1. Revolving-field analysis of asymmetric
three-phase machines and its extension to
single- and two-phase machines
Bhag Singh Guru
Indexing terms: Equivalent circuits, Machine theory, Magnetic flux, Induction motors, Stators
Abstract: The revolving-field theory as applied to single-phase induction motors is extended to develop a
concise yet comprehensive theory for asymmetric three-phase induction motors. Each phase is represented by
its equivalent circuit, not only to allow for uneven direct transformer interactions, due to asymmetric
locations of phase windings, but also to account for the different number of turns, wire size, winding factor
etc. each phase may have. The accuracy of the theory was confirmed by actual measurements on symmetric
and asymmetric three-phase induction motors. Computed and test data on some motors are included for
illustration. It is also shown that the general three-phase development can be easily applied to determine the
behaviour of two- and single-phase induction machines, a useful feature for unified computer-aided design, by
eliminating one and two phase windings, respectively. Not only the procedural details for determining the
performance of single- and two-phase induction motors are given, but comparisons of numerical and test
results on some output entities are included as well. Though the comparisons among theoretical and test
results are made on fractional horsepower motors, the types being built at Universal Electric, the author is
certain that the development presented here is equally applicable to all sizes of induction motors.
1 Introduction of suitably chosen components of current and voltage.
Further adaptation of this technique to three-phase
A symmetric polyphase induction motor, operating on a induction machines with unequal windings and arbitrary
balanced polyphase power supply, is defined as an spatial displacements in addition to unbalanced line
induction motor whose windings are evenly displaced in voltages may require a large number of voltage and current
space and excited by the currents having the same dis- transformations. At this time, such a development, if there
placement in time phase as the windings have in space is any, is not known to the author. Another analytical
phase. Consequently, the internal interactions of such an method would be the direct extension of the revolving-
induction motor are simple, and can be easily characterised field approach as applied for capacitor motors with
by one primary circuit and a corresponding secondary windings not in quadrature.3 Even though this method
circuit based on the revolving-field theory. On the other requires only three unknown quantities, the torque
hand, the simple crossfield analysis of such motors involves equation by itself is quite involved.
three times as many circuits as there are phases. Hence the All the foregoing methods, in the author's opinion,
polyphase induction motors are universally analysed on the satisfy only one particular requirement, that is, if the
basis of the revolving-field theory. However, quite often the development is carried out for a three-phase motor with
stator core punchings designed for a certain pole configur- asymmetric windings, it would require comprehensive
ation are also used for another pole configuration, and in changes to make it useful for an asymmetric two-phase
that case, if the number of stator slots per pole pair is not motor and vice versa. Stated differently, separate computer
a whole number, the phase windings would have to be programs are needed to analyse different-phase motors.
arranged at resultant angles other than the conventional This, in turn, increases the cost of maintaining these
angle for such motors. This results in a spatially unbalanced programs as 'permanent files' on a time-sharing machine. To
motor even though the windings are identical. The actual circumvent the situation, we can seek refuge in Kron's
performance of an asymmetric motor is bound to differ generalised theory. 4 ' 5 This theory, as Kron states in his
from the conventional motor, and therefore necessitates a book,s has to be extended further to encompass the entire
different theoretical approach. As far as the author is range of asymmetric machines. Moreover, an extension of
aware, no attempt has ever been made in this direction. Kron's theory requires the in depth understanding of
Therefore, the primary aim of this paper is to develop a tensors, symmetrical components, three-phase/two-phase
systematic method for the analysis of asymmetric three- and inverse transformations. An average user, however, may
phase induction motors. In addition, the effects of not be able to satisfy some or all of these demands.
