is necessary to find the sign of  sRf — (2 — s)i?j. This term pre-multiplies sin<f> in equation 14 and can be  expanded to:  cgn  (  4fr/s]  I  fo/sf+ixm+Xi]  1  (2 - s ) ^ [ r / ( 2 - s)] 2  2  (24)  [r /(2 - s)] + [a; +a;i] J 2  2  2  2  m  For 0 < s < 1, all quantities in the above equation are positive and the expression is easily reduced to:  sgn{sR - (2 - s)R }= sgnjs - (2 - s ) } = - 1 2  f  Consequently,  b  2  (25)  sin<f> must be positive to achieve minimum power dissipation and  so efficiency is optimized with respect to phase when:  <f> = 17  7T/2  (26)  2.4.2 Auxiliary Winding Current Magnitude To find the simultaneous solution of equations 19 and 20 it is necessary to eliminate the Lagrange multiplier, A. These equations are easily manipulated to yield: dP  ,dP  dl ' I-dl e  dl '  dl  m  m  a  (27)  a  The optimal auxiliary winding current can now be determined as a ratio of the required main winding current. Defining this ratio as k = I /Im a  and letting  <f> = 7r/2, the above equation becomes:  ( n + Rf + Rb) + a(R - R )k f  (R  - R ) + a(R  f  b  [R + r c  + R )k  f  b  + a (Rf  + R )]k +  2  la  a (R  h  - R )k + a(R  2  f  b  f  (28)  a(Rf-Rb)  b  + R) b  After cross-multiplication and simplification this equation can be written in quadratic form: 3_RfRb_  4a  Rf  +  +  a  ,  R  ^  +  r  ^  -f- R  k  2  b  Rf — Rb (R + n Rf + Rb c  RR  4a „  f  b  a  - a n) k  (29)  „ + ari = 0  Rf + Rb  If stator losses are assumed to be zero, this equation reduces to a k 2  implies that I  m  — al  a  2  — 1 which  and, as noted previously, the machine is balanced. When  stator losses can not be neglected the optimal value for k is given by: k  -B  ± VB  2  2A 18  - AAC  (30)  where: A  =  ^Jj3L.  i  Kf  + a(R + r ) c  + K  la  b  =Srl^ «- )  B Because  +ri  aVi  (31)  is defined as the ratio of the magnitude of two vector quantities, only  positive values are allowed. By inspection, for 0 < s < 1, A is positive and C is negative so —AAC  > 0 and y/B  2  — AAC > \B\. The optimal choice for fc  is therefore given by  k  -fl + v / T O T  =  ( 3 2 )  2.4.3 Main Winding Current Magnitude It now remains to satisfy the constraint given by equation 21. With <j> = TT/2, this relationship can be written as:  (R  - R ) (I  2  f  b  + a / ) 2  m  2  + 2a(R  + R )l I  f  b  m  a  =  ™  Tu  + *f  ( ) 33  After substitution of the previously computed value for fc the solution is:  T m  =  I  ^'(l-s)[(l  T  U  M  +  P  >">  + a k )(R -R )+2ak(R 2  2  f  b  f  n  + R )}  K  4  N }  b  2.4.4 Stator Frequency The final step in the optimization is to determine the optimal stator frequency. Equations 26, 32, and 34 define the optimal main and auxiliary winding currents 19  assuming fixed output power and a constant stator frequency.  Because the  above relationships are strongly dependent on slip, an analytical solution for the optimal stator frequency is intractable and a numerical solution is the only remaining option. Such an optimization requires knowledge of the equivalent circuit parameters and an experimental determination of these parameters, is presented in Appendix A . These parameters are used throughout the remainder of this work.  2.5 Optimization of Starting Performance Starting performance is optimized by assuming that a known value of starting torque is required. The objective is therefore to minimize the input power required to achieve this value of torque.  The analysis given in the preceding section  optimizes efficiency by minimizing losses with a constraint of constant mechanical power. Because slip is assumed constant throughout the analysis this constraint is equivalent to one of constant air-gap power. Torque and air-gap power are related by equation 11 and so the optimal starting conditions are only a special case of the solution to equations 18 through 21. The optimal phase angle between main and auxiliary winding currents is therefore as before:  <j> = T T / 2  20  (35)  At standstill 5 = 1 and the forward and backward rotor resistances of equation 1 reduce to: x r  R =R = f  m  b  r\ + (x  2  +  m  (36)  2  xiY  The optimal ratio of magnitudes between main and auxiliary winding currents, as defined by equation 32, can therefore be reduced to: x r m  2  + n r\ + (x  m  +  xif  (37)  k = a x r  + (R  2  m  2  c  ru) r\ + (x  +  m  + xi)  2  The required main winding current is given by:  r\ + (x  m  \  + xiY  (38)  2akx r m  21  Tu>.  2  3 Simulation Results Computer simulations are provided to validate theoretical results as well as to predict optimal operating conditions for specified load conditions. The equations describing the model were implemented in Matlab® using the parameters derived in Appendix A . It is assumed throughout these simulations that the motor is supplied by a variable frequency drive and that there is no limitation on stator frequency.  3.1 Validation of Theoretical Results Theory predicts that, given any load condition, efficiency can be maximized by proper choice of auxiliary winding current (both magnitude and phase) and stator frequency. These results are verified by computing the optimal operating conditions at rated speed and for several values of torque. Efficiency is plotted as each of these three parameters are varied about their optimal values. The first result predicted by theory is that efficiency is maximized when the auxiliary winding current leads the main winding current by 90°. Figure 2 shows a plot of efficiency vs. auxiliary winding phase and it is apparent that, as predicted, efficiency is maximized when the auxiliary and main winding currents are in quadrature. 