/

ANALYTICALAND EXPERIMENTALSTUDYOF

HIGH PHASEORDER INDUCTIONMOTORS

EUGENEA, KLINGSHIRN

DEPARTMENT OF ELECTRICAL ENGINEERING

CLEVELAND STATE UNIVERSITY

CLEVELAND, OHIO 44115

AUGUST, 1983

PREPARED FOR

NATIONALAERONAUTICSAND SPACEADMINISTRATION

LEWIS RESEARCH CENTER

UNDER GRANT NAG 3-108

FINAL TECHNICAL REPORT

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I_IOUCTION MOIQR_ tin.iT F_:chnic_I RePort

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ANALYTICAL AND EXIERIMENTAL STUDY

OF HIGH PHASE ORDER INDUCTION M3TORS

Eugene A. Klingshirn

Department of Electrical Engineering

Cleveland State University

August, 1983
TABLE OF OONTENTS

TABLE OF CONTENTS .........................

LIST OF FIGURES ...........................

l_ge

t

v

LIST OF TABLES .......................... vii

SU_9 t_RY .............................. 1

CHAPTER 1 - INTRODUCTION ..................... 3

Section I-I Purpose of Investigation ............ 3

1-2 Literature Review ................ 4

1-3 General Analysis Methods ............ 5

1-4 Description of HPO _,i3zors............ 6

1-5 Connections and Winding Terminology ....... 6

1-6 Use of Three-Phase Groups ............ 7

CHAPTER 2 - SYMMEFRICAL CC_DONENT METHOD _ND CIRCUITS ....... Ii

Section 2-1 Introduction .................. II

2-2 Inductance Values for Equivalent Circuits .... 12

9-3 Sequence Circuits ................ 13

2-4 Simplification of the Positive and Negative

Sequence Circuits ................ 14

2-5 Effect of Rotor Skin Effect on Rotor Parameters. 14

2-6 Sequence Circuits for HPO Motors ........ 15

2-7 Solution of Sequence Circuits .......... 15

2-8 Torque and Losses from Sequence Circuits .... 19

2-9 Analysis of S12_ Motors, Steady State Case . . . 19

2-10 Sequence Circuits for Certain Common Operating

Modes ...................... 20

-i-
Page

2-ii Case I; S12_ Motor with Balanced Sinusoidal

Source ....................... 22

2-12 Case II; S12_ Motor with Intergroup Unbalance .... 23

2-13 Case III; S12_ Motor with One Line Open ....... 25

2-14 Case IV; Motors with Periodic, Non-Sinusoidal

Sources ....................... 27

2-15 Discussion of Non-Sinusoidal Performance ...... 28

CHAPTER 3 - OTHER TRANSFO_TION _,{ETHODS ............... 32

Section 3-1 Introduction .................... 32

3-2 General _-B-Y and d-q-y Circuits .......... 32

3-3 Equivalent Circuits for e-B-7 Components ...... 33

3-4 Transient Solution Using _-8-Y V#riables ...... 35

3-5 Equivalent Circuits Based on _l_iple d-q-7

Components ..................... 36

3-6 _-B Equivalent Circuits for Sine Wave Case ..... 38

3-7 _-B Circuits for Harmonics ............. 40

3-8 Multiple d-q-y Circuits for Sine Wave Case ..... 40

3-9 Summary of Equivalent Circuits ........... 41

CHAPTER 4 - FILTERS DESIGNED FOR HPOMOTORS .............. 45

Section 4-1 Stator I2R Losses Due to Unbalance or Harmonics... %5

4-2 Filters to Reduce Stator 12R Losses ......... 46

4-3 A Filter Example for a S12_ Motor .......... 47

4-4 Other Versions of Filters for S12_ Motors ...... 48

4-5 Example of Filter to Reduce Harmonics in S12¢ Motor. 48

4-6 Rotor I2R Losses for Above Example ......... 52

4-7 Stator I2R Losses .................. 52

4-8 Total Harmonic Losses ................ 52

-ii-
Page

CHAPTER 5 - EXPERIMENT._L EQUIPMENT .................. 55

Section 5-1 The _btor Used ................... 55

5-2 60 Hz Power Source ................. 56

5-3 Measurement of Current and Power .......... 57

5-4 Motor Temperature _a_rements ........... 57

5-5 _o_or Slip Measurements ............... 58

5-6 _bror Tests on Non-Sine Wave Source ......... 58

5-7 Experimental Filters ................ 58

5-8 Impedances for Experimental Filters and _ror .... 60

CHAPTER 6 - MOTOR OPERATION WITH ONE LINE OPEN ............ 62

Section 6-1 Introduction .................... 62

6-2 Locked Rotor Tests ................. 62

6-3 Full Load Tests with One Line Open ......... 65

6-4 Efficiency Measurements ............... 65

6-5 Full Load Slip _leasurements ............. 67

S-S S_ry of Full Ix)ad Tests ............. 69

CHAPTER 7 - _URRENTS IN THE EXPERIMENTAL MOTOR WITH SOURCE

UNBALANCE ......................... 70

Section 7-1 Introduction .................... 70

7-2 Effect of Intergroup Vol_age Unbalance ....... 70

7-3 Conclusions Regarding Current Unbalance ....... 72

CHAPTER 8 - EXPERI,'_ENT.%L MOTOR ON NON-SINUSOIDAL VOLTAGE;

WITH FILTER ........................ 74

Section 8-1 Test Setup ..................... 74

8-2 Experimental ResulZs ................ 75

-iii-
P_e

CHAPTER9 - OONCLUSION........................ 78

Section 9-I Results of the Investigation ............ 78

9-2 Suggestions for Further Investigation ........ 79

REFERENCES .............................. 81

APPENDIX A - TRANSFOP_ATION MATRICES ................. 83

Section A-I Symmetrical Component TransformaZion _trix

for General n-Phase Case .............. 83

A-2 S_trical Component Transformation, 12-Phase . . . 84

A-3 The Symmetrical Component Transformation, S12¢ . . . 85

A-4 The Two-Phase (eBY) Transformation _trix ...... 86

A-5 The dq Transform_rion for _%chines with

_ultiple Three-Phase Winding Groups ......... 87

APPENDIX B: S_TRICAL CO_X)NENT RESISTANCES AND INDUCTANCES .... 90

Section B-I Stator Windings and Parameters ........... 90

B-2 Transforn_tion to Symmetrical Component

Inductances ..................... 92

B-3 _,_gnetlzing and Leakage Inductances ......... 93

APPENDIX C: FILTER DESCRIPTION __ND ANALYSIS ............. 97

Section C-I Filter Description ................. 97

C-2 Analysis of the General Filter Type ......... 97

C-3 Five-Phase Filter Example .............. 98

C-4 Phase AdJacent Scheme ................ 103

C-5 Filters for S12¢ and S18¢ Motors .......... 105

C-6 The Wye-Del_a Filter for S12¢ _%btors ........ 108

-iv-
Figure

I-I

1-2

1-3

2-1

2-2

2-3

2-4

2-5

2-6

2-7

2-8

2-9

3-1

3-2

3-3

3-4

3-5

3-6

4-1

4-2

5-1

5-2

6-1

LIST OF FIGURES

Title Page

Schematic Diagram of a 2q-Phase _chine ............ 8

Five-Phase or Semi lO-Phase Stator ............... 8

Semi 12-Phase Stator Connections .... •............ I0

Sequence Circuits for a Polyphase Motor ............ 17

Simplified Sequence Circuits .................. 17

Stator Connections for an n-Phase _{otor ............ 17

Stator Connections for a S12_ _otor .............. 24

Phasor Voltages for S12% Motor ................. 24

Positive Sequence Circuit ................... 24

S12_ Voltages with Intergroup Unbalance ............ 24

Sta_or Connections for S12_ Motor with Open Phase ....... 26

Equivalent Circuit for $12_ Motor with One Line Open ...... 26

Circuits for $12_ Motor with _-8 Variables ........... 34

Equivalent Circuits for $12_ Motor Based on _ul_iple

d-q-x Circuits ......................... 37

_-8 Circuits for $12_ Motor, Phasor Form ............ 39

Phasor Form of Mul_iple d-q-7 Circuit for $12_ Motor ...... 42

Modified Voltages for _Itiple d-q-_ Curcuits of S12_ Motor . . 43

Con_ponent Circuits for d-axis Based on Superposition Method . . 44

Single Pulse I_,_ Waveform and its Harmonics .......... 50

Harmonic 12R Losses with Single Pulse P_ ........... 53

Power Transformer Connection for S18_ Source .......... 59

Steel Core Used for Harmonic Filter Element .......... 59

Locked Rotor Torque with One Line Open ............. 63

-V-
Figure Title Page

6-2 Full Load Efficiency with One Line Open ............ 66

6-3 Full Load Slip with One Line Open ............... 68

8-I Oscillograms of Volzage and Current with 6-Step Source

and S18¢ _btor ......................... 76

A-I Phasor Arrangement for 12-phase System .............. • 88

A-2 Geometric Arrangement for _ltiple d-q-y Sets ......... 88

B-I Diagram of an n-Phase Stator .................. 90

C-I Diagram and Inductances of Filter Transformer ......... 99

C-2 Phasor Diagram for Positive Sequence Currents of a

Five-Phase Motor ........................ 99

C-3 Five-Phase Filter, Phase Opposite Type ............. I00

C-4 Five-Phase Fil_er, Phase Adjacent Type ............. 104

C-5 Nine-Phase or S18¢ Filter, Phase Opposite Type ......... 106

C-6 Nine-Phase or S18¢ Filter, Phase Adjacent Type ......... 107

C-7 812¢ Filter, 6-Transformer Type ................ 109

C-8 S12¢ Filter, 3-Transformer Type ................ 110

C-9 Wye-Del_a Filter for 812¢ Motors ....... . ........ Ill

-vi-
Table

2-1

2-2

2-3

2-4

5-1

5-2

7-1

LIST OF TABLES

Title Page

Symmetrical Component Sequence Circuits for n-Phase Motor . . . 18

Symmetrical Component Sequence Circuits for S12¢ _btor ..... 21

Harmonic Circuits for S12¢ Motor ................ 29

Harmonic Circuits for S18¢ _tor ................ 30

}.4otor Voltage and Current Ratings ......... _ ..... 56

Sequence Reactances for Filters and Motors ........... 61

Results of Voltage Unbalance on S12¢, 6 PoleMotor at

Full Load ........................... 71

Reactances for S18¢ Filter ................... 74

Ha_nonic Currents ...... " ................. 75

-vii-
S_l_Y

Induction motors having more than three phases have been investigated

to determine their suitability for electric vehicle applications. The

objective is to have a motor with a current rating lower than that of a

three-phase motor. The name chosen for these is high phase order (HPO)

motors. [_ors having six phases and nine phases were given the most

attention.

It was found that HPOmotors are quite suitable for electric vehicles,

and for many other applications as well. They have characteristics which

are as good as or better than three-phase motors for practically all

applications where polyphase induction motors are appropriate. Unlike

three-phase motors, they can be used with one phase open. In comparison to

three-phase motors, they have lower I2R loss and smaller torque pulsations

due to harmonic currents in the rotor when the source is not sinusoidal.

Some of the analysis methods are presented, and several of the

equivalent circuits which facilitate the determination of harmonic currents

and losses, or currents with unbalanced sources, are included. The means

for finding the appropriate winding inductances needed for cal_,lations are

explained, and the effects of these inductances on performance are

indicated.

The sometimes large stator currents due to harmonics in the source

voltages are pointed out. Filters which can limit these currents were

developed as part of this project. An analysis and description of these

-1-
-2-

filters is included. Whenfilters are used, i_ is possible to have these

HPOmotors supplied by non-sinusoidal voltage sources such as inverters and

the motors can perform practically as well a they would on a sinusoidal

source with comparable efficiency and smoothness of operation. These

filters are selective in limiting certain harmonic or sequence _irrents

without affecting the fundamental frequency, positive sequence currents

which produce useful zorque.

Experimental results which confirm and illus_rate much of the theory

are also included. These include locked rotor test resul_s and full-load

performance wizh an open phase. Also shown are oscillogran_ which display

the reduction in harmonic currents when a filter is used with the

experimental motor supplied by a non-sinusoidal source.
CHAPTERi. INTRODUCTION

i-i Purpose of Thislnvestigation

In the process of designing and building inverters and motors for

electric vehicles, the use of relatively low voltage imposes the need for

high current when three-phase motors are used. This sometimes requires

Zhat multiple power transistors or thyristors be used in parallel in the

inverter. An even more serious problem is that the motor stator winding

requires conductors of large cross section. This makes it difficul_ to

wind the stator, and the large cross section also results in increased statot I2R loss due to skin effect (eddy currents) in the large conductors.

This is especially the case with the harmonic frequency currents produced

by inverter sources.

It is apparent that the current per phase is lower if more phases are

used. This reduces the number of parallel power devices required, and also

reduces Ehe required conductor cross section. Based on this obvious advantage, the investigation on characteristics of motors having more than three

phases was undertaken. The name suggested for these is high phase order

(HPO) motors.

Early in the investigation, several advantages of HPOmotors over tile

usual three-phase motor were recognized. Some of these advantages are:

-Lower current per phase for a given voltage rating.

-Able to start and run wi_h one line open.

-Elimination of more of the space harmonics from the air gap flux.

-Reduction in rotor harmonic I2R loss with inverter sources.

--3--
-4-

-Reduction of torque pulsations with inverter sources.

-Filters can be designed to reduce stator harmonic currents

and their I2R losses with inverter sources, and these

filEers do not affect fundamental Eorque-producing

currents.

1-2 LiEerature Revie#

Several papers have been published which present some of the characteristics of HPO motors. In reference I, a comparison of the harmonic

losses is made for three-phase, nine-phase, and 27-phase motors supplied by

an inverter. That study shows reduced rotor I2R loss, and an increase in

sr_tor 12R loss when more phases are used.

Nelson and Krause (ref. 2) derived a model for motors having multiple

sees of three-phase windings. They then did a simulation on six-phase motors supplied by a 6-step voltage source inverter. Their study reveals the

elimination of the sixth harmonic torque pulsation when the six-phase motor

with 30 degree winding displacement is used. Also, the reduction in rotor

harmonic current and the increase in stator harmonic current is shown.

A study of motors having many phases was carried out by Jahns (ref.

3). The abiliEy of these motors to start and to run with one phase open or

shorted out was brought out. The improved reliability of inverter mlpplied

HPOmotors was demonstrated.

An analysis of a six-phase motor supplied by a current source inverter

was made by Andresen and Bieniek (ref. 4). Their work brings out the reduction in torque pulsations and the reductions in rotor 12R lass.

The early analytical work and experimental resul_s of this project

have been reported in references 5 and 6. These include a description of
-5-

the machines, the relationship between symmetrical componentsand time harmonics, the effect of coil pitch on inductances, the results of speedtorque tests, and the currents with non-sinusoidal voltage sources.

All of the above references pertain to induction motors. There is

also considerable activity in the design of synchronous motors having six

phases. These are generally used with current source Inverters. Their

lower torque pulsations and lower 12R losses makethegn attractive, especially for very large variable speed motors.

