THE PROCEEDINGS OF THE INSTITUTION OF ELECTRICAL ENGINEERS

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1 The Institution is not, as a body, responsible for the opinions expressed by individual authors or speakers. An example of the preferred form of bibliographical references will be found beneath the list of contents. THE PROCEEDINGS OF THE INSTITUTION OF ELECTRICAL ENGINEERS EDITED UNDER THE SUPERINTENDENCE OF W. K. BRASHER, C.B.E., M.A., M.I.E.E., SECRETARY VOL PART C. NO. 8. SEPTEMBER The Institution of Electrical Engineers Monograph No. 283 U Jan A SIMPLIFIED ELECTROMAGNETIC THEORY OF THE INDUCTION MOTOR, USING THE CONCEPT OF WAVE IMPEDANCE By Professor A. L. CULLEN, Ph.D., B.Sc, and T. H. BARTON, Ph.D., B.Eng., Associate Members. {The paper was first received 8th July, and in revised form 30th October, // was published as an INSTITUTION MONOGRAPH in January, 1958.) SUMMARY Electromagnetic field theory is applied to the analysis of performance of an induction-motor rotor when it is exposed to a rotating magnetic field of constant amplitude. An idealized model of the rotor is taken and the analysis is simplified by the application of the concept of wave impedance. The model rotor is capable of simulating the effects of tooth-top and zigzag leakage fluxes, but end effects are neglected. The equations of performance thus obtained are shown to be identical with those obtained by the conventional theory when the latter includes skin effect. The model motor used by Mishkin {Quarterly Journal of Mechanics and Applied Mathematics, 1954, 7, p. 472) is shown to give results widely different from those obtained in practice, since it neglects the zigzag leakage fluxes. The analogy between the tangential force acting on the rotor and radiation pressure is indicated, and it is shown that a change of variable familiar in microwave theory leads to the theory of the variable-speed induction motor developed by Williams and Laithwaite [Proceedings I.E.E., Paper No U, June, 1956 (104, A, p. 102)]. LIST OF PRINCIPAL SYMBOLS a = Depth of tooth-top region of rotor. b = Depth of rotor bars. b s = Width of rotor bars. B m = Maximum magnetic flux density. d = Width of opening between tops of teeth on rotor. E z = Electric field strength in equivalent uniform rotor. f x = Force per unit volume on equivalent uniform rotor. F = Force per unit area of rotor surface. H x, H y Magnetic field components in equivalent uniform rotor. J z = Current density in equivalent uniform rotor. p s = Slot pitch. R = Rotor-bar resistance. s = Slip. Correspondence on Monographs is invited for consideration with a view to publication. Prof. Cullen is Professor of Electrical Engineering, University of Sheffield. Dr. Barton, who was formerly in the Department of Electrical Engineering, University of Sheffield, is now in the Department of Electrical Engineering, McGill University, Montreal. X Q = Equivalent standstill rotor-bar reactance. x, y Rectangular co-ordinates. Z 01, Z o2 = Wave impedances in tooth-top and rotor-bar regions respectively. a = Skin effect parameter: a" 1 = skin depth. fi = Phase-change coefficient of flux wave. Y\>Y2 = Propagation coefficients looking into rotor through tooth-top and rotor-bar regions respectively. [x 0 = Absolute permeability of free space. fx r Relative permeability. fju x, fjl y, (x x, fjl y = Absolute permeabilities of equivalent uniform anisotropic media. (1) INTRODUCTION The induction motor, like all other electrical devices, must obey the laws of electricity and magnetism as formulated in Maxwell's equations of the electromagnetic field. However, a practical induction motor is so complicated in its geometry that an exact solution of these equations is quite impracticable, even if the effects of non-linearity and hysteresis in the iron are ignored. A working theory of the induction motor has, however, been developed by making use of a.c. circuit theory. The parameters introduced into this theory can be evaluated theoretically by individual field calculations, usually of an approximate character. In a rather indirect sense, this theory can be regarded as an approximate solution of Maxwell's equations, for the a.c. circuit equations themselves follow as a consequence of Maxwell's equations, subject to certain definable limitations. Recently, however, an approximate solution of Maxwell's equations for an idealized model of an induction motor has been obtained by Mishkin. 1 His model is linear and replaces the practical toothed structure of stator and rotor by continuous though inhomogeneous regions, having appropriately averaged resistivities, and different permeabilities in directions parallel to, and perpendicular to, the air-gap. The effect of end windings is taken into account in the theory. The analysis, though approximate, leads to results which are exceedingly complicated, so that VOL. 105, PART C, NO. 8. the connection between Mishkin's equations and the usual 1958: The Institution of Electrical Engineers [ 331 ] 12