22  1  1 f =  6 0  Hz  . . . .u  ri .oo. r\ = \J  s = C  .024  adraturj  0  p.u.  1  T = 1.0  !  T = 0.75  1  T = 0.5  p.L  p.u.  1  O  1  T = 0.25  p.L  ;  1 0  2 0  4 0  6 0  8 0  100  120  140  160  180  P h a s e [Deg]  Figure 2: Efficiency vs. Auxiliary Current Phase Angle Peak efficiency is predicted when the magnitude of the auxiliary winding current is in fixed proportion to that of the main winding.  This constant of  proportionality can be computed from equation 32 and at rated speed the optimal value is k = 0.38. Figure 3 plots efficiency against the auxiliary winding current in per unit of main winding current illustrating that peak efficiency is obtained at the predicted value. The final variable in the optimization is slip.  Because the rotor speed is  fixed the optimal value of slip determines the optimum stator frequency. Because of the complicated relationship between slip and efficiency, an analytic solution could not be obtained and so there is no numerical value to be compared with the 23  80 75 70 65 T = 0.5 p.u.  -60 ! 55 50  T = 0.25 p.u  45 40 35  0.1  0.2  0.3  j  i_  0.4 0.5 0.6 Magnitude [p.u.]  0.7  0.8  0.9  1  Figure 3: Efficiency vs. Auxiliary Current Magnitude results shown in figure 4. A numerical optimization indicates that peak efficiency is obtained when s = 0.024. Computer simulations reveal that the optimal slip depends only on rotor speed and is independent of torque.  The same is true  of auxiliary winding current magnitude and this can be seen by examination of equation 32.  24  I  1  i  f = 60 Hz  r\ — \ 1  i  1  .2  t  T=  1.0 p.u  phi == 90 deg  T == 0.75 p.  T == 0.5 p.u  (  O i  d  II .!».[  T == 0.25 p.u.  II / 0  i  0.01  0.02  0.03  0.04  0.05 0.06 Slip [p.u.]  0.07  Figure 4: Efficiency vs.  Slip  0.08  0.09  0.1  3.2 Optimal Winding Currents As noted in the previous section, when properly normalized, optimal operating conditions depend only on rotor speed and are independent of torque.  The  following figures plot these optimal conditions against rotor speed. It should be noted that because slip is one of the variables in the optimization, stator frequency is not constant and is determined by the rotor speed and the optimum slip. Figure 5 plots the optimum auxiliary winding current as a fraction of the main winding current. Note that this ratio is relatively constant for practical values of 25  i  1  Normalized auxiliary current i s i n d e p e n d e n t s torque. Slip and stator frequency are as s l o w n in figures 6 and 7.  k = 0.3 8  0.3  0  0.1  0.2  0.3  0.4  0.5 0.6 Rotor Speed [p.u.]  0.7  0.8  0.9  1  Figure 5: Optimal Auxiliary Current Magnitude vs. Rotor Speed rotor speed and that at rated speed k = 0.38. This value is consistent with that shown in figure 3. The slip which provides the highest efficiency, as determined by numerical optimization, is plotted against rotor speed in figure 6. At rated speed the optimal slip is s = 0.024 as is also shown in figure 4. Stator frequency can be computed from the optimum slip and is plotted against rotor speed in figure 7. Very litde error is introduced by linearizing this relationship and the least-squares fit is given by u> m u s  m  + 0.026.  26  i  Slip is indep sndent of torque. Auxiliary current andistator frequency are as shown in figures 5 and 7.  s = 0.02 4 i  •  0  0.1  0.2  0.3  0.4 0.5 0.6 Rotor Speed [p.u.]  i  i  i  0.7  0.8  0.9  1  Figure 6: Optimal Slip vs. Rotor Speed  Stat or freque>ncy is independcsnt of tor que. Aux liary cur ent and slip are as s nown in figures £ and 6. 0.8 h  0.6 h  0.4 h  0.2\-  Q !  0  1  1  1  0.1  0.2  0.3  I  I  I  0.4 0.5 0.6 Rotor Speed [p.u.]  I  I  I  I  0.7  0.8  0.9  1  Figure 7: Optimal Stator Frequency vs. Rotor Speed 27  80  i  1  1  1  1  1  1  r  1  T = 1.0 p.u.  70  T = 0.75 p.u.  60  T = 0.5f3.U. 50  T. = .0.25 p.u....  c 40 LU  30 Aux liary cur ent, slip and sta or frequ(sncy are as show 1 in figur is 5 thro ugh 7.  20  10  0.1  0.2  0.3  0.4 0.5 0.6 Rotor Speed [p.u.]  0.7  0.8  0.9  Figure 8: Maximum Efficiency vs. Rotor Speed Although the optimal operating conditions are independent of torque, the maximum efficiency is related to torque.  Under no-load conditions, the total  torque produced by the motor is exactly equal to friction and windage torque. Conversely, when the motor is heavily loaded, friction and windage torque is only a small fraction of the total torque produced.  Consequently, efficiency  increases with load torque until electrical losses become dominant. The maximum efficiencies for selected values of load torque are plotted against rotor speed in figure 8.  28  3.3 Optimal Winding Voltages Terminal voltages can be determined by applying the impedance matrix of equation 3 to the results derived in the preceding section. Main winding voltage is plotted against mechanical speed in figure 9.  Stat or frequency and auxiliary i voltage magnitL de and phase are as she)wn in figjures 7, 11 and 12. T = 1.<D p.u.  .  0.8 6 a, CD oi  10.6 o > c 0.4  T = o.:25 p.u.  0.2  0  0.1  0.2  0.3  0.4 0.5 0.6 Rotor Speed [p.u.]  0.7  0.8  0.9  1  Figure 9: Optimal Main Voltage Magnitude vs. Rotor Speed  The V/Hz ratio can be computed from the main winding voltage and the optimal stator frequency determined in the previous section.  