Solid-Rotor and Composite-Rotor Induction Machines

Introduction

The most elementary type of rotor for the polyphase. . J. induction machine is the solid-iron rotor, which offers

Solid-Rotor and Composite-Rotor advantages in ease of manufacture, in high torque per

Inductian Machines ampere at standstill, in withstanding high rotationalstresses, and in operating in unusual environments. Theinteraction between the eddy currents induced in the solid

MULUKUTLA S. SARMA, Senior Member, IEEE cylindrical rotor steel structure and the revolving field ofGORAKH R. SONI, Student Member, IEEE the air gap produces the electrodynamic torque. RecentlyWest Virginia University there has been wide interest in the possible use of suchMorgantown, W.Va. 26506 solid-rotor motors for ultrahigh-speed inverter drives with

suitable high stator supply frequency as well as variable-speed drives for conventional frequencies and speed ranges.Finzi and Paice [1] have concluded that the solid-rotor

Abstract machine has characteristics that are particularly suitable for

The feasibility of constructing composite rotors in the laboratory solid-state power controls.

for use on a 60- to 120-Hz supply has been established and te Operational and military requirements demnd thatelectrical equipment be small and lightweight. Very effi-

experimental results with such rotors have been published for the cient lightweight electrical power supplies have been devel-first time. Results obtained from the tests made on a three-phase oped with a primary frequency of 2000 and 3000 Hz. Theinduction motor with different solid-iron as well as composite rotors output of these high-frequency systems is generally trans-are presented. These are compared with the theoretically predicted formed by solid-state static frequency changers to 60 or

characteristics based on existing analytical as well as numerical 400 Hz. However, these reduced frequencies may not

analysesm always be availa-ble and it becomes desirable to developthree-phase induction motors that can operate on theprimary frequency of the power supply. At present,solid-rotor induction motors are used for this purpose. Inthis context, Wilson et al. [71 have presented a novel ideaof an aerospace composite-rotor induction motor.

The difficult problem of analyzing the solid-rotorinduction motors and developing lumped-parameter equiv-alent circuits wherever possible has occupied designers andresearch workers for many years. Since conventionalinduction machine theory has proven inadequate for suchmachines, the need has arisen for improved methods ofinvestigation. McConnell and Sverdrup [2] have developedan equivalent circuit, based on the magnetization character-istic of a square-wave form and a two-dimensional analysis.Heller and Sarma [3] have studied the electromagnetic fieldin solid iron, which eventually led to the development of anequivalent circuit published by Heller [4], who furtherextended the analytical method for considering the endeffects. A rather sophisticated numerical method of electro-magnetic field analysis has been developed by Wilson [5]whfle taking into account the field-dependent rotor mag-netic permeability. Sarma and Erdelyi [6] have presentedan approximate three-dimensional numerical analysis forthe solid-rotor induction motor.A composite-rotor construction, wherein the outer

stratum consists of a small number of layers of variousconductivities and permeabilities, has recently been con-sidered. The theoretical linear two-dimensional analysis ofsuch composite-rotor induction motors based on electro-magnetic field theory is presented by Wilson et al. [7], [5]under certain simplifying assumptions. It has been shown

Manuscript received September 12, 1971. through theoretical numerical solutions that the efficiencyM.S. Sarma is now with the Department of Electrical Engineering, of the composite rotor as compared with a solid-iron rotorInstitute of Technology, Banaras Hindu University, Varanasi, India- is increased considerably, that the torque-speed character-

FLEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-8, NO. 2 MARCH 1972 147

istic of the composite-rotor motor rises steeply and isnearly flat between 20- and 100-percent slip, and that theoptimum design approaches the characteristics of a multi-ple-cage motor, while operating temperatures are consid-erably reduced.Now that there are several methods of analysis for the

solid-rotor and composite-rotor induction motors varyingfrom equivalent circuits to sophisticated digital computerprograms, it is intended to perform several experiments ona three-phase induction motor with various rotors, comparethe test results with those predicted from selected theoret-ical methods, and arrive at some conclusions as to thereliability and limitations of the available theoreticalmethods. Literature survey indicates that there are no testresults published so far on composite-rotor inductionmachines.

Brief reviews of McConnell's theory [21, Heller's theory[4], and Wilson's theory [5] are given below, because theexperimental results will be compared to those predictedfrom their theories.

