CHAPTER 3

LEAKAGE REACTANCE CALCULATIONS

3.1 Introduction

In Chapter 2, while considering the fundamental

energy conversion process in a L~I, the attention was

necessarily focussed on the active region of the machine.

Inevitably, even in the idealised theoretical model, there

exist some leakage fields that link either the stator or

rotor circuits only. The leakage reactances assume

importance when the applied voltage and power factor

characteristics of the current-fed machine are examined,

or when the machine is run from a constant-voltage source.

There has not been any significant contribution in this

as~ect of LIMs except the paper by Allin et al30 in which

the leakage reactances of a laboratory LIM model is experi-

mentally obtained. It is probable that questions regarding

the leakage reactances were pushed into the background as

the emphasis in LIM research so far has all along been on

the current-fed mode of operation10 ,22-24. In addition,

exhaustive work had been previously done on the leakage

reactances of different types of conventional round rotor

induction motors (Alger, P.L. 31 ), and it is qUite possible

that some of the formulae could be modified and used for

38

the LIMs. Yet it is considered desirable to explore the

possibility of developing analytical expressions for the

leakage reactances of LIMs keeping the geometry of the

machine more or less intact for the following reasons:

1. The stator winding overhang leakage reactance of

the longitudinal-flux LIM is relatively large on account

of the larger pole pitch as shown by Laithwaite1 ,3.

2. A topological cylindrical equivalent of the double­

sided LIM does not exist conceptually, since only the outer

surface of the round rotor is available for induction.

3. It is recognised that the sheet secondary constru­

ction of the LIM facilitates a simplified field theoretic

formulation of the leakage reactance problem.

The elements of leakage reactance that are generally

taken into consideration for a conventional induction motor

can be broadly classified as 31

1. Primary winding overhang leakage reactance

2. Primary slot leakage reactance

3. Secondary leakage reactance

4. Air-gap leakage reactance

Some of the physical features characterising the

round rotor induction motor and giving rise to the above

leakage paths are not present in the type of linear

induction motors studied in this investigation.

39

Specifically, the aluminium sheet rotor construction

results in a low secondary leakage reactance even for

thick plates as shown by Laithwaite 1 • Analogous to the

terminology of conventional induction motors, the airgap

leakage reactance is brought about by the differential,

zig-zag and peripheral leakage fluxes. Again the unslotted

sheet rotor configuration permits ampere-turn balance of

the primary currents in the opposite gap surface at every

point, thereby eliminating the differential fluxes. For

the same reason, the zig-zag leakage fluxes also vanish

almost entirely.

It may be noted that in the previous chapter the

performance of the linear induction motor is predicted by

solving for the vector potential along the stator surface

so that the airgap leakage flux is automatically included

in the analysis of the active region. Consequently the

important components of the leakage flux that have to be,

separately accounted for are the coil end-turn leakage and

the primary slot leakage fields 25 •

3.2 Stator Overhang Leakage Reactance

In general, the coil end configuration of the stator

overhang region along with the geometry of the end ring

assembly gives rise to a complicated three-dimensional

pattern of end leakage flux that is very difficult to

determine exactly. One redeeming feature, however, is the

40

truely leakage nature of the overhang flux so that it can

be treated independent of the main field. In fact, as

Alger31 has pointed out, the calculation can be made as

if the stator core were removed and the winding projections

on either side brought together with the flux paths

entirely in air. Hence for the sake of analysis it is

assumed that the overhang field is unaffected by the

active ,region and rotor currents. This implies that the

longitudinal end effects and transverse edge effects of

the rotor can be neglected, affording a simplified treat-

mente

Alger31 has shown that for a double layer winding

with diamond shaped coils, the overhang currents can be

resolved into the axial (z) and peripheral (x) components,

which obey sinusoidel variation in both x and z directions.