These results are  plotted in per unit of rated V / H z against rotor speed in figure 10. The optimal V/Hz ratio increases very sharply at low speeds in order to compensate for the 29  3 Stat or frequs ncy and auxiliary i voltage magniti de and phase a re as sh<3wn in ficJures 7, 11 and 12.  2.5  2  '« 1.5  T = 1.0 f 3.U. T = 0.75 p.u. T = 0.5f).U. T = 0.25 p.u.  Q\ 0  I  I  I  I  0.1  0.2  0.3  0.4  I  I  0.5 0.6 Rotor Speed [p.u.]  I  I  I  0.7  0.8  0.9  I 1  Figure 10: Optimal Main V / H z Ratio vs. Rotor Speed proportionally higher voltage drop across the stator resistance. This low-frequency voltage boost is a common feature of commercially available drive systems^ ^. 1  At near rated speed and torque, because a linear model has been assumed the optimal per unit V / H z ratio predicted by theory exceeds unity. In practice, operating under these conditions would result in saturation of the magnetic circuit and the efficiency would be much lower than predicted. The practical significance of this figure is therefore that the V/Hz ratio should be reduced when the machine is operated under light load conditions. When increased torque is desired, the V/Hz ratio should be increased but should only exceed its rated value at very low speed. 30  i  1  ncy, m a nwindir ly  xiliary  vviuay  wint ing pha! ;e are as shown n figures 7, 9 an  a 11.  0.2  0  0.1  0.2  0.3  0.4  0.5 0.6 Rotor Speed [p.u.]  0.7  0.8  0.9  1  Figure 11: Optimal Auxiliary Voltage Magnitude vs. Rotor Speed The optimal auxiliary winding voltage is plotted as a fraction of the optimal main winding voltage in figure 11 and it is apparent that this ratio is nearly constant for practical values of rotor speed. Figure 12 plots the relative phase of the auxiliary winding voltage against mechanical speed. Again, for practical values of mechanical speed, the phase is nearly constant.  31  h  74 72 701  0  : i  0.1  \  :  :  i  i 0.3  i i i 0.4 0.5 0.6 Rotor Speed [p.u.]  0.2  :  •: i 0.7  i  0.8  .  :  i  0.9  1  Figure 12: Optimal Auxiliary Voltage Phase Angle vs. Rotor Speed  32  4 Experimental Results 4.1 Apparatus 4.1.1 Motor Experimental measurements  were made on a Franklin Electric model  2143084416 submersible motor rated for 1 HP at 3450 RPM. This rating is given assuming that the motor is immersed in water and is therefore adequately cooled. For the purposes of experimental work this cooling requirement was met by mounting the motor in a drum filled with water.  4.1.2 Load A heavy duty 24V automotive alternator was directly coupled to the motor under test. The alternator was prepared by machining shoulders on the end bells. These shoulders were concentric with the alternator shaft and were designed to accept bearings. With this arrangement, when mounted in the test fixture, both the alternator shaft and housing were free to rotate. Torque sensing was accomplished by locating a load cell between the alternator housing and a stationary part of the test fixture, preventing rotation of the alternator housing. The load cell output was therefore proportional to the total torque produced by the motor under test. A fitting was located on the alternator housing to accept a torque wrench for the purpose of calibrating the load cell. 33  The rectifiers were removed from the alternator and its windings were shortcircuited. With this connection the total power generated by the alternator was dissipated by the alternator windings. A n external resistive load bank would have been preferable as it would have reduced heating in the alternator but the resistance required to achieve a reasonable power transfer was too low to be practical. By applying a field current of approximately 2A the alternator produced a braking torque equal to the rated torque of the motor under test.  4.1.3 Supply All of the measurements reported here were made at line frequency and with a 208V three phase supply. Variacs were used to adjust the magnitudes of the main and auxiliary winding voltages and an adjustable phase shifting transformer was used for variable phase measurements. The majority of tests were run with a fixed phase of approximately 90° between the main and auxiliary winding voltages. This phase shift was accomplished by Scott-connected isolation transformers.  4.1.4 Instrumentation All electrical quantities were measured using a Data Precision model Data 6000A waveform analyzer. Isolation amplifiers were used to interface the main and auxiliary winding voltages and currents to the analyzer's four inputs. The analyzer was programmed as shown in Appendix C to compute the following quantities: 34  v  •  Main Winding RMS Voltage  •  Main Winding RMS Current  •  Main Winding Average Power  •  Auxiliary Winding RMS Voltage  •  Auxiliary Winding RMS Current  •  Auxiliary Winding Average Power  •  Auxiliary Winding Voltage Phase  •  Frequency  The mechanical quantities recorded were average torque and rotor speed. A stroboscope was used to determine the rotor speed and average torque was measured with a load cell mounted on the alternator, connected to a bridge amplifier and calibrated to read directly in N-m.  4.2 Sources of Error 4.2.1 Modelling Error Numerous assumptions are made in the derivation of the model used to describe the motor under test.  These assumptions include a linear magnetic  circuit and a sinusoidal winding distribution. Experimental data presented in the following figures show that these assumptions could be expected to introduce significant error.  