Copyright 2007 by Ronald De Four All rights reserved. No part of this publication may be reproduced in any material form without the written permission of the Copyright Owner, except in accordance with the provisions of the Copyright Act 1997 (Act No. 8 of 1997) or under the terms of a License duly authorized and issued by the Copyright Owner.

VECTOR ANALYSIS OF A THREE-PHASE STATOR

Ronald De Four Department of Electrical & Computer Engineering, The University of the West Indies

St. Augustine, Trinidad rdefour@eng.uwi.tt

Emily Ramoutar University of Trinidad & Tobago, Point Lisas Campus, Trinidad

eramoutar@tstt.net.tt

Juliet Romeo Department of Electrical & Computer Engineering, The University of the West Indies

St. Augustine, Trinidad jnromeo@hotmail.com

Brian Copeland Department of Electrical & Computer Engineering, The University of the West Indies

St. Augustine, Trinidad bcopeland@eng.uwi.tt

Abstract

Vector analysis is widely used for the analysis, modeling and control of electrical machines

excited with sinusoidal supply voltages. However, since the presentation of the theory to

field, it is not evident that any attempt had been made to justify the existence and location of

vector currents and voltages and the equality of scalar and vector current magnitudes. This

paper presents the development of an equivalent electric and magnetic circuit to show the

production of a magnetic vector current from a scalar electric current in a winding, provide

justification for the equality of scalar and vector current magnitudes and support the

existence and location of voltage vectors. In addition, the development of the equivalent

electric and magnetic circuits of a three-phase stator would provide a platform for vector

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analysis of electrical machines excited with non-sinusoidal and dc supplies as is the case with

brushless dc machines.

Keywords: Vector Analysis; three-phase; stator.

1. Introduction

Vector analysis of three-phase electrical machines was developed by Kovacs and Racsz in [1]

and is widely used in the modeling, transient analysis and control of these machines. The

technique developed in [1] has been documented in many books of reputed authors. [2-5], and

has many advantages over other methods, some of which are: a reduction in system equations,

easier to control the machine, a clear conceptualization of machine dynamics and easier

analytical solution of dynamic transients of machine variables [6] However, the detailed material

presented by Kovacs in [7] for the development of the vector analysis theory of a three-phase

stator raises some issues of great concern.

This paper is intended to present the issues of concern in the vector analysis theory presented in

[7] and provide a basis for the existence of current and voltage vectors associated with the three-

phase stator of an electrical machine, through the development of an equivalent electric and

magnetic circuit.

2. Space Vector Theory Issues

The material presented by Kovacs in [7], for the development of the vector analysis theory of a

three-phase stator, raises the following issues of concern:

(1) It was reported by Kovacs in [7] that a current flowing through a stator phase winding

produces a related current vector along the windings magnetic axis. In addition, if the

instantaneous value of the winding current is given by i, the magnetomotive force (mmf)

produced was given by

NiF = (1) where, N represented the number of turns in the winding. The mmf was reported to be a vector

quantity, lying on the magnetic axis of the winding and given by

iNFrr = (2)

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which presents the existence of a current vector ir

which is collinear with the mmf Fr

. However,

the current flowing through the winding was also regarded as a vector. Hence, the mechanism

and process by which a scalar current flowing through the winding produces a vector current

along the windings magnetic axis were not clearly presented.

(2) It was stated that the magnitude of the scalar current i in the winding was equal to that of the

vector current ir

lying on the windings magnetic axis, but no proof was given for this equality.

(3) The current vectors produced by each phase winding of a three-phase, two-pole stator,

whose phase windings were separated from each other by 120 electrical degrees, were added

vectorially to produce the resultant current vector, which was represented by

iaiaii cbarrrr 2++= (3)

where, i rr represents the resultant current vector, ia , ib and ic are the instantaneous values of

currents in phase windings a, b and c respectively. Vectors ar and ar2 are unit position vectors representing the position of magnetic axes for windings b and c respectively. It was then inferred

that the resultant voltage vector could be produced in a similar manner by the vector addition of

the voltage vectors produced by each phase winding and given by

vavavv cbarrrr 2++= (4)

where, vrr is the resultant voltage vector and va , vb and vc are the instantaneous values of

phase voltages for windings a, b and c respectively. Although the above equation exists, it was

not proven how scalar supply phase voltages were transformed into vector supply phase voltages

and how these vector supply phase voltages lie on the magnetic axes of the windings.