unbalanced terminal voltages are also included to make the
development more general. However, note that the well In literature, almost, all the time, a symmetric three-
known method of symmetrical components1 may be used phase induction motor is represented by an analytically
to analyse a symmetric (equal windings) three-phase simple yet 'tricky' equivalent circuit. An equivalent circuit,
induction motor running on an unbalanced power supply. in the author's opinion, should be based upon the physical
This method has also been extended for the analysis of picture as opposed to a 'trick' circuit. In a three-phase
asymmetric two-phase induction machines2 by making use motor, the flux produced by one phase induces voltages by
direct transformer action in the remaining phases and vice
versa. An exact representation of this machine, would
Paper T308 P, first received 9th May and in revised form 24th require an equivalent circuit for each of its phases and an
November 1978 account for the mutual interactions of the fluxes rather
Dr. Guru is with the Universal Electric Company, 300 East Main
Street, Owosso, Michigan 48867, USA than a single yet tricky circuit. Although the former may
ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1 37
0140-1327/79/010037 + 08 $01-50/0

2. increase the number of equations needed for an analytical displaced at an angle from the r.p.w. can effectively be
solution, it certainly helps the engineer's understanding and represented by two windings connected in series, one in
thereby enables him to attack more complicated problems phase with r.p.w., the other in quadrature. The straight-
that he would otherwise hesitate to approach analytically. forward extension of this principle states that a given set of
This paper therefore represents each winding by its equival- windings arranged arbitrarily in space can be made fully
ent circuit and then accounts for the direct transformer equivalent to another arbitrary set of paired quadrature
flux linkages among the windings. Since each phase is rep- windings. It is, however, obvious that if two or more
resented by its equivalent circuit, the asymmetric three- windings are in space phase, that is when they are arranged
phase induction motor can be analysed by making use of a so that their magnetic axes coincide, a direct transformer
simple method published recently.6 By doing so, the final type of coupling exists between their fluxes. In the present
set of equations are not only useful for analysing symmetric case, the direct-transformer linkages exist among the fluxes
or asymmetric three-phase motors, but also enable the of r.p.w. and tf-axis components of the remaining offset
engineer to use them effectively for single- and two-phase phase windings on one hand, while such coupling exists
induction machines, as will be shown later. Therefore, the among the fluxes of q-axis components of the offset
present analytical approach does away with the necessity of windings, on the other. As usual, the voltages are induced in
having separate computer-aided design programs for single- the main and crossfield axes of the rotor due to trans-
two- and three-phase machines. This in turn considerably former action and rotation, respectively. Once this concept
reduces the cost of maintaining the 'permanent files' for is clearly understood, the development of the theory
regular use on a time-sharing computing machine. quantitatively is basically simple and straightforward.
The present method, however, involves the concept that
any given winding can be made equivalent to two mutually 3 Assumptions
orthogonal windings. Although the author employed such
terms as 'd-axis' and 'q-axis' to explain the concept for lack The following assumptions are imbedded into the theoreti-
of better names, it is different from the so-called ld- g-axis' cal development that follows in subsequent Sections:
technique used by other authors. The latter applies to the (a) All the windings follow concentric distribution and
transformation of the voltages and currents while the therefore, in comparison to the fundamental flux, the
former to the windings. Utilising this concept, each winding effects of other space-harmonic fluxes are negligible.
of a polyphase motor may be looked upon as two separate (b) Friction, windage, and surface losses are all con-
windings mutually perpendicular. However, the two sidered to vary as the square of the speed. These losses are
windings have to be assumed to be connected in series regarded as an extra load on the shaft and are therefore
to have the same effect as the original winding. This subtracted from the output.
concept is explained further in Section 2. In Section 3, (c) The magnetic-circuit parameters of the offset phase
some of the basic assumptions generally accepted in motor windings are all expressed in terms of the magnetic circuit
theory are stated. Section 4 explains the treatment of core reactances of the reference-phase winding (r.p.w.).
loss. The general theoretical development for three-phase (d) For direct transformer action, the local mutual
induction motors is presented in Section 5. The systematic leakage and magnetising fluxes couple perfectly for the
and efficient implementation of this general development transformed windings both on the d- and the ^r-axes.
for single- two- and three-phase induction motors is
explained in Section 6. The numerical examples are also 4 Treatment of core loss
included. Finally, the highlights of this development are Often, while calculating the performance of an induction
summarised in Section 7. motor, very approximate methods3 are used to account for
the core loss. Since these methods do not interfere with
2 Conceptual background performance equations, they are often acceptable. To be
more exact, however, the core loss may be handled by
Since the concept of treating a winding as a set of two introducing an equivalent resistance, either between rx and
mutually perpendicular windings has already been xx as suggested by Trickey,9 or in shunt with the magnet-
published6 most of the details are omitted here. The ising branch.4 For fractional horsepower motors the author
method of decomposing a winding into two orthogonal found no appreciable difference between these two equiv-
windings is needed here to account for uneven direct trans- alent representations. In this development, however, the
former couplings among mutual fluxes of all the windings, former method is followed. Even though the equivalent-
since actual windings are not symmetrically arranged. A resistance concept that accounts for the core loss adds to
summary of the concept is given below. the complexity of the performance equations, it does
If two mutually perpendicular axes, direct axis or d-axis narrow down the differences between the theoretical and
for short, and quadrature axis or g-axis, are oriented in such the test results. Also, for large induction motors, the former
a fashion that the axis of one of the three phase-windings assumption would lead to better correlation between the
coincides with the d-axis, then the positions of the other theory and tests than the latter.