Although the voltage supplied to the motor had very low  harmonic content, the winding currents showed a high level of odd harmonics. 35  Figure 13 shows the main and auxiliary winding voltages with a vertical scale of 100 V/div for a load of approximately 0.5 N-m. The corresponding winding currents are shown in figure 14 with a vertical scale of 2 A/div. Third harmonics due to saturation effects are particularly evident in the auxiliary winding current. Saturation effects are also apparent in the main winding current although the rectangular shape is somewhat unexpected.  1  l.oov  -QQV  FreqC 1}=59.97 Hs  j-O.OOs  VrmsC 15=2.000 V  5.oog/  f± STOP  VrmsC25=2.345 V  Figure 13: Main & Auxiliary Voltage  One unusual feature of the motor under test which may introduce additional error is the presence of a liquid filled stainless steel jacket between the rotor and stator.  Although there is no experimental evidence in support of this claim,  depending on the thickness of the jacket and the conductivity of the liquid, eddy current losses in the space between the rotor and stator could be rather 36  1  21.3)200??  200ff  j-O.OOs  f  /  -1  \  VII-  \  I  • 1 • 1  :  F r e q C O = 6 0 . O B HE  :  :  ji.  VrmsC1)=443.9mV  /  • iv | .  •r/iiN^  \  \N  V  \  V.J : :  7i  \ \  /=  \  r  \ \  • llll-J  y  f± STOP  5.00g/  1r  \  :  : : •_ VrmsC25=106.2mV  Figure 14: Main & Auxiliary Current high. In conventional motors there are no conductive materials in this space, and consequently no eddy current losses.  4.2.2 Measurement Error With the exception of the load cell, error introduced by instrumentation should be negligible. Voltages and currents measured by the isolation amplifiers and waveform analyzer compared well with values displayed by conventional metering. The largest source of measurement error is expected to be due to the load cell. Because the motor and alternator generated significant heat, the entire test fixture including the load cell was subjected to temperature cycling of up to 30° C during the course of experiments. Such large changes in the load cell temperature could be expected to introduce significant experimental error. 37  4.3 Efficiency Measurements Figure 15 plots efficiency against auxiliary winding phase. The main winding voltage is supplied with 90% of rated voltage and the auxiliary winding voltage is set to 120% of the main winding voltage. Measurements at load torques of 0.5 N-m and 1.5 N-m correspond to 25% and 75% of rated torque respectively. The measured efficiencies compare quite well with simulation results and the maximum efficiency occurs when the auxiliary winding current leads the main winding current by slightly less than 90°. This result is predicted by the model as shown in figure 12.  Simulated T = 0.5N m  M e a s u red XXX X X + + + + +  T = 1.5N m  +  ^-^ _ -yr  x  ' v.  _  X •v.  ' '  X  X  X  oi  60  1  1  70  80  1  1  90 100 Voltage P h a s e [Deg]  1  1  110  120  Figure 15: Dual Winding Efficiency vs. Auxiliary Voltage Phase Angle 38  Efficiency as a function of auxiliary winding voltage magnitude is shown in figure 16.  The main winding voltage is fixed at 80% of rated voltage and  the auxiliary winding voltage is varied between 90% and 130% of the main winding voltage.  Auxiliary winding phase is fixed at 87.5°.  Although the  differences between measured and predicted efficiencies are slight and may be within experimental error the peak efficiencies occur at unexpected values of auxiliary winding voltage.  Figure 11 shows that peak efficiency should be  obtained when the auxiliary winding voltage is 120% of the main winding voltage. The data presented in figure 16 indicates that the optimum auxiliary winding voltage is between 105% and 115% of main winding voltage. 100 90 80  Simulated  Measured  T = 0.5 N.m  X X X )< X X  T = 1.5 N.m  + + + ^V +  +  70 ^  60  >s  c 2 o E  50  m  40  X  X._  (  >  x  x  x  x  X  X  X  ; x - -x-  30 20 10 0.9  0.95  1.05 1.1 1.15 Auxiliary Voltage [p.u.]  1.2  1.25  1.3  Figure 16: Dual Winding Efficiency vs. Auxiliary Voltage Magnitude 39  Figure 17 shows a plot of efficiency vs.  main winding V / H z ratio. For  this experiment the auxiliary winding voltage was adjusted to 110% of the main winding voltage and the auxiliary winding phase was fixed at 87.5°. There are significant differences between measured and predicted efficiencies and at the two values of torque shown the peak efficiencies are higher than predicted by theory. In addition these peak efficiencies occur at lower V/Hz ratios than predicted by theory. This is most likely due to saturation effects which are not accounted for by the linear model used in the analysis.  100 90 80  Sim jlated  Measured  T = D.5 N.m  X X X X X + + + + +  T = 1.5N.m  70  + +  • 60 c  50 X  X  •  + +  +  H  X . X..x.x.  40  K  X  30 20 10 0  _l  0.1  L.  0.2  0.3  0.4 0.5 0.6 V / H z Ratio [p.u.]  0.7  0.8  0.9  1  Figure 17: Dual Winding Efficiency vs. V/Hz Ratio  Other authors have reported efficiency improvements by operating at reduced 40  V/Hz ratios and thereby reducing core losses within the machine . This particular [6]  machine is designed to be very compact and has a very small diameter in comparison to its length. Due to its compact design the machine could be expected to be well into saturation when operated at rated V/Hz. To check this supposition similar tests were run with only the main winding of the machine excited and this data is presented in figure 18. Once again the effect of varying the V/Hz ratio is dramatic and peak efficiencies occur at lower V/Hz ratios than predicted by the model. 