(4) And finally, the multiplication of a scalar voltage differential equation for a phase winding

by a unit position vector representing the position of the magnetic axis of that winding, although

mathematically correct, fails to show how these scalar voltages are physically transformed into

vector quantities. For example, Kovacs in [7] presented Eq. (5) which represent the scalar

voltage differential equation for winding b as

dt

dRiv bbbb

+= (5)

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where, vb , ib and b are the scalar phase supply voltage, scalar phase current and scalar phase flux linkage respectively and Rb the stator phase resistance for winding b. Eq. (5) was then

multiplied by ar , the unit position vector indicating the direction of the positive magnetic axis of winding b, producing Eq. (6).

dt

daRiava bbbb

+= rrr (6)

Eq. (6) is mathematically correct, however, no physical steps were taken by Kovacs in [7] to

show how scalar voltages Ri bb and dtd b in Eq. (5) were transformed to vector quantities in Eq.

(6).

It was also reported by Kovacs in [7] that the vector method is a simple but mathematically

precise method that makes visible the physical background of the various machine phenomena.

This is absolutely correct, however, the issues presented in (1) to (4) above, has failed to use the

physical background of the various phenomena in the development of the vector quantities and

equations. As a result, the full power and benefits of the vector method in the analysis of three-

phase machines were not realized.

Holtz in [8-10] has employed and presented the space vector theory developed by Kovacs and

Racz [1] in the development of vector equations for electrical machines. However, the issues

raised in (1) to (4) were not addressed. In fact, Holtz summarized the space vector notation as

introduced by Kovacs and Racz by stating it represents the sinusoidal field by a complex vector.

He added, it is postulated that the causes and effects of such field, namely the currents and

voltages, also have the property of space vectors owing to existing formal properties [10].

3. Equivalent Circuits of a Three-Phase Stator

A cross section of the stator windings of a two-pole, three-phase machine is shown in Fig. 1. The

phase windings are shown to be displaced from each other by 120 degrees and the positive

direction of current flowing through each winding is upwards through the non-primed side and

downwards through the primed side of each winding. Using this convention of current flow

through the windings, positive magnetic axes were developed for each phase winding, along

which all magnetic quantities exists.

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Fig. 1 Stator Windings of a Two-Pole, Three-Phase Machine

The analysis of electromagnetic systems has traditionally been performed with the production of

two circuits, an electrical circuit for electrical analysis and a magnetic circuit for magnetic

analysis [11]. However, quantities in the electrical circuit affect quantities in the magnetic circuit

and vice versa. As a result of the dependence of both electrical and magnetic circuits on each

other, the development of an equivalent circuit containing both electrical and magnetic quantities

would prove to be very useful in the analysis of electromagnetic systems. Since the three-phase

stator is an electromagnetic system, then the development of an equivalent circuit containing

both electrical and magnetic quantities would be a powerful tool in the vector analysis approach

of this electromagnetic system.

For this analysis, one phase winding of the three-phase stator, winding aa' was selected for

analysis. This winding is represented by its center conductors and current flow through the

winding is in the positive direction as shown in Fig. 2(a). The phase winding possesses resistance

which is an electrical quantity and inductance which is both an electrical and a magnetic

quantity.

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The winding resistance Ra being an electrical quantity is removed from the winding together

with dt

diL aa , which is an electrical voltage. These two quantities Ra and dt

diL aa are placed on

the left (electric) side of the circuit with the supply voltage V a . The winding with its magnetic

quantities and electric current is on the right side of the circuit of Fig. 2(a). This process was

undertaken in order to separate the electrical quantities from the magnetic quantities. The electric

scalar current ia , which leaves the electric circuit flows through the winding and produces

vector magnetic field intensity H ar along the positive magnetic axis of winding aa' as shown in

Fig. 2(a). The magnitude of H ar , is obtained by a Amperes Circuital Law and is given by

laNi aa , where la is the length of the path of H a

r and N a is the number of turns of winding aa'.