2 332 CULLEN AND BARTON: A SIMPLIFIED ELECTROMAGNETIC THEORY OF performance equations of the induction motor are not readily discernible. His analysis ignores the large part played by toothtop and zigzag leakage flux in a practical machine. This omission leads to performance equations in which a principal component of leakage reactance is that of the rotor-slot, which, as is well known, is exceedingly dependent upon the slip. As a consequence, a current locus which diverges considerably from the circle diagram is given in a particular numerical example, and Mishkin attributes the divergence to skin effect. In fact, skin effect would be negligible over a practicable speed range in the example taken. Such divergence as is observed in practice is primarily attributable to saturation of the leakage-flux paths. The purpose of the present paper is to present a more elementary electromagnetic theory of the induction motor in which, for simplicity, end effects are neglected, but which pays due attention to the part played by tooth-top leakage fluxes. The use of a simpler model facilitates comparison with conventional theory and moreover makes possible an exact solution of the problem. The solution of the field equations is simplified, and given a clear physical significance, by making use in the analysis of the concept of wave impedance, which has for many years been a corner-stone of microwave theory, but which, so far as the authors are aware, has not previously been invoked in the theory of electrical machinery. A further point of interest is that a simple change of variable, familiar in microwave circuit theory, leads at once to the theory of the brushless variable-speed induction motor of Williams and Laithwaite. 3 It is shown, furthermore, that the theory of the squirrel-cage induction motor is closely related to the theory of radiation pressure and to the theory of electromagnetic surface waves. The force acting on the rotor of such a machine can in fact be legitimately described as the radiation stress due to a surface wave incident on the rotor at a complex angle of incidence of the form T7-/2 + jifj, where \\s is much greater than unity. (2) CIRCUIT ANALYSIS The purpose of this Section is to state briefly the results of the circuit method of analysis of the induction motor, for comparison with the field-analysis results which are obtained in the following Section. We consider a motor in which the flux density in the air-gap is a sinusoidal travelling wave of maximum value B m. The timeaverage tangential force Fper unit area of rotor surface is given by sr F = 2TT PS where to 0 is the stator frequency; s, the slip; T, the pole pitch; p s the slot pitch; R, the resistance per unit length of the rotor bars; and X Q, the rotor-bar reactance per unit length at standstill. If R and X o can be regarded as constants, eqn. (1) gives the relationship between torque and slip for an induction motor quoted in elementary textbooks. In fact, however, R and X o depend on the rotor frequency because of skin effect. Formulae allowing for skin effect are given by Alger. 4 For our case and our notation they can be written _ a sinh lab + sin lab a Q b s cosh lab cos lab v fa 1 a sinh lab sin lab\ *o ^o^o I ~j ' E u ^, -z r i \ct sojqfiq GQb s cosh 2ab cos 2ao/ where a 0 is the conductivity of the rotor bars and The rest of the notation is given in Fig. 1. (2) (3) V7\ ^ IRON COPPER Fig. 1. Slotted linear model motor. The term co 0 /x 0 (a/c/) in eqn. (3) is the reactance due to tooth-top leakage, which is dependent solely on the total conductor current and is unaffected by a non-uniform current density. Eqns. (2) and (3) can, of course, be expanded in powers of ab, and the familiar formulae for R and X o neglecting skin effect can easily be derived in this way. To a first approximation, we find that 1 <j o bb s (2a) These formulae are applicable if lab < 1.. (3a) (3) FIELD ANALYSIS The first step in applying the field method of analysis