4) BS

liH

Magnetic Intensity

-Bs

McConnell's Equivalent Circuit

The magnetization characteristic for a solid rotor hasbeen assumed, as shown in Fig. 1. Although this idealiza-tion is quite extreme and is a limiting nonlinear treatment,and its use in the field analysis introduces certain incon-sistencies, a number of realistic results are obtainedpractically by its use. In the analysis [21, it can be seen thatthe radial depth of the flux penetration depends on themagnitude of the peak Bmax of air-gap flux density and on

the pole pitch r and does not depend on the rotor slipfrequency. As the rotor slips, any one point within the beltalternately acquires the values +Bs and -B, and at any

given instant the regions of opposite flux density are

separated by interfaces, which start from the rotor surfaceand have the shape of one quadrant of a sine wave. Theinterfaces sweep through the belt at a velocity that isproportional to the slip. The magnitudes of the instan-taneous eddy current density increase with slip, but theirrelative distribution does not vary. Finally, when the loadcomponent of stator current is tied together with the slip, itwill be seen that the equivalent circuit of the rotor has a

power factor that does not depend on the slip.The per-phase equivalent circuit of the polyphase induc-

tion motor with solid rotor, resulting from the abovetreatment [2], is shown in Fig. 2 for the case of balancedoperation. RS and Xs represent the stator winding resistanceand leakage reactance, respectively; Rm and Xm representthe resistive and reactive components of the magnetizingshunt impedance; E is the air-gap voltage, and V is thestator applied voltage per phase; RL is the rotor equivalentcircuit resistance, and XL is the equivalent rotor reactanceat the line frequency, corresponding to a given per-unitvalue of slip S.

Both the quantities RL and XL are inversely pro-

portional to the slip as well as the air-gap voltage E. The

power factor is

RLcos OL = 0.895

ZL(1)

and the impedance ZL is simply 1.12 RL. The followingexpressions result from the equivalent circuit:

FE E2Scurrent per phase IL = =

bf, amperes (2)ZL

and

mI2RL 0.063 mpE3Storque T = bf2 Jjoules (3)

where m is the number of stator phases and p is the numberof poles. The voltage E can simply be related to the appliedvoltage V through the values of RS and X, and theequivalent circuit. The phasor diagram may easily be drawnnoting that the angle by which IL lags E is a constant 26.60.The constant b is related to the motor data by

57.2 mL2 (kp I kd lN)3 Bspb-

pr(4)

where L is the core length, kp 1 and kd l are the pitch anddistribution factors for fundamentals, N is the stator turnsper phase in series, p is the resistivity, and r is the polepitch.

Heller's Equivalent Circuit

Heller's theory (41 is also based on an idealizedmagnetic characteristic shown in Fig. 1. It recognizes theexistence of two different fluxes, OR and Os. Os is a

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS MARCH 1972

Fig. 1. Ideal ized magnetic characteristic.

Fig. 2. Per-phase equivalent circuit for the solid-rotor machine based on McConnell's theory.

rx

148

Fig. 3. Phaser dia-gram of magneticfluxes (Heller'stheory).

Fig. 4. Phaser diagram of voltages(Heller's theory).

Fig. 5. Per-phase equivalent circuit for the solid-rotor machine based on Heller's theory.

Rs Ij~~X IL RL

i 6~~~~~~'l

transversal air-gap flux that links with the stator windingonly, but not with the rotor. For small values of fre-quencies the flux Os becomes negligible as compared to ObR.The total magnetic flux 01, given by the vectorial sum ofOR and Os, links with the stator winding while OR is theflux linking both the stator and the rotor. The phasordiagram of the magnetic fluxes is shown in Fig. 3. Thecorresponding voltage phase diagram is drawn in Fig. 4,where Vs is the voltage induced by flux Os, VR is thevoltage induced in the stator winding by flux OR, and V1 isthe stator terminal voltage. IL represents the load com-ponent of the stator current that compensates for the MMFof the rotor. The internal power factor, cos OL, iS shown tobe 0.915. The analysis is further continued, and it isestablished that both the primary and secondary currentsremain finite for the limiting infinite value of the fre-quency. Thus it is maintained that the rotor impedancedoes not increase without bounds for larger values of thefrequency. It may be recalled that the rotor impedancetends to infmity as the frequency tends to infinity inMcConnell's model [2]. However, in the normal range of60-Hz frequencies, both RL and XL are directly pro-portional to the frequency, as in McConnell's theory.Further, it is shown that the rotor impedance is not aconstant and is inversely proportional to the rotor flux OR.The expressions for RL and XL show their dependency onslip as well as the air-gap voltage. In a certain range offrequencies, both vary inversely proportional to the squareroot ofE and S. It should be pointed out here that both RLand XL vary in proportion to the reciprocals ofE and S inMcConnell's model for all frequencies. Based on the aboveconsiderations, an approximate equivalent circuit is devel-oped, as given in Fig. 5, including the stator winding

resistance, leakage reactance, as well as the magnetizingshunt impedance. Expressions for Xq, XL, and RL in termsof motor data are developed in [4]. It is shown thatappropriate simplified expressions are to be used dependingon the range of frequency. The stator leakage reactance Xsincludes the usual differential leakage, slot leakage, andtheend-connection leakage. It is further pointed out [4] thatthe drop in the cross-reactance should be really caused bythe current IL above and not by I . Since the errorincluded is rather small, a simple equivalent circuit is drawnwithout making the circuit unduly complicated.