1-3 General Analysis Methods

Methods for analyzing HPO motors are generally extensions of the methods used for the _n Three-phase motors. The methods include some which

transform the machine voltage and current variables (called phase variables) to an equal number of variables i_ another reference frame. Two of

the most well-known of these are The transformation to the syYmnetrical component reference frame, and the transformation zo the direct-quadrature, or

d-q reference frame. When the reference frame is stationary, the d-q variables reduce to the so-called two-phase, or a-8 variables. For motors

having more than %_o phases, there must, of course, be additional variables

for _he d-q set or the _-8 set to mee_ the need for a sufficient number of

variables.

The textbook authored by White and Woodson [ref. 7] has an entire

chapter on the general machine having n stator phases, and m rotor phases

where the above transformation methods are presented. Most of the theoretical work included in this report is based on the White-Woodson book.
-6-

In this report, the sy-_netrical componentmethod is used extensively.

Other methods are briefly presented only for comparison, or are mentioned

as techniques for the analysis of transient type problems not attempted in

this investigation.

1-4 Description of HPO Motors

High Phase Order (HPO) motors may be defined as ac motors having more

_han three phases. These n_y be induction motors, synchronous motors, or

reluctance-synchronous motors. This report presents the analysis and behavior of induction motors only.

Induction motors may be built with any number of phases, but the number of slots in the stator laminations must be such that a symmetrical

winding can be accommodated. To give an example, a four-pole motor wi_h 12

phases requires a_ least 48 stator slots. For motors in the i0 _o 20 HP

range, using more than about 48 slo_s becomes somewhat impractical.

i-8 Connections and Windin_ Terminology

The winding terminology is based on the phase belt angle 8, and the

number of phase belts per pole which is given the symbol q. The relationship between zhese is

q : zso/B

where 8 is the phase belt angle in electrical degrees.

The basic machine s_ator has 2q coil groups symmetrically arranged

with 8 degrees separating them as in figure I-i.
-7-

If the machine has n phases, and if n = 2q, the name for the connection is an n-phase or a 2q-phase connection (for example, a 12-phase connection). Then, if q is a whole number, there can be n = q phases, and

this comes about when one of the two coils on each magnetic axis is deleted. The name given to this connection is the semi 2q-phase connection

(for exan_le, a semi 12-phase connection which has six phases).

If q is an odd number, the semi 2q-phase arrangement is derived from

the 2q phase type by deleting alternate coils around the stator (coils b,

d, .... n-1 in figure 1-1). The semi 18-phase (S18_) arrangement is an example of this, and it has nine coils 40 ° apart.

If q is an even number and a multiple of three, the connection uses

the first q/3 coils of the 2q-phase arrangement, deletes the next q/3, includes the next q/3, etc. The semi 12-phase (S12_) is an example. Finally, if q is an odd number and not a multiple of 3, the connection uses the

first set of q coils taken sequentially (a, b, c, .., q), and omits all

others.

In this report, semi 12-phase and semi 18-phase will be abbreviated

S12_ and S18_. Figures 1-2 and 1-3 show some of the arrangements and illustrate both star and mesh connections. In Figure 1-2, five-phase or semi

lO-phase connections are shown for star and for mesh.

1-6 Use of Three-Phase Groups

The most practical arrangement should consist of several phase groups,

each of which is either a wye or a delta connection of three phases. These

three-phase groups, if isolated from each other, cannot carry third

harmonic currents and this is a distinct advantage. In most of the analyt-
-8-

2q

• ° ° a

d c

Figure i-I. Schematic diagram of

a 2q-phase machine

e e

d

a

c c a

b

(a) (b) b

Figure 1-2. Five-Phase or Semi lO-Ph_se Stator.

(a) Star-connected (b) Mesh connected
-9-

ical work and all of the experimental work carried out in

only the three-phase, the $12¢, and the $18¢ connections were used.

1-3 shows the S12¢ arrangements.

For low voltage motors where the number of turns per coll tends

this project,

Figure

to be

small and the conductor cross-section tends to be large making winding difficult, the delta connected groups, shown in Figure l-3(b), would be better

since, for a given voltage and current rating, when c_ared to a wje connection, they would use 43 times as _ny turns of conductor with 1/43 rimes

the cross section.
-I0-

c2

c_ nl to n2 connection __

w_k - is op._ti._on..a L . -_

_ "b2

bl

(a)

cl c2

a

4/ b2 a2

bl

(b)

Figure I-3. Semi 12-Phase Stator Connections

(a) Star-connected phase groups

(b) Mesh-connected phase groups
CHAPTER 2. SYMmeTRICAL _NENT METHOD AND CIRCUITS

2-1 Introduction

In a great many applications, motors operate at

speed and with a periodic applied voltage or current.

practically constant

These steady state

situations lend themselves to an analysis based on the use of phasor forms

for voltages and currents. The motor may then be represented by equivalent

circuits using resistances and reactances. At a given speed, the slip is

known and the equivalent circuit parameters are all known. Then, for a

given applied voltage (or current), the circuit solution gives the currents

(or voltages), the torque, the I2R losses, and the power input and output.

This is the conventional, well-known method for determining the performance

of three-phase motors.

For HPOmotors, this steady state phasor approach is an attractive

method. The symmetrical cor_onent method provides the basis for constructing the set of simple equivalent circuits which are applicable to these motors operating from balanced or unbalanced, sinusoidal or non-sinusoidal

sources. In this last case, tile Fourier series method is used to separate

the non-sinusoidal applied voltage or current into its fundamental and harmonic frequency con_ponents. Then, the phasor approach applies to each of

these separately, and results can be combined according to the principle of

superposition.

-II-
-12-

2-2 Inductance Values for Equivalent Ci[cuits

The symmetrical component method requires that the machine variables

(phase volrm4_es and currents) be transformed to the sequence variables

using the appropriate matrix transformation. The matrix required is the

inverse of matrix [A] given in Appendix A. In this transformation process,

the n_chine resistances and inductances are transformed to the sequence resistance and inductances. It is assumed that the machine has a balanced

n-phase stator winding as described in Appendix B. The relationship between the stator resistance and inductance matrices and their symmetrical

con%x)nent forms is described in Appendix B. The stator resistance matrix

is a diagonal matrix and i_ is unchanged when it is transformed ro the symmetrical component resistance matrix.

The s_a_or inductances include _he self inductance of each phase and

the mutual inductances between phases. Appendix B gives the details for

finding the sequence inductance values from the srator inductances. The

result is the diagonal stator sequence matrix [L s] given by the following"
-13-

I

0

1

3

[Ls]= •

n-2

n-i

0

Lo

P N

1 2 n-2 n-i

LI+L m

L2

• • • • b

L2

LI+L m

In the above n_zrix, the various inductances are"

L o

L1

Lm

L2

L3

Q

Li

zero sequence inductance

leakage inductance for positive sequence (also for negative

sequence)

magnetizing inductance

inductance for sequence two (also for sequence n-2)

inductance for sequence three (also for sequence n-3)

inductance for sequence i (also for sequence n-i)

2-3 Sequence Circuits

The symmetrical component method leads to a set of equivalent circuits; one circui_ for each of the n sequences of an n-phase motor. Since

#

the sequence resistance matrix and _he sequence inductance matrix are
-14-

both diagonal matrices, the sequence circuits have no coupling between them

and can therefore be solved as independent circuits.

No matter how many phases a polyphase motor might have, it is only the

positive and the negative sequence circuits which have coupling between the

stator and the rotor portion of each of these circuits. Therefore, only

these two sequence circuits include rotor elements. The rotor circuit configurations applicable to the positive and the negative sequence circuits

of three-phase motors apply to the following HPO motor sequence circuits as

well. Since these sequence circuits for three-phase motors are well known,

they can be presented here without derivation. These circuits for the positive and negative sequence are shown in Figure 2-1 on page 17.

2-4 Simplification of the Positive and Negative Sequence Circuits

It is convenient to simplify the sequence circuit of figure 2-1(a) by

reducing the parallel combination of magnetizing impedance j_L m and rotor

impedance (rr/S + jmL r) to the equivalent impedance (Rf + jXf). Details of

this are given in reference 8, pages 547-564. The subscript "f" here means

"forward," indicating the forward revolving positive sequence field which

produces forward txlrque. The simplified positive sequence circuit is shown

in figure 2-2(a). The same kind of a parallel circuit reduction can be

made for the negative sequence circuit of Figure 2-1(b). The resul_ is

shown in Figure 2-2(b). The subscript "b" means "backward".

2-5 Effect of Rotor Skin Effect on Rotor Parameters

The effective rotor resistance and inductance are dependent on the

frequency of rotor currents, and on the rotor bar geometry and material.
-15-

Reference 9 gives derails on pages 265-271. This should be taken into account when Ehe values of rr and L r are used to calculate Rb and X b for _he

negative sequence circuit. Also, in a later section, harmonic frequencies

will be applied and this rotor frequency effect should be taken into account in calculating Rf, Xf, Rb, and Xb.

2-6 Sequence Circuits for HPO Motors

In general, the number of independent sequence circuits is equal to

the number of phases. A _wo-phase motor requires only the positive and the

negative sequence circuit discussed in Section 2-3. A zhree-phase motor

requires one additional circui_ called the zero sequence circuit. It consists of the resistance aad zero sequence reactance of the staror only.

Then, for more than three phases, additional circuits similar to the zero

sequence cirouir are required. Table 2-1 shows all of the circuits for an

n-phase motor.

2-7 Solution of Sequence Circuits

The symmetrical component method is generally used to analyze motors

operating at constant speed and with periodic applied volLages. Phasor

forms of voltages and currents are used. The phase volLages and currents

for a general n-phase motor stator are shown in Figure 2-3.

procedure to be used for a situation where applied voltages

is to first transform them to the sequence voltages by using

ate symmetrical component transformation matrlx [A] -I given in

This is expressed as

The solution

are specified

the appropriAppendix A.
-16-

[vs] = [A]-l[vn¢] (2-1)

or written out, with the bar above each voltage symbol to indicate that it

is a phasor quantity,

" m

V o

vp

V2

V3

I

V i

• I

VN

= [A]-I

Va

Vb

t. .,

(2-2)

The _,bscripts are o for the zero sequence,

P for the positive sequence (sequence I),

2, 3, ...i for sequences two, three, ...i,

and N for the negative sequence (sequence n-l).

These sequence voltages are then used with the appropriate sequence

circuits shown in Table 2-1 to solve for the sequence currents• If the actual phase currents are wanted, they can be found using matrix [A] to

transform from sequence currents, giving line] = [A][I s] (2-3) or, written

out and again using the bar above each symbol for current to identify it as

a phasor,
-17-

r s _L 1 _L r

_I rr/s

(a)

r _L. _L

S I r

_ rr2-S

(b)

Figure 2-1. Sequence Circuits for a Polyphase Motor,

(s is the slip).

(a) Positive sequence

(b) Negative sequence

r s wL 1 Xf

(a)

r _LI Xb

S

(b)

Figure 2-2. Simplified Sequence Circuits

(a) Positive sequence

(b) Negative sequence

R b

Figure 2-3.

i V ÷ n

i_ " " _n___n

- _ra

" _" a

c

Stator Connections for an n-Phase rv:otor
-18-

TABLE 2-1. SYMMETRICAL COMPONENT SEQUENCE CIRCUITS

FOR n-PHASE MOTOR

(Constant speed is assumed, and phasor voltages and currents are used)

Sequence

Number

1 or P,

Positive

Sequence

2

(n-I)

or N,

Negative

Sequence

Sequence Circuit

r s _L o

r s _L 1 Xf

Rf

r s _L 2

r s _L i

r s _L 1 X b

Torque

0

syn

= Tf

0

0

IN2R b

¢

= T b

Stator

I2R Loss

2

I 0 r s

2

I r

p s

2

12 r s

2

I. r

1 s

2

I N r s

Rotor

I2R Los_

0

SIp2R

0

0
-19-

Ib

Tc

In

2-8 Torque and Losses fr_n Sequence Circuits

If torque or losses are wan_ed, it is not necessary to solve for the

actual phase currents. Equations for torque and the stator and ror_or I2R

losses are given in Table 2-1 in zerms of the sequence currents. The total

motor torque is then Tf-T b. The total stator I2R loss is _he sum of _he

individual sequence circuit losses as found in Table 2-1. The same is true

for to_al rotor 12R loss. This is true because the transformation matrix

[A] given in Appendix A is "power invariant," meaning that _he to_al loss

using all sequence variables is equal to the to_al loss using machine variables.

2-9 Analysis of Sl2_Motors, Steady S_ate Case

I_ is expected _ha_ _he 812¢ version of HPOmotors will be the mos_

popular• These will be _he subject of _he next several sections. The S12¢

motor is derived from a 12-phase moor by deleting alternate pairs of

stator windings from _he 12-phase motor. See Appendix A. In the symmetri-
-20-

cal componentcircuits, this eliminates all even numbered sequences frora

the 12-phase sequence sets. The sequence circuits for the S12¢ motor are

shown in Table 2-2. The torque and the I2R losses are found in the same

manner as in Table 2-i. The matrix [A] for transforming between phase

variables and sequence variables for a S12¢ mozor is given in Appendix ._.

2-10 Sequence Circuits for Certain Conmlon Operating Modes

There are several modes of operation of S12_ motors which are _enable to simple solutions when the symmetrical component method is applied.

Among these are:

Case I

Case II

- Balanced S12%operation on sinusoidal source.

- Sinusoidal S12# source consisring of two balanced

three-phase sources, but the two sources either:

(Case II-a) are displaced _/6 radians from each

other, but with unequal magnitudes

(magnitude unbalance),

or

(Case II-b) have equal magnitudes, bur are

displaced =/6 + _ radians from

each other (angle unbalance).

Case III- Sinusoidal S12¢ source, and with one line open.

Case IV - Non-sinusoidal, balanced S12_ source.

For case IV, the Fourier series expansion for the source voltages or

currents give the fundamental and harmonic frequency components, each of

which forms a balanced S12_ sinusoidal source. For each of these, one of

the several sequence circuits is applicable with an appropriate frequency

and value for slip.
-21-

TABLE 2-2. SYMMETRICAL COMPONENTSEQUENCE CIRCUITS

FOR S12_ MOTOR

(Constant speed is assumed, and phasor quantities are used.)

Sequence

Number

I or P,

positive

3

5

7

9

Ii or N,

Negative

Sequence Circuit

r s _L 1 Xf

r s _L 3

.= . ,

r s _L 5

r s _L 5

r s _L 3

r s mL 1 X b

Torque

I

__2Rf

msyn

= Tf

0

0

0

0

IN2R b

syn

= T b

Stator

I2R Loss

2

I r

p s

2

1

3 r

S

2

I r

5 s

2

17 r s

I92rs

IN2rs

Rotor

I2R Loss

SIp2Rf

0

0

I

0

0
-22-

These special cases are covered in the following sections.

