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Three-Phase Asymmetrical Induction Motor Fed by a Single-Phase Distribution

System - Detailed Mathematical Model

R.G. de Mendona, MSc; L. Martins Neto, Dr; J.R. Camacho, PhD.UFU/DEENE/LabMaq

P.O.Box: 593 - Campus Santa MnicaCEP: 38400-902 - Uberlndia MG Brazil

jrcamacho@ufu.br

Abstract - This paper focuses a time domain model, a three-phaseasymmetrical induction motor fed by a single-phase distribution

system. A low cost solution for a common problem in Brazilian rural

areas. The main effort in this case is to obtain a feedback on the

design of asymmetrical three-phase induction motors with a single-

phase feeding who intend to be a low cost and useful option for rural

applications. Modeling was validated by the design and construction

of a 2 HP asymmetrical three-phase induction motor, in this case

theoretical analysis are compared with experimental results. This

comparison will allow us to obtain conclusions on this machine global

behavior under full-load operation.

Index terms- Induction Motor, Model, Asymmetry.

I. INTRODUCTION

Countries with extensive rural industry and landextensions where the main characteristics of electrical

power consumers are a low monthly kWh consumption;low number of consumers per km of rural network; lowcoincidence factor. Add to that scenario the limitedfinancial resources to be applied in rural electrification

programs, for this purpose Brazil is taken as an example. Inthis case Electrical Utility Companies, in order to make themost with low resources available need to expand theirrural network with a single-phase distribution system. It isclear then, the rural energy consumer has a urgent need forenergy in his farm with the purpose of a better workenvironment, a profitable rural property and all theresources given by electrical apparatus. The mentionedconsumer will be locked with the single-phase system

peculiarities. The use of typical three-phase loads, being anexample the induction motor with power above 12.5 HP.Due to its size, it has its utilization restricted to a three-

phase rural consumers only. The substitution in rural areasof a single-phase network by a three-phase network inmany cases can be financially prohibitive. Alternatives tonetwork upgrading could be the use of single to three-phasestatic or rotary converters[3], or to use a morecontemporary and lower-cost alternative, the three-phaseasymmetrical induction motor fed by a single-phasesource[5]. Those references take in consideration a similar

problem with the use of the frequency domain modeling.However, a very important question not addressed in

previous publications is the time domain linear analysis,considering magnetic saturation, under dynamic operatingconditions. In this case it will be possible to analyze theelectromagnetic torque oscillations for the asymmetricalthree-phase induction motor with single-phase feeding.

As in references [1], [6], [7], [8] and [9], theasymmetrical three-phase induction motor is developedfrom the symmetrical three-phase induction motor. This is

made only by making different the number of turns in eachstator phase. Generically speaking this motor hasunbalanced magnetomotrice forces (mmf) in each phase.This unbalancing allows the decomposition of such mmfdistributions in such a way that can be obtained thedistribution of static and rotary magnetic fields, where thelater ones are of positive and negative sequences. The

positive sequence rotary magnetic field, BE1, according tothe symmetrical induction motor operating principle, also

produces indirectly a rotary magnetic field, BR1, with thesame sequence of the machine rotor. The same happen withnegative sequence rotary magnetic field, BE2, producing amagnetic field BR2, where the rotating field is opposite to

the machine rotor speed. Figure 1 illustrate this affirmative.

Figure 1 - Representation of rotating magnetic fields.

With the observation of Figure 1 one can reach thefollowing conclusions in steady state: the two pairs of magnetic fields BE1, BR1 and BE2, BR2

have a constant displacement angle between them,therefore the alignment resulting electromagnetic torqueis constant;

the angular position between the two pairs of magneticfields BE1 and BR2 and BE2, BR1 is a periodical variable,therefore the resulting electromagnetic torque alignmentis oscillatory.

The main proposal in this paper is the mathematicalmodel for the asymmetrical induction motor coveringmechanical and electrical transients, and also the

unbalancing factor already defined in previous references.From the modeling can be defined the design philosophy

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for the asymmetrical motor. Therefore, building aprototype a comparison of theoretical and experimentalresults can be made and will validate the presented model.

II. MATHEMATICAL MODEL

For the sake of simplicity, the development ofmathematical equations will be made from a genericwinding consisting of two phases "i" and "j". In thefollowing paragraphs this concept will be extended to thesingle phase fed asymmetrical machine. Machine equationscan be written based in Figure 2, which shows a transversecut of machine stator windings.

Figure 2 Coil distribution for the phase "j" windings.