For small values of slip, it is shown that the current IL isnearly proportional to the per-unit slip 5, and the torque isapproximately

T _ kjS - k2S2 (5)where k1 and k2 are constants.

In order to include the end effects of the finite solidrotor, the analysis is further continued, and a correctionfactor for the rotor length has been developed such that thetwo-dimensional theory may still be applied for practicalengineering results. Using the theory developed, rotorlosses, torque, as well as the related data at different valuesof slip, can be calculated with reasonable accuracy. If therotor ends are provided with heavy short-circuiting rings,the theory based on an infinitely long rotor is generallysatisfactory.

Wilson's Theory [51

A mathematical model is set up for the electromagneticfield analysis and the partial differential equations, as wellas the boundary conditions, are described starting from

SARMA AND SONI: SOLID- AND COMPOSITE-ROTOR INDUCTION MACHINES 149

Maxwell's equations, taking into consideration the motionof the rotor. As a first step, a digital computer program hasbeen developed based on the linear two-dimensional fieldtheory for the calculation of performance characteristics ofthe solid-rotor induction motor, i.e., torque, phase current,power factor, power output, and the magnetic flux distribu-tion as functions of the slip at different relative permeabil-ities and conductivities.

Then a composite-rotor construction is considered forperformance improvement. The partial differential equa-tions of this arrangement are developed and the boundaryconditions at the interfaces discussed. A digital computerprogram for the explicit solution of the equations based onthe linear two-dimensional analysis is presented [5] to findthe performance characteristics of two- and three-layercomposite-rotor structures.

To include the end effects of a finite rotor, a three-dimensional linear treatment is developed next. Con-siderably involved analytical work has been carried on and acomputer program is written to predict the performance ofthe solid-rotor induction motor.

Finally, the nonlinear two-dimensional analysis of thesolid-rotor induction machine is attempted, and a sophisti-cated digital computer program is developed with consid-erable ingenuity. The nonlinear magnetization curve of thesolid-rotor iron is assumed to be single valved. In thenonlinear region where saturation prevails, the partialdifferential equations are transformed to partial differenceequations. The vector potentials are found at large numbersof discrete points. The grid system of the analysis isdescribed and the difference algorithms established. In thenonlinear region a unique set of vector potentials satisfyingthe difference equations is found by relaxation iterationswhose convergence is monitored continuously. Line relaxa-tion techniques are used, and a novel method has beendeveloped for satisfying the boundary condition caused bythe periodicity of the fields. In the unsaturated iron regionthe linear solution has been used. The interface between thesaturated and unsaturated regions is moved to take care ofany new calculated saturation conditions. For a betterconvergence of the iterative procedure, the reluctivities(which are the reciprocals of the permeabilities) have beenunderrelaxed by a new method of finding an optimumunderrelaxation factor. It is shown that saturation hasconsiderable influence on the performance characteristics.The magnetic flux distribution for different values of slip iscalculated and the skin depth is shown to have increasedcorresponding to the degree of saturation. The flow chartsand detailed listings of the computer programs are given int5]. They are too involved and space consuming to bediscussed here.

Experimental Study

The performance characteristics of different solid-rotorand composite-rotor induction machinres have been investi-gated experimentally by using the stator of a commercial

three-phase squirrel-cage induction motor, whose pertinentmachine data is given below:

Generalrated power output: 1/3 hprated line voltage: 220 voltsrated frequency: 60 Hznumber of phases: 3number of poles: 4

Stator winding dataconnection: starnumber of turns per phase connected in series: 408winding pitch factor (fundamental): 0.985winding distribution factor (fundamental): 0.960winding resistance per phase at 25°C: 7.8 ohms

Machine Dimensionsstator core length: 0.0508 meterpole pitch: 0.0698 meterphysical airgap length: 2.92 X 109 meterstator bore diameter: 0.0889 meterrotor diameter: 0.0883 meterratio of active rotor length to pole pitch: 0.73.