is to formulate a model of an induction motor which is simple enough to make such an analysis possible. (3.1) Formulation of Idealized Problem Wefirstignore the curvature of the air-gap and take a linearized cross-section of the actual machine as shown in Fig. 1. The effect of stator slots is also ignored. To facilitate analysis, the toothed structure of Fig. 1 is replaced by the continuous structure comprising regions 1 and 2 of Fig. 2. It is assumed that the v Fig. 2. Uniform linear model motor. (4) rotor core, region 3, and the stator are of infinite permeability and zero conductivity. Region 1 represents the air-gap and tooth tops of Fig. 1, and therefore its conductivity is taken as

3 THE INDUCTION MOTOR, USING THE CONCEPT OF WAVE IMPEDANCE 333 zero, whilst region 2 represents the conductor region of Fig. 1 setting out the relevant equations side by side, as Booker has and has a mean conductivity of done in dealing with other electromagnetic problems. 2 a = T a o ( 5 ) Induction motor Transmission line The effective permeability, on the other hand, is more difficult to evaluate. It is clear, in the first place, that the average reluctance of a path parallel to the rotor surface and passing through the slots is much greater than that in a direction perpendicular to the rotor surface. Thus the continuous region 2 in Fig. 2, which corresponds to the slotted region in Fig. 1, must be anisotropic in its magnetic properties. Similar remarks apply to region 1 in Fig. 2, which corresponds to the tooth-top region in Fig. 1. An exact determination of the effective permeabilities is very difficult, but if fringing is neglected, simple magnetic-circuit ideas can be applied to give very crude approximations. For region 1: ^=ja>i* x H x >f* (o +.P) Ez dy V J<OVyJ There is clearly a one-to-one correspondence between the following quantities E r -+ V H x ja>fj, x (12) 7>y >' Dy YV For region 2: H-'x - Ho j Ps H-y - Ps... (6) Since Z and Y represent respectively the series impedance and shunt admittance per unit length of the transmission line, we can construct for the induction motor a transmission line equivalent which represents exactly the relationship between E z and H x in region 2. An elementary section of this line is shown in Fig. 3. fly - Ps - (7) In these equations fx r is the relative permeability of the iron. This will in practice be of the order of 4000, and has been taken as infinite elsewhere in formulating eqns. (6) and (7). It is also taken as infinite in region 3, which represents the unslotted rotor core. (3.2) Field Analysis Maxwell's equations in region 2 are: (8) Sy orsy naa/ws (9) lh y - ^.... (10) 17 In these equations a time factor e-" 0 ' is assumed, and to = S<JO Q is the angular frequency of the rotor currents. The co-ordinates x and y are measured in a co-ordinate system fixed to the rotor. Derivatives with respect to z have been set equal to zero since end effects are neglected, and current flow is assumed to be wholly in the z-direction, so that H z has been put equal to zero. Displacement currents have also been neglected. Allfieldquantities vary with x as e~j& x, whereft = 2TT/A = TTT; A, the wavelength of the travelling wave, is equal to twice the pole pitch. Subsituting y/3 for T^fax in (9) and (10) and eliminating H y between these two equations, we get If a solution of eqns. (8) and (11) for E z can be found the problem is solved, for H y can easily be found from E z using eqn. (9). Thus (8) and (11) are the basic equations. Their solution is easily found by making use of the wave-impedance concept and employing a transmission-line analogue. Corresponding quantities in the analogue are most easily seen by Fig. 3. Element of equivalent transmission line. The characteristic wave impedance and the propagation coefficient for the ^-direction can be written down by inspection (13) The suffix '2' refers to region 2 of the induction-motor model in Fig. 2. By exactly the same argument, we can set up a transmissionline model for region 1. The characteristic wave impedance and the propagation coefficient can be obtained from eqn. (13) by replacing fx x by fx x, fx y by [i' yi and putting cr = 0. This gives 4 (14)