2-11 Case I. S12¢ Motor with a Balanced Sinusoidal Source

For chis case, the s_ator voltage and current variables are the phasor quantities shown on the szator connection diagram for the S12¢ motor in

Figure 2-4. The phasor voltages for this balanced situation are shown in

Figure 2-5. In that figure, the set of phase vol_agesVal-gbl-Vcl form a

balanced three-phase set with abc sequence. This will be called phase

group number I. The phasor set Va2-?b2-?c2 is also a balanced three-phase

set, called phase group 2, and it is displaced 30 ° from phase group 1 as

shown in Figure 2-5.

For case I, these vol_ages must form a balanced S12¢ set and, therefore, all six voltage n_gnitudes are equal so that

% = Jvall--IV_l = I%1] = IVb_.l= IVoll= IVo21

When these volzages are transformed to sequence voltages according to SeeZion 2-7, they are as follows

vp= j_v,

and all other sequence voltages are zero. Therefore, the positive sequence

circuit alone is required, and iz is shown in Figure 2-6. The equations

listed for She positive sequence circuit in Table 2-2 give the total torque

and I2R losses. Also, the phase current in line al may be found as

1"a_= Tpl¢'8"-
-23-

and the set of phase currents will be balanced S12¢ with phasor relationships like those shown for voltages in Figure 2-5.

2-12 Case II. S12¢ Motor with Intergroup Unbalance

This case is similar to Case I, with each of the two three-phase

groups forming a balanced three-phase group. However, balance between the

groups is lacking. In Case If-a, volt.age m_4_nitudes are

v¢l : 19aii: l_bll: IVoll

v¢2 = IV_! = lVb21: I_c21

v¢1 _ v¢2

This is shown in Figure 2-7(a), and is called intergroup magnitude unbalance. Then, for Case II-b, the voltage magnitudes are all equal, but the

phase group 2 set of voltages is ¢ radians more than 7/6 radians (30 ° ) behind phase group 1. This is shown in the phasor diagram of Figure 2-7(b),

and is called intergroup angle unbalance.

The equations giving sequence voltages for Cases II-a and II-b are as

follows:

For Case II-a:

For Case II-b:

Vp = ¢r_Vav e, where Vav e = 1/2(V¢1 + V¢2)

V7 = ¢r_ AV, where AV = 1/2(V¢1 - V¢2)

All other sequence voltages are zero.

Vp = 4_ v¢/=i2

vv =/_ v,_/2 L_/2-_/2

All other sequence voltages are zero.

t
-24-

cl c2

Vcl - y

bl_

Figure 2-4. Stator Connections for a S12¢ Motor.

_cl _c2

Vbl

r s _L 1 Xf

Figure 2-5. Phasor Voltages

for 812_ Motor.

Figure 2-6. Positive Sequence

Circuit.

Vcl

c2

_ __/_ _ Val

Vb2 _/ _ Va2

(a) Vbl

Figure 2-7.

m

VcI_; 2

_Val

(bl b2 /Vb I _ Va2

S12_ Voltages with Intergroup Unbalance.

(a) Magnitude unbalance (b) Angle unbalance
-25-

The equations for Case ll-b involve approximations, but are quite accurate for E small (less than about i/i0 radian).

As indicated above, only the positive sequence and the 7th sequence

circuits of Table 2-2 are used for Case II. The equation listed there for

positive sequence torque is the total torque. The I2R losses consist of

the positive sequence losses as for Case I, plus the 7th sequence losses.

If, for Case If, the positive sequence vol_age is assumed to be the same as

in Case I, then the torque will remain the same. But the I2R losses will

increase by the amount of 7_h sequences losses. This addizional loss is

the resul_ of the unbalance, and causes additional heat to be generated in

the stator winding.

For Case If, the actual phase currents ,my be found from the sequence

currents according to:

Tal = i/¢6(Tp + ][7), and Tal, Tbl, Tel, form a balanced 3-phase set

withi_l--lqil

_a2= I/j_(Tp- rT) andr_ rb_ Tc_aiso_o_ a _l_nced3-phase

set with I¢2 = l_a21

2-13 Case Ill. S12¢ Motor wi_h One Line Open

For Case Ill, the source is a balanced, sinusoidal S12¢ voltage

source. Line 8/ is open, sinmlated by opening switch SW in Figure 9-8.

The equivalent circuit in Figure 2-9 is based on the symmetrical component

transformation. In this circuit, the vol_age V¢ is the line to neutral

voltage magnitude for the balanced S12¢ volr_4_e source. The circuit _y be
-26-

-cl rc2

=_n_e__ / _ \_

Figure 2-8. Stator Connections for S12¢ Motor: showing

switch SW for operation with phase al open,

and switch NSW for neutral connections. The

source provides V¢ volts per phase.

_L 1 r s

"TN

t

_N

R b

SW

rs _ rs

[ [

¥7 ,}Y5

mL 5

r s _L 1

- +

Xf

f

Figure 2-9. Equivalent Circuit for S12_ Motor

with One Line Open. Switches SW

and NSW correspond to those in

Figure 2-8.
-27-

solved for all the sequence currents, which are indicated on the circuit.

(Note that _9=jI3 ). The total torque is (Tf-Tb) where Tf and Tb are as

given in Table 2-2. The I2R losses are also as given in that table.

If SW is closed as for balanced operation, this circuit reduces to

that of Case I. But when the switch is open to simulate one open phase the

negative sequence circuit is introduced along with the o_her sequence

circuits in parallel with i_. The effect of these parallel circuits for

sequences 5, 7, (and also sequences 3 wi_h 9 if neutral switch NSW is

closed) is to shun_ current away from _he negative sequence circuit, thus

reducing IN and the negative sequence torque Tb. The overall effec_ is

that, for _his motor, torque reduction at a given slip can be small, even

though one phase is open. For machines having small values for the 5_h and

3rd sequence inductances (L5 and L3), this reduction in torque can be quite

small. Experimental resul_s are given in Chapter 6.

2-14 Case IV _orors with Periodic, Non-Sinusoidal Source

For Case IV, the voltage (or current) source is non-sinusoidal, typically an inverzer source. Bur, based on the assumption of constant speed,

the phasor volzages and currents and equivalent circuits based on slip and

impedance may be used separately for the fundamental component, and for

each harmonic of the Fourier series.

Each harmonic component, as well as the fundamental, requires one of

the sequence circuits to represent the motor. The slip and the reactances,

and the rotor frequency which affects rotor bar skin effect, must be based

o_] the rotor speed and on the frequency of The particular Fourier series
-28-

component. After each significant harmonic has been used, the principle of

superposition can be used to arrive at final results for currents, torque,

and losses for the non-sinusoidal case. Reference I0 gives dezails of this

method as applied to three-phase mozors.

Table 2-3 has been prepared for S12_ motors to summarize the details

of the appropriate equivalent circuit for each harmonic component, and

Table 2-4 is similar, but for 9-phase and S18# motors.

2-15 Discussion of Non Sinusoidal Performance

The equivalent circuits presented in Tables 2-3 and 2-4 make it possible to compute the individual harmonic current magnitudes if the harmonic

voltages are first found for any given voltage waveforrn produced, for example, by a voltage source inverter (VSI). If the source is a current

source inverter (CSI), then the harmonic currents are found directly from a

Fourier series analysis of the inverter current waveform. In either case,

the currents produce I2R losses as given in the two tables. The induetances in the equivalent circuits are the individual sequence inductances

defined and used for the symmetrical con_onenr method. The following are

important observations.

-Only the fundamenzal component of curren_ produces torque,

which is equal to

T= P

2 _

where p is zhe number of poles on the mozor.

¢
-29-

TABLE 2-3. HARMONIC CIRCUITS FOR S12¢ MOTOR

(j is any integer, and _ = 2_ )

Harmonic

Order h

I, or

fundamental

3,9,15,

21, ....

(6j + 3)

5,7,17,

19,...,

( 12j-6_ )[

II, 13,

23,...,

(12j±l)

Circuit Diagram

r s _L 1 Xf

r s h_L 3

Stator

12R

ll2rs

Rotor

I2R

SIl2Rf

Comment

s is the slip,

I 1 is the same

as the positive

sequence cur- 0

rent Ip.

lh=O for delta

phase groups, or

r s h_L 5

r s huL 1 h_L r

__I rr

2

I h r s

Ih2rs

lh2rs

0

0

lh2rr

for wye with

neutrals

separate.

Correct Lr and

r to account

r

for skin effect

using rotor

frequency

fr = 12Jfl
-30-

Harmonic

Order h

1

3,15,21,

33,. -• ,

(9j_+6)

5,13,23,

31, ••• ,

(18j-9i_4)

7,11,25,

29,. ••,

(18j-9+_.2)

9,_7,

45,. ••,

:(18j_+9)

17,19,

( 18j+i )

TABLE 2-4.

(j is any integer,

HARMONIC CIRCUITS FOR S18¢ MOTOR

and _ = 2_f I)

Circuit Diagram

r _I Xf

rs

r s h_L 5

rs

-rrs hmL 1 h_L r

_h_ _h rr

i ,

Stator

I2R

I12rs

2

I h r s

Ih2r s

Ih2rs

Ih2rs

Ih2rs

Rotor

I2R

Sll_ f

0

0

0

0

Ih2r r

Comuent

s = slip

lh=O for delta

phase groups,

or for wye

with neutrals

seDarate.

19=0 for delt_

or for wye

with neutrals

separate.

Current Lr and

r r to account

for skin

effect using

rotor frequency fr=18Jfl •
-31-

-There are harmonic currents in the rotor only for those harmonics

listed in the last line of each table. In general, these are

harmonic order numbers given by

h=2qj _+ I

where q is number of phase belts per pole, and j is any integer.

-Only the harmonics which produce rotor currents can produce the

pulsating components of torque. These pulsating torque components

are at a frequency

fp = 2qfj Hz

where f is fundamental frequency and j is any integer.

-The average torque for the harmonics which produce no rotor currents

is zero, and it is very nearly zero for those producing rotor

current.

-Certain harmonics are limited by stator impedance alone. The

inductance, which is part of that impedance, is influenced by sL_tor

coil pitch. Some of these inductances can have a very small value.

The result is tha_ some s_a_or harmonic currents can be quite large,

but an external fil_er can be added to greatly reduce these

currents. (For details see Chapter 4.)
CHAPTER 3. OTHER TRANSFOI_IATION METHODS

3-1 Introduction

Motors operating with rapidly varying speed, or with sudden changes in

input voltage are examples of transient operation. Cases such as these

lead to a set of differential equations with time-varying coefficients.

The most c_nmon method used to simplify these equations is to transform the

machine voltage and current variables _o the direct-qua/rature-zero (d-q-y)

reference frame. This subject has been fully developed and the literature

contains any good explanations of the method applied to three-phase motors

(Ref. 7 and ii). This general zheory can be extended to HPOmotors and

Chapter I0 in Reference 7 covers this. The theory therefore need not be

repeated here. The zransfor_ztion _trices involved are given in Appendix

A. A brief discussion of the methods begins in the next section, and

equivalent circuits are included.

The presen_atio_ of these general circuits is included mainly for

background purposes. The general transient solutions were not pursued in

this investigation. However, these general circuits will be reduced to

their sinusoidal steady state forms. They are useful as alternatives to

the symmetrical co,_onent circuits presented in Chapter 2, and are closely

related ro them as will be shown.

3-2 General a-b-y and d-q-y Circuits

Several versions of methods using the machine variables tranformed to

the d-q-y variables are found in literature. In particular, those dealing

-32-
-33-

with HPOmotors are found in references 2, 3, and 7. Use of a stationary

reference frame is most often the best choice for induction motors. In

that case, the d-q-y variables for the stator are identical to what are

generally called the two-phase, or a-B-y variables. White-Woodson (ref. 7)

uses this in the analysis of motors having any number of stator phases, and

calls them the "generalized two-phase" components. A discussion of these

for HPOmotors follows, with a simplification of the Whire-Woodson notation.

For a two-phase motor, the stator variables are is and i 8, and va and

For three-phase motors the zero order or y variable is also needed.

v B•

Then, for more than three phases, the a and 8 variables will occur as m1!-

tiple pairs. These, along with one or two of the y variables for some

cases, must give a total of n variables for an n-phase motor. The variables will be identified as aI and 81 forming pair I, a2 and 82 forming

pair 2, etc; plus YO and Yn/2 if required.

3-3 Equivalent Circuits for _-8TY Components

The general d-q-y theory leads to equivalent circuits as a pictorial

way of interpreting the method. An example of this is shown in Figure 3-1

for a S12_ motor. Circuits for a S18_ motor would be similar, but also require a pair of circuits for e7 and 87 and one y circuit consisting of rs

and L0 in series. The inductance values used in these circuits are the

same as those used in the symmetrical component circuits of Chapter 2. Only

the _I and 81 circuits include rotor currents idr and iqr. The

othe_ circuits have stator currents only. Therefore, only the el and 81

circuits can produce torque. Also, the _I and 81 circuits must include the
-34-

# lal

ibl_ vbl

Zc2

Vc2 (a) Stator

Connections

rs L1 Lr rr (p/2).,miBl_L_

_al IkLm _dr

(p/2 )_m(Lm+Lr)iqr

r s L 1 L r

-

iBl i

qr

¢-L m vB1

- i,

(p/2 )%Lmial (p/2 )_m(Lm+Lr )idr

(b) a-B

Circuits

rs L 3

va3

rs L 3

zB3

vB3

r s L 5

za 5

va5

r s L5

i

B5

VB5

s

Notes:

I.) _ is rotor

m

speed in

mechanical

radians per

sec.

2.) p is number

of poles

3.) If neutrals

are connected,

then ia3=iB3=0.

Figure 3-1. Circuits for S12¢ Motor with a-B Variables.
-35-

speed voltages shown as part of their rotor circuits. These result in

cross coupling between the two circuits in the general transient solution.

The close similarity between the al and 81 circuits on one hand, and

the positive and negative sequence circuits on the other is apparent.

Likewise, the similarity between other a-8 pairs and the symmetrical component circuits is noted.

3-4 Transient Solution Using cr-8-y Variables

The matrix [T] given as in reference 7, page 567 is used to transform

between the actual phase variables and tile general two-phase or a-8-y variables. This is expressed by the transformation equations

[VaSy] = [T]-l[vabc] (3-1)

and [iabc] = [T][ia8 _] (3-2)

For the S12_ motor, and using the machine model and variables shown in Figure 3-1(a), equation 3-1 becomes

b

"Val 1 Val

Va3 [ Va2

i

Va 5 ] = [Ts12_]-I Vb 1

I

v 85 ] Vb2

v 83 _ Vcl

v 81 j Vc2

(3-3)

where [Ts12¢] -I is as given in Appendix A.

The a--8 voltages found fr_a equation 3-3 are applied to the separate

circuits shown in Fig. 3-1(b). The currents in these circuits may then be
-36-

found by solving the set of simultaneous differential equations. In the

general transient solution where speed is a variable, a torque equation is

also required. After the currents are found, they may be transformed to

the actual phase currents by means of equation 3-2. For the S12¢ example,

the expanded form of that equation is

ial

Ia2

ibl

_b2

_cl

ic2

= [Ts12,] i_5 (3-4)

185

183

181

D .