phase windings can be expressed in terms of electrical
degrees circumferentially offset from the reference-phase
winding (r.p.w.). Thus the flux produced by the r.p.w. has 5 Theory
only a d-axis component while the flux produced by any In a symmetric three-phase induction motor, the number of
other phase-winding can now be resolved into its com- turns, wire sizes etc. are the same for each of the three
ponents oriented along d- and ^-axes, respectively. Since phases and follow identical distribution. Moreover, any two
the flux produced by a winding is a measure of its total phases are offset circumferentially by 120 electrical degrees
effective conductors, it can likewise be said that the in space. For the present development, however, let us
winding can be resolved into its components whose axes consider that the effective number of conductors, wire
coincide with the d- and q-axes. In other words, a winding sizes etc. are different for each of the phases in addition to
38 ELECTRIC POWER APPLICATIONS, FEBRUAR Y 1979, Vol. 2, No. 1

3. the arbitrary displacement of the 2nd- and 3rd-phases by 62 ductors for d- and q-axs components of the said phase
and d3 electrical degrees in space with respect to the windings to the effective number of conductors in the
lst-phase, as shown in Fig. 1. A and B may be defined as lst-phase winding, respectively, are:
the ratios of effective number of conductors in the 2nd-
and 3rd-phase to the effective number of conductors in the d-axis component of 2nd-phase: Ad = Acosd2 (1)
lst-phase, respectively. If kwl and Nlt Kw2 and N2, and
q-axis component of 2nd-phase: Aq = A sin 62 (2)
kw3 and N3 are the winding factors and the number of
turns for 1st- 2nd- and 3rd-phase, respectively, then A = d-axis component of 3rd -phase: Bd = B cos 0 3 (3)
(kw2N2)l(kwlNx) and B = (kw3N3)/(KwlN1). It is how-
ever, obvious that for a symmetric motor, kwx = kw2 = kw3, g-axis component of 3rd phase: BQ = B sin 0 3 (4)
yV, =N2 = N3, d2 = 120° and 0 3 = 2 4 0 ° . In other words, For an unbalanced three-phase power supply, the magni-
A=B=l. tude of the voltage may be different for each of the phases.
To simplify the theoretical development to some extent Furthermore, each phase may not be 120° apart in time.
but still maintain the generality of the problem, let us Assuming the lst-phase voltage as a reference, the 2nd- and
consider that the axis of the lst-phase winding coincides 3rd-phase voltages may then be given as:
with the d-axis. This, therefore, is the reference-phase
winding. The 2nd- and 3rd-phase windings can now be 2nd-phase voltage: V2 = C2Vt exp(jt2) (5)
decomposed into their d- and q-axis components as
3rd-phase voltage: V3 = C 3 K j e x p ( / f 3 ) (6)
shown in Fig. 2. The ratios of effective number of con-
where C2, C3 are the constants of proportionality, and t2
and t3 are the phase-shifts for the 2nd- and 3rd-phases,
respectively, with reference to the lst-phase supply. For a
balanced three-phase supply, it is obvious that C2 = C3 —
1, t2 — 120° and t3 = 240° in time. The asymmetric three-
phase induction motor of Fig. 2 can now be represented by
an equivalent circuit. Such a circuit is shown in Fig. 3. The
equivalent circuit parameters for the 2nd- and 3rd-phase,
respectively, are defined in terms of the lst-phase para-
meters and the effective winding ratios. The resistances /y el ,
rfe2 and Ayc3, as shown in Fig. 3, are introduced to account
for the core loss in each of the three phases. This network
representation involves six unknowns. However, by the
straightforward application of The"venin's theorem, 8 the
equivalent circuit of Fig. 3 may be transformed into
another modified equivalent circuit as shown in Fig. 4.
