vector analysis of a three-phase stator

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, provid

© Copyright 2007 by Ronald De Four

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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

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 winding’s magnetic axis In addition, if the

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

produced was given by

F = Ni (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

Fr N ir

= (2)

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 winding’s 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 winding’s 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

irr=i a+ari b+ar2i c (3)

where, irr represents the resultant current vector, i

a , i b and i c 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

vrr =v a+arv b+ar2v c (4)

where, vrr is the resultant voltage vector and v

a , v b and v c 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

d R i

λ +

= (5)

where, v b , i b and λb are the scalar phase supply voltage, scalar phase current and scalar phase

flux linkage respectively and R b 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

d a R i a v

ar b= r b b+r λb

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

show how scalar voltages i b R b and

dt

dλb

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

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

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

with

dt

di

L a a , which is an electrical voltage These two quantities R a and

dt

di

L a a 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 i a, which leaves the electric circuit flows through the winding and produces

vector magnetic field intensity Hra along the positive magnetic axis of winding aa' as shown in Fig 2(a) The magnitude of Hra, is obtained by a Amperes Circuital Law and is given by

la

N

i a a

, where l a is the length of the path of Hra

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

The magnetic field intensity Hra

produces flux density Bra

, which is collinear with Hra, and

whose magnitude is given by µ| Hra|

, where µ is the permeability of the medium in which Bra exists Flux density Bra produces flux φr

a , which is also collinear with Bra and whose

magnitude is given by Bra A a

, where A a is the cross-sectional area of concern The flux linking winding aa' is given by λra, which is collinear with φra and whose magnitude is given by

a

a N

φr And the flux linkage λra produces current vector ira whose magnitude is given by

a

a L

λr

which is collinear with λra , where L a is the inductance of winding aa' Hence magnetic

quantities Hra

, Bra, φr

a, λra and current vector ira all lie along the positive (+ ve) magnetic axis

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

current vector ira 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 i a is of same magnitude as the vector magnetic current ira The separation of electric and magnetic circuits is

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

The vector magnetic current ira

, on entering the electric circuit, produces a scalar current i a in

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

vector magnetic current ira 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 i a and ira of same magnitude

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

Equivalent Circuits (b) Electric and Magnetic Equivalent Circuits

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

voltage

dt

di

L a a which opposes the current i a Since the magnitude of i a and ira are equal, and

ira lies along the winding’s magnetic axis, then the voltages

R

i a a and

dt

di

L a a can be referred

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

of ira R a and

dt i d

L a a

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

vector Vra along the magnetic axis of winding aa' Applying Kirchhoff’s voltage law to the

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

dt

di L R i

V a = a a + a a for the electric side, (7) and

dt

i d L R i

r r

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 i a of an electromagnetic system, leaves the electric circuit and enters the magnetic circuit, it is converted into a vector quantity ira 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 i a R a and

dt di

L a a can be

referred to the magnetic axis of the winding becoming ira R a and

dt i d

L a a

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

sum of the scalar voltages i a R a and

dt

di

L a a in the electric circuit results in the scalar supply

voltage, then, the sum of the vector voltages ira R a and

dt

i d

L a a

r 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 Hrres

, flux density Brres, flux φr

res, flux linkage λrres, current vector irres

, stator resistance voltage drop irres R s, inductance voltage

dt

i

d

L s res

r

and supply voltage Vrresare given by the vector addition of their phase variables

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

Hrres =H a+arH b+ar2H c (9)

Brres=B a+arB b+ar2B c (10)

φr =φ +arφ +ar2φ (11)

λrres=λa+arλb+ar2λc (12)

irres=i a+ari b+ar2i c (13)

irres R s=i a R a+ari b R b+a2i c R c (14)

dt i d L a dt i d L a dt i d L dt i d

r r

r r r

r

2

+ +

vrres=v a+arv b+ar2v c (16)

where, R s=R a =R b=R c and L s=L a=L b=L c

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 R s and L s respectively

The application of Kirchhoff’s law to the vector voltages on each magnetic axis yields,

dt

d R i

v a= a a+ λa

dt

d a R i a v a

λ +

=

r (18)

dt

d a R i a v a

vrc= r2 c= r2 c c+ r2 λc

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)