Construction of Different Rotors

A solid rotor was constructed out of cold-rolled SAE1045 steel. Its diameter is so chosen as to keep the air-gaplength the same as in the original squirrel-cage inductionmotor. It is known that field pulsations due to the presenceof open slots in the stator are the cause of parasitic surfacelosses in nonlaminated rotor structure. Increasing thestator-to-rotor air gap by a certain ratio would practicallyeliminate these losses. However, this aspect of the problemhas not been considered in this experimental work. Theaxial length of the rotor was chosen equal to the stator corelength. It was far from infinite as assumed in two-dimensional analysis; in fact, it was barely equal to 0.73times pole pitch. It is known that the use of two thickcopper end plates would greatly reduce the rotor endeffects by offering an external path for the rotor-inducedcurrent to close. In such a case the assumption of aninfinite-length rotor and the two-dimensional analysiswould be more acceptable. However, the rotor in thisexperimental work was not fitted with the copper endrings.

Four two-layer composite rotors were later constructed.The first solid-iron layer was made of SAE 1045 steel, andthe second thin layer was made of commercial copper,0.007, 0.03125, 0.0625, and 0.25 inch thick, respectively.In each of the above cases, the air-gap length was the sameas that in the original machine. The lowest thickness ofcopper was deposited on the solid-iron surface by thefamiliar electrolytic process. The other thicknesses ofcopper were achieved by shrink-fitting appropriate sizes ofcopper sleaves onto the solid-iron rotor and suitablemachining on the lathe. It should be pointed out here thatsuch a technique of construction works for operation of themotor on supply frequencies ranging from 60 to 120 Hz.However, it may not be suitable for ultrahigh-speedoperation at high supply frequencies of the order of 3000to 4000 Hz. In such cases special metallurgical processes

IEEE TRANSACTIONS ON AEROSPACIE AND ELECTRONIC SYSTEMS MARCH 1972150

EXPERISE EAL

-w ~THELLER'S THEORY

I- -

NMcCONNELL'S THEORY

o/o8/oo.. WILSON'S 2 D-LIN.,,/ Z/ EREL. PER. 100

.,2_--.--- -r -

7 , - - - - - - - - - r - - - - -WILSON'S 2 D-LIN.

ffi i~~~~~L REL. PER. 400

0.0

.50.05

HELLER'S THEORY

._-..______ McCONNELL'S THEORY

- =--______ l EXPERIMENTAL

-/ JUWILSON'S 2 D-LIN.-- / I REL. PER. - 100

L..._WILSON'S 2 D-LIN.- EREL. PER. - 400

0.1 0.2 0.3 0.4 0.5O 0.1 0.2 0.3 0.4 0.5

P.U. SLIP

Fig. 6. Performance characteristics at 220 volts and 60 Hz of the

solid-rotor induction motor.

P.U. SLIP

Fig. 7. Performance characteristics at 160 volts and 60 Hz ofthe solid-rotor induction motor.

Fig. 8. Performance characteristics at 220 volts and 110 Hz of thesolid-rotor induction motor.

0.2p

Or

IR.1

EXPE RIMENT*AL

_- IIELLER'S THEORY

=- --- - ---McECONNELL'S THEORY

WILSON'S 2 D-LIN.

IEREL. PER.

-

---- ---------T

-l~WLSON'S 2 D-LIN.

I I t I REL. PER. - 4000 0. 0.2 R3 I

O 0.1 0.P2 0.3

P.U. SLIP

would have to be considered in -order that the compositerotor and its inner interfaces would withstand the associ-ated temperature rises and rotational stresses.

Then the three-layer composite-rotor construction was

considered. One such rotor was made with the first thicklayer of solid iron (SAE 1045 steel), the second thin layerof 0.03125-inch-thick copper, and the third layer of0.03125-inch-thick iron (SAE 1045 steel) closest to the airgap of the machine. The shrink-fitting technique was againsuccessfully utilized. The final diameter of the compositerotor was so arranged to retain the same stator-rotor air gapas in the original machine.

Thus one solid rotor, four two-layer composite rotors,and one three-layer composite rotor were made for pur-

suing the experimental study.

Tests

A. Solid- Rotor Induction Machine

No-load and locked-rotor tests were carried out on thesolid-rotor machine with a sinusoidal 60-Hz supply. Byusing the conventional theory of classical induction motors,the stator resistance, stator leakage reactance, as well as themagnetizing shunt impedance, were calculated. These valueswere eventually used in the equivalent circuits.

Load tests were performed at different supply voltagesand frequencies using an electric dynamometer. The var-

iable supply frequencies were obtained from operating a

medium-sized regular induction motor as a frequencyconverter. The results corresponding to only three cases are

typically reported here in view of the space limitations.Even though various performance characteristics were

plotted [8], only the horsepower versus per-unit slipcharacteristics are presented here for discussions andcomparisons, corresponding to the following power supplyconditions: 1) 220 volts, 60 Hz; 2) 160 volts, 60 Hz; and 3)220 volts, 110 Hz. These characteristics are shown in Figs.6 through 8.