Fig. 1 Asymmetric three-phase induction motor with 2nd- and
Srd-phase windings displaced at d2 and d3 electrical degrees from Note the terms rm,, rm2 and rm3, and Vml, Vm2 and Vm3.
the lst-phase winding These are the modified resistances and the voltages for the
3-phase windings and are given as:
d -axis
r
mi — rpirfei/(rpi (7)
v
mi = (8)
where the subscript / = 1, 2, or 3 corresponds to 1st- 2nd-
or 3rd-phase values of both the resistances and supply
voltages, respectively. rpl, rp2, and rp3 are the winding
resistances for 1st- 2nd- and 3rd-phase, respectively.
As explained earlier, and is now obvious from Fig. 2, in
all there are three windings arranged along the d-axis and
two windings arranged along the q-axis. Since the currents,
as shown in Fig. 4, in all the windings oriented either along
the d-axis or the q-axis flow in a sense in the same direction,
the mutual interactions among them are considered to be
positive. Since the d- and 4-axes are in quadrature, the net
flux produced by the windings oriented along one axis, say
the d-axis, does not show any direct transformer linkage
with the flux produced by the windings concentric with the
other axis, ^-axis, and vice versa.
q -axis Thus, the voltage induced in the lst-phase winding by its
own leakage flux and also by direct transformer coupling of
the leakage fluxes of the d-axis components of the 2nd- and
3rd-phase windings is
Fig. 2 Equivalent representation of an asymmetric three-phase
induction motor of Fig. 1 +AdJ2+BdI3)xl (9)
The 2nd- and 3rd-phase windings are shown decomposed into d- and Similarly, the voltages induced in the d-axis components of
(/-axis components the 2nd- and 3rd-phase windings by the leakage fluxes are
ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1 39

4. = /(A + AdI2 + BdI3)Adxx (10) forward: Ef2d = (h + AdI2 + BdI3)AdZf (16)
= /(A + A dI2 + BdI3)Bdxx 00 backward: Eb2d = (/, + AdI2 + BdI3)AdZb (17)
In a similar manner, the voltages induced in the g-axis (b) For the <i-axis component of the 3rd-phase:
components of the 2nd- and 3rd-phase windings by their
leakage fluxes are forward: End = (/, + AdI2 + BdI3)BdZf (18)
£», = j(AqI2+BqI3)AqXl (12) backward: Eb3d = (/ t + AdI2 + BdI3)BdZb (19)
#3<, = j(AqI2+BqI3)BqXl (13) Similarly, the voltages induced in the g-axis components of
the 2nd- and 3rd-phase windings by the forward- and
Voltages induced in the lst-phase by its forward and backward-revolving fields set by them, respectively, are:
backward revolving fields as well as by direct transformer (a) For the q-axis component of the 2nd-phase:
linkage of the two revolving fields set by the d-axis com-
ponents of the 2nd- and 3rd-phase windings are forward: Ef2q = (AqI2 + BqI3)AQZf (20)
forward: Efld = (/, +AdI2 + BdI3)Zf (14) backward: Eb2q = (AqI2 + BqI3)AqZb (21)
backward: Ebld = (/, + AdI2 + BdI3)Zb (15) (b) For the g-axis component of the 3rd-phase:
Likewise, the voltages induced in the d-axis components of forward: Ef3q = (AqI2 + BqI3)BqZf (22)
the 2nd- and 3rd-phase windings by the field components,
backward: Eb3q = (Aqf2 + BqI3)BqZb (23)
which are set by them as well as by direct transformer
action of the revolving fields set by the lst-phase winding, We are now in a position to express mathematically the
are: speed voltages induced in the d-axis components of the
(a) For the d-axis component of the 2nd-phase: windings by the forward- and backward-revolving fields of
"fid
a.