100  90 80  i Simulated T = D.5 N.m  i Measured X X X X + + + +  T = 1.5 N.m  X +  70  -rr  60 +  c 50 LU  X X-"  0.1  0.2  0.3  x_.x__>  0.4 0.5 0.6 V / H z Ratio [p.u.]  +  +  •lc~ X "  0.7  +  + HI-  X  X  0.8  <  +  X  +  X"  0.5  Figure 18: Single Winding Efficiency vs. V / H z Ratio  By inspection of figures 17 and 18 it becomes clear that there is a relationship 41  between optimal V/Hz ratios and load torque for both single and dual winding excitation. Both figures show that at 25% of rated torque, efficiency is optimized when the V / H z ratio is between 50% and 60% of rated value. At 75% of rated torque the optimum V/Hz ratio is between 75% and 85% of rated value. Based on these two data points, a linear approximation to the relationship can be written as:  V m  0.4 + 0 . 5 - ^  Vrated  (39)  J- rated  Figure 19 compares measured efficiencies for single and dual winding excitation. In both cases the main winding voltage is adjusted according to equation 39. The  upper data set represents auxiliary winding excitation at 110% of main  winding voltage and a phase difference of 87.5°. There is clearly an efficiency improvement of approximately 5% and this improvement is consistent with that predicted by the model.  42  Simulated  Measure d  Single Winding Excitiition  X X X  -•-  Dual Winding Excitat on  X X  + + + + + •  ——•" _ - •* -x~ ' "  +  _  X  X  + / f  r »' t /' •/  i  1 0.1  0.2  0.3  0.4 0.5 0.6 Mechanical Power [HP]  0.7  0.8  Figure 19: Efficiency vs. Mechanical Power  43  0.9  5 Conclusion The objective of this analysis was to determine operating conditions which provide the highest possible efficiency for single-phase induction motors equipped with an auxiliary winding. The analysis was based on a model which is widely accepted and has been used by many authors^ ' ' ' 10  13  9  1 2  \ The computer simulations  which have been presented confirm that the analysis correctly determines optimal operating conditions for motors which can be accurately described by the model. Experimental measurements made with a submersible single-phase induction motor indicate that the effects predicted by theory are present and that it is possible to improve efficiency by excitation of the auxiliary winding. However, the peak efficiency does not occur under exactly the same conditions as predicted by theory and it appears that the linear model used for the analysis is not capable of accurately describing the motor under test.  The model assumes that the  permeability of the magnetic circuit is independent of flux density and that skin depth effects in both the rotor and stator are negligible. Depending on the type and construction of motor, both of these approximations could introduce significant error. The highest accuracy could be expected for motors designed for high efficiency as these motors operate in the linear portion of the magnetization characteristic.  On the other hand, the model could not be expected to accurately 44  predict the performance of compact motors designed for high starting torque. Increased starting torque is achieved by taking advantage of deep bar effects so that the rotor resistance is strongly dependent on slip frequency and is therefore very high during starting^ ]. At constant slip, stator frequency and slip frequency 12  are proportional and consequently rotor resistance increases with stator frequency. Compact motors typically have a smaller magnetic circuit cross section and so are further into saturation when operated at rated V/Hz. The theory's ability to deal with modelling error could likely be improved by experimentally determining sets of parameters which depend on stator frequency and flux density. Provided voltage is adjusted in proportion to stator frequency, the flux density could be assumed constant and the dependence of model parameters on skin depth effects could be determined.  The effects of changes in flux  density could be determined by varying the V / H z ratio at constant frequency. Although such an approach would limit the utility of the analytical work which has been presented here, if the equations were used in conjunction with computer simulations an accurate prediction of motor performance should be possible. Despite the differences between experimental results and computer simulations the measured and predicted efficiency improvements compare quite closely. Experimental results presented in figure 18 show that, depending on operating point, considerable gains in efficiency are possible by operating with single winding excitation and properly choosing the V/Hz ratio to match the load. Because 45  of the ease with which this could be accomplished this should certainly be taken advantage of before resorting to dual winding excitation. For this particular motor, when operated at rated speed and 25% of rated torque, a 10% increase in efficiency was achieved by operating at approximately 60% of rated V/Hz. This optimal V/Hz ratio is consistent with that predicted by theory as shown in figure 10. Equation 39 can be used to approximate the optimal V/Hz ratio as a function of load torque.  