4 334 CULLEN AND BARTON: A SIMPLIFIED ELECTROMAGNETIC THEORY OF In order to complete the analogy, we must consider boundary conditions. At the interface between regions 1 and 2, E z and H x must be continuous. Thus, in the analogue, V and / must be continuous at the junction of the two dissimilar transmission lines representing regions 1 and 2. This continuity follows from Kirchhoff 's laws. In region 3, H x must be zero everywhere since the permeability is infinite. In particular, H x = 0 at the boundary between regions 3 and 2. In the analogue we must have / = 0 at the corresponding point. The transmission line for region 2 is therefore open-circuited at the corresponding end. The opencircuit condition can alternatively be obtained by noting that, since fx = oo in region 3, the associated characteristic wave impedance is infinite, and this is the 'termination' of region 2. Fig. 4 shows the transmission-line analogue of the interaction >L. v o.c. Fig. 4. Equivalent transmission line system. region of the induction motor, regions 1 and 2 of Fig. 2. The sending-end and receiving-end voltages of a transmission line terminated by an impedance Z r are related as follows: 02 We have now found the form of the field distribution in region 2, but the absolute magnitude is not yet determined. We take as datum the maximum value of the magnetic flux density B yi in the air-gap. Since relativistic effects can be ignored at the low velocities involved, this quantity is unaltered by transformation to co-ordinates fixed to the stator, and can be equated to B m introduced in the circuit analysis. The air-gap flux density can be written ^i = (20) Since the normal component of magnetic flux density is continuous at an interface, eqn. (20) also applies just inside region 1. But in region 1 eqn. (9) applies if fx y \s replaced by /Ay. Also B y = \n'yh y. Hence Combining (19) and (21) we have CO Vi (21) ^i cosh B / Z \ " (cosh y t a cosh y sinh y^ sinh y 2 b J \ ^02 '.... (22) This is the exact solution for the rather highly idealized model we have chosen to consider. To calculate the tangential force on the rotor, we start with the force per unit volume exerted in the ^-direction by the magnetic field on the current. This is f x =- J z B y If J z and By are interpreted as complex amplitudes rather than instantaneous values, the time-average force density is x = c/t-^j z Ji y {* *) Using eqn. (21) and remembering that J z = oe z, we have (24) V s = V R (cosh yl + ^ sinh yl) Using the equivalences set out in (12) and referring to Fig. 4, the relationship between E zi and E z2 can be written Ez\ = E 2l (cosh y,a + -^ sinh y x aj.. (15) Here, Z 2 is the input wave impedance of region 2, which, by analogy with an open-circuited transmission line, is Z 2 = Z 02 coth y 2 b (16) Thus E zx = E z2 (cosh y x a + ~ tanh y 2 b sinh y^a) (17) Making use of the well-known formula for the voltage distribution on an open-circuited transmission line, we can express the electric field strength E z at any distance from the interface between regions 2 and 3 in terms of E z2 as follows: 1 z Combining (17) and (18), we have cosh cosh y 2 b (18) E 2 = E Ml cosh y? (19) cosh y x a cosh y 2 b + ~ sinh y x a sinh y 2 6 Substituting (22) in (24) and remembering that \B yx \ = B m, we get f - - R 2 2 P cosh y 2 cosh y* cosh y x a cosh y 2 b + sinh y y a sinh y 2 b -^02 (25) Tofindthe force per unit area of rotor surface we must integrate (25) with respect to between the limits 0 and b. The result is acob^ y 2 sinh y 2 b cosh y*b y* sinh y*b cosh y 2 b 2 cosh y x a cosh y 2 b + ~- -02 sinh y x a sinh y 2 b (26) This equation can be put into a form more convenient for practical application. In the first place, for all reasonable values of y x and a, it is perfectly adequate to write cosh y x a 1 sinh y x a y Secondly, y 2 can be approximated as (27)

5 THE INDUCTION MOTOR, USING THE CONCEPT OF WAVE IMPEDANCE 335 But, from eqns. (6) and (7), we see that /X X CT = ju. o cr o. Thus y 2 can be expressed in terms of the parameter a defined by eqn. (4), as follows, y 2 =a(l+y) (28) Substituting (27) and (28) in (26), putting j8 = TT/T and simplifying, we get _ OOQTB%, GQbsS sinh 2ab + sin 2ocb 2TT PS 2a 2rj 2 (cosh 2<xb cos 2ab) + 2-r) (sinh 2ab sin 2a.b) + (cosh 2ab + cos where (30) It is not immediately obvious from eqn. (29) that the field-theory result agrees with that obtained by circuit analysis. However, if eqns. (2) and (3) are substituted in (1), the resulting formula can be reduced exactly to eqn. (29). It is at first sight remarkable that this exact agreement should be found between the two formulae, for approximations have been made in deriving both, and the approximations are not obviously equivalent. However, a deeper examination of the problem shows the