3-5 E_uivalen_ Circuits Based on _l_iple d-q-y Components

In Appendix A, the matrix [D] is given. It is used to transform from

the phase variables of a machine having mul_iple three-phase windings to

mul_iple sets of d-q-y variables. While the principle involved applies to

motors wi_h 6, 9, 12, 15...3k phases, the equivalent circuit _urns out to

be simple only for the S12¢ case. Therefore, only the S12¢ case is presented here. The equivalent circuits shown in Figure 3-2 surlnarize the

method. The volzage variables may be transformed from the a-b-c reference

frame to the d-q-y reference frame. Then, with these voltages applied to

the circuits in Figure 3-2, the solution for currents is carried out. Note

that in _he general transient case, this involves _he numerical solution of

a se_ of non-linear differential equations, including also a torque equation not listed here. After solving for the currents in the d-q-y refer-
-37-

L

v

÷

Vdl

r s L 5

(LI-L 5) r r L r

r L- I --'-_•

p _

+ ids m dr (_)_m_ iqs

Xd2 =

Vd2

÷', (Lm r i

_m 2 qr

(a)

+

Vql

r s L5 /P

(LI-L5) rr L r _/ _m

ioi i--F-- -_ --_ ._

qs [ qr [Pi,,, Lm

+L )" (Lm r Xdr

2

(b)

r S L 3

Zyl

Vyl

r s L 3

i

Vy2 y2

J

(c)

Figure 3-2. Equivalent Circuits for S12¢ Motor Based

on Multiple d-q-y Circuits

(a) d-axis circuit

(b) q-axis circuit

(c) y-axis circuits
-38-

ence frame, they are transformed to the a-b-c currents to complete the solution. These transformations are sum_narized as"

[Vdqy] = [D]-l[vabc] and [iabc] = [D][idq Y]

or

F di] vai

Vql I Vbl

I

!V_il = [D]-i Vci

i

iVd21 Va2

! q21 vb2

I v_2I Vc2

ial

ibl

and icl

in2

Ib2

ic2

: [D]

l idl

lql

iyl

id2

h2

iy2

3-6 a-SEquivalent Circuits for Sine Wave Case

When the volzages and currents are sinusoidal and the steady state has

been reached with speed practically constant, phasor forms of the variables

may be used for the equivalent circuits. The equivalent circuit of Figure

3-1 retains the same form except that instantaneous voltages and currents

are replaced by phasors, and also the inductive reactance associated with

each inductance may be found. The circuits can then be solved as complex

number algebraic equations rather than differential equations.

For certain quasi-balanced voltage sources such as those specified in

Section 2-10, Case I, Case II, and Case IV, the phasor equivalent circuits

can be simplified and appear as shown in Figure 3-3. Note that for these

quasi-balanced cases, the speed voltages do not appear in the equivalent

circuits. Speed is implicitly included when the value of slip s is specifled, and values for Rf and Xf are computed according to Section 2-.4.
-39-

c 2

ci T

b 1

rs _L 1 Xf

- Rf Va 1

r s _LI

I81

81

Xf

Note: These circuits apply

to the quasi-balanced cases

I, II, and IV only.

Currents _a3 and I_3 are

zero if neutrals are not

connected.

r s _L 3

Ia3

Va3

r s mL5

r s _L 3

_83

r s _L 5

i85

V

85

Figure 3-3. a-8 Circuits for S12¢ Motor, Phasor Form
-40-

3-7 _-8 Circuits for Harmonics

Case IV presented in Section 2-10 is an important case to consider.

Ir applies to motors supplied by inverters. The harmonic currents resul_-

ing when a motor is supplied by a voltage source inverter are of concern

since they resulz in I2R loss, bur produce no useful torque.

When the non-sinusoidal source is balanced, the relationship between

currents in the a-circuits and the 8-circuits of Fig. 3-3 is as follows

T81 = -jTal

T B3 = -jTa3

T85 = -iT5

(3-5)

Therefore, all currents are known if the e-currents are found.

For Ehe S12¢ motor for which Fig. 3-3 is applicable, the fundamenEal

and harn_nics II, 13, 23, 25, ... produce curren_ in the el and 81 circuits, and _hese produce rotor currents and torque. The harmonics 3, 9,

15, 21, ... produce currents in the _3 and 83 circuits, and harmonics 5, 7,

17, 19, 29, 31, ... produce currents in the a5 and 85 circuits, and none of

these produce rotor currents or _orque. These observations are consistent

wi_h those which can be made from Table 2-3 where the symmetrical conponen_

circuit for each harmonic is shown.

3-8 Multiple d-q-y Circuits for Sine Wave Case

In a manner exactly the same as explained in Section 3-6, _he equivalent circuits shown in Fig. 3-2 may be modified to accommodate phasor

voltages and currenEs for the sinusoidal steady state cases. Again, for

special cases I, If, and IV given in Section 2-10, the circuits become
-41-

quite single and are shown in Fig. 3-4.

In order to use the principle of

represented by those shownin Fig. 3-5.

Superposition, Zhe circuits maybe

The voltages shownzhere are based

on voltages Vdl, Vd2, _ql, and Vq2 of Fig. 3-4, and are given by

Vd' = i/2(Vdl+ Vd2)

Vd" = i/2(?dl - Vd2)

Vq' = 1/2(Vql + Vq2 )

vq= i/2(Vql- v-q2)

(3-6)

Then, in Fig. 3-6, the separate circuits for voltage components V d' and Vd"

are shown. Iz is obvious from _hese circuits that Vd' produces rotor current and torque, while Vd" produces currenE circulating through zhe sta_or

ilmpedance elements only, with no rotor current or torque. When balanced

non-sinusoidal vol_ages are applied to a S12¢ motor, voltage componen_ Vd '

results from the fundamental and harmonics Ii, 13, 23, 25, ..., and ?d" is

zero for these. BuT harmonics 5, 7, 17, 19, ... result in voltage corcgx)-

nent Vd", and give a zero value for Vd'. This again illustrates the same

observations made in Section 3-7.

3-9 _y of Fquivalent Circuits

While the symmetrical component circuits presented in Chapter 2 are

the most versatile for the steady sta_e cases, o_her me_hods such as those

based on m-8-y variables or multiple d-q-y variables may also be used for

_he quasi-balanced cases, including the important non-sinusoidal case.

These latter me_hods are also useful for the general transient solutions.
-42-

r s _L 5

Idl

r _L 5

S

_(LI-L 5 )

2

-

_Lm/2

a

_L /2

r

¢_r 12

S

l

r s _L 5

_L /2

r

rr/2

S

r s _L 3 r s _L 3

Figure 3-4. Phasor Form of Multiple d-q-y Circuit

for S12_ Motor, Cases I, II, or IV.
-43-

÷

• r s _L 5

r s _L 5

Id2

_( LI-L 5 )

_L /2 2 r

ds dr

_ _Lm/2

1

I rr/2

S

(a)

r s wL 5

"_'_ co(Ll_L5 )

Iql 2 _Lrl2

r s wL 5 __--

I c_Lm/2

q2

(b)

r _L 3

S

, ,

rs _L 3

y2

(c)

r /2

r

s

Figure 3-5. Modified Voltages for Multiple d-q-y Circuit

of S12¢ Motor.

(a) d-axis

(b) q-axis

(c) y-axis
-44-

rs _L5

_(LI-L 5 )

idl 2 _Lr/2

(a)

r s _L 5

_ -_-

I'll

+_ r s _oL=

_(LI-L 5 )

2 _Lr/2

l_s=O

rr/2

s

(b)

Figure 3-6. Component Circuits for d-axis Based on

Superposition Method.

(a) _" ffio d

(b) V'_--0

U
CHAPTER 4. FILTERS DESIGNED FOR KPO MOTORS

4-1 Stator I2R Losses Due to Unbalance or Harmonics

In Section 2-2, the various stator sequence inductances are defin@d.

They are involved in the stator impedances in the sequence circuits as

shown in Tables 2-1 and 2-2. The positive and negative sequence circuits

have the positive sequence inductance L 1 for the stator. For all of the

other sequence circuits, _he stator inductance is one of the other sequence

inductances Li, where iel. Also, unlike the positive and negative sequence

circuits which have both stator and rotor impedances to limit the current,

these other sequence circuits have only the stator impedance to limit the

current. As a consequence, any kind of unbalance that produces sequence

volr_%ges of the non-positive and non-negative sequence kind can result in

relatively large sequence currents. The only effect these currents have on

motor performance is the detrimental effect of added I2R hearing losses in

the sEator.

A very similar situation exists when motors are supplied fr_n nonsinusoidal voltage sources, such as voltage source inverrers. Tables 2-3

and 2-4 show the equivalent circuits for S12_ and S18# motors, respectively, for the various time harmonics. These circuits are similar to the sequence circui£s of Table 2-2. The positive sequence inductance L1 is used

for the fundamental frequency circuit in Tables 2-3 or 2-4, and this is the

essential torque-producing circuit. Inductance L1 must also be used in the

circuit at _he bottom of those tables, and that circuit applies

-45-
-46-

to certain time harmonics as listed with it in each table.

produce stator and rotor 12R losses, but the torque they produce

ally very small.

The equivalent circuits shown in Tables 2-3 and 2-4 for all

harmonics involve only the s_ator impedance which consists of

resistance and a sequence inductance as shown in the tables.

rents for these harmonics are limited by stator impedancealone,

be large. Reference 5 explains how the coil pitch used for

These harmonics

is generthe other

the stator

Since curthey can

the stator

winding affects the values of sequence inductances. For certain pitch values, one or more of the sequence inductances can be very small so that the

impedance for certain harmonics can be small, and large harmonic currents

can result. This causes excessive heating of the stator winding by these

non torque-producing harmonics.

4-2 Fil_ers to Reduce Stator 12R Losses

From the description in Section 4-1 above, it is clear that, except

for positive sequence inductance LI, all sequence inductances should be

large so tha_ the non _orque-producing sequence or harmonic currents can be

kept small to reduce their I2R losses. In this way, less heating and higher motor efficiency can be achieved. With HPOmotors, it is easy to increase certain sequence inductances without increasing the positive sequence inductance. The devices used for this purpose will be called harmonic filters. They are described in de_ail in Appendix C where connection

diagrams and tables which list the connections for the various filter _ypes

are shown. Also included there are the inductance values for those filters.
-47-

4-3 A Filter Example for a S12¢ Motor

A S12¢ motor with a filter can be used as an example. If no neutral

connections are used, the currents for sequence numbers and time harmonics

which are multiples of three will be zero, and r_herefore need not be included. As Figure C-7 in the Appendix indicates, one version of a practical filter for a S12¢ motor has for its sequence inductances:

for positive and negative sequence, LIF = 0

for sequence 3 and sequence 9 , L3F = 3N2p (4-i)

for sequence 5 and sequence 7, LSF = I_N2P

where N is the number of turns on a filter coil and P is the permeance of

its magnetic core.

When the filter is included with

quence inductances shown in Table 2-2

ponding sequence inductance for the

(L3+L3F) , L5 is replaced by (L5+L5F),

inductance applicable to harmonics 5,

the motor, each of the stator sc_

or 2-3 is increased by the corresfilter so that ['3 is replaced by

etc. Thus, for this $12¢ motor, the

7, 17, 19,...in Table 2-3 will be

L5+LsF, and this can be made as large as desired by choosing values of N

and P for the filter. It follows that currents for these harmonics can be

made as .small as desired. But the positive sequence inductance is not affected by the filter and therefore, torque will not be affected. Also, the

currents for harmonics ll, 13, 23, 25,...wili not be affected.

The overall effect of the filter can be a substantial reduction in

stazor I2R losses. For a $18¢ motor-operating from a non-sinusoidal

source, all harmonics below the 17th can be greatly reduced when a filter
-48-

is used. In contrast to this, a filter for a three-phase motor could not

reduce any of the non-triplen harmonics without reducing torque-producing

current as well.

4-4 O_her Versions of Filters for 812_ Motors

Messrs. Tad W. Macie and Darius Irani designed a filter for a 812_ motor in abou_ 1964. It used exactly the same principle as the one used in

the exan_le of the previous section. Their design with three transformers

can produce the same filtering effect as the six transformer version used

in the previous section. It is described in Figure C-8 in Appendix C. For

the same to_al per phase inductance value, it requires a value of the quantity N2P which is twice as large as the six transformer version. Its advantage is that fewer transformers and interconnections are required, although each transformer is somewhat larger.

There is another possible arrangement of small transformers which can

be used as a fil_er for the S12_ motor. It uses six transformers, each

with %w_o coils. The six primary coils are in series with the six stator

leads, one coil for each lead. The secondaries are interconnected with

three in delta and three in wye. The connection diagram and equations are

given in Appendix C. This type of filter has relatively small inductances

for a given value of N2P and is therefore inferior compared _o the threecoil versions described.

4-5 Example of Filter to Reduce Harmonics in S12# Motor

When inverter-supplied motors are to be operated aE variable speed by

varying frequency, it is necessary to maintain a constant ratio between
-49-

fundamental voltage and frequency to produce constant air gap flux. The

pulse width modulation (PWM) schemehas been widely used with three-phase

motors ro achieve this constant volts per hertz relationship. Various

strategies have been used to achieve low harmonic currents and small torque

pulsations with three-phase motors. These strategies usually require high

switching rares and the associated switching losses in the inverter.

It would be advantageous to use a PWMwavefonn which requires a low

switching rate. The lowest possible switching rate results if the P_d wave

has only one pulse per half cycle, such as that shown in figure 4-1(a).

The amplitudes for the fundarnental, and for a few of the harmonics of this

wavefor_n are plotted in figure 4-1(b) as a function of the pulse width 8.

The fundamental voltage is controlled by varying 6 so r_at constant vol_s

per hertz can be maintained over a wide frequency range.

If this simple waveform is used with a three-phase motor, the relatively large 5th and 7th harmonic con_x)nents of voltage produce 5th and 7_h

hanmonic currents. These currents produce a sometimes troublesome 6th harmonic torque pulsation, and also produce excessively large I2R losses in

the stator and rotor.

HPOmorors inherently suppress some of the torque pulsations which are

usual for three-phase inverter supplied motors, including some of the

largest of these. For example, a $12_ motor has no 6th harmonic torque

pulsation. The 12th harmonic torque pulsation which is due to the llth and

13rh harmonic remains, but is often small enm_h in amplitude and high

enough in frequency so that it is not troublesome.
-50-

0

I 3_r/2 2_r

! : ]

['i Trl 2 7r a_t

(a)

20

6 in degrees

Fundamental

Harmonics

(b)

Figure 4-1. Single Pulse PWM Waveform and its Harmonics

(a) Line to line voltage waveform

(b) Harmonic amplitudes
-51-

Bu_ the 5th and 7_h harmonic stator currents to a S12_ motor would be

extremely large for this single pulse I_%! waveform since volL_ge amplitudes

are large for most values of 6, and also stator impedance limiting these

currents can be quite low as explained in section 4-1. However, these harmonic currents can be greatly reduced by using a filter as described in

section 4-2, and very good performance can be achieved with this simple PWM

waveform. To show the very low I2R losses which can be achieved using the

single pulse PW%i waveform for a HPO motor with filter, the following example is presented.