In addition to what can be achieved by properly adjusting the V / H z ratio, further efficiency improvements are possible by appropriate choices of auxiliary winding voltage magnitude and phase. Although it is the current phase which must be in quadrature to obtain peak efficiency, for all practical purposes it is sufficient to maintain the auxiliary winding voltage in quadrature with the main winding voltage.  For the motor under test, experimental results presented in figure 3 reveal that peak efficiency is obtained with the auxiliary winding voltage set to approximately 110% of the main winding voltage as shown in figure 15. When the machine is operated with these conditions on auxiliary voltage magnitude and phase, an additional 5% increase in efficiency is achieved at rated speed and over a wide range of load torques as shown in figure 19.  In a practical application this efficiency improvement would likely be offset by 46  losses within the auxiliary winding power conditioning unit and careful analysis would be required to determine if there would be any benefit to retrofitting such a power converter. The author therefore recommends that any project undertaken to improve the efficiency of systems using solar powered inverter driven submersible single-phase motors such as the Franklin Electric model 2143084416 be completed in two stages. The first stage should be to experimentally determine optimal V/Hz ratios for single winding excitation under a number of operating conditions. These ratios should then be correlated to rotor speed and torque and the inverter software should be modified to adjust the modulation index depending on operating point. Stator frequency and winding current should be sufficient to determine operating point as these quantities are very closely related to rotor speed and torque. This solution would not involve any hardware changes to existing systems and would therefore be a very inexpensive means of improving efficiency. If additional improvements are desired after modifying the system to operate at optimal V/Hz ratios, the first step should be an economic analysis. Because the potential gain in efficiency is quite small, the cost of an auxiliary winding power converter may not be justified by the savings in solar cells. In this instance, the motor under test is rated at one horsepower and at rated load reaches its peak efficiency of approximately 65%. At rated load an increase in efficiency of 5% would therefore reduce the electrical input power by 82 Watts. Assuming that 47  the solar cells can be obtained at a cost of $5 per Watt, potential savings would be $410 less the cost of additional power electronics required for the auxiliary winding power converter. If such savings are large enough to be of interest the proposed implementation described in Appendix B could be applied.  48  Appendix A Equivalent Circuit Parameters The method suggested by VeinottJ ] is used to compute equivalent circuit 13  parameters based on D C winding resistance, locked-rotor and no-load tests. The usual approximations made in order to obtain equivalent circuit parameters from these tests neglect the effects of magnetizing reactance on the locked-rotor impedance. It is claimed that Veinott's method provides improved accuracy by introducing a slight modification to account for these effects. This method does not provide all of the parameters required to model a singlephase induction motor with both main and auxiliary windings. The two additional required parameters model the auxiliary winding stator leakage reactance and the effective turns ratio between main and auxiliary windings. The manufacturer was able to provide the turns ratio and the leakage reactance was approximated from locked-rotor measurements, turns ratio and rotor leakage reactance.  Vim  [Hz]  h m  [V]  [A]  v  h a  PLa  [V]  [A]  [W]  La  [W]  30  27.3  5.00  114  39.2  2.50  91  40  30.5  5.02  117  41.1  2.47  87  50  33.6  5.01  121  44.0  2.53  96  60  37.3  5.04  124  45.9  2.49  91  Table 2: Locked-Rotor Measurements 49  V  Io  Po  [A]  [W]  0  [Hz]  [V]  .  [RPM]  30  104  4.61  165  1790  40  138  4.63  215  2392  50  173  4.68  265  2990  60  206  4.73  335  3590  Table 3: No-load Measurements During no-load measurements both line current and input power increased sharply as the rated V/Hz ratio was approached. This effect is to be expected and is due to saturation of the magnetic circuit. While efficient operation at higher than rated V/Hz is not possible, it was expected that reducing the V/Hz ratio could improve efficiency. No-load measurements were therefore made at 90% of rated V/Hz in order to obtain a linearized model which could more accurately predict performance over the anticipated range of V/Hz ratios. The equations which must be applied to the locked-rotor and no-load data are  [13].  w  Lr  P  T  2  -  (40)  r\  'W  Lm  Q  X = xi + x  =  2  Y -l o —  A  (41)  T  Jin  2  x  (42)  T -L  2  X  m  —- —  50  V  o  Y A  (43)  Si. [mH]  [Hz]  ru  w.  ["]  [mH]  £2. Wl  [ft]  [mH]  a  [p.u]  30  2.63  7.9  11.9  18.9  2.08  7.9  1.234  40  2.63  7.8  11.9  22.3  2.16  7.8  1.234  50  2.63  7.4  11.9  16.7  2.35  7.4  1.234  60  2.63  7.4  11.9  18.3  2.42  7.4  1.234  Table 4: Equivalent Circuit Parameters In a squirrel-cage single-phase induction motor there is no way to separate rotor and stator leakage reactances and it is common practice to assume them to be equal^ ' \ Under this assumption equations 42 and 43 can be solved to yield: 13  9  X l = x = X (l - y/l - XlX ^j 2  0  (44)  0  and:  x = X - xi m  (45)  0  The auxiliary winding leakage reactance is approximated by neglecting the effects of magnetizing reactance and computing the input reactance from the auxiliary winding locked-rotor measurements. This input reactance is the sum of rotor and stator leakage reactances.  