Taking in consideration two generic motor i ej the following equations can be written:

v r id

dti i ii= +.

(1)

.j

j j j

d

v r i dt

= + (2) i ii i ij j di iL i L i L i= + +. . . (3)

j jj j ij i dj jL i L i L i= + +. . . (4)

where:vi , vj - voltages at phases i and j;ii , ij - currents at phases i and j;i , j - magnetic fluxes of phases i and j;ri , rj - resistance for phases i and j;Lii , Ljj - self inductances for phases i e j, withoutleakage;Lij - mutual inductances between phases i e j", and;

Ldi , Ldj - leakage indutcances for phases i and j ,respectively.The magnetomotrice force of phase j is given by:

( ) ( )[ ]Fmm N n k k i h hj h j j p j h d j h j j = 2 1

. . . . . . .cos . (5)

where:kpjh , kdjh - step and distribution factors, respectively, for

the hth harmonics, given equations (6) and (7).

k hp j h j=

sen . .

2

(6)

j h

sen . .2

.sen .

Rj

j

dRj

j

h n

k

n h

=

(7)

The magnetic field density distribution Bjh produced byFmmjh() is obtained through the Ampere's Law and theresult is given by equation (8). In this case the iron'smagnetic circuit reluctance is neglected when comparedwith air-gap's reluctance, considered uniform.

( ) ( )[ ]B N n k k i h hj ho

j j p j h d j h j j

= 2 1. . . . . . . .cos . (8)

where o is the air's magnetic permeability and is the air-gap's radial length.

In order to obtain the magnetic flux of phase "j" in phase"i", ijh , must be obtained the phase "j" magnetic flux whoembraces phase "i". Therefore, starting from equation (8),can be obtained the mentioned magnetic flux and then theflux coupling between phases ijh , given by equation (9).

( )[ ] i j h i j j

w i h

i jk N N i

k k

h h= 1 2. . . .

.

cos .

w j h (9)

kp L R n ni j o

1 42

= .. . . . . .

.

(10)

k k kp i h d i hw i h = . (11)k k kp j h d j hw j h = . (12)

where:L - rotor cylinder length;R - air-gap radial length;2p - machine pole number;ni , nj - coil number of phases i and j , respectively;

Ni , Nj - turns number of phases i and j , respectively;

kdih , kdjh - distribution factor of phases i e j ,respectively;kpih , kpjh - step factor of phases i e j , respectively.

It is important to mention that ijh doesn't include thephase mutual leakage flux.

The harmonic inductance of order h between phases ie j, Lijh , can be obtained through equation (13).

Li j h = i j h

ji(13)

Therefore, substituting (9) in (13) the followingexpression can be obtained:

( )[ ]L k N N

k

hh

ij h i j

ijh

i j= 1 2. . . .cos

(14)

k k kijh wih wjh= . (15)The total magnetic flux coupling of one phase, "i" as an

example, i, which can be split in the sum of contributionsof magnetic flux coupling which embraces stator and rotor,i, and the dispersion flux, di . Therefore, we have:

i di i= + ' (16) di i iL i= . (17)

where:Li - dispersion inductance of phase i.

The dispersion inductance can be supposed constant andtherefore can be written:

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L k Ni di i= .2 (18)

where:kdi - magnetic circuit dispersion permeance.

Accordingly with reference [11], the flux coupling ican be obtained through the superposition of couplingharmonic components of all phases ijh. Therefore, can bewritten:

i ijhhj

' = (19)From equations (13) and (19) we have:

i ijh jhj

L i' .= (20)from equations (16), (17) and (20) we obtain equation (21):

i i i ijh jjh

L i L i= + . . (21)

Finally, substituting equation (21) in equation (1) can beobtained the full voltage equation

v r i Ldi

dtL

di

dti

dL

dti i i ii

ijh

j

j

ijh

jh

= + + +

. . . . (22)

Once defined the magnetic flux coupling generalequation and voltage equation, those can be extended to theasymmetrical three-phase machine. In the matrix form wehave:

[ ] [ ] [ ] = L I. (23)where:

[ ]

=

a

b

c

A

B

C

'

'

'

(24) [ ]I

I

I

I

I

I

I

a

b

c

A

B

C

=

'

'

'

(25)

[ ]

=

44362616

44352515

44342414

363534332313

262524232212

161514131211

00

00

00.

LLLL

LLLL

LLLL

LLLLLL

LLLLLLLLLLLL

kL (26)

onde:

Lk

hKaah

hL11 2= + , L

k

hKbbh

hL22 2= + ,

L

k

h Kcch

hL33 2= + , Lh

RRh

Kh

kL '3

2cos1244 +

=

,

Lb

k

hhabh

h12 2

1 2

3=

. .cos .