A three-dimensional solution for the leakage field is

obtained by representing the winding overhang by a thin

continuous current sheet on either side of the stator

core level with the stator-airgap interface. Also, in

the following analysis, only the space fundamental of the

stator current distribution is considered, while the field

quantities have sinusoidal time variation at supply

frequency ~ as in eqn. 2.6.~

Referring to fig. 3.1 which shows the winding dis-

position along with the co-ordinate axes, the fundamental

41

current distribution in the stator due to the three phase

double layer winding is given by (Alger31 )

h = 2 6. cos ~ x amp/m

where ~ is the peak value of the linear current density

per layer of the primary and is given by

6. =3j2~T1

P lamp/m

where I is the primary phase current (r.m.s.) and .~ is

the pole pitch.

Consider a point P in the overhang region with co­

ordinates P (x,y,z). By Alger31 there is an outgoing

current 10

and a return current 1r at this point the

linear densities of which are given by

10 6 ( ~

'¥ z )= cos x -2' Zo

and I 6 cos ( p x + 't.. z )= --rr 20

where i' is the coil span in radians.

The transverse (z) and longitudinal (x) current

density components are given by

= = 2 6. cos f3 x cos 1\ z

I II I

) / ))J))/ / / / / '

/ .//// // / / / /

/ / / / /, // / / /

/ //

P(lI".I)

Fig.3.1(a) Winding disposition

y

y. I

X(1)1---' --.-- '--.;..,~;;....--. - .-

Fig.3.1 (b) Longitudinal view of stator

y

I;'.'.,',

StatorI'ramerln.

(),/erhan.CU"ren\.h ••,

Fig.3.Hc) End view of stator

43

and Ix = (I -I) cote = 2 ~ sin ~ x sin "Z cote (3.6)o r

and e is the angle of bend of the end"'V2Zo

turn measured from the positive x direction as in fig. 3.1(a).

where

The approach for determining the overhang leakage

reactances25 is essentially to solve for the field distri-

bution generated by the transverse and longitudinal compon-

ents of the end turn currents by formulation of the appro-

priate boundary value problems. Expressions for the leakage

reactive power flow are derived SUbsequently in terms of

the Poynting vector.

Due to certain constructional details of the disc

drive, discussed in Chapter 4, the two sides of the over-

hang on either side of the stator core are to be treated

slightly differently. vlliile the overhang at the outer

periphery of the (curved) stator block is placed entirely

in air, that at the inner side has an iron backing ring

which is part of the stator frame assembly holding the

laminations together. Also for each overhang current

sheet there are two regions of field propagation25 • In

the following, the airgap region between the stators LS

denoted by (1) and the back air region by (2).

The field equations in the regions (1) and (2) for

the outer overhang in the absence of rotor currents are

= 0

+ = 0, o<y < CX)

44

along y = 0, by symmetry about the xz plane

and = + along y =g

considering only the transverse component at the moment,

with I z given by eqn. 3.5.

The general solution of eqn. 3.7 for region (1)

consistent with eqn. 3.5 is given by

AZ1 (x,y,z) = Ai (y) cos ~ x cos1\ z

Substitution of AZ1 in eqn. 3.7 results in the

ordinary differential equation

= 0

where 11 2 =

45

The solution o~ eqne 3.11 that satisfies the boundary

condition 3.8 is given by

:: C1 cosh " Y

where C1 is an arbitrary constant.

The normal component of the flux density

BY1 (x,y,z)a

:: - bx AZi (x,y,z)

:: C1 (6- cosh "l y sin p x cos AZ

The tangential component of the magnetic field

strength

HX1 (x,y,z) = 1-""0

o-oY AZ1 (x,y,z)

Cit)= - sinh'" Y cos f! X COB 1\ ZtJ.