Because the rotor leakage reactance is  specified with respect to the main winding it must be referred to the auxiliary winding by the turns ratio. The auxiliary winding stator leakage reactance is approximated by^ ^: 13  ^"vSf-W) ~  a2x2  51  <46)  These parameters determine the electrical characteristics of a single-phase induction motor but can not model the effects of friction and windage on the mechanical performance. Friction and windage losses can be approximated from the no-load measurements and equivalent circuit parameters. This is done by assuming that friction and windage losses are given by the difference between measured input power and resistive losses as computed from equivalent circuit parameters and measured current.  P  f w  = P  0  - ( r i + R f  + R )l b  2 0  ^Bu  m  (47)  Regression analysis on experimental data reveals that the appropriate values of the required constants are: B = 0.24  (48) n = 1.12 when w  m  is specified in rad/s.  This data is presented in figure 20.  The manufacturer of the motor was able to provide a plot of experimentally determined performance characteristics and this is displayed in figure 21. This data is used as a benchmark to determine the validity of the equivalent circuit model as well as the parameters used in the model. Figure 22 shows the results of a computer simulation run under the same conditions as the experimental measurements. The agreement between the two sets of data is quite good, indicating that the model and parameter set accurately describe motor performance.  52  Figure 20: Friction and Windage Losses vs. Rotor Speed  53  R-ankSn EteotHo T E S T  60 HERTZ RUN MFD PAGE 119 DATE 1- IS-^2.  VOLTS 230 ST. MFD 118.6 TEST PR 2647 APPR B Y ^ J ^  R FRAME A MODEL T RPM I S.F. N TYPE G  HP VOLTS PHASE HERTZ  4 INCH SUB 2143081216 3450 1.4 C6IR  1 230 1 60  TEST W ONE MOTOR NOT GUARANTEED AS MINIMUM PERFORMANCE 8  T  M  , lz—ti—ai—U FULL LOAD TORQUE BREAKDOWN TORQUE LOCKED ROTOR TORQUE  0 0  ill U 17.9 LB-IN 49.7 LB-IN 43.4 LB-IN  U  U  1.6  2.0  HORSEPOWER  Figure 21: Measured Motor Characteristics  54  MOTOR PERFORMANCE CHARACTERISTICS [W] [%] [A] 95 13 3250 RPIV PF^ AMP s EFF TO  v vA 1 1  o  1  0  /,  0.5  1  1.5 HORSEPOWER  Figure 22: Simulated Motor Characteristics  55  90  12 3000  85  11 2750  80  10 2500  75  9  2250  70  8  2000  65  7  1750  60  6  1500  55  5  1250  50  4  1000  45  3  750  40 2  500  35  250  1  2.5  Appendix B Proposed Implementation When used for photo-voltaic pumping applications, the single-phase induction motor studied here is commonly powered by a single phase inverter. Because the voltage provided by the solar cells is relatively low, the inverter is usually an H-bridge with trapezoidal or sine-coded pulse width modulation. If sine-coded modulation is used, a third harmonic component is sometimes added to increase the magnitude of the fundamental component. Some changes to this approach will be required in order to supply the auxiliary winding. One option would be to provide a second H-bridge for the auxiliary winding. However, because the main and auxiliary windings share a common neutral, the auxiliary inverter would have to be isolated from the main inverter and the cost of providing this isolation is likely to be prohibitive. Without isolation, a three phase inverter is the only practical way to provide the required phase difference between main and auxiliary windings. As shown in figure 23 the auxiliary winding voltage which can be provided is limited by the required main winding voltage and the relative phase.  The  maximum fundamental phase voltage which can be provided by a single leg of a sine-coded P W M inverter is given by:  56  Figure 23: Voltage Utilization of Three Phase Inverter With this limitation on phase voltage, the main and auxiliary winding voltages are limited to: V  dc  V  m  =  — V1 y £  /  -  C O S OL  (50)  K = -yVl-cos/? The phase displacement required to generate the desired main winding voltage is given by:  a = cos" (l - 4-jJf)  (51)  1  Because the relative phase between main and auxiliary winding voltages can be written as:  «+ =  —--^  (52)  the phase displacement required for the auxiliary winding phase is given by:  p=  2(TT  -  <f>) - a =  2(TT  57  -  <f>) - cos" ^1 1  4-p|^  (53)  The maximum available auxiliary winding voltage, as a function of main winding voltage and relative phase, is therefore: V  dc  1  — COS  2<f> + cos-  1  (1-4^|  (54)  dc When the main and auxiliary winding voltages are in quadrature 4> =  7r  /2  and this relationship simplifies to the Pythagorean Theorem.  V  2  m  + >  V  2  'a  4  V  =W)  As the main winding voltage approaches \f2V h p  (55)  the voltage available for the  auxiliary winding phase is very low. For the motor under test the power factor at rated load is approximately 70% as shown in figure 21. The winding currents therefore lag terminal voltages by approximately 45° and a portion of the auxiliary winding phase shift could be provided by a series capacitor as shown in figure 24.  