A typical 20 HP, 60 Hz, squirrel cage induction motor is chosen to illustrate the performance attainable when the 812_ arrangement is used. The

voltage source inverter is assumed to consist of two separate three-phase

inver_ers wi_h their outputs 30 degrees out of-phase as required for the

S125 source. Each inverter is assumed to produce a single pulse P_,! waveform whose line-to-line voltage is as shown in Figure 4-1(a). For this

waveform, the fundamental voltage amplitude and some of the harmonic voltage amplitudes are as shown in figure 4-1(b), showing amplitudes expressed

as a fraction of the 60 Hz fundamental voltage. The 12R losses were computed for this motor for frequencies from about 5 to 60 Hz. It was assumed

_hat 6 '_as adjusted to maintain a constant volts per hertz value _hroughout

the range. A filter of the type described for $12# mor_ors in Appendix C

was included for the computations. The fifth

inductance of this filter was taken to be ten times the s_ator

ductance of _he motor; that is, 15F = 10L 1 •

sequence

leakage in-
-52-

Resulzs of this analysis are plotted in Fig. 4-2. Since the heating

and efficiency of three-phase motors operating from a 6-step variable voltage inverter source have been widely publicized and are well-known, curves

for them have been included in Figure 4-2 for comparison.

4-6 Rotor 12R Losses for the Above Example

As shown in Figure 4-2(a), _he rotor loss for the S12¢ motor is very

small, remaining less than 10% of the fundamental rotor loss at full load

for any frequency above 15 Hz. This frequency corresponds to 1/4 of base

speed. For low speed, _his harmonic loss increases and it reaches 75% of

full load rotor loss at 5 Hz. This rotor loss characteristic is typical of

HPOmotors, and is no_ affected by the filter.

For c_nparison, the dashed curve shows the rotor I2R loss for harmonics when the a 3-phase motor is used with a 6-s_ep waveform. The "S12¢ motor on single pulse P_,{ has less rotor loss than the 3-phase motor on its

6-step waveform, down to about 13 Hz.

4-7 S_ator 12R Loss

Figure 4-2(b) shows the stator harmonic loss for the S12¢ motor _ising

zhe filter, and using the single pulse waveform. This loss remains less

than 10% of the fundamental stator 12R loss at full load for frequency

above 12 Hz, and is 50% of full load loss at 5 Hz. Again, the dashed curve

shows the 3-phase motor loss on a 6-s_ep waveform for comparison.

4-8 To_al Harmonic Losses

When the stator and rotor I2R losses are combined as in Fi_Ire 4-2(c),

it is evident that harmonic losses are very low for speeds above 1/4 of
-53-

0

o9

o9

0

Z

Z

0

cO

0

Z

0

0

0

I.-I

80

6O

40

20

0

0

4O

2O

0

0

60

40"

(a) HAP_ONIC LOSS IN ROTOR

S12_ motor, PWM

,

i0 20 30 40 50 60

Frequency, Hz

_2_ (b) HARMONIC LOSS IN STATOR

motor, PWM

ase motor, 6-step wave

| | | | | | L | i | I |

I0 20 30 40 50 60

20-

0

0

(c) TOTAL HARMONIC LOSS

S12qb motor, PWM

i I ! J I

I0 20 30 40 50 60

Figure 4-2. Harmonic I2R Losses with Single Pulse PWM
-54-

base speed if the 812# motor is used on the single pulse P_4 inverter

source. This is with a filter having the sequence inductance 15F ten times

the stator leakage inductance. A lower filter inductance requiring a filter of reduced size could be used with only a modest increase in harmonic

losses. According to the analysis above, the moor operates quite well on

a simple waveform which allows adjustment of voltage amplitude as frequency

is varied. The inverter would have switching losses comparable to those in

a 6-step inverter. Above about 1/4 base speed, motor efficiency and heating can be practically the same as when the motor operates fr_n a sine wave

source.

By using Do to four pulses per half cycle for the I_M waveform when

speed is low, these excellent characteristics could be extended down _o

I/i0 base speed or below, and switching losses would remain low.

To summarize, HPOmotors with filter are superior to three-phase motors using PWM waveforms, or using 6-s_ep waveforms. They can easily be

made to operate practically as well as on a sinusoidal source. Experimental results for a S18_ motor with a filter are presented in Chapter 8.
CHAPTER 5. EXPERIMENTAL EQUIPMENT

5-1 The }_otor Used

The experimental results were obtained using a squirrel cage induction

motor built by G.E. for use in educational laboratories. Its rating &s a

4-pole motor on 60 Hz. is 2 H.P. All of its 36 stator coils have their

leads brought out to a 72-terminal panel. By externally interconnecting

these coils in different ways, it was possible to get the following combination of phases and poles:

3-phase, 4 or 6 poles

Semi 12-phase, 4 or 6 poles

Semi 18-phase, 4 poles

The current rating was assumed to be 12 a_ps for the three-phase connections with parallel wye, and 6 amps for the S12¢ and S18¢ connections.

Voltage ratings depend on connections, and are listed in the following

table.

The motor was coupled to a dc machine used as a cradled dyn_ter.

This made it possible to measure the torque produced by the motor. Some of

the impedances of the motor are included in Table 5-2 on page 61.

-55-
-56-

TABLE 5-1 Motor Voltage and Current Ratings

Connection

3¢, 4P, parallel Y

3¢, 4P, series Y

S12¢, 4P, Y-Y

S18¢, 4P, Y-Y-Y

3¢, 6P, parallel Y

3¢, 6P, series Y

$12¢, 6P, Y-Y

Razed

Current

Amps

12

6

6

6

12

6

6

Voltage Bating, Volfs

Line to line for

Line to neutral

63.5

126

63.5

43.3

63.5

126

66

each 3-phase group

I' "

ii0

9.2O

ii0

75

Ii0

220

114

5-2 60 Hz Power Source

Many of the tests were n_de using a 60 Hz sinusoidal source. A special set of power transformers were used to supply either S12¢ or 9¢ ($18¢)

voltages from the 3-phase 60 Hz lines in the laboratory. There were three

of these transformers, each razed 3 KVA. The $12¢ motor was supplied by

connecting one of its three-phase winding sets directly to the 3-phase

source, with its other three-phase winding set supplied by the transforr._rs

connected delta Zo wye to give the necessary 30 degree phase displacement.

For $18¢ operation, the three transformers were connected as shown in

Figure 5-1. They were designed with the proper taps on their windings to

give the nine equally spaced points on the circular locus as shown in that

figure.
-57-

5-3 h_asurement of Current and Power

The line currents in the six or nine leads to the S12¢ or S18¢ motor

were measured using conventional iron-vane type anm_ters. They were accurate to about ± I/i0 ampere. This accuracy was considered adequate to demonstrate the effects of loading, and particularly the effect of operation

with an unbalanced source or with one line open. Comparative readings were

taken with and without the filters, and results are given in _napter 6.

The power input to the motor was determined by measuring the power

taken from the three-phase source using two wattmeters connected between

the source and the phase conversion transformer primary. The power as measured was therefore the power input to the motor plus transformer losses.

Transformer core loss was determined by measuring transformer power input

with the motor disconnected. The transformer I2R losses were measured by a

short circuit test. Then, for each motor test, the motor power input was

found by subtracting these known transformer losses from the indicated

watt_neter power taken from _he three-phase source.

5-4 Motor Temperature Measurements

The temperature of the stator winding was monitored by a thermocouple

fixed to the end-_urn on one stator coil. Temperature was read using a

"DigitenTp" digital me_er. For all tests involving full load power input,

slip, or torque; the temperature was held nearly constant as follows:

For 6-pole load tests, temperature was 60 ° C ± 3°

For 4-pole load tests, _emperature was 58 ° C ± 3°

For locked rotor _es_s, temperature was 30 ° C ± 4 °
-58-

5-5 Motor Slip Measurements

The motor slip speed was measured during load tests by visually observing the apparent shaft speed when it was illuminated by a stroboscopic

light source. This strobe light was synchronized with the same 60 Hz

source which supplied the motor, given 30 flashes per second for 4-pole motors and 20 flashes per second for 6-pole motors. It was possible to actually count Ehe apparent number of shaft revolutions (slip revolutions) in

an interval accurately timed wi_h a digital stop watch.

5-6 Motor Tests on Non-Sine Wave Source

A rotary switch mechanism was constructed using an

driven at variable speed by a d-c motor.

ing eight adjacen_ bars for each segment.

source through slip rings. Then, nine equally spaced brushes in contact

18-bar commutator

Two segments are formed by joinThese were supplied from adc

with the conm_ta_or provided a 6-step waveform for the S18¢ motor. The

waveform of motor voltage and current were recorded by means of a dual beam

oscilloscope with a camera attached.

5-7 Experimental Filters

Chapter 4 describes vari_is types of filters which can limit some of

the unwanted sequence or harmonic currents. Two of the filters described

there were built and tested with the experimental motor. Details of these

are presented here.

The filter requires a number of magnetic cores, each having two or

three coils. Each laminated core has dimensions shown in Figure 5-2. The

permeance (P) of each core is approximately 17 microhenries a_ low flux
-59-

3-phase

Source

c2

!

/

I

I

cl

\

Figure 5-I. Power Transformer Connection for S18¢ Source

3 i/8"

2_5'_ 5"__ l_" "_-_ -- -_

J

1

i

Figure 5-2. Steel Core Used for Harmonic Filter Elements
-60-

density, and this value decreases as core flux increases because of magnetic saturation.

For the S18¢ filter, nine of these cores are required. Each has three

coils as shownin Figure C-I in Appendix C. The m_nberof turns was chosen

to be N = 8 for each auxiliary coil. Then, for the filter based on the

phase opposite schemeand described in Figure 0-5, the value of k had to be

1.879. Based on this, the main coil required %Nor 1.879 X 8 = 15.032

turns. Therefore, a coil having 15 turns was used. The nine transformers

were connected according to the Connection Table in Figure 0-5 to form _he

S18¢ filter.

For the S12_ filter, the wye-delta type of filter shown in Figure C-9

was built. It used six of the magnetic cores already described. The primary on each filter transformer had N = 16 turns. The value chosen for the

turns ratio was a = 11/16. Three of the transformer secondaries therefore

had Ii turns and these were w-ye-connected as shown in Figure C-9. The

other three transformers required iI/-3 = 19.05 turns, and 19 turns were

used for the delta-connected secondaries.

5-8 Impedances of Experimental Filters and Motor

The sequence impedances of the motor with its vari_Is pole and phase

combinations, and also the sequence impedances of the two filters were

found. To measure these impedances, the filter was connected between the

motor and the phase-transforming power source described in Section 5-2. By

interchanging the appropriate leads at The S12¢ or S18¢ source, the voltages applied to the filter-motor combination were fifth sequence for the

S12_ motor, and fifth and then seventh sequence for the S18¢ motor. It was
-61-

found necessary to keep currents below about 2.5 amperes to avoid saturation in the filter cores. These low currents were obtained by using a low

source voltage. Measurements were made for each phase current, the volLage

across the filter in series with each motor lead and the voltage from each

motor phase to neutral. Then voltage divided by current gives the impedance in ohms, from which reactance can be calculated. Results are tabulated in Table 5-2.

TABLE 5-2 Sequence Reactances for Filters and Motors

Motor

Phases/Poles

S12¢/4P

S12¢/6P

S18¢/4P

Sequence

Number

5 and 7

5 and 7

5 and 13

7 and ii

Reactance at 60 Hz

ohms per phase

Motor Filter

1.69

1.56

1.96

3.53

Apparent Values for

Fil_er Co_re, _H

N_p P

17.5

16.2

16.4

12.6 +

i.79

0.45

1.51

_nall

4500*

4100.

1050.*

804**

+saturation may be cause of lower value

*N = 16 turns

**N = 8 _urns

The motor reactances recorded in Table 5-2 vary for the different sequences. One reason for this variation is the influence of coil pitch as

explained in reference 5. Attention is called to the low val_e of reactance for sequences 5 and 7 in the

sequence 7 and ii in the Sl8¢motor.

ductance is very small.

6 pole S12¢ motor, and also for

In each of _hese cases, sequence in-
CHAPTER 6. MOTOR OPERATION WITH ONE LINE OPEN

6-1

ate,

might be due to a blown fuse or a faul_y inverter source.

Introduction

It is useful to consider motors which are sometimes required to operat least temporarily, with one phase unable to carry current. This

It is well known

that three-phase motors can continue to

will generally overheat in a short time.

one phase open.

run with one phase open, but they

They have no starting torque with

HPOmotors can be stated and can be run wi_h one phase open, and their

running characteristics can be nearly as good as with all phases carrying

current. The equivalen_ circuit applicable to this situation is given for

the S12# motor in Figure 2-9, and the analytical solution for currents,

torque, losses, and efficiency can be based on that symmetrical component

circuiz. The 2HP experimental motor was tested with one line open and results are reported in the following sections, and also in Reference 6.

6-2 Locked Rotor Tests

The motor was tested as a S12_ type with its rotor locked. Torque

measurements were made with one-half rated voltage applied. The sequence

filter described in Chapter 4 was used in some tests to determine its effect on _orque. Results of these tests are shown in the bar graphs of

Figure 6-1(a) and (b) where locked rotor torque is expressed as a percentage of that produced when all lines carry current and no filter is

used. The locked rotor torque for the motor connected for _hree-phase hal-

-62-
0

4m

0

cq

o

0

o"

0

0

o _

o

o

o

o

i00- 3-phase

8O

60

4o

2@

0

-63-

balanced S12_

_/J

_JJ

_/J

_JJ

_fJ

_JJ

w_J

•- |

(a)

_ without filter

with filter

FsSI2_ with one line open

eparate-Neutrals-joined _

i. '11

. i 11

. = |J •

I00_

80_

I

6o7

I

4o7

20"

0

I

"/2

rj. ,

"/i

r.S" _

"/,4,

(b)

m

/F

f/

fJ

//

F/

#-/

//

f/

/J

Z_

"//,

riJ

. m ,

vii

3-phase

I00. .__

80-

60 _

40,-

20-

O_

balanced S18_

-'_ ,_-_ , ¢_

i

:7"Z

1

(c)

_sS185 with one line open-- epar at e-Neut r a_- j o ined

/J

Figure 6-1. Locked Rotor Torque with One Line Open

(a) 4-pole, S12_ motor

(b) 6-pole, S12# motor

(c) 4-pole, S18_ motor
-64-

anced operation is also shown in the figures for comparison. Figure 6-I(c)

has a similar set of bar graphs for the S18_ connection of the motor.

These results show that, for balanced S12¢ or S18_ operation, locked

rotor torque is slightly greater than for the three-phase connection.