, Lc

k

hhach

h13 2

1 2

3=

. .cos .

,

( )Lk

hhaRh

h14 2= .cos . , L k

hhaRh

h15 2

2

3= +

.cos

,

Lk

hhaRh

h16 2

2

3=

.cos

, Lb c

k

hhbch

h23 2

1 2

3=

.. .cos .

,

Lb

k

hhbRh

h24 2

1 2

3=

. .cos

, ( )Lb

k

hhbRh

h25 2

1= . .cos . ,

Lb

k

hhbRh

h26 2

1 2

3= +

. .cos

, Lc

k

hhcRh

h34 2

1 2

3= +

. .cos

,

Lc

k

hhcRh

h35 2

1 2

3=

. .cos

, ( )Lc

k

hhcRh

h36 2

1= . .cos . ,

b

a

Nb

N= , c

a

Nc

N= .

where:k machine's magnetic circuit constant;

Na, Nb, Nc turns number for phases a, b and c ofmachine's stator, respectively.

For the three-phase asymmetrical machine with single-phase feeding, can be observed the presence of anadditional capacitor between terminals "B" and "C", withthe task to solve the machine's starting problem, andimprove the performance in nominal steady-stateconditions, as can be seen in Figure 3.

Figure 3 Schematic diagram for the three-phase asymmetrical inductionmachine stator with single-phase feeding.

Through a complete analysis of electrical andmechanical equations for the asymmetrical machine, basedon well established equations for a generic three-phaseinduction machine, we have:

v r id

dt

a a aa= +.

(27)

v r id

dtb b bb= +.

(28)

v r id

dtc c cc= +.

(29)

With Figure 3 as our reference, the following can bewritten:

v v va c= (30)v v vCap b c= (31)

i Cap dvdt

iCapCap

b= = . (32)

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( )i i ic a b= + (33)

Mathematical manipulation of equations from (27) to(33), give us:

( )v r r i r i ddta c a c bac= + + +. . (34)

( )v r r i r id

dtCap b c b c abc= + + +. .

(35)

where: ac a c= bc b c=

For the short-circuited rotor we have:

0 = +r id

dtR AA' . '

'(36)

0 = +r i ddtR B

B' . ' ' (37)

0 = +r id

dtR CC' . ''

(38)

In order to solve the system of equations from (34) to(38) it is necessary to relate the magnetic flux couplingwith stator and rotor currents. For an asymmetricalmachine, the following can be written:

[ ]

a

b

c

A

B

C

a

b

c

A

B

C

L

i

i

ii

i

i

'

'

'

. '

'

'

=

(39)

where[ ]L is the inductance matrix from equation 26.From equations (33) and (39) can be obtained equation

(40).

ac

bc

A

B

C

a

b

A

B

C

L

i

i

i

i

i

'

'

'

. '

'

'

=

1 (40)

where:

[ ]

=

332515

332414

332313

2524232212

1514131211

1

00

00

00.

AAA

AAA

AAA

AAAAA

AAAAA

kL(41)

The terms in matrix [L1] are:A L L L11 11 13 332= +. ; A L L L L12 12 23 13 33= + ;

A L L13 14 34= ; A L L14 15 35= ; A L L15 16 36= ;A L L L22 22 23 332= +. ; A L L23 24 34= ;A L L24 25 35= ; A L L25 26 36= ; A L33 44= .

Where the values of Lij are the terms of the matrix inequation (26).

The machine's mechanical equations are introduced as inthe following equations:

. Rdw

Tm Tc J dt

= (42)

RR

dw

dt

= (43)

where:J - inertia moment of rotating parts;wR - machine's angular speed;R - angular displacement, in mechanical

degrees;Tm - electromagnetic torque;

Tc - load torque.The electromagnetic torque is given by:

[ ][ ]

Tmp

i i i i i id L

d

i

i

i

i

i

i

a b c A B C

a

b

c

A

B

C

=

4. ' ' ' .

'

'

'

(44)

where:p - pole number;[L] - inductance matrix in equation (26); - angular displacement in electrical degrees.

In order to obtain a system of equations whichrepresents the functional behavior of an asymmetricalinduction machine with single phase feeding, it is enoughto mix the electrical equations from (34) to (39) with themechanical equations from (42) to (44).