o
a
'm2
r
m3
-3d
-f3d
-b3d
10
Fig. 3 Equivalent circuit of an asymmetric three-phase induction Fig. 4 Thevenin 's equivalent of Fig. 3
motor based upon revolving field theory
40 ELECTRIC POWER APPLICATIONS, FEBRUAR Y 1979, Vol. 2, No. 1

5. the g-axis components of the said windings and vice versa. Zl3 = BdZs ~JBQZd
After some simplifications, the expressions for these
voltages are:
Z23 = sabzs + JCabZd
Z31 — BdZs + /BqZd (40c)
A', = -j(AqI2+BqI3)Zf (24)
(25)
Z32 = SabZ s-jcabzd
E* = }(Aqf2+BqI3)Zb
z33 — r
m3 -B2Zs
E3 = ~KAqI2 + BqI3)AdZ, (26)
with
lU = j(AqI2 + BqI3)AdZb (27)
zd zb zs — Zf ' Zb T jx 1 (41)
Es = -j(Aqf2 + Bqh)BdZf (28)
Sab = AdBc + AqBq Cab = AqBd-AdB
(29)
Once again, eqn. 40a gives the Z-coefficients for single
£"7 = / ( / , + AdI2 + Bdf3)AqZf (30) phase motors, whereas eqns. 40a and 40b are for two-phase
motors. Needless to say, all the above equations for the
^s = " / ( A + ^ d / 2 + BdI3)AqZb (31)
Z-parameters are needed for symmetric or asymmetric
£9 = /(/i + i4 d / 2 + Bdh)BqZf (32) three-phase induction machines. Eqns. 37—39 may be
rewritten in a concise matrix form as
ff,o = ~J(h + Adh + Bdh)BqZb (33)
[Z] M = [V] (42)
Simultaneous equations for the voltages in the three circuits where [Z] is a complex impedance matrix of dimensions
of Fig. 4 can now be written by summing up the voltages in 3 x 3 whose elements are given in eqn. 40 in terms of
each of the three stator windings, obtaining known motor-circuit constants. [/] and [V] are the
column vectors for the circuit currents and the Thdvenin
1/ — f. r + / • ' . . -I- F , • + F.. . 4- F 4- P- f^& equivalent of the impressed voltages, respectively. The
'ml •'I'ml ' Lj
d ' '-'fid ' ^bld ' lj
1
-'2 Wv circuit currents can then be determined from the following
equation:
r +
Vm2 ~ h m2 ^2d + Ef2d + ^b2d + E3 + E4
2q [/] = [ Z p t K ] (43)
E-,n + Ef2q + Eb2q + En + E8 (35)
i +
^63d + Es + E6 where [Z]"1 dentoes the inverse of the [Z] matrix.
+E9+ E10 (36) The actual phase currents can now be determined from
l3q
the following relation:8
Eqn. 34 is for the referenced-phase winding and can be used Ipi = (Vt + Iirfei)l(rpi + rfei) (44)
to analyse a single-phase induction motor running on the for / = 1,2, and 3, respectively.
main winding only, by equating the 2nd- and 3rd-phase The total power input in phase / is given by
currents involved implicitly to zero. Adding eqn. 35, we can
obtain the performance of two-phase symmetric/asymmetric (45)
motors with balanced/unbalanced terminal voltages at any The forward and backward torques can be calculated
speed of rotation. Remember that a permanent-split capaci- independently for each one of the three phases. This can be
tor motor is only a special case of an unbalanced two- done by calculating the forward and backward voltages
phase induction motor. Further, these equations can also be induced in each of the windings from eqns. 14—33, and
used to determine the locked-rotor torque and the starting multiplying each voltage by the conjugate of the actual
performance of a single-phase induction motor whose current flowing in the winding of concern and using only
auxiliary-phase is disconnected at a predetermined speed. It the real part of the product. Following this procedure, and
is, however, obvious that all the above equations are needed after some simplifications, the torques developed in
to predict the performance of an asymmetric or a sym- synchronous watts for each of the three phases are:
metric three-phase induction motor.
(a) For the lst-phase:
By making direct substitutions for the induced voltages,
eqns. 9—33, and after some rearranging, eqns. 34—36 can by forward field: 7>, = Re [IfI?2f] (46)
be rewritten in the general form as
by backward field: Tbl = Re [IbI*Zb] (47)
V = IZx +I2Z12 + I3Z3 (37)
(b) For the 2nd-phase:
=
V 2 hZ2 +/)Z 2 2 +/3Z23 (38)
by forward field: 7>2 = Re{/fI2*Zf (Ad +fAq)} (48)
32 Z33 (39)
by backward field: Tb2 = Re {IbI2*Zb (Ad-jAq)} (49)
where the Z-coefficients are
(c) For the 3rd-phase:
= r +Zs (40a)
by forward field: 7} 3 = Kt{IfI*Zf (Bd + jBq)} (50)
Zn - A dZ s -jAqZd
by backward field: Tb3 = Re{IbI3*Zb(Bd-jBq)} (51)
Z2x = AdZs+jAqZd (40b)
where the symbol * indicates the conjugate of the com-
Z72 = rm2 +A2ZS plex quantity involved and
ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1 41