Also, torque with a balanced source is very slightly reduced when the filter is included. This reduction would be due to the small resistance and

leakage reactance of the non-ideal filter transformers. I_ is obvious from

these tests that the filter has almost no effect on torque when the source

is balanced, as predicted by the theory.

The bar graphs in Figure 6-1 which pertain to the motor with one line

open also confomn to what the theory predicts. In Chapter 2, section 2-13,

it was pointed out that, for HPO motors, the reduction in torque when one

line was open would be small, and that the amount of reduction would depend

on the value of sequence inductance l_ and L3 when neutrals are joined, or

L5 only if neutrals are separate. This reduction will be smallest when

these inductances are small. The bar graphs verify this. The 6-pole S12¢

motor has a value of 16 much smaller than that of the 4-pole S12_ motor.

With one line open and without the filter, the 6-pole motor with its very

small L_ has torque reduced from 100% to 76.6% with neutrals separate. In

contrast to this, the 4-pole motor with its larger value of i6 gives a corresponding reduction of 100% to 55.5%. These reductions are only slightly

less when the neutrals are joined.

When the filter is added to these motors, the values of L5 and L3 increase considerably. The effect is to cause a greater reduction in

locked rotor torque when one line is open; from 100% to 6L_ for the 6-pole
-65-

motor, and from 100%to 46%for the 4-pole motor. The filter is therefore

detrimental to torque production whenone line is open, as would be expected from the theory.

Similar results are shown in Figure 6-I(c) for the $18_ motor _hich

has a very small value for LT.

6-3 Full Load Tests with One Line Open

The motor was tested from a sinusoidal, SO Hz source at full load for

Ehe several phase and pole number combinations, wizh and without filter, on

a balanced source, then with one line open. For the 4-pole motors, torque

was maintained constant at 8.33 N-n for all tests. For _he S-pole motors,

the torque value was a conszant 12 N-m. The bar graphs in Figures 6-2 and

6-3 show results for efficiency and slip, respectively. As with locked rotor tests in Ehe previous section, the performance is compared with rhe 3-

phase motor version.

6-4 Efficiency Measurements

According to these experimental

ficiency as a HPOmotor on balanced

resul_s shown in Figure 6-2, _he efsource is slightly lower than the 3-

phase version. This unexpected result may be due _o the 12R losses in the

additional wiring required, or to unaccounted for losses in the power

transformers needed for the HPOmotors, or to the fact that the HPO sources

were not perfectly balanced. It is known that any volLage unbalance results in additional I2R losses due to the sequence currents.

When one line is open, the bar graphs in Figure 6-2 indicate thaz

efficiency drops. But this drop in efficiency is very small for the 6 pole
-66-

80 _3-phase

b.

70-

60-

_i i

balanced S12_

./

./

fJ_

without filter

_with filter

(a)

--S12_ with one line open--

separate-Neutrals-joined

Z

0

Z

>.

Z

8°I

70]"

I

1-

60 -

!

"ill I

 ,2t

I

!

r/j

r/j

//" I

_'/j

(b)

/21

f_

fA

f_

/A

_7_

r/2

rf,

fj.,

f_-j

fJ-A

i j,

fJJ

80 .

3-p.h_se

70 '

p

60 _

L

balanced S18_

ffJ

r/j

f/J

--S18¢ with one line openseparate-Neutrals-joined

(c)

!

!

I

I

I

J J1

/Jl

fA

/W1

JJl

//I

i

Figure 6-2. Full Load Efficiency with One Line Open

(a) 4-pole, S12_

(b) 6-pole, S125

(c) 4-pole, S18_
-67-

version (Figure b) and less than that for the 4-pole version (Figure a).

This is again due to the shunting effect of the sequence impedance branches

as seen in the applicable equivalent circuit in Figure 2-9. The current

shunted away from the negative sequence circuit flows through the sequence

paths. The lower the sequence inductance values, the more current is diverted to the sequence branches. The negative sequence I2R loss becomes

much less while I2R loss due to this diverted current is only slightly

more. The net effect is a reduction in loss.

When a filter is added, sequence inductances other than the positive

and negative su_uence inductances are increased, less current is diverted

through the sequence branches so that negative sequence current and iCs associated I2R losses are greater. The efficiency with one line open therefore is less when the filter is included. The bar graphs of Figure 6-2

verify this.

6-5 Full Load Slip Measurements

Full load tests are described in Section 6-3. The slip was measured

and Figure 6-3 shows the slip in percent for each case. Except for the

S18¢ motor [Figure 6-3(c)], slip on a balanced source without a filter is

slightly less for the HPOmotors tested as compared to the 3-phase case.

Then, the inclusion of the filter causes a slight increase in slip because

the filter is not quite ideal.

Without the filter, opening one line causes a significant increase in

slip for the 4-pole motor which has a rather large value for L5 , but a much

smaller increase in slip for the 6-pole motor which has a very small value

of 15. Again, this is explained by the diversion of current fr_n the nega-
I0-

8

6

4

2

0

3-phase

-68-

balanced S12¢

rI

I

i

¢/•

fi,

f/,

r_gj

(a)

ffl

without filter

with filter

S12¢ with one line open--

eparate-Neutrals-joined

' _ --

ff_

//,

z

o

z

p-.

O3

I0

8

6

4

2

0

77;

f/•

//2

f_'2

,-//

/ /

. =

v/j

(b)

.f

r/

./

z/

./

//,4

cFA

/Jl

m

v/A

//2

r/A

f/_

i/1

I0'

8

6

4

2

0

3-phase balanced S18¢

ff.

f/_

r/j

Figure 6-3.

(c)

_sSI8_ with one line open- 1

eparat__/_e-Neutrals-j--oined I|

/F

J /,

, lJ

Full Load Slip with One Line Open

(a) 4-pole, S12¢

(b) 6-pole, S12¢

(c) 4-pole, $18¢
-69-

rive sequence branch, thus reducing negazive sequence torque and its effect

on slip. When the filter is added, L5 is increased and results show the

predictable increase in full load slip with one line open.

6-6 Summary of Full Load Tests

The imporCant observations which _heory predicts and experimental

_ests verify are: I) The filter has almos_ no effect on operation of the

motor when the source is balanced, 2) a very low sequence inductance value

can resul_ in a very small deterioration in full load performance (slip and

efficiency) when one line is open, and 3) _he filter has a detrimental effect on full load performance when one line is open.

Locked rotor tests show s_nilar results, wi_h the least reduction in

torque when a line is opened if sequence inductance is small.

HPO motors can be designed to take advantage of these excellent characteristics with one line open. The effects noted in the experimental motor can be even more dramatic if the motor is larger and therefore has its

sequence impedances less influenced by the resistive component. Then sequence impedance can be made very low by the proper choice of coil pitch,

and this will enhance the performance with one line open.
CHAPTER7. CURRENTSIN EXPERIMENTALMOTORWITHSOURCEUNBALANCE

7-1 Introduction

It is common to have at least a small amount of unbalance among the

phase volzages of a polyphase system. Induction motors can be sensitive to

such unbalance which produce sometimes large current unbalance and additional 12R losses. HPOmotors can be especially sensitive to unbalance in

the volEage source when one or more of their sequence inductances are

small. A sequence/harmonic filter can be used to reduce the amount of current unbalance, zhereby reducing the sta_or I2R loss somewhat. However,

the filter might not give much improvement in overall efficiency, as experimenzal rests will show.

7-2 Effect of Intergroup VolzageUnbalanc 9

Section 2-12 in Chapter 2 defines intergroup voltage unbalance and the

quantities AV and ¢ associated with it. The experimenr_l motor was subjected to voltages having different amounts of voltage magnitude unbalance

AV, and also different amounts of angle unbalance ¢. Results are summarized in Table 7-1 for the S12¢, 6-pole motor. The 4-pole motor was also

tested, bur because of its higher value for sequence inductance 15, current

unbalance _as less than for the 6-pole version with its small value of L5.

The test results reported in Table 7-1 are for the motor at full load.

In the table voltages V¢I and V¢2 are the magnitudes of the line to neutral

source vol_ages for motor phase group 1 and 2, respectivelE. The angle

-70-
-7-1-

TABLE 7-1 Resullis of Voltage Unbalance on S12¢, 6P Mol;or at Full Load

Case Fi Iter

No. Used

1 No

2 Yes

3 No

4 No

5 Yes

Volts tO Neutral

V¢l V¢2

63.2 62.9

63.4 62.9

63.3 63.8

61.5 63.2

61.0 62.8

Vol_

.15

.25

-.25

-.85

-.90

%* de_. deE. I¢I

.23 29.2 -.8 6.53

.40 29.2 -.8 6.10

-,40 30.6 .6 5.63

-1.40 30.7 .7 4.85

-1.45 30.7 .7 5.75

*Expressed as percent of Vav e, where Vav e = i/2(V¢i + V¢2 )

**Expressed as percent of lay e, where Iave = 1/2(1¢1 + I¢2)

Phase Current,

I¢2

5.30

5.86

6.17

7.13

6.36

AI

%**

i0.0

2.0

-5.0

-19.0

-5.0

is equal zo (30 ° + ¢). Wizhin phase group I, volZages are substantially

balanced three-phase, and the same is true for phase group 2. As the

theory in Section 2-12 shows, the kinds of unbalances used here result in

7th sequence currents in addition to positive sequence.

For case 1 and 2 in Table 7-1, the source is simply the delta-wye

power transformers used to give 30 degrees displacement between phmse

groups 1 and 2, wish one phase group supplied directly from the three-phase

line. The small magnitude unbalance in voltages, and the phase shift of

29.2 instead of 30 degrees are the. result of impedance drops in the nonideal power transformers. The considerable unbalance in currents seen for

case I, where I¢i and I¢2 are 10% above and below the average current, respectively, is a consequence of the unbalance in voltages and the small sequence inductance for 7th sequence (L5). For case 2, the filter

described in Figure C-9 in Appendix C, and having reactance values tabular-
-72-

ed for the Sl2_/6Pmotor in Table 5-2, was added. As Table 7-1 shows, the

filter reduced the current unbalance from 10% to only 2%. However, the

power input actually increased by about 5 watts (out of about 1800 watts

total) when the filter was added. Reducing the 7th sequence current with

the filter must have decreased motor power input slightly. Apparently, the

I2R losss in the added filter was greater than the reduction in motor I2R

loss.

For case 3 in Table 7-1, the power transformers were modified by the

addition of a one-turn coil on each, and this coil connected in series with

the appropriate transformer secondary gave a phase shift _ = 30.6 ° instead

of 29.2 °. Source voltage V_2 was also slightly increased. The combination

of these &_all voltage changes gives a current unbalance with I_l less than

the average by" 5%. Finally, with the one-turn coil reconnected, the source

voltage V#I was reduced for cases 4 and 5. With this larger value of AV,

and with _ = 30.7 °, current unbalance was -19% without the filter, and reduced to -5% with the filter. Again, the power input was practically the

same whether the filter was used or not.

7-3 Conclusions Re_ardin _ Curren_ Unbalance

For the intergroup unbalance studied here for the experimental motor,

the following observations are made:

i. For motors having small values for sequence inductances, large

current unbalance can occur for voltage magnitude unbalance over

about 1%, or for angle unbalance more than about 1 degree.

2. Filters are effective in reducing the current unbalance.
-73-

3. The current unbalance caused by 7Zh sequence curren_ in the S12¢

motor does not affect torque and causes only a sligh_ increase in

stator I2R loss.

4. Including a filter can actually increase rather than decrease the

total I2R loss.

In su_nary, HPO n_tors can be operated successfully on somewhat unbalanced sources, even though the current unbalance might be rather large. It

is possible ro design

ficienrly large value

ance.

the motor wi_h coil pitch chosen to give a suffor sequence inductances _o control curren_ unbal-
CHAPTER 8. D(PERIMENT.%L RESULTS USING FILTER WITH 6-STEP

WAVEFORM OF VOLTAGE

8-1 Test Setup

The 2HP motor connected as a S18¢, 4-pole motor was supplied from the

rotary switch described in Section 5-6. This provided a balanced 6-step

voltage ro the motor with a fundamental frequency of 17 Hz. A camera

mounted on a dual-beam oscilloscope was used to record the waveform of

for line-to-neutral vol_age, and also the line current.

A filter of the phase-opposite type described in Figure C-5 of Appendix C was included between the source and the motor for one of the tests.

Based on the reactances of this filter recorded in Table 5-2, its fifth sequence inductance is 5.2 mH, and this same inductance value applies to the

fifth harmonic. Its seventh sequence inductance is 9.4 mH, and this value

applies ro the seventh and eleventh harmonics. Based on this information,

the filter reactances for the various harmonics were as given in Table 8-I.

The motor reactances are also included.

TABLE 8-1. Reactances for 818¢ Filter

Harmonic

1

5

7

ii

Frequency

Hz

17

85

119

187

Reactance, Ohms

2.2

Small

Small

_or Filter

0

2.8

7.0

U.O

-74-
-75-

8-2 Ex_Experimental Results

The Sl8_mo_or was run at no load using the 17 Hz 6-step voltage waveform. The oscillogr_ns of line to neutral vol_age and line current are

shown in Figure 8-1. Figure 8-1(a) shows these without the filter, and the

large harmonic content of the current is evident. Then Figure 8-1(b) shows

resulzs when the filter is included between the same 6-step voltage source

and the motor. The voltage waveform in Figure 8-1(b) is taken at the motor

terminals, and is more nearly sinusoidal than the 6-step source voltage because of the filter. The current waveform with the filter clearly sh_vs

that the harmonics have been s1_pressed.

The current waveforms for Figure 8-1 were analyzed to determine the

amplitude of each frequency component, and results are given in Table 8-2.

TABLE 8-2. Harmonic Currents

Harmonic

5

7

ii

Current; A_s _4S

Without Filter With Filter

1.73

0.68

1.20

0.70

1.76

0.40

0.12

0.08

As Table 8-2 indicates, the fundamental current is not significantly affected by the filter. The fifth harmonic is reduced somewhat by the filter. The seventh and eleventh harmonic currents are rather large without

the filter because motor reactance for these harmonics is small. But when

the filter is added, these currents are reduced by a factor of

I
-76-

6-step source voltage,

and also motor voltage

line to neutral

Line current to

motor, no load

(a)

Motor voltage,

line to neutral

Line current to

motor, no load

(b)

Figure 8-1. Oscillograms of Voltage and Current

for S18_ Motor with 6-Step Source.

Frequency is 17 Hz.

(a) Without filter

(b) With harmonic filter of Figure C-5
-77-

about ten due to the substantial increase in reactances produced by the

filter.

The results here clearly demonstrate that by using a filter with an

HPOmotor, harmonic currents can be made small.
CHAPTER 9. CONCLUSIONS

9-1 Results of the Investigation

It has been demonstrated that inducZion motors having more than three

phases are entirely practical. They meet the need for electric vehicles

where the high current per phase for three-phase motors has been a problem.

They have several other advantages which can be important for electric vehicles and for many other applications as well.