agreement is not unreasonable. The traditional analysis is founded upon the superposition theorem, in that the fields produced by the rotor currents are considered separately from the externally applied field which produces these currents. Further there is the axiom, usually unstated, that the tangential force on the rotor due to the interaction of its currents and their own field is zero. This follows from the magnetic uniformity of the rotor and stator in the tangential direction. Hence it follows that the only relevant force is that between the radial component of the externally appliedfieldand the conductor cross-section. Because of the high permeability of the teeth it can be assumed that the flux density is independent of depth, and hence the force per conductor per unit width is equal to the product of flux density and total conductor current, and this in fact is the manner of derivation of eqn. (1). In applying the field theory the position is complicated since the resultant of the externally applied and the self-generated fields is considered. However, using the previous axiom it is evident that the answer must be identical with that obtained by considering only the externally applied component. The term ^/(Jco/Jiy) in eqn. (13) determines the decay of the externally applied field in the rotor slot region. This is easily seen by putting a = 0 in that equation; there are then no rotor currents and no magnetic field other than the externally applied field. Neglecting ^/jcofjiy in comparison with a, as in eqn. (28), is therefore equivalent to neglecting the decay of the externally applied field in the rotor slot region. The approximation (27) is equivalent to neglecting the decay of the externally applied field in the tooth-top region. Hence the exact equivalence of the two results. (3.3) Numerical Example To illustrate the formulae numerically we have taken a machine with the following rotor dimensions: Tooth top depth Slot depth.. Slot width.. Slot pitch Slot opening.. a = 1 cm b = 2cm b s = 1 cm p? = 2 cm d= 0-5cm The apparently excessive tooth top depth has been chosen to make some allowance for the zigzag reactance which would occur in a normal machine. In Fig. 5, torque/slip curves at constant air-gap flux density have been plotted. Curve (a) is the result obtained from the field theory or from the usual eqn. (1) J) o r I / / / V / (C)/ s s ^ SLIP Tb) (0) Fig. 5. Torque/slip characteristics. (a) Double-layer theory. (b) Neglecting skin effect. (c) Single-layer theory. together with (2) and (3). Curve (b) is obtained from (1) using (2a) and (3a); i.e. it neglects skin effect. A deep-bar rotor has been selected to emphasize the wellknown importance of skin effect in such cases; the standstill torque is approximately doubled. (3.4) Comparison with Mishkin's Analysis In Mishkin's paper 1 the rotor is represented by a much simpler model than that used in the present paper, though he has, of course, taken end effects into account and dealt with the stator as well as the rotor. Without going into so much detail, however, we shall show that his model of the rotor is too simple to represent the facts accurately, and in fact it greatly over-exaggerates skin effect. Mishkin's analysis does not take proper account of the shape of the slot or of the fact that the rotor bars are embedded in the rotor and are separated from the rotor surface by a non-conducting (ideally) mixture of iron and air. Whilst we have taken account of this fact by using a double-region model for the electromagneticfieldanalysis with one region (No. 2) to represent the conductor region and a second region (No. 1) to represent the tooth-top region, Mishkin has used a single region to represent both. Our formulae can easily be applied to a single-region model by putting a = 0. The effective conductivity of the single layer is easily obtained, and the average values JT^ and ]T y can be obtained from [x x and fx x, and fx y and [Ay, by combining reluctances for horizontal and vertical flux paths in parallel and in series respectively. When this is done, the single-layer theory yields the result shown as curve (c) in Fig. 5. It is clear that skin-effect is much exaggerated. It should be noted that the Mishkin results can be made to fit the case of a practical machine if the permeability of the air-gap in the x-direction is increased so as to allow for the effects of tooth-top and zigzag leakage fluxes. (4) FIELD THEORY OF THE BRUSHLESS VARIABLE- SPEED INDUCTION MOTOR A simplified theory of the very elegant motor described by Williams and Laithwaite 4 can be derived by the methods used