6. If = /, + I2 (Ad -jAq) + I3(Bd -jBq) (52) studied in the subsequent paragraphs.
In the first example, a typical 230V, 60 Hz, 0-4 h.p.,
Ib = h+h(Ad+jAQ)+I3(Bd+jBq) (53) 4-pole symmetric three-phase induction motor rated at
The overall forward and backward torques developed, in 1650r/min is considered. The windings in each phase
synchronous watts, including friction and windage losses consisted of two concentric coils, the inner coil was wound
are: n over 6 teeth and had 57 turns, the outer coil was wound
over 8 teeth with 76 turns of 24 gauge wire. The per phase
gross forward torque developed = 7} = £ 7},- (54)
1= 1
values of the motor constants obtained from design con-
n siderations are reproduced below.
gross backward torque developed = Tb = £ r w (55)
1=1 r, + / x , = 12-401 +/5-77412
where « specifies the number of phases. For a three-phase
motor, n = 3. rfe = 822-8 £2
The actual torque developed by the motor, in synchron- r2 +jx2 = 5-319+/2-653 ft
ous watts, is
xm = 88-601 ft
T = Tf-Th (56)
F&W = 8-70W
The secondary or the gross output power can be calculated
by multiplying eqn. 56 by the per unit speed of the motor. The laboratory measurements were taken by connecting the
Subtract the friction and windage losses to obtain the motor to a balanced three-phase supply. The test data and
useful power output. theoretical predictions on various output quantities are
given in columns 1 and 2, Table 1. Note that these theoreti-
6 Procedural details cal results in this case can also be obtained by making use
6.1 Three-phase motors
of well known equations for symmetric three-phase
induction motors based on a single equivalent circuit. It is,
The suggested procedure to predict the performance of however, necessary to increase the values of the motor
three-phase induction motors is as follows: constants given above by 50%, except for the stator-
(a) From the known number of turns and the winding winding resistances.
factor for each of the three phases, determine A and B. A provision was made to add extra windings in series
(b) From the known locations of the 2nd- and 3rd-phase with each phase of the above motor to test the motor with
windings with respect to the lst-phase winding, calculate stator windings having an unequal number of turns. The
the d- and 4-axis components of these windings, using eqns. tested and computed results, as given in columns 3 and 4 of
1—4. However, for a symmetric motor, set 62 = 120° and Table 1, are for a motor with 69 and 92, 66 and 88, and 57
0 3 =240°. and 76 turns in the inner and the outer coils, for the lst-
(c) Assuming the lst-phase voltage as a reference, define 2nd- and 3rd- phase windings, respectively.
the 2nd- and 3rd-phase voltages in terms of their magni- The last example involves a three-phase induction motor
tudes and time-phase displacements. In other words, define with unequal turns in each phase and with arbitrary spacing
C2 and C 3 , t2 and t3, respectively. For a balanced three- of the phases. The number of turns and coils were same as
phase supply, C2 = C3 = 1, and t2 = 120° and t3 = 240°. in the previous example, whereas the 2nd- and 3rd- phase
(d) Calculate Thdvenin's equivalent of resistances and were displaced at 100 and 200 electrical degress, respect-
voltages by making use of eqns. 7 and 8, respectively. ively, from the lst-phase. The measured values and theor-
(e) From the known constants of the motor, compute etical predictions are given in columns 5 and 6, Table 1.
the Z-parameters as given in eqns. 40a, 40b and 40c. In all the foregoing cases, as is obvious from Table 1, test
(/) Use a computer subroutine to invert the complex data are in close agreement with theoretical predictions.
impedance [Z] matrix.' This, therefore, confirms the accuracy of the present
(g) By making use of eqn. 43, calculate the complex method as applied to both symmetric and asymmetric
values of the circuit currents for the modified equivalent three-phase induction machines.
circuit of Fig. 4. From eqn. 53, it can readily be shown that for a sym-
(h) The actual phase currents can now be determined metric three-phase supply, the backward field current Ib is
using the relation given in eqn. 44. identically zero. Consequently, there exists no backward-
(/) Calculate the input power and the gross torques revolving field in the motor, as expected at the outset.
developed by the forward and backward fields from eqns.