They can be more reliable than three-phase motors because of their

ability to start and run with one phase open (or shorted). It has been

shown that by choosing coil pitch so that one or more of the sequence inductances is small, the sEarting and running characteristics of these HPO

motors wi_h one phase open can be almost as good as for balanced operation.

However, when these inductances are small, large current unbalance resul_s

if source voltage is unbalanced.

An important application of H_Omotors is to systems where the source

is an electronic inverter, either the current source or the voltage source

type. When supplied by a current source inverter and in comparison to

three-phase motors, HPOmotors will have certain of the lower frequency

torque pulsations eliminated, and will also have lower I2R loss produced by

harmonic currents in the rotor because currents for certain harmonics cannot flow in the rotor.

When supplied by a voltage source inverter, these same characteristics

of small pulsating torque and small rotor I2R losses apply. However, those

-78-
-79-

harmonics eliminated from the rotor will produce larger currents in the

stator, causing an increase in stator 12R loss. This can be easily overcome by using one of the harmonic filters developed as part of this investigation. The result can then be a motor which, when supplied by a nonsinusoidal voltage source, can be practically as efficient and smooth running as a motor supplied by a sinusoidal source. This use of harmonic filters is considered to be an especially important result of the project.

While motors having any large number of phases would perform quite

well, practical considerations limit the number to six or nine. Choosing

multiples of three is important because the well-known advantages of

three-phase _e or delta connections can be retained. With more than about

nine phases, the number of stator slots and the number of stator leads

becomes excessive, filters if used require too _ny components, and the

improvement over six or nine phase types is slight.

9-2 Suggestions for Further Investigation

A complete evaluation of a system consisting of a voltage source inverter, a harmonic filter, and a motor having six or nine phases would be

the next logical step. This should include the design of the components of

the system, and the building and testing of prototypes. One area of concern is the size, the weight, and the cost of a filter to be included.

Since the filter reduces the harmonic losses in the motor, the motor itself

could be somewhat smaller than without a filter. Therefore, the overall

weight of the system need not be increased much by the inclusion of a

filter. Also, the use of a filter could reduce the complexity as well as
-80-

the losses in the inverter so that the overall cost of the system including

a filter could be comparable to that of a conventional system.
RNt_RENCNS

1. McLean, G.W., Nix, G.F., and Alwash, S.R., Performance and Design of

Induction Motors with Square-Wave Excitation, Proceedings of IEEE, Vol.

116, No. 8, August, 1969, pp. 1405-1411.

2. Nelson, R.H. and Krause, P.C., Induction Machine Analysis for Arbitrary

Displacement Between Multiple Winding Sets, IEEE Transactions on Power

Apparatus and Systems, Vol. PAS-93, May/June, 1974, pp. 841-848.

3. Jahns, T.M., Improved Reliability in Solid-State AC Drives by Means of

Multiple Independent Phas____eDrive Units, IEEE Transactions on Industry

Applications, Vol. IA-16, _y/June, 1980, pp. 321-331.

4. Andresen, E. and Bieniek, K., 6-Phase Induction Motors for

Current-Source Inverter Drives, Conference Record from Industry

Applications Society Meeting in Philadelphia, October, 1981, pp.

607-618.

5. Klingshirn, E.A., High Phase Order Induction Motors, Part I -

Description and Theoretical Considerations, IEEE Transactions on Power

Apparatus and Systems, Vol. PAS-I02, January, 1983, pp. 47-53.

6. Kllngshirn, E.A., _ _Phase Order Induction Motors, Part II -

Experimental Results, IEEE Transactions on Power Apparatus and Systems,

Vol. PAS-I02, January, 1983, pp. 54-59.

7. White, D.C. and Woodson, H.H., Electromechanical Energy conversion,

John Wiley & Sons, New York, 1959.

8. Fitzgerald, A.E., Kingsley, Charles Jr., and Kusko, Alexander, Electric

Machinery, Third Edition, McGraw-Hill, New York, 1971.

-81-
-82-

9. Alger, P.L., Induc¢ion Machines, 2nd edition, Gordon and Breach, New

York, 1970.

I0. Klingshirn, E.A. and Jordan, H.E., Polyphase Induction Motor

Performance and losses on Non-Sinusoidal Voltage Sources, IEEE

Transactions on Power Apparatus and Systems, Vol. PAS-87, March, 1968,

pp. 624-631.

Ii. Krause, P.C. and Thomas, C.H., Simulation of 8_trical Induction

Machine, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-84,

November, 1965, pp. 1038-1053.
APPENDIXA

TRANSFOP_AT ION MATRICES

A-I .S_trical Component Transformation _trix for General n-phase.Case

The symmetrical component transformation matrix for n phases is:

a

0

1

b 1

c 1

n 1

2 n-I

1 1 . . 1

a-I a-2 a-( n-I )

a-2 a-4 • • a-2 (n-1)

¢

a-(n-l) a-2(n-1) . . a -1

where a = e j2n/n

The inverse of maZrix [A] is equal to its conjugate transpose giving

a b c n

0

2

n-1

[A]-I = i

Cn

1

m

1

a2

1 a (n-l) a 2(n-l) •

1

a(n-l)

a2(n-1)

• I:I

-83-
-84-

NoZe that sequence 1 is the positive (P) sequence,and sequence (n-i) is the

negative (N) sequence.

A-2 S_netriqal .Component Transformation, 12-Phase

To illustrate the symmetrical component transformation, the matrix [A]

is given for 12-phases, with the phases labeled as four sets of three phases

each.

[A12 ]= i

0 1 2 3 4 5 6 7 8 9 i0 ii

al 1 1 1 1 1 1 1 1 1 1 1 1

a2 I a-i a-2 a-3 a-4 :-5 -i a5 a4 'a3 a 2 a

a3 1 a-2 a-4 -I a4 a2 1 a-2' a-4 -1 a4 a2

• i

a4 1 a-3 -i a3 1 a-3 -i a3 1 a-3 -i a3

bl 1 a"4 a4 1 a"-4 a4 1 a"4 a_ 1 a-4 a 4

b2 1 a-5 a2 a-3 a4 a-I -i a a''4 a3 a-2 a5

b3 1 -i 1 -i I -1 1 -i 1 -1 1 -1

b4 I a5 a-2 a3 a"4 a -I a-I a4 a-3 a2 a-5

c 1 1 a4 a -4 1 a 4 a _4 1 a 4 a -4 1 a 4 a -4

L ,I

c2 1 a3 -1 a-3 1 a 3 -1 a -3 1 a 3 -1 a -3

c3 1 a2 a4 -1 a "4 a -2 1 a 2 a4 -1 a "4 a -2

c4 1 a a2 a 3 a4 a 5 -i a -5 a -4 a -3 a -2 a -I

where a = eJ"/6 = 1 / 30 °
-85-

A-3 The Symmetrical Component Transformation, S12#

The S12¢ stator is derived from the 12-phase machine by deleting phase

sets 3 and 4 from the 12-phase version. The corresponding rows of its symmetrical component matrix are therefore deleted. Also, all even numbered

columns must be deleted from the 12-phase matrix. Finally, the resulzing

matrix must be multiplied by /2 to preserve power invariance in the _ransformation. The resulting sym_netrtcal component matrix for the S12¢ machine

is,

[As12¢] =_..!.__ I

¢6

1 3 5

al 1 1 1

a2 a-I a-3 a-5

bl a-4 1 a4

b2 a-5 a-3 a-I

c i 4 I a-4

c2 a3 _-3 a3

where a = e"j_/6 = 1/30 °

7 9

1 1

a5 a3

a-4 _-4

a-3 a3

ii

1

(x

_4

a 5

_-4

a-3

The inverse of this matrix is its conjugate transpose given by
-86-

[As12¢] -I = i

7dI

3

5

7

9

Ii

al a2 bl b2 cl c2

1 a a4 a5 a-4 a-3

1 a3 1 a_ I a3

i _5 -4 _ _4 _-3

1 a-5 a4 a-I a-4 a 3

1 a-3 1 a-3 1 a-3

1 a-I a-4 a-5 a4 a3

A-4 The Two-Phase (aB%) Transformation _trix

The general matrix for transforming from phase variables to general

two-phase or aBy variables can be found in reference 7 on pages 565-568.

When applied to the S12# motor, the n_trix is

[Ts12¢ ] =..._..i

?3

al

a2

bl

b2

Cl

c2

a I a3 a5 B5 B3 81

i i I 0 0 0

/_/2 0 -/_/2 1/2 i 1/2

-i/2 I -1/2 -_/2 0 _/2

-_/2 0 _/2 1/2 i i/2

-i/2 I -i/2 /_/2 0 -_/2

0 0 0 -I 1 -I

Its inverse is equal to its transpose.
-87-

[Ts12¢] -1 =_!_1

¢3

a 1 a2 b 1 b2 c 1 c2

a 1 1

a 3 1

a5 1

B5 0

B3 0

81 0

_/2 -1/2

1/2

-1/2 0

0 1 0 i 0

-/312 -1/2 /312. -112 0

1/2 -/3/2 1/2 ¢"_/2 -i

I 0 i 0 1

/3/2 1/2 -,,"3/2 -i

A-5 The dq Transformation for _chines with _Itiple Three-Phase Winding

Groups

Reference 2 presents a transformation to represent the multiple sets of

three-phase windings by multiple sets of direct-quadrature (d-q-y) windings.

This is an extension of the d-q-y method commonly used for three phase machines.

The S12¢ machine will be used as an example.

represented as two three-phase sets in Figure A-2.

Its stator currents are

Set number i (al-bl-C I)

is resolved into components dl-ql-Yl, where dI and ql are shown along the d

and q-axls directions in Figure A-2. Likewise, set number 2 (a2-b2-c 2) is

resolved into d2-q2-72. The y-components are defined in exactly the same

way as in the standard dq7 representation of 3-phase machines.

This graphical construction leads to the transformation matrix
-88-

c2

cl c3

b3 -- al

bl

a4

Figure A-I. Phasor Arrangement for 12-Phase System

d-axis

q-axis

cl

-112_ ,,

P

I /3/2

bl

al

c2

b2

ql-axis q2-axis

d2-axis

Figure A-2. Geometric Arrangement for Multiple d-q-y Sets
-89-

[D] :

al

b 1

Cl

a2

b2

c2

dl ql Y1 d2 q2 Y2

0 i//_

1//'2

0 0 0

-1/2 /_/2 o o o

-1/2 -_s/2 1//_ o o o

0 0 0 /S/2 1/2 i//_

0 0 0 -/S/2 1/2 1/¢°2

0 0 0 0 -1 1//"2"

The _ factor makes this a power invariant transformation.

[D] is its transpose.

[D]-1 : ¢_

dl

ql

YI

d2

q2

Y2

al

i

0

1//_

0

0

0

b I Cl a2 b2

-1/2 -1/2 0 0

The inverse of

c2

0

0 _s/2 -/_/2 o o

I/_ i/_ o o o

o o /_/2 -/_/2

0 0

1/2

i/,/_

1/2

i//'_

0

-1

1/,/_
APPENDIX B

SY_,_TRICAL (D_fi°ONENT RESISTANCES AND INDUCTANCES

8-i 8tator Windings and Parameters

The basic n_chine considered here has a uniform air gap

salient pole type) and the stator has a balanced n-phase winding.

(is nonFigure

B-I shows the schematic dikgram for the spatial arrangement of windings.

The stator phases are identified as a, b, c, ..., n. These are spaced with

an angle _ = 2_/n.

q

c

Figure B-I. Diagram of an n-phase stator.

Each phase has a resistance rs ohms.

tance matrix, which is a diagonal matrix:

This leads to the stator resis-

[R_] =

a

°

a r s

b

c

n

b c n

r s

r s

r s

-90-
-91-

Also, each phase has a self inductance; these are all equal for a balanced n-phase winding such _hat

Laa = Lbb = Lcc - .... = Lnn

There is a mutual inductance between each pair of phases, such as Lab, Lac,

Lbc, etc. Because of syms_try, all pairs of phases separated by the same

angle have the same mutual inductance. Thus Lab --Lbc, Lac = Lbd , etc.

All of the stator inductances r_y be displayed in matrix form as

a

b

C

(n-l)

n

a b c (n-l) n

Laa Lab La c - • • . Lac Lab

,,,, , i , , ,,

Lab Laa Lab La d Lac

Lac Lab Laa Lad

58. C

Lab

@

l_.d Laa Lab

, i ,,,,

Lad .... Lab usa

Note that there are several characteristics

trix:

of this highly symmetric ma-

-It is symmetric about both diagonals

-Each successive row is the same as the row above, but shifted one

column to the right• This is called "circulant symmetry".

-Any one row includes all of the inductance values involved. Thus,

the entire matix is specified if any one row is given.
-92-

B-2 Transformation to S_metrical Component Inductances

The equations for transforming the stator inductance values to the

symmetrical component sequence inductances are given in some detail in reference 7, page 582. The equations take advantage of the symmetry observed

for the stator inductance matrix. They lead to values for the sequence inductances LO0, LII, _2, -''', L(n-l)(n-l)" It will be convenient to simplify the notation by using a single subscript instead of the double.

Also, the equations given there for sequence inductances n_y be reduced to

either of the following:

The value for the inductance of the ith sequence is:

If n_nber of phases n is an odd number,

L i = Laa + 2{Lab cos i_ + Lac cos 2i6 + .... + La(n/2+l/2)

cos(n/2-1/2)_} (B-I)

and if n is even,

L i = Laa + 2{Lab cos i_ + Lac cos 2i_ + .... + L(a)(n/2)cos(n/2-1/2)_ }

+ La(n/2+l ) (B--2)

In these equations, _ = 2w/n, and i = O, i, 2, ..., (n-l).

There must be n sequence inductances for an n-phase stator. However,

equation B-I or B-2 will verify that L(n_i) = L i for any of the n posslble

values for i and therefore, there can be only n/2 different numerical values for these inductances if n is even, and (n+l)/2 different values if n

is odd. The rea_ining values are repeated such that L(n_l) = L I, L(n_2) =

L2, etc.

The sequence inductance n_trix displays these inductance values in a

diagonal matrix which is, for the general n-phase case.
-93-

[Ls]=

0

1

2

3

I

n-3

n-2

n-1

0 1 2 3 " "

L1

h2

53

e

L3

n-2) n-1 )

L 1

B-3 Magnetizing and Leakage Inductances

Each of the self and mutual inductances may be divided into two components; that due to the air gap flux, and that due to leakage flux. This

makes it possible to write the stator inductance nmtrix as

[L¢] = [U¢'] + [L¢"] (B-3)

where [L_'] is the air gap inductance matrix consisting of the inductances

due to air gap flux only, and [L¢"] is the leakage inductance matrix due to

leakage fluxes only.

For the symmetric, uniform air gap machine, and assl_ing a sinusoidal

distribution of air gap flux, the first row of the air gap inductance matrix is

a

[L'¢(a)] = [ _a

b c (n_l) n

...... =¢Laa cos
-94-

or slightly simplified

!