The magnetic field intensity H ar produces flux density Ba

r , which is collinear with H ar , and

whose magnitude is given by || H ar , where is the permeability of the medium in which Bar

exists. Flux density Bar produces flux ra , which is also collinear with Ba

r and whose

magnitude is given by AB aar , where Aa is the cross-sectional area of concern. The flux linking

winding aa' is given by ra , which is collinear with r

a and whose magnitude is given by

aa Nr

. And the flux linkage ra produces current vector i ar whose magnitude is given by a

aLr

which is collinear with ra , where La is the inductance of winding aa'. Hence magnetic quantities H a

r , Bar , ra ,

ra and current vector i a

r all lie along the positive (+ ve) magnetic axis

of winding aa' and are spatial vector quantities possessing both magnitude and direction. Since

current vector i ar leaves the magnetic circuit, a similar current vector i a

r must also enter the

series connected magnetic circuit of Fig. 2(b). In addition, since the electric and magnetic

circuits are connected in series, this implies that the scalar electric current ia is of same

magnitude as the vector magnetic current i ar . The separation of electric and magnetic circuits is

shown by the dotted vertical line I in Fig. 2(b).

The vector magnetic current i ar , on entering the electric circuit, produces a scalar current ia in

the electric circuit, and the scalar electric current ia on entering the magnetic circuit, produces a

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vector magnetic current i ar on the magnetic axis of winding aa' as shown in Fig. 2(b). Hence

the magnetic axis of winding aa' completes the electric circuit making ia and i ar of same

magnitude.

Fig. 2 Equivalent Circuits of Winding aa' (a) Electric and Magnetic- Electric

Equivalent Circuits (b) Electric and Magnetic Equivalent Circuits

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If ia is changing, then the effect of the magnetic circuit on the electric circuit is seen in the

voltage dt

diL aa which opposes the current ia . Since the magnitude of ia and i a

r are equal, and

i ar lies along the windings magnetic axis, then the voltages Ri aa and dt

diL aa can be referred

to the magnetic axis of winding aa' without changing their magnitudes. The vector summation

of aa Rir and

dtidL aar

along the magnetic axis of winding aa', produces the supply voltage

vector V ar along the magnetic axis of winding aa'. Applying Kirchhoffs voltage law to the

electric and magnetic sides of Fig. 2(b) yields,

dt

diLRiV aaaaa += for the electric side, (7)

and

dtidLRiV aaaaa

rrr += for the magnetic side. . (8)

The production of an equivalent circuit containing electric and magnetic quantities for the

electromagnetic system represented by one phase winding of a three-phase stator, clarifies the

issues raised in (1) to (4) above. A summary of the benefits gained from the above analysis

utilizing the equivalent circuit containing electric and magnetic quantities as it relates to the

issues raised in (1), (2) and (4) are as follows [12-13].

(5) It shows, when a scalar current ia of an electromagnetic system, leaves the electric circuit

and enters the magnetic circuit, it is converted into a vector quantity i ar of the same magnitude

as the scalar current. This is as a result of the series nature of the electric and magnetic circuits

resulting in the same magnitude of both scalar and vector currents. The location of the current

vector is along the magnetic axis of the winding, because all magnetic quantities are located on

its magnetic axis.

(6) Since the vector current is of the same magnitude as the scalar current and this current

vector lies on the magnetic axis of the winding, then, scalar voltages Ri aa and dtdi

L aa can be

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referred to the magnetic axis of the winding becoming aa Rir and

dtidL aar

respectively, with

these vector voltages being of same magnitude as their scalar counterparts. In addition, since the

sum of the scalar voltages Ri aa and dtdi

L aa in the electric circuit results in the scalar supply

voltage, then, the sum of the vector voltages aa Rir and

dtidL aar

results in the vector supply

voltage, which is of the same magnitude as the scalar supply voltage.