Figure 24: Auxiliary Winding Compensation  For this application the optimum compensating reactance can be defined as the value which minimizes the required link voltage over the desired range of 58  speeds and torques. The optimum slip is independent of torque but varies with rotor speed as shown infigures4 and 6 respectively. Consequently the impedance matrix, which has slip dependent terms which are presented in equation 1, depends only on rotor speed. As a result the choice of compensating reactance is also independent of torque. It is possible to show that for a compensating reactance X  the quantities V  c  a  and a in figure 24 are given by:  + V cos (9 + <f>)] + [V sin (9 + <f>f  V = \][XJ a  2  a  a  a  (56)  and: _! (X I c  a = cos  + VaCOs(9 + <j))\  a  (57)  0  j-9  ^  The required link voltage V^ can therefore determined by solving: c  V  2  V = "T^Wl - cos 2a + cos- [ 1 1  a  (58)  i-^-  Given that:  cos (7 + c o s X) = X cos 7 —  \J\-X sin 7  -1  2  (59)  equation 58 can be rewritten as a quadratic in V^ : c  A{Vl)  + B(V )+C  2  2  d c  =0  (60)  where: A = [1 - cos (2a)]  2  = -[1 - cos (2a)] [(2V )  2  B  m  C = (2V f m  - 2(2V ) (2V ) 2  m  2  a  59  + (2V )  2  (61)  1  a  cos (2a) +  (2V f a  The  required link voltage can therefore be determined by applying the  quadratic formula to equation 60.  When rated voltage is applied to the main  winding, and the auxiliary winding voltage magnitude and phase are chosen to provide optimum performance, the link voltage is as shown in figure 25. There are two capacitances which result in the same minimum link voltage. Although the smaller capacitor would be less expensive, the link voltage increases abruptly for small deviations from this optimum value. Consequently the larger value would be a more practical choice.  450  400 350 Vdc = 325 V C1 = 13.0 uF C 2 = S2.7 uF  300 250 TO  s  O > :200 C  V m = 230 V V a = 276 V  150  phi = 5!8:i cleg  100  50  10  12  14  16  18 20 22 Capacitance [uF]  24  26  Figure 25: Link Voltage vs. Run Capacitance 60  28  30  The values shown in figure 25 are only optimal at rated stator frequency but the approach could be extended to determine a single value of capacitance which minimizes link voltage over the range of anticipated stator frequencies.  61  Appendix C Waveform Analyzer Program This program is written for the Data Precision model Data  6000A waveform  analyzer. Input buffers are sampled in lines 10 through 40. Instantaneous power is computed as a point by point multiplication of voltage and current for the main and auxiliary windings in lines 50 and 60 respectively. Cursor positions for positive zero crossings of the main and auxiliary winding voltages are measured in lines 70 and 80 and this information is used in lines 100 through 120 to compute the phase of the auxiliary winding voltage relative to that of the main winding. 10 20 30 40 50 60 70 80 90 100 110 120 130 14 0 150 160 170 180 190 2 00  VM = U N I T ( B U F . A 1 , 2 , 0 , 1 0 0 , 2 ) IM = U N I T ( B U F . A 2 , 2 , 0 , 1 0 , 4 8 ) VA = U N I T ( B U F . A 3 , 2 , 0 , 1 0 0 , 2 ) IA = UNIT(BUF.A4,2,0,10,48) PM = VM*IM PA = V A * I A DSPL(SC:CRSP(BUF.A3)) DSPL(NX:SE:CRSP(BUF.Al)) F = FREQ(VM) PHI = 360*XDELTA*F PHI = PHI-180 PHI = U N I T ( P H I , 2 , 0 , 1 , 4 1 ) DSPL(F) DSPL(PHI) DSPL(RMS(VM)) DSPL(RMS(IM)) DSPL(MEAN(PM)) DSPL(RMS(VA)) DSPL(RMS(IA)) DSPL(MEAN(PA))  62  References [1]  Bernard Adkins. The General  Theory  of Electrical  Machines.  Chapman &  Hall, London, 1957. [2]  Philip L . Alger. Induction  Machines:  Their  Behaviour  and Uses, 2nd Ed.  Gordon and Breach Scientific Publishers, New York, 1970. [3]  A . Vandenput et al. Run capacitor optimization in single-phase induction motors. IEEE  [4]  Applications  Conference  Proceedings,  1986.  E . R. Collins Jr. et al. Single-phase induction motor adjustable speed drive: Direct phase angle control of the auxiliary winding supply. IEEE Trans, on Industrial  [5]  Industrial  Applications,  Vol. 35, No. 6, pp. 246-252,  1988.  H . Huang, E.F. Fuchs, and J.C. White. Optimal placement of the run capacitor in single-phase induction motor designs. IEEE Trans, on Energy Conversion,  Vol. EC-3, No. 3, pp. 647-652,  September,  1988.  [6]  Joseph D. Law and Thomas A . Lipo. A single phase induction motor voltage controller with improved performance. IEEE Trans, on Power Electronics, Vol. PE-1, No. 4, pp. 240-247, October, 1986.  [7]  Terrance M . Lettenmaier, Donald W. Novotny, and Thomas A . Lipo. Single phase induction motor with electronically controlled capacitor. IEEE on Industrial  Applications,  Vol. 35, No. 6, pp. 169-174,  [8]  Jerrold E . Marsden and Anthony Tromba. Vector Calculus. 1981.  [9]  Leander W. Matsch and J. Derald Morgan. Electromagnetic mechanical  Machines,  Trans,  1988.  W. H . Freeman,  and  Electro-  3rd Ed. John Wiley & Sons, Toronto, 1986.  [10] Wayne J. Morrill. The revolving field theory of the capacitor motor. Trans. AIEE,  Vol. 48, pp. 614-629,  April, 63  1929.  [11]  J.M.D. Murphy and F . G . Turnbull. Power Electronic  Control  of AC  Motors.  Pergamon Press, Toronto, 1989. [12]  Gordon R. Slemon and A . Straughen. Electric  Machines.  Addison-Wesley,  1980. [13]  Cyril G . Veinott. Theory  and Design  of Small Induction  Motors.  McGraw-  Hill, New York, 1959. [14] Herbert Vickers. The Induction London, 1953.  Motor,  64  2nd Ed. Sir Isaac Pitman & Sons,  

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