[L_#(a) ] = Laa

a b c

l 1 Icos 6 I cos 26l

When equation B-I or B-2 is applied

the sequence components, the

(n-l) n

..... Icos 26

to these air gap inductances to find

result gives for the sequence inchactance matrix, air gap component;

0

1

[L,s] = 2

3

(n-l)

0

0

1 2 3 (n-l)

n/2 _a

0

0

imm

n/2 L_a

(B-4)

=In/2 L'

Or simply, L'i L0

a_ for i = 1 and (n-l)(

[

for i _ 1 or (n-1)J

!

The inductance L' I = n/2 Laa is called the magnetizing in_actance,

L, 1 = n/2 L_a' (B-5)

When equation B-I or B-2 is applied to the leakage inductance matrix

[L"¢], the resulzing sequence leakage inductance matrix is
-95-

It [L s] =

0

1

2

8

(n-2_

(n-l'.

Lo,,

LI"

n t!

B

L _r

L 1"

(B.-6)

The sequence inducZance matrix is then found by adding the air gap and

leakage components from equations B-4 and B-6, or

[Ls] = [L's] + [Ls" ]

0

1

2

3

[Ls]--.

0

(n-2

(n-i

1 2 3 • • • n-2) (n-l)

Lm+L 1"

L2"

J

I_+L I"

(B--7)
-96-

The primes and double primes may now be deleted without ambiguity giving

the stator sequence inductance .mtrix as

0

1

2

3

[Ls] = •

(n-3_

(n-2:

(n-1

0

ro

1 2 3 • ° • (n-3) (n-2) (n-l)

im+L 1

L2

L3

#

L 3

L2

Lm+L 1

where [0 is the zero sequence inductance

Lm is the magnetizing inductance

L 1 is the positive and negative sequence leakage inductance

L 2 is the sequence 2 and sequence (n-2) leakage inductance

L i is the sequence i and sequence (n-i) leakage inductance.
APPENDIX C

FILTER DESCRIPTION AND ANALYSIS

C-i Filter Description

The filters for HPO motors consist of small transformer type devices

connected in series with the stator leads. The general version which can

be applied to any motor having more than four phases uses n magnetic cores

for an n-phase motor. Each core is wound with three coils. These coils

are interconnected so that each of the stator leads has three of these

coils from three separate cores in series with it. The selection of which

specific line currents pass through the three coils on each core and the

turns ratios for these coils are based on the requirement that the fil_er

should not increase the positive sequence inductance. The analysis of

these filters is given in the next section.

C-2 Analysis of the General Filter _pe

Figure C-I shows one of the n identical transformersused for the general type of filter. As the figure shows, each core has two identical

coils with N turns on each. Let these be called the auxiliary coils. Each

core also has one coil with k times N, or kN turns, and this will be called

the main coil. The core has a permeance value P which is dependent on core

dimensions and permeability of the core steel. This is expressed as

p = (c-i)

£

where u is permeability

-97-
-98-

A is cross sectional area of the core

£ is length of the core

Based on magnetic circuit theory, the self and mltual inductances for

the various coils are dependent on the numberof turns on the coils and the

core permeance. The inductances of the transformer in Figure C-I are given

with the figure.

A filter for an n-phase motor is made up by combining n of these

transformers. Since the objective is to first of all have a zero value for

the positive sequence inductance, three of the phase currents and a particular value of the turns ratio k must be selected so that the r_gnetomotive

force (M_) will be zero for each core when only positive sequence currents

are used. Further explanation is based on the following five phase example.

C-3 Five-Phase Filter Example

The phasor diagram for the positive sequence currents of a five-phase

system is shown in Figure C-2. If the current of phase a passes through

the main coil on the transformer of Figure C-l, its MMF can be. cancelled by

having either currents Ic and Id or Ib and Ie through the two auxiliary

coils. The former case will be called the phase-opposite scheme because

currents Ic and Id are nearly opposite Ia in the phasor diagram of Figure

C-2. The latter case will be called the phase-adjacent scheme since Ib and

Ie are adjacent to Ia.

Figure C-3 shows the connection diagr,am for the entire filter for a

five-phase application using the phase-opposite scheme. As this diagram
-99-

X

y _

kN

N

Z

i Sz

Coil x-x has kN turns and is

the main coil.

Coils y-y and z-z have N turns

and are the Auxiliary coils.

The winding inductances are:

Self inductances 2 2

M-ain c--_? _ '=k N P

xx

Aux. coils,Lyy=Lzz=N2p

Mutual inductances

Main to Aux., L =L =kN2p

xy _z

Aux. to Aux., L =N P

yz

where P=permeance of the core.

Figure C-I. Diagram and Inductances of Filter Transformers

e

d

i

b

_-a

Figure C-2. Phasor Diagram for Positive Sequence

Currents oi a Five-Phase Motor
-I00-

I

a

I

C

N

T1 Ib

Ib

2

Arrows show

direction of

currents from

source to motor.

e

T

T

3

I

e

CONNECTION TABLE

For 5-Phase Filter

(Phase opposite type)

Transf.

T1

T2

T3

T4

T5

I Phase Current

Main

EN N N

a

b

C

d

e

Auxiliary

C

d

e

a

b

d

e

a

b

C

I

a

I d

Row "a" of normalized

phase inductance matrix is

[L#F(a)] a b c d e

N2p =[k2+2 1 1 i 2k 2k 1 I

The normalized sequence

inductance matrix with

k=1.618 is

0 1 2 3 4

Figure C-3. Five-Phase Filter, Phase Opposite Type
-I01-

shows, for transformer T I , the total _JF around the core is

F 1 = kNI a + NI c + N_ d (C-2)

This MMF must be zero if the positive sequence inductance for the filter is

to be zero. For this five-phase case, it is zero if F 1 = 0, and this is

satisfied if k = 1.618.

Figure C-4 shows the five-phase filter connected to use the phaseadjacent scheme. For this case, _ is

F 1 -- kN_ a - NI b - NI c (C-3)

Setting this equal to zero requires that k = .618.

In either of the two schemes, the connections for the other four

transformers are similar to transformer TI, as Figures C-3 and C-4 show.

All of the information shown in these connection diagrams can be displayed

in a Connection Table such as those given with Figures C-3 and C-4. For

the filter of Figure C-3, the Connection Table indicates by the entries in

the first line that transformer T 1 has Ia through its main (kN-_urn) coil,

and Ic and Id through the two auxiliary (N-turn) coils, all with additive

polarity. The second line gives similar information for T2 , etc.

With these connections, the matrix of inductances for the complete

filter can be written using the various self and mutual inductance values

given with Figure C-I. The result is the normalized filter inductance matrix given by
-102-

a

b

C

d

e

a b c d

k2+2 1 2k 2k

1 k2+2 L 2k

2k 1 k2+2 1

2k 2k 1 k2+2

1 2k 2k 1

e

1

2k

2k

1

k2+2

(C-4)

Since this matrix has circulant symmetry,

information. The matrix may therefore be simplified by

row (row a) only as

row a alone contains all of the

writing its first

a b c d e

To find the sequence inductance values for the filter, equation B-I

from Appendix B may now be used. It is convenient to use it as an alternate

to the method given in Section C-3 for finding k. The method is based on

the fact that the positive sequence inductance LIF, found by using i = 1 in

equation B-l, must be zero. For the five-phase case, equation B-I uses 6 =

360/5 = 72 °, and values for Laa, Lab , and Lac come from equation C-4.

Thus, Llf = N2p_+2) + 2(cos 72 ° + 2_cos 144°)} = O, and this gives k =

1.618 as expected. Then, the use of equation B-I with i = 0 and i = 2

gives the renmining sequence inductances iOF and L2F. (Recall that L4F =

LIF and L3F = L2F since, in general, L(n_l) F = LiF). The resulting

normalized sequence inductance matrix for the five-phase fil_er using the

phase-opposite scheme is
-103-

 [ujF] = 2

4

0

13.1

1 2 3

0

This diagonal matrix may be written more conpactly as

0 1

1 [LSF] = diag[ 13"I I 0

2 3 4

(c-5)

To sun_mmize, for the five-phase example, the

posite scheme gives sequence in_ictance values:

L0 = 13.1 N2p H.

L1--L4=O

L2 -- L.3-- 4_12P H.

filter using the phase opC-4 Phase.Adjacent Scheme

Figure C-4 shows connections for the five-phase filter example using

the phase adjacent scheme described in Section C-4. The Connection Table

is given with Figure C-4. The negative signs in the last two colmmns of

I

that table indicate that, if current through the main coil is into the dot-
-104-

e

I

a

kN

Ib

T2

Ib

Arrows show

directions of

currents from

source toward

motor.

I d I C

I

C

I d

CONNECTION TABLE

for 5-Phase Filter

(Phase. adjacent type)

Transl.

T 1

T 2

T 3

T 4

T 5

ain

Phase Current

kN

a -b

b -c

c -d

d -3

e -a

[ Auxiliary

N N

-e

-a

-b

-C

-d

I,

Row "a" of the normalized

phase inductance matrix is

[L_F(a)] a b c d. e

N2p = I k2+2 !-2kl 1 I 1 I-2k'[

The normalized sequence inductance

matrix with k=0.618 is

0 1 2 3 4

N2p

Figure C-4. Five-Phase Filter, Phase Adjacent Type
-105-

ted terminal, then the currents through both auxiliary coils must be out of

the dotted terminals. An equivalent statement is that the mu_ml inductance between the main coil and either auxiliary coil is negative, and this

explains the negative terms in the inductance matrix [LiF(a )] given with

Figure C-4.

The sequence inductance matrix [LsF] is determined using [LiF(a)] and

equation B-I from Appendix B in the same way as explained for the example

in Section C-3. The result is the sequence inductance matrix given with

Figure C-4. In this case, with k = 0.618, sequence inductances are:

L0F = 1.91 N2p

LIF = L4F = 0

-- = 5 N2p

This phase-adjacent scheme gives a somewhat larger value of

sequences 2 and 3 than does

Section C-3.

inductance for

the filter using the phase-opposite scheme of

C-5 Filters for S12# and S18# _otors

The extension of the filter design method to more phases is obvi_is.

Resul_s are m_rized for the S18_ case in Figures C-5 and C-6.

For S12_ motors, a variation of the method is used. First, the filter

is set up for a 12-phase case where 12 transformers would be used. Its sequence inductances can be found in the usual manner. But then, Just as the

S12_ motor resul_s When certain phases of the 124 design are deleted,

the six transformers associated with the deleted phases are removed from

the filter. The previously determined sequence inductances for the odd-
-106-

c2

cl

c3

b3

al

b2

a2

a3

Schematic Diagram of

a 9-Phase or S18_ Stator

CONNECTION TABLE

For 9# or S18_ Filter

L(Phase opposite type)

Phase Current

Transf. Main

kN

T1 al

T2 a2

T3 a3

T4 bl

T5 b2

T6 b3

T7 cl

T8 c2

T9 c3

Auxiliary

N N

b2 b3

b3 cl

cl c2

c2 c3

c3 al

al a2

a2 a3

a3 bl

bl b2

Row "al" of the normalized phase inductance matrix is

[LcF( al )]

N2p

al a2 a3 bl b2 b3 cl c2 c3

--12+k2111 °i°l2k12klol°lIB

The normalized sequence inductance matrix for k = 1.879 is

1 3 5 7 9 II 13 15

L_s_J- _L 0I0,_1,,0_1_-0_f_-0_J_-0_l,,_ lo,_l N2p

17

0

Figure C-5. Nine-Phase or 818¢ Filter, Phase Opposite Type
-107-

6DNNECTIONTABLE

For 9¢ or S18¢ Filter

(Phase Adjacent Type)

Phase Current

Transl.

T1

T2

T3

T4

T5

T6

T7

T8

T9

Ma'i'n

kN

al

a2

a3

bl

b2

b3

cl

c2

c3

N N

-a2 _3

-a3 _I

42 _3

43 41

_I 42

_2 43

_3 _I

-al _2

Row "al" of the normalized phase inductance matrix is

al a2 a3 bl b2 b3 cl c2 c3

N2P

The normalized sequence inductance matrix with k=1.532 is

1 3 5 7

N2p

9 ii 13 15 17

Figure C-6. Nine-Phase or $18¢ Filter, Phase Adjacent Type
-108-

numbered sequences are unchanged in this process and become the

the 812¢ motor. The results are summarized in Figure C-7.

In about 1965, Messrs. Tad Macie and Darius Irani developed

for a S12_ motor which is based on the same principle as presented

appendix. Their version requires only three transformers.

diagram and inductances are shown in Figure C-8.

values for

a filter

in this

The connection

C-6 The Wye-Delta Filter for S12# Motors

Another type of filter which can be used for S12_ motors only was investigated. It is based on the phase shift inherent in a wye-delta transformer connection. It uses six transformers, each having t',o windings.

The connection diagram is shown in Figure 0-9. The primary winding of each

transformer has N turns, and one of these windings is in series with each

stator lead. The secondaries of three of the transformers each have aN

turns, where a is any convenient number. These three secondaries are connected in _-]e. Then, the secondaries on the other three transformers each

have ¢r_ aN turns, and they are connected in delta. Finally, the wye and

delta groups are connected together to complete the connection. With connections as shown in Figure C-9, the positive sequence inductance is practically zero, and the inductance for sequence five is i_F=N2p.

For the same total amount of copper for windings and steel for the

core, L5F for this filter is only about 1/3 that of the filter using three

coils per transformer as described in Figure C-7. The three-coil version

is obviously the better one to use.

I
-109-

T 4

al

al

T 2

a2

cl

Arrows show

direction of

currents from

source to motor

c2

bl

CONNECTION TABLE

For S125 Filter

(6-Transformer Type)

Phase Current

Transl.

Main

kN

T1 al

T2 a2

T3 bl

T4 b2

T5 cl

T6 c2

Auxiliary

N N

b2 -a2

cl -al

c2 -b2

al -bl

a2 -c2

bl -cl

Row "al" of the normalized

phase inductance matrix is

al a2 bl b2 cl c2

[L@F(al)] =Ik2+2 I-2k I-I I 2k I -ii 0 ]

N2p

The normalized sequence inductance

matrix with k-1.732 is

1 3 5 7 9 11

N2p

Figure C-7. S12_ Filter, 6-Transformer Type
-II0-

b2 lal

' This version of a O

T _ filter was devised

1 _ al by Tad W. Macie

and Darius Irani.

.li e

cl

• Q

T 2

cl

c2

a2 bl "bl

CONNECTION TABLE

For S12@ Filter

(3-Transformer Type)

Phase Current

Transf

TI

T2

T3

Main

kN

al

bl

cl

Auxiliary

N N

b2 -a2

c2 -b2

a2 -c2

Row "al" of the normalized

phase inductance matrix is

[L_F(al)]

N2p

al a2 bl b2 cl c2

The normalized sequence inductance

matrix for k=1.732 is

1 3 5 7 9 II

[LsF]

N2p

Figure C-8. S125 Filter, 3-Transformer Type
-IIIal

bl

T1

a2

b2

c2

vr3a_

T5

T6

Figure C-9. Wye-Delta Filter for S12_ Motors