B. Composite-Rotor Induction Machines

Load tests were also conducted on the two-layer as wellas three-layer composite-rotor induction motors and againthe output horsepower versus per-unit slip characteristicsare shown in Figs. 9 through 12. The supply line voltagewas fixed at the rated value of 220 volts, and load testswere repeated at different frequencies. The performancecurve corresponding to the operation of the solid-rotorinduction machine is also shown for the respective cases inFigs. 9 through 12 for ready comparison purposes. Theexperimental characteristics corresponding to only thefollowing power supply conditions are depicted here: 1)220 volts, 60 Hz; 2) 220 volts, 110 Hz. The quantity AIshown in the Figs. 9 and 10 represents the thickness-of thecopper layer in the two-layer composite-rotor construction.The Al and A2 shown in Figs. 11 and 12 stand,respectively, for the thicknesses of the iron layer and thecopper layer of the three-layer composite rotor.

It should be pointed out here that no experimentalresults could be obtained with the two-layer compositerotor in which the thickness of the copper layer was 0.25

SARMA AND SONI: SOLID- AND COMPOSIT E-ROTOR INDUCTION MACHINES

0.20.

C 0.1

I11 . 11 I A I n L f) 11

0.4 U.!5

is I

0.4[

0.3

20.

,3/oX | WILSON'S 2 D-LIN. THEORYAl - 0.03125 INCH

REL. PER.-

400

EXPTL. (&I - 0.007 IENH)

EXPTL. (SOLID ROTOR)0.1 g X_ _1- SHOWN FOR COMPARISON

WILSON'S 2 D-LIN. THEORYAl - 0.007 INCHEEL. PER. -400

0 0.1 0.2 0.3 0.4 0.5

P.U. SLIP

Fig. 9. Horsepower versus slip characteristics of the two-layercomposite-rotor induction motor at 220 volts, 60 Hz.

Fig. 10. Horsepower versus slip characteristics of the two-layercomposite-rotor induction motor at 220 volts, 110 Hz.

WILSON'S 2 D-LIN. THEORY_ F ==Al 0.03125 INCH

EXPT. (41 - 0.03125 INCH) _ - R_EEL. PER. - 400

gw

0.

p 0.1

0. I .

EXPTL.l1 - A2 - 0.03125 INCH

WILSON'S 2 D-LIN. THEORYREL. PER. (LAYER 1) - 100REL. PER. (LAYER 3) - 500

EXP`L... (SOLID RarOR)SHOWN FOR COMPARISON

O 0.1 0.2 0.3 0.4 0.5

P.U. SLIP

Fig. 11. Horsepower versus slip characteristics of the three-layercomposite-rotor induction motor at 220 volts, 60 Hz.

Fig. 12. Experimental characteristics of solid-rotor, two-layerand three-layer composite-rotor induction motors.

3-LAYER COFP. ROTOR2V20V, 60N

41 - A 2 . 0.03125 INCH

0.3I

:t° 0.2

t0

0.1 0.2 0.3 0.4 0.5

P.U. SLIP

inch. The line currents drawn from the supply as well as thetemperature rises were too excessive to permit any resultsto be taken.

Typical test results with different rotors and supplyconditions are shown in Fig. 12 for purposes of readycomparison.

Theoretically Predicted Characteristics

A. McConnell's Theory

The stator resistance was measured to be 7.8 ohms perphase using the conventional methods. The familiar no-loadtest and lock-rotor test yielded values for the magnetizingimpedance and the stator leakage reactance. Rm wascalculated to be 440 ohms; Xm to be 1.56 fi ohms; and X,to be 0.315 fi ohms, where fi is the supply frequency.Load tests were then performed and analyzed. The designconstant b over the range of test voltages and frequencieswas found to be varying from 36 to 47. An average value of42 is chosen for b to draw the equivalent circuit. The valueof b calculated from the machine data using (4) is 37,which is about 12 percent lower than the average value

chosen from the experimental results. The dependence ofRL upon E was essentially substantiated. The equivalentcircuit based on McConnell's theory for the experimentalsolid-rotor induction motor was developed, as shown in Fig.13. Performance characteristics for three different supplyconditions based on the above approximate equivalentcircuit are drawn for purposes of comparison in Figs. 6through 8. Several other characteristics such as inputcurrent, power factor, and efficiency at various supplyvoltages and frequencies have been included in [8].