45—55. The actual torque developed can now be computed 6.2 Two-phase motors (permanent-split capacitor motors)
from eqn. 56. In the true sense, there are not that many two-phase
(/) Determine the forward and backward I2R losses due induction motors being built toda>. The reason, of course,
to the forward and backward fields by multiplying the is that the two-phase power source needed for the normal
forward and backward torques developed by s and 2s, operation of such motors is virtually an extinct species.
respectively. However, a permanent-split capacitor motor may be treated
(k) Obtain net power developed, total losses, horsepower as a two-phase motor with unequal windings. The pro-
etc. by employing standard relations. cedure outlined below may be used to determine its
Since it is not possible to cite all practical possibilities, performance.
theoretical and test data on three different motors are
(i) Calculate A from the known turns and winding factor
for each of the two windings. Set B equal to zero.
t A subroutine to solve this matrix is available from the author on (ii) Use eqns. 1 and 2 to determine the d- and q-
request
42 ELECTRICPOWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1

7. Table 1: Theoretical and test results on three-phase, permanent-split capacitor, and single-phase induction motors
Three-phase motors Two-phase motors (p.s.c.) Single-phase
motor
Column 1 2 3 4 5 6 7 8 9 10 11 12
Test Theory Test Theory Test Theory Test Theory Test Theory Test Theory
/p..A 1-45 1-476 0-93 0-902 3-21 3221 0-73 0-734 0-76 0-761 0-97 0-960
/ P J ,A 1-47 1-476 0-67 0-678 1-12 1053 0-38 0-383 0-26 0-258 — -
'P3.A 1-45 1-476 2-10 2085 3 54 3-658 — — — — — —
Pint.™ 150 0 156-3 83-7 80-44 172-5 177-5 160-7 160-7 151-7 153-1 181-2 179-7
Pirn.™ 1520 156-3 80-3 77-24 84-7 82-85 820 81-57 60-3 59-33 - -
Pim.W 1515 156-3 240-2 243-1 380-1 384-1 _ — — — — —
Pout. W 298-9 3063 255-5 252-5 237-9 235-5 158-6 157-8 139-1 173-7 114-7 112-4
T, kgm 0-1764 0-1808 0-1508 0-149 0-1404 0139 00936 0 0931 00821 00812 0-967S) 00664
Efficiency 0-66 0-65 0-63 0-63 0-37 0-37 0-65 0-65 0-66 0-65 0-63 0-63
components of the 2nd-phase (auxiliary phase for a perma- 6.3 Single-phase motors
nent-split capacitor motor).
The starting performance of single-phase motors can be
(iii) Define the magnitude and the phase angle for the obtained by using the method explained in Section 6.2,
2nd-phase voltage reference to 1st-phase voltage. In other since starting the motor employs both the windings. How-
words, define C2 and t2. For a two-phase motor with ever, when the motor is running at its rated speed on the
balanced two-phase terminal voltages, C2 = 1 and t2 = 90° main winding only, its performance can be obtained by
(time). However, for a permanent-split capacitor motor modifying the procedure of the previous subsection as
(p.s.c.), set C2 = 1 and t2 = 0°. follows:
(iv) Calculate from eqns. 7 and 8, the The'venin's equiv- (a) Determine The'venin's equivalent of resistance and
alent of resistances and voltages. Note that for a p.s.c, the the voltage from eqns. 7 and 8, respectively, for the main
capacitor impedance should be added to the auxiliary- winding.
winding's resistance for all theoretical purposes except for (b) Set I2 = h = 0 and A = B = 0.
obtaining the primary loss for that winding. (c) Compute Zn from eqn. 40a.
(v) Compute the Z-parameters from eqns. 40a and 40b.