III. HOW TO DESIGN AND BUILD A PROTOTYPE

The design of a three-phase asymmetrical is made froman ordinary three-phase symmetrical induction motor, theonly difference are the changes made in the originalwindings with the stator and squirrel cage rotor magneticstructure being preserved. Initially, a 2 HP, 4 poles,220/380 V, ordinary commercial three-phase inductionmotor was tested in the laboratory in order to obtain itsequivalent circuit parameters, as given in Table 1, in thismotor the equivalent circuit is the same for each phase.With the parameters and the motor model, a number ofresults were obtained for phase turns number andcapacitance Cap, for each case torque oscillations weremeasured, and the result which showed lower oscillation, atnominal condition for torque and speed was:

Nb = 0.60Na ; Nc=1.40Na ; Cap=40F

IV. DIGITAL SIMULATION

With the machine's equivalent circuit data, the turnsnumber relationships and the capacitance value, theoretical

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simulation results can be obtained considering themachine's spatial harmonics or only the fundamental (firstharmonic), as can be seen in Figures 4, 5, 6 and 7.

Table 1. Equivalent circuit parameters for the three-phase inductionmachine.

Stator and Rotor Phase Data da defesa(referred to the stator)

Stator Resistance 3,800,03Rotor Resistance 3,010,03

Blocked Rotor Reactance 3,100,03Magnetization Reactance 75,150,7

Figure 4 - Asymmetrical three-phase induction motor torque with single-phase feeding. - First harmonic only.

Figure 5 - Asymmetrical three-phase induction motor torque with single-phase feeding. - First harmonic only.

Through figures 4, 5, 6 and 7, can be observed that theasymmetrical three-phase induction motor torque withsingle-phase feeding shows satisfactory torque oscillationsunder nominal load conditions. Can be also observed thatthe inclusion of machine's spatial harmonics doesn't changemuch its behavior concerning torque oscillations.

V. EXPERIMENTAL RESULTS

Since confirmed the asymmetrical three-phase inductionmotor good behavior with single phase feeding, a prototypewas built and tested under load conditions, the objectivewas to validate the theoretical results using the laboratory

experimental results as the real case. Therefore, the testbench in the lab was assembled as shown in Figure 8.

Figure 6 - Asymmetrical three-phase induction motor torque with single-

phase feeding. - Up to the 30th harmonic.

Figure 7 - Asymmetrical three-phase induction motor torque with single-phase feeding. - Up to the 30th harmonic.

Figure 8 Bench test assembled in order to measure the variables for theasymmetrical three-phase induction motor.

Figure 9 shows the asymmetrical three-phase inductionmotor torque with single phase feeding in steady-stateunder nominal load conditions.

VI. SYMMETRICAL AND ASYMMETRICAL MOTORS -COMPARISON

As a verification only, to check the torque oscillationspresented by the asymmetrical three-phase induction

motor, analysis is made for the same power symmetricalthree-phase induction motor, fed by a three-phase voltage

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system with a perfectly acceptable voltage unbalance inelectrical systems of around 2% between phases.

Figure 9 Asymmetrical three-phase induction motor torque at nominalconditions with single-phase feeding.

Figure 10 Symmetrical three-phase induction motor torque at nominalconditions.

VII. CONCLUSIONS

From the analysis of Figures 4, 5, 6, 7 and 8 can beobserved that results obtained from simulations matchalmost perfectly to the results obtained from experimentaltests, and the model used is perfectly validated. ComparingFigures 9 and 10, can also be observed that even with theasymmetrical induction motor submitted to a normal

voltage unbalance from the source, torque oscillations,produced by this unbalance, are close to the torqueoscillations presented in the symmetrical three-phaseinduction motor from normal voltage unbalance in thesource. Therefore, it is demonstrated that the validation ofthe model presented, and also the satisfactory performanceof the asymmetrical three-phase induction motor undersingle-phase feeding.

VIII - BIOGRAPHY

Roberlam G. de Mendona - Mr. Mendona was born in Itabuna, Bahia,Brazil in May 6th, 1969. He did his BSc in Electrical and ElectronicEngineering from UNIVALE - Universidade do Vale do Rio Doce,

Governador Valadares, MG, Brazil in 1991. Finished his MSc degree inElectrical Engineering in 1997 at UFU - Universidade Federal deUberlndia, MG, Brazul. He is a lecturer from CEFET - Centro Federal de