B. Heller's Theory

The values of stator resistance, stator leakage reactance,and magnetizing shunt impedance found earlier were used.For the particular range of test frequencies used in thisexperimental study, the rotor impedance and the crossreactance were calculated using the appropriate expressionsgiven in [4]. The corresponding rotor terms are seen tovary directly proportional to the supply frequency, andinversely proportional to the square root of slip as well asair-gap voltage. The constant appearing in the expressionfor RL is found to vary from 6.39 to 8.3 from tests results,


n1 I n1 ) I In In1 cn

0.41

I I

0.3

00

152

7.8.JA. O0.315f,).~ RL frl

R~~L S

V at fl 440-l L SE-(1.56 1)

Fig. 13. Equivalent circuit of the solid-rotor induc-tion motor based on McConnell's theory.

Fig. 14. Equivalent circuit of the solid-rotor induc-tion motor based on Heller's theory.

7.8fl. (.315f, A (0.0072fI) .

E

and an average of 7.7 has been chosen. The approximateequivalent circuit based on Heller's theory was thendeveloped, as shown in Fig. 14, for the solid-rotorinduction motor under investigation. For the range of testfrequencies used, it was found that the effect of the crossreactance was negligible for the investigated motor. How-ever, it might be quite significant if one is operating atfrequencies of the order of 3000 to 4000 Hz. Outputcharacteristics for three different supply conditions basedon the above approximate equivalent circuit are shown forpurposes of comparison in Figs. 6 through 8.

C. Wilson's Theory [51

While McConnell and Heller thought of developingapproximate lumped-parameter equivalent circuits for theprediction of the solid-rotor induction motor, Wilsondeveloped digital computer programs for the analysis. Onlythe computer program based on two-dimensional linearanalysis has been applied for the solid-rotor inductionmachine. With the supply conditions of 220 volts and 60Hz, theoretical performance characteristics have been ob-tained for two different values of relative permeability (100and 400) of the solid-iron rotor. Typical horsepower versusslip curves corresponding to these cases are shown in Fig. 6.Similar characteristics obtained for different supply con-ditions have been drawn in Figs. 7 and 8.

Later the computer program developed for the two-dimensional linear analysis of the composite-rotor induc-tion motor was used for obtaining the theoretical perfor-mance curves of the experimental induction motor withdifferent two-layer rotors as well as the three-layer rotor.For the cases with two-layer rotors, the relative permeabil-

ity of the solid-iron rotor is chosen as 400; the conductivityof the solid iron is taken to be 5 X 106 mho/m; and theconductivity of the copper layer has a value of 5.8 X 107mho/m. Characteristics with the two-layer rotors are shownin Figs. 9 and 10.

In the case of the three-layer rotor, two values of therelative permeability (50 and 100) of the iron layer that isnearest to the air gap of the machine are tried, while that ofthe solid-iron rotor is kept at 500. Corresponding theoret-ical performance curves are given in Fig. 11.

Discussion of Results

An induction motor with solid-iron rotor should notsimply be considered as equivalent to a squirrel-cageinduction motor with high rotor resistance or a deep-barsquirrel-cage machine with frequency-dependent rotorresistance. The mechanism of flux penetration into thenonlinear magnetic material of the solid-iron rotor dependsgreatly on saturation effects. The dependence of rotorimpedance upon the air-gap flux densities has been verified.

The predicted results based on McConnell's and Heller'sequivalent circuits agree reasonably well with the exper-imental results, as can be seen from Figs. 6 through 8. Inthe range of 0.2 to 0.5 per-unit slip, the agreement appearsto be within 20 percent of the test results. The disagree-ment may be attributed to the end effects in view of thefinite dimensions of the rotor (ratio of active rotor lengthto pole pitch is 0.73) as well as the practical nonlinearmagnetic properties of the solid-iron rotor.

Theoretical predictions based on McConnell's theory aswell as Heller's theory involve a certain number of implicitassumptions such as the idealized magnetic characteristic,constant internal power-factor angle, and undistorted flux-density distribution. Starting from the assumption ofsinusoidal air-gap flux density, McConnell and Sverdrup [2]concluded that even for the case of sinusoidal supplyvoltage and ideally distributed stator winding the air-gapfield really has significant space harmonics; nevertheless,performance calculations ignoring this effect are of con-siderable engineering value. An analysis of the effects onthe motor of time-varying harmonic voltages in the supplyis not easy because the magnetic nonlinearity precludes theuse of the superposition principle.

Wilson's two-dimensional linear analysis gives differentresults depending upon the value chosen for the relativepermeability of the solid-iron rotor. When the machine isoperating under conditions of high saturation, this linearanalysis cannot be obviously applied and relied upon. Insuch cases, one has to apply the nonlinear analysis programdeveloped by Wilson. In view of the complexity of thecomputer program and limitations on computer time andmoney, it has not been applied for the experimental motor.However, one can obtain considerable information re-garding the general trends of the performance character-istics even from the linear analysis program. As expected,the rotor with relative permeability of 100 yields higher


horsepower than the one with relative permeability of 400.The predicted results are comparatively closer to the testresults in Figs. 7 and 8, because the supply conditionscorrespond to the operation of the motor under lesssaturated conditions. The effects of nonlinearity andsaturation evidently gives rise to higher horsepower outputat higher values of slip, as may be seen from Fig. 6.