6 336 CULLEN AND BARTON: A SIMPLIFIED ELECTROMAGNETIC THEORY OF THE INDUCTION MOTOR in this paper. We shall not go into details, but merely indicate the main features of interest. In the first place, it is well known in microwave theory that certain types of discontinuity in waveguides can be treated theoretically by first examining the corresponding discontinuity in a parallel-plate line carrying a TEM wave, and then transforming the solution back to the waveguide case by simple substitutions. For example, 5 the susceptance of a capacitive diaphragm in a rectangular dominant-mode waveguide can be obtained from the corresponding strip-line formula by replacing A by \. This transformation arises from the fact that a 3-dimensional problem with a harmonic field variation in say the z-direction leads to a wave equation which is formally identical with a 2-dimensional wave equation if the wavelength is properly chosen. Physically, the waveguide mode can be constructed from obliquely-travelling plane waves, the velocity of each of which measured normal to its wavefront has the constant value c. The phase velocity of the composite wave is, however, increased to c sec 6, where 6 is the angle between the plane-wave direction and the axis of the waveguide, and sec 6 = \ g jx. Exactly the same argument can be applied to the induction motor, and the speed increase associated with obliquely travelling air-gap flux waves follows at once. (5) CONCLUSIONS It has been shown that the concept of wave impedance can be used in developing a simple electromagnetic field theory of the induction motor which leads to identically the same formula for the torque-slip relationship as the conventional circuit analysis with skin-effect corrections. Thus the same equation may be regarded either from a circuit point of view as an approximate solution of the actual problem, or from a field point of view as the exact solution of an idealized problem. An inadequacy in Mishkin's earlier electromagnetic theory of the induction motor has emerged, and the need for a 2-region model of the rotor has been demonstrated. The fundamental connection between the action of an induction motor and the phenomenon of radiation pressure should be mentioned. The closest analogy is with the tangential stress due to radiation obliquely incident on an absorbing surface. This effect has been observed by Barlow and Poynting when light falls obliquely on a blackened surface. In the induction motor, the incident wave is not a simple plane wave but a surface wave. The principle is the same, however. In both cases the incident field sets up currents which are acted upon by the magnetic field of the incident wave to produce a mechanical force. That the forces are so different in magnitude in the two cases tends to conceal the underlying unity; this is essentially due to the enormous difference in wave velocity in the two cases. For the induction motor at standstill F = P\v, where P is the power delivered to the rotor and v is the velocity of the flux wave. The radiation stress due to light, on the other hand, is of the order of Pic, where c is the velocity of light. Bearing in mind that in the optical case P could hardly exceed 1 watt, the difference in the magnitudes of the forces is easily understandable. Finally, having established a link between microwaves and induction motors, it is suggested that some of the theoretical methods developed for microwaves might usefully be examined by the machine designer. In particular, the variational methods for dealing with discontinuities, and the various techniques for handling periodic structures of various kinds, might find new application in the field of electrical machinery. (6) REFERENCES (1) MISHKIN, E.: 'Theory of the Squirrel-Cage Induction Motor derived directly from Maxwell's Field Equations', Quarterly Journal of Mechanics and Applied Mathematics, 1954, 7, p (2) BOOKER, H. G.: 'The Elements of Wave Propagation using the Wave-Impedance Concept', Journal I.E.E., 1947, 94, Part III, p (3) WILLIAMS, F. C, LAITHWAITE, E. R., and PIGGOTT, L. S.: 'Brushless Variable-Speed Induction Motors', Proceedings I.E.E., Paper No U, June, 1956 (104 A, p. 102). (4) ALGER, P. L.: 'The Nature of Polyphase Induction Machines' (John Wiley and Sons, New York, 1952), pp (5) LEWIN, L.: 'Advanced Theory of Waveguides' (Iliffe), p. 37.