(7) In addition to showing the process by which scalar voltages are referred to the magnetic

circuit of the electromagnetic system, the analysis provides a scalar and a vector voltage

differential equation as shown in Eqs. (7) and (8). If scalar analysis is being performed, then the

scalar voltage differential equation is utilized, while, if vector analysis is being performed on the

electromagnetic system formed by the stator, then, the vector voltage differential equation would

be utilized.

4. Magnetic and Electric Vectors of a Three-Phase Stator

The application of the above technique to the three-phase, two-pole stator shown in Fig. 1, whose

phase windings are displaced from each other by 120 electrical degrees, produces the magnetic

and electric quantities of each phase along the phase magnetic axes as shown in Fig. 2.

Each magnetic or electric phase variable can now be added vectorially to produce the resultant of

that variable. Hence the resultant magnetic field intensity H resr , flux density Bres

r , flux rres , flux linkage rres , current vector i resr , stator resistance voltage drop sres Rir , inductance voltage

dtidL ressr

and supply voltage V resr are given by the vector addition of their phase variables

shown on the magnetic axes of Fig. 3, which yield:

HaHaHH cbaresrrr 2++= (9)

BaBaBB cbaresrrr 2++= (10)

++= cbares aa rrr 2 (11)

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++= cbares aa rrr 2 (12)

iaiaii cbaresrrr 2++= (13)

RiaRiaRiRi ccbbaasres 2++= rr (14)

dtid

Ladtid

Ladtid

Ldtid

L ccb

ba

ares

s

rr

rrrr 2++= (15)

vavavv cbaresrrr 2++= (16)

where, RRRR cbas === and LLLL cbas === .

In Eqs. (9) to (16), ar and ar2 are unit vectors representing the position of the positive magnetic axes of windings bb' and cc' respectively and the magnetic and electric variables on the right

hand side of these equations are the instantaneous values of these variables for the particular

winding. In addition, the stator resistance and inductance of each phase winding are represented

by Rs and Ls respectively.

The application of Kirchhoffs law to the vector voltages on each magnetic axis yields,

dt

dRiv aaaa

+= (17)

dt

daRiavav bbbbb

+== rrrr (18)

dt

daRiavav ccccc

+== rrrr 222 . (19)

Eqs. (16) to (19) show that both the phase vector supply voltage and the resultant voltage vector

of the three phase windings were obtained by vector addition of vector voltages that exist on the

axes of the phase windings of Fig. 3 which addresses the issue raised in (4).

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Fig. 3 Magnetic and Electric Quantities of Two-Pole, Three-Phase Stator Shown on Magnetic

Axes of the Windings

5. Conclusion

The development of equivalent magnetic and electric circuits for a phase winding of a three-

phase stator produced electrical quantities on the electric side of the circuit and magnetic

quantities on the magnetic side. The boundary separating the electric and magnetic circuits is

interfaced by the scalar electric current on the electric side and the vector magnetic current on the

magnetic side of the equivalent circuit. This indicates that a scalar electric current flowing

through a stator phase winding produces a vector magnetic current on the magnetic axis of the

winding, with the reverse taking place when the vector magnetic current enters the electric

circuit. The equality of scalar electric current and the magnitude of vector magnetic current are

also realized by the manner in which the electric and magnetic circuits are connected. This

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equality in scalar electric and vector magnetic current magnitudes permit the referral of scalar

electric voltages, including the supply voltage, to the magnetic circuit and lying on the magnetic

axis of the winding. The absence of this equivalent electric and magnetic circuit and the above

results derived from it has not inhibited growth in the area of transient analysis and modeling of

electrical machines under sinusoidal excitation. However, its introduction to the analysis of

electrical machines under sinusoidal excitation reveals the background of the various phenomena

occurring in the machine. In addition, the introduction of the equivalent electric and magnetic

circuits of a three-phase stator would provide a platform for vector analysis of electrical

machines excited with non-sinusoidal and dc supplies as is the case with brushless dc machines.

References

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12. De Four R., A Self-Starting Method and an Apparatus for Sensorless Commutation of

Brushless DC Motors WIPO Publication No. WO 2006/073378 A1 July 13, 2006.

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