Educao Tecnolgica, Jata , GO. Currently he is pursuing his DoctoralDegree in Electrical Engineering at UFU, his area of interest is ElectricalMachines.Luciano Martins Neto - Dr. Martins Neto was born in Botucatu, SP,Brazil on May 22nd, 1948. He has a Doctoral degree in MechanicalEngineering from Escola de Engenharia de So Carlos at Universidade deSo Paulo (USP), So Carlos, Brazil since 1980. Worked as a lecturer atFaculdade de Engenharia de Lins, Lins, SP, Brazil, at Escola de

Engenharia de So Carlos ( USP), So Carlos, Brazil and at the ElectricalEngineering Department (UNESP - Universidade Estadual Paulista) atIlha Solteira, SP, Brazil. He is currently working as a Senior Lecturer atUniversidade Federal de Uberlndia, MG, Brazil. His areas of interest areElectrical Machines and Grounding.Jos Roberto Camacho - Dr. Camacho was born in Taquaritinga, SP,Brazil on November 3rd, 1954. Completed his PhD degree in the Electricaland Electronic Engineering Department at the University of Canterbury,Christchurch, New Zealand, in August 1993. He is a Senior Lecturer atUniversidade Federal de Uberlndia where he works since February 1979.Dr. Camacho is a Researcher-Consultant of CNPq (Brazilian NationalCouncil for Scientific and Technological Development) and worked as acollaborator-member of Brazilian Committee of CIGR-JWG 11/14-09(Unit Connection). His areas of interest are Dynamic Simulation,Electrical Machines and HVAC-DC conversion.

IX. REFERENCES

[1] - Richard, Jean-Claude, A Comparative Study Between Single-Phaseand Asymmetrical Three-Phase Induction Motors, Master's Dissertation(in Portuguese), UFU - 1993.[2] - Alvarenga, B.P., Model for the Computation of Torque for anInduction Machine Including Winding and Saturation Effects, Master'sDissertation (in Portuguese), UFU - 1993.[3] - Chaves, M.L.R., Design and Assembling of Static Systems forFeeding Three-Phase Loads from Single-Phase Networks, Master'sDissertation (in Portuguese), UFU - 1987.[4] - Martins Neto, L., Single-Phase Feeding of Asymmetrical Three-Phase Induction Motor, I Electrical Energy Distribution InternationalSeminar, (in Portuguese), Belo Horizonte, MG, Brazil, October 1990.[5] - Alwash, J.H.H. & Ikhawan, S.H.; Generalised Approach to theAnalysis of Asymmetrical Three-Phase Induction Motors, IEE Proc.

Eletric. Power Appl., Vol 142 No. 2, March 1995.[6] - Martins Neto, L., Mendona, R.G., Camacho, J.R. & Salerno, C.H. ;The Asymetrical Three-Phase Induction Motor Fed by Single PhaseSource: Comparative Performance Analysis. IEEE - IEMDC, InternationalEletric Machines and Drives Conference. Milwaukee, Wisconsin, May1997.[7] - Martins Neto, L., Mendona, R.G., Camacho, J.R. & Andrade, D.A.;Asymmetrical Three-Phase Induction Motor Under Single Phase Feeding:Oscillating Torque, Theoretical and Experimental Analysis - HarmonicEffects, ICEM-98, International Conference on Electrical Machines.Turkey, Istanbul, September 1998.[8] - Martins Neto, L., Mendona, R.G., Camacho, J.R. & Andrade, D.A.;Single Phase and Asymmetrical Three-Phase Induction Motors: AComparative Steady-State Analysis Under Single-Phase Feeding. ICEM-98, International Conference on Electrical Machines. Turkey, Istanbul,September 1998.

[9] - Mendona, R.G & Martins Neto, L; Comparative PerformanceAnalysis: Oscillating Torque of Symmetrical and Asymmetrical Three-Phase Induction Motors, (in Portuguese), Proceedings of XII BrazilianAutomatic Control Conference XII CBA, Vol. I, pp.243-247-September 14-18, 1998 - Uberlndia, MG, Brazil.[10] - Martins Neto, L., Salerno, C.H., Bispo, D. & Alvarenga, B. P.;Induction Motor Torque: An Approach Including Windings AndSaturation Effects, International Conference on Electrical Machines inAustralia - ICEMA. Adelaide, University of South Australia , September1993.[11] - Martins Neto, L., Camacho, J.R., Salerno, C.H. & Alvarenga, B.P.;Analysis of a Three-Phase Induction Machine Including Time and SpaceHarmonic Effects: The A, B, C Reference Frame; PES-IEEE Transactionson Energy Conversion, Volume 14, Nr. 1, March 1999. Article number:PE-154-EC-0-10-1997.