(d) Calculate the current I as follows:
Note that eqn. 40c is not needed in this case because there
are only two-phase windings. Therefore, set/ 3 = 0. A = Vml/Zn
(vi) Steps (vi)-(xi) are essentially the same as those given
in Section 6.2. Compute the power input, the forward and Other quantities of interest can be obtained by following a
the backward torques with n = 1. similar procedure given in the previous subsections, of
Two permanent-split capacitor motors were actually built course, overlooking nonessential equations.
to verify the accuracy of the present method as extended to One of the p.s.c. motors was selected for the testing
two-phase induction motors. One motor was built with an purposes as a single-phase induction motor. At its rated
auxiliary winding displaced in quadrature with respect to speed, 1650r/min, the auxiliary winding was cut off. The
the main winding, while the other with auxiliary arranged test data and the theoretical predictions on various output
at 60 electrical degrees from the main. Since the same quantities are given in the last two columns of Table 1. The
windings were used for both motors, the following motor feasibility of using the present development to analyse
constants are applicable to either of them: single-phase induction motors running on the main winding
only is indicated by the close relationship between the
rpx + jxx = 32-255+/ 35-49 ft measured and theoretically predicted values.
Therefore, it can be concluded that a single-analysis
rn = 6884-05 ft computer program based on the equations presented in this
paper may be used effectively to analyse single two- and
r 2 + / x 2 = 36-453+/21-238 ft three-phase induction motors. In addition, the phase
xm = 735-544 ft windings may be either equal or unequal, and the spacing
among the phases may be arbitrary.
A = 1-32107
7 Conclusions
rp2 = 135-194ft
This paper provides, first, a basic theory to predict the
capacitance = 3 05 mF behaviour of an asymmetric three-phase induction motor
using revolving-field theory, secondly, extensions of the
These 230 V, 60 Hz, 4-pole permanent-split capacitor above for single- and two-phase induction motors, and
motors were tested at 1650r/min. The tested and com- thirdly, detailed step-by-step analysis procedures for such
puted results on quadrature and nonquadrature motors are motors. A computer program was developed based upon
given in Table 1, columns 7 and 8, and 9 and 10, respect- the present development and is being used extensively by
ively. It is obvious from these results that the correlation the Application Department to analyse such induction
between the tested and computed values is excellent and machines as split-phase, permanent-split capacitor with
thereby approves the applicability of the general method to either quadrature or nonquadrature windings and sym-
permanent-split capacitor motors, and in turn to other such metric or asymmetric three-phase motors. To demonstrate
motors that use two windings for their operation at the the effectiveness of the present theoretical approach,
rated speed. however, actual measurements and theoretical predictions
ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1 43

8. on various output quantities are reported. In each case 9 References
given in this paper, and almost all others not included here,
1 FORTESCUE, C. L.: 'Method of symmetrical coordinates
a remarkable correlation exists between theory and actual applied to the solution of polyphase networks', Trans. Amer.
tests. Needless to say there is a reduction in the cost of Inst. Electr. Engrs., 1918, 37, pp. 1027-1115
maintaining a single permanent file on a time-sharing 2 LYON, W. V., and KINGSLEY, C. Jun.: 'Analysis of unsym-
system instead of three separate files that otherwise would metrical machines', ibid., 1936, 55, pp. 471-476
be needed to predict the behaviour of single- two- and 3 PUCHSTEIN, A. F., and LLOYD, T. C: 'Capacitor motors with
windings not in quadrature', ibid., 1935, 54, pp. 1235-1239
three-phase induction machines. 4 ALGER, P. L.: 'The nature of induction machines' (Gordon and
Breach, 1965), chapters 5 and 11
5 KRON, G.: 'Equivalent circuits of electrical machinery' (Wiley,
8 Acknowledgments New York, 1951)
6 GURU, B. S.: 'Revolving-field analysis of capacitor motors with
The author would like to thank J. Postema for his comments nonquadrature windings', Electr. Mach. & Electromech., (to be
on the manuscript and for several useful discussions during published)
the development of this paper. The author is grateful to H. 7 VEINOTT, C. G.: 'Theory and design of small induction motors',
(McGraw-Hill, New York, 1959), Chap. 9, pp. 167-175
Penhorwood, Director Engineering Division, for granting 8 GURU, B. S.: 'Two-equation analysis of a capacitor motor by
permission to publish this paper. The author also owes his cross-field theory', Electr. Mach. & Electromech., 1978, 2, pp.
appreciation to C. Yeiter for expertly typing the manu- 147-153
script and to D. Austin in drafting the Figures for this 9 TRICKEY, P. H.: 'Iron loss calculations on fractional horse-
paper. power induction motors', Trans. Amer. Inst. Electr. Engrs.,
1959,pp. 1663-1669
44 ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1