As Finzi and Paice [ 1 pointed out, the peculiardependency of the rotor parameters upon the air-gapvoltage and frequency imparts very desirable features to thesolid-rotor motor if speed control by varying input voltageis considered or if problems of motor starting from invertersupplies arise. In the latter case, either with reduced voltagestart or with increased frequency start, the degradation ofstarting torque is much less severe than for conventionalmotors.

Equivalent circuits developed by the use of the con-ventional circuit concept have convenient forms that areinterpretable and easy to visualize. Although all phenomenaoccur in space and time, and these are field phenomena, itis possible to approximate these by time relations only,summarizing the actual physical constants by means ofparameters, preferably of constant value. However, anaccurate knowledge of the magnetic field distribution is ofgreat importance in design optimization. As a result, acloser examination of the magnetic field becomes moreimportant than ever before. The "field theory" as opposedto "circuit theory" is becoming increasingly important inmany engineering problems. Wilson's approach for theanalysis of solid-rotor and composite-rotor inductionmachines yields more insight regarding the distributions ofthe magnetic field as well as the eddy currents in the rotor.

The primary factor playing a key role in the perfor-mance of solid-rotor and composite-rotor induction motorsis the penetration depth of the currents present in therotor. It is necessary to attempt to make the best use of theregions in which these torque-producing currents exist toobtain a more efficient rotor. The problem of optimizationof the composite rotor may be explored in the near future.

Equivalent circuits for the prediction of performance ofcomposite-rotor machines have not yet been developed. Ithas been verified that the effect of the composite-rotorcircuit cannot adequately be represented by a singleimpedance ZL. The composite-rotor circuit should con-ceivably be characterized by a series of impedances cor-responding to the different layers. These impedances wouldof course be dependent on the frequency, air-gap voltage,and slip. The determination of their functional forms wouldbe too complicated.

At the present stage, only the two-dimensional linearanalysis developed by Wilson et al. [71 is available in theliterature for the prediction of performance of a com-posite-rotor induction machine. Figs. 9 and 10 show thecomparison between the theoretical and experimentalresults for typical cases of two-layer composite-rotormachines. It is seen that the agreement is better when thethickness of the copper layer is increased and when themotor is operated under less saturated conditions.

As suggested in the conclusions of Wilson et al. [71, alayer of high permeability was placed nearest to the air gapand a layer of high conductivity was placed next to it onthe solid-iron part of the rotor. The highly permeable layerserves to increase the flux density at the rotor surface andshould be chosen to be relatively thin, thereby allowing thehighly conductive region to lie nearer the rotor surface inregions of higher current density. An excessive increase inthe conductivity and/or the thickness of the highly con-ductive layer may result in a deterioration of the per-formance characteristics due to the increase in reluctance ofthe magnetic circuit. However, some increase in thethickness results in an increase in the slope of thehorsepower versus slip curve at low values of slip, as can beseen from Figs. 9 through 11.

In view of the fact that the iron layer nearest to the airgap of the motor would be highly saturated compared tothe solid-iron part, its relative permeability was given twodifferent values (50 and 100) while that of solid iron waskept at 500. Results for the three-layer rotor based on thecomputer program are shown in Fig. 11 along with theexperimental results. It may be concluded that the motor isin fact highly saturated.

Fig. 12 indicates clearly that the horsepower versus slipcharacteristic of the composite-rotor motor rises steeplycompared to that of the solid-rotor machine. The currentsdrawn from the supply with the three-layer rotor werefound to be consistently lower than those with thetwo-layer rotor. Thus, it is possible to design a three-layercomposite rotor which retains the desirable performance ofthe two-layer construction in the operating range and whichlimits the current at high values of slip. The efficiency ofthe composite rotor as compared with a solid-iron rotormay be increased considerably with optimum design.

Conclusions

The usefulness of three different analytical approachesto predict the performance of solid-rotor induction motorshas been established by comparing the theoretical andexperimental results. Within the scope of the limited testsconducted, it is concluded that the equivalent circuitsdeveloped by McConnell and Heller for the solid-rotormachine yield reasonably good results in the normalworking range. Wilson's approach provides more informa-tion about the magnetic-field distribution as well as theeddy currents within the rotor. His method of analysis isparticularly useful for optimizing the designs of composite-rotor induction machines which would decidedly havebetter characteristics than those of the solid-rotor motorfor certain aerospace requirements. While the tests couldonly be conducted here over a frequency range of 60 to120 Hz, in view of the limited available generationequipment, it would be highly desirable to develop theexact technology of depositing the copper and iron layersfor the construction of aerospace composite-rotor induc-tion motors and perform the tests at high supply fre-quencies such as 3000 to 4000 Hz.