Now, for the back air region (2), the solution of

eqn. 3.7 is of the form

AZ2 (x,y,z) = C2 e- t'\ y cos is x cos A Z

which satisfies the physical requirement that AZ2~ 0

as y ... + 00

46

The normal component of the flux density

BY2

(x,y,z)o= - ~ AZ2 (x,y,Z)

= (3 C2

e - IJ y sin ft x cos A. z

Also the tangential component of the field strength

HX2 (x,y,z) 1= -~o

AZ2 (x,y,z)

I') 02= - - e- '1 y cos f3 x cos ~ zfJo

Equating By1 (x,g,z) and BY2 (x,g,z) for continuity

of normal component of flux density along the plane y = g

= C1 e1) g cosh ~ g

By eqn. 3.9 which relates the tangential components

HX1 and HX2 at y = g,

C1 ~- sinh t'l g cos (!> x cos 1\ z =

tJ o

cos ?\ z + 2 A cos f3 x cos A Z

Eliminating C2 in the last eqn. by using 3.14 and

solving for C1 ,

47

2 A Po=

t') (sinh t') g + cosh') g )

and =2 A f!~ e ''1 g cosh ~ g

t') (s inh ~ g + cosh r') g )

Hence the final expressions for the distribution of

the magnetic vector potential in regions (1) and (2) are25

AZ1 (x,y,z) =

and

2 A J.Locosh !} Y , cos ~ x cos 1\ z

~ ( sinh 1 g + cosh ~ g )O<'y< g

AZ2 (x,y,z) = 2 A J.l.cosh r') g e- '1 (y-g) cos ~ x cos A z

~ ( sinh ~ g + cosh ~ g )

y> g

The reactive power flow leaving the current sheet

per unit normal area into the regions (1) and (2) are

given by the outward drawn Poynting vectors32 ,33 81 and

82 evaluated at y =g.

-48

*51 = - 0.5 EZ1 HX1

() AZ1 (x,g,z) * (x,g,z)1 e>Az1(3.19)= 2 1-40 ~t ay

aAZ2 (x,g,z) * (x,g,z)1 ~ A Z2 (J. 20)and 52 = - 2po "b t ~ Y

While 8 2 represents reactive power density that is

entirely leakage in nature, part of 51 crosses into the

secondary representing a mutual effect25 • The latter

component denoted by 5d can be obtained by replacing d

for g in eqn. 3.19 and subtracted out from 51 to give the

true leakage25,31~ Consequently the stator leakage is

given by

s = (3.21)

The expressions for 51' 52 and 5d are given by

2j t-t>oU)o~2 cosh II g sinh '1 g cos2 ~ x cos 21\ z

~ ( sinh ~ g + cosh ~ g )2

···(3.22)

2J V.~oD,.2 cosb2 1] g cos2 fl x cos2 i\ z

., ( sinh 1] g + cosh II g )2

49

~ { sinh ~ g + cosh ~ g )2

where the negative sign indicates inductive reactive power.

~he above procedure needs a modification for the

overhang side provided with an iron back-up ring25

(fig, ~.1c). \lliile eqns. 3.7 and 3.8 hold without change,

AZ2 * 0 in eqn. 3.9. It is now required to solve for

Az1 only in the airgap region with the boundary condition

= along y = g

where I z is given by 3.5 as before.

The solution of Az1 satisfying eqns. 3.7, 3.8 and

3.25 is straight forward and is given by

AZ1 (x,y,z) = 2 6 eocOsh t] y cos " x cos " z1 sinh Y) g

•••• (3.26)

The rest of the derivation including correction

for the mutual field entering the rotor remains the same

as in eqns. 3.19 and 3.21. Denoting the corresponding

Poynting vectors by Si and SJ with S~ = 0, the reactive

50

volt-ampere per unit area of overhang is given by

8' =8' - 8 I1 d

where 81. = - 2j tJ.o~oa2 cosh rt g sinh '1 g cos 2 tsx cos 2 l\z

~ sinh2 '1 g

···(3.28)

and 8 f ~ _ ,gj fJo(.l.)o ~ cosh 11 d sinh tl d cos 2 lex cos2~ z

d r, Sinh2 '1 g

Let xi denote the leakage reactance per phase due to

the transverse component of the overhang current. Equating

the 3 phase reactive power with the surface integral of

the Poynting vector over P pole pitches

3 12 x = P1

Z "r~ 0 ~'( 8 + 8' ) dx dz

z=o x=o

~ cosh!J g sinh., g - cosh!) d sinh" d + c.o~h2 '1 ~

( ., ( sinh t'1 g + cosh yt g ) 2

+cosh'1 g sinh ~ g - cosh., d sinh'1" ~ f'