Acknowledgkment

The authors express their thanks to Dr. W.W. Cannon,Chairman of the Department of Electrical Engineering atWest Virginia University, for his encouragement and sup-port given in carrying out the experimental investigation.Special thanks are due to W.B. Moore for his help inconstructing the rotors.

References

11 L.A. Finzi and D.A. Paice, "Analysis of the solid iron rotorinduction motor for sohid-state speed controls," IEEE Trans.Power Apparatus and Systems, vol. PAS-87, pp. 590-596,February 1968.

12] H.M. McConnell and E.F. Sverdrup, "The induction machinewith solid iron rotor," AIEE Trans. Power Apparatus and

Systems, vol. 74, pp. 343-349, June 1955.L31 B. Heller and M.S. Sarma, "The electromagnetic field in solid

iron," Acta Tech. (Prague), no. 6, 1968.141 B. He-ler, "Synthesis of high speed homopolar alternators and

theory of solid rotor electrical machines, phase DD, pt. III,"USAMERDC Rept., Fort Belvoir, Va., March 1969.

151 1.C. Wilson, "Theory of the solid rotor induction machine,"Ph.D. dissertation, University of Colorado, Boulder, August1969.

161 M.S. Sarma and E-A. Erdelyi. "Synthesis of high speedhomopolar alternators and theory of solid rotor elecicalmachines, phase DD, pt. IV," USAMERDC Rept., Fort Belvoir,Va., March 1969.

171 J.C. Wilson, R.E. Hopkins, and E.A. Erdelyi, -Aerospacecomposite-rotor induction motors," IEEE Trans. Aerospace(Supplement), vol. AES-3, June 1965.

181 G.R. Soni, "Experimental study of solid-rotor and composite-rotor induction motors," M.S. thesis, West Virginia University,Morgantown, 1971.

Mulukuta S. Sarnia (S'65-M'68-SM'69) was born in Andhra Pradesh, India, onNovember 26, 1938. He received the B.S. degree in electrical engineering and the M.S.degree in electrical machine design, both from Banaras Hindu University, Varanasi, India,in 1958 and 1959, respectively, and the Ph.D. degree in electrical engineering from theUniversity of Colorado, Boulder, in 1968.

In 1960 he was selected as first of the two Hindu-Hitachi Scholars from India and wassent to Hitachi Ltd., Japan, for higher technical training in the field of electrical powerstation equipment. Since 1961 he has worked as a Pool-Officer of the Government ofIndia attached to the Indian Institute of Technology, Kharagpur; as a Lecturer inelectrical engineering at the Indian Institute of Technology, Madras; and as a ResearchAssistant, Visiting Assistant Professor, and a Postdoctoral Research Associate at theUniversity of Colorado. From 1969 to 1971 he was at West Virginia University,Morgantown, as an Assistant Professor. He has been the author of a number of technicalpapers, and a consultant to electrical equipment users and manufacturers, as well aselectric utility companies. He is now with the Department of Electrical Engineering,Institute of Technology, of Banaras Hindu University. His areas of interest areelectromechanics and energy conversion, electromagnetic field theory, power systems,and computer-aided analysis and design.

Dr. Sarrna is a member of the Institution of Electrical Engineers (London), theInstitute of Electrical Engineers of Japan, the Tensor Society of Great Britain, theAmerican Association of University Professors, the American Society for EngineeringEducation, Eta Kappa Nu, Tau Beta Pi, and Sigrna Xi. He is a chartered ElectricalEngineer in Great Britain.

Gorakh R. Soni (S'70) was born in Hamachal Pradesh, India. He received the B.S.degree in electrical engineering from Banaras Hindu University, Varanasi, India, in 1962.Since July 1970 he has been working towards the M.S. degree at West Virginia University,Morgantown.

He worked as a Development Engineer with M/S Air Conditioning Corporation Ltd.,Calcutta, India, from 1962 to 1965. In 1965 he was selected by the All IndiaManufacturer's Association, Bombay, and was sent to West Germany for advancedtraining in the manufacturing of air conditioning equipment. During his stay of 1½ yearsin West Germany, he also took advanced courses in industrial engineering with REFA,Darmstadt, West Germany, Upon his return to India, he worked with M/S AirConditioning Corporation Ltd. in different capacities. His interests include electro-mechanics, air conditioning, and power systems.


Documents

Solid-Rotor and Composite-Rotor Induction Machines