'1 Sinh2 ') g ~

where T' =z l'

o ~

z=~ x=o

51

Carrying out the above integration and substituting

for 6, in eqn. 3.30 in terms of I,

3 l.\)oP'cZo(1+kz) T2

K;

P'11:'

~ cosh!) g sinh" g - cosh., d sinhr'\ d + cosh2Y') g

{ ( sinh ~ g + cosh ~ g )2

+ cosh!\ g sinh !') g - cosh!) d sinh !' d ~"002 ~S1 '1 g

where k z = sinX

t

and is zero for full pitch coils.

Eqn. 3.31 can be rewritten in the form25

3 woP'oozo (1+kz ) T2

K;

Py\T

~ sinh r'\ (g-d) cosh ,,(g+d) + cOSh2 'J g

{ cosh 2'1 g + sinh 2 " g

+sinh ~ (g-d) cosh 1 (g+d)

Sinh~ g

ohms

52

The analysis for the longitudinal component of over-

hang current parallels the above derivation except that

the variables of interest in the Laplace's equations for

regions (1) and (2) are AX1 and Ax2 • The boundary value

problem characterised by magnetic lines of force mapped

on the y-z plane, is described by

+ + = 0, o < y < 00 (3.34)

~Ax10 along 0 (3.35)CY = y =

andOAx1 oAX2

Ix• (3.36)

~ y = bY along Y = g,

where Ix is given by eqn. 3.6.

Equating the normal components of flux density

at y = g

a Ax1 (X,g,z)

o Z =

The tangential components of the field strength

are related by eqn. 3.36 and following the steps leading

to eqns. 3.17 and 3.18 one gets25

Ax1 {x,y,z) =

53

2 A ~ cosh '1 y s in ~ x sin" z cot e (3. 37)')( cosh '1g + sinh')g )

and l c~sh ~ g e- ~ (y-g)AX2 {x,y,z) = - 2 A U

1"0 If (cosh 1) g + sinh Y) g )

sin ~x sin 1\ z cot 9

The Poynting vectors 81 and 82 representing the

reactive power density are

*81 = Ex1 Hz1 / 2

a AX1 (x,g, z) *1 ~ AX1 (x,g,z)

= 2 t-Lo at oy

() AX2 (x, g, z) *and 8

21 ()Ax2 (x,g,z)

= - 2 f-lo at ay

The final expressions for 81 , 82 and 8d corresponding

to the longitudinal component of overhang current are

=

2. 2

- J U w Af'"'o 0 .

2 21 cosh Y) g cot e{ ~ (cosh r')g + Sinh'1g )2

I S =d. 2 ~

-2j Mt;t.Qo A t sinh t') d cosh f) d cot2e ~'1 (cosh W'\ g + sinh 'l g )2 ~

2 . 2sin ~ x sin A,Z

and S =

For the overhang region provided with a back-up ring

AX2 = 0 and analogous to eqn. 3.26, the vector potential

in the airgap is given by

AX1 (X,y,Z) = 2 6. f-iocosh '1 y sin @:x sin~z cot er'} sinh r; g

The corresponding power density vectors in tune

with eqns. 3.27 a~d 3.28 are given by

SI =1

and S 1 ;::d 2 " 2

-JuU3Arel 0

2~ sinht"\dcosh'1dcot et Yj sinh2 t') g l

55

The true leakage reactive power density is now

8 1 ;:: S 1 - 8 11 d

Let x 2 denote the leakage reactance per phase of the

stator due to the longitudinal component of the overhang

current~ Equating the 3 phase reactive power with the

surface integral of the Poynting vector over the current

sheet, with 8 and SI given by eqns. 3.42 and 3.45,

=z

P ~ 0

z=o

T

~x=o( 8 + S r ) dx dz

;::

2­cot2e~ cosh I) g sinh l) g-cosh !) d sinh!) d+cosh 'J g

~ y) (sinh '1 g + cosh ry g) 2

+ cosh l) g sinh!) g-cosh !) d sinh f) d ~I 2 ~

~ sinh Y). g

7:'.

~ -sin2 ~ x sin2

"l\ z dx dz (3.46)x=o

56

Evaluating the double integral and substituting

for ~ in terms of I from eqn. 3.2, the final expression

for x2 is obtained as25

3 ~o~oZo(1-kz) T2 ~ cot

2 eP'lT

~ sinh!) (g-d) cosht) (g+d) + COSh2ng( cosh 2 t')g + sinh 2 " g

+ sinh" (g-d) cosh'l (g+d)

sinh2 '1 gohms.

Finally the stator overhang leakage reactance per

phase is given by the sum of xi and x2

3.3 Slot Leakage Reactance

There are standard formulae available in the

literature3i for calculating the slot leakage reactances.

In this section the slot leakage reactance is calculated

for the upper and lower coil sides of the double layer

winding separately, because of the inherent wnbalance of

the stator winding disposition, with half-filled slots

at the ends of the stator.

57

Consider a single open slot shown in fig. 3.2 with

coil sides P and Q carrying currents I p and IQ respect­

ively in the direction normal to the paper.

The current Ip flowing in coil side P produces a

flux density varying linearly along its height hi. The

flux density at an elemental height dy, y from the bottom

of coil P is obtained by considering the current enclosed

by the lines of force. Neglecting the reluctance offered

by the iron, the flux density variation is given by

b (y)tJoNp I p Y

=x 2hi w

In the region h2 of the slot, the flux density is

constant and is given by fJC)N p I p / 2w.

The total flux linkage of the coil P due to its own

current is therefore given by

+ dy

where 2a is the width of the stator.

The self inductance Lp is now given by setting I p

equal to unity in the above equation.

•h, : 9·4 m m

hz =1.4mm

h J = ·4 mm

2 w = 9·525 mm

59

phases n1 n2 ns

A 13 11 3

8 15 9 3

C 15 9 3

n, - upper coil sIdes

n2 - lower co il sIdes

ns - numbe r of slots

containing only one phase

Slots cor. tom ut u a I c 0 i I sides ~

A 8 - 8 I B C - 6 I CA - 6.

Fi g. 3.2

Slot and coil side details

59

The current I Q flowing in coil side Q sets up flux

density varying linearly along its height, while in the

slot region above Q, the flux density is constant, being

given by ~oNQ I Q / 2w.

Hence the total flux linkage of the coil Q due to

I Q is given by

dy

The self-inductance LQ is now given by putting

I Q = 1 so that

l!.N~ a ~"N: a hiLQ = ( hi +h2+h:; ) + (:;.49)w :;w

To obtain the mutual inductance MPQ' let Q carry a

current IQ• The field due to IQ is uniform in the region

h1 and h 2 and is given by

While the entire flux in region h2 links the coil P9

in fUll, the linkage in hi is linear in y. Hence the

mutual inductance MPQ' given by the flux linkage of P

due to unity current in Q is equal to

60

( + (3.50)

Let the phase A consists of n1 upper coil sides and

n2 lower coil sides. Also let n8 denote the number of

slots containing only A phase and� the number of slots

containing A aJld B phases •

. The self-leakage reactance of phase A is given by

=

and the mutual reactance between A and B phases is

=

The other elements of the slot reactance matrix,

namely xb, xc, xac and �care similarly obtained.

(3.51)

(3.52)

Combining the above with the final results of Section

3.2, the stator leakage reactance matrix is given by

[x) • (3.53)

61

3.4 Summary

The problem of calculating the leakage reactances

in a double-sided, sheet secondary type Linear Induction

Motor is considered in this chapter. The leakage fields

assume importance when the complete terminal characteri-

sation of the machine is desired, as discussed in the

subsequent chapters. In this sense, the results here

complement those of Chapter 2. The three-dimensional

field model enables computation of the 3 phase reactive

power flow in the coil end winding region and facili-

tates calculation of the 3 x 3 leakage reactance matrix.

It may be noted that the derivation does not employ any

empirical formula or factor and the final expressions

for the stator overhang and slot leakage reactances can

be evaluated using normal design data.