carters coefficient

7/25/2019 Carters Coefficient

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626

In our previous analysis, we neglected the effects of slots on the stator and rotor. As it

turns out, the effects of slots can be readily incorporated into the analysis by replacing

the air gap gwith a modified air gap g. In particular, for the case of the stator slots,

the modified air gap is calculated as

=g gcs (B-1)

where csis the stator Carters coefficient. We will now derive this result as well as a

value for cs.

The derivation of (B-1) begins with consideration of Figure B-1. This figure depicts

the developed diagram over a small range of position wcorresponding to one-half ofa stator slot width plus one-half of a stator tooth width. Thus

w w wss st = +1

2

1

2 (B-2)

where wssis the stator slot width and wstis the stator tooth width, both measured at the

stator/air-gap interface.

Let us first consider the situation if we ignore the slot. In this case, it can be shown

that the flux flowing across the air gap in the interval wmay be expressed as

= +0

2

l

gw wss st ( ) (B-3)

where l is the length of the machine and is the magnetomotive force (MMF) drop

between the stator and rotor at that point. Because the slot is unaccounted for in (B-3),

this expression is in error, because part of the flux (2) will have to travel further. Our

goal will be to establish a value gsuch that

Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk,Scott Sudhoff, and Steven Pekarek. 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc.

CARTERS COEFFICIENT

APPENDIX B


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CARTERS COEFFICIENT 627

=

+0

2

l

gw wss st ( ) (B-4)

is correct, or is at least a good approximation.

To this end, let us calculate the flux, including the effects of the slot. To this end,

it is convenient to divide the flux into two components,

= +1 2 (B-5)

The first term is readily expressed as

10

2= w l

g

st

(B-6)

The second term is more involved. At a positionz(see Fig. B-1), the distance from the

rotor to the stator along the indicate path is g+z/2. Thus, the field intensity along this

path may be estimated as

Hg z

=+

/ 2 (B-7)

The flux 2may be expressed as

20

2

==Bldz

z

wss/

(B-8)

Substitution of (B-7) into (B-8) and noting that the fields are in air yields

202

1 4= +

l w

g

ss

ln (B-9)

The final step is to add (B-6) and (B-8) and to equate the result to (B-4). The result is

(B-1), where

Figure B-1. Carters coefficient.

stator

tooth

rotor

g

sswstw

w

1f 2f

z


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628 CARTERS COEFFICIENT

cw w

wg w

g

sss st

stss

= +

+ +

41

4

ln

(B-10)

Observe that g, g, and cscan all be functions of position (as measured from the stator

or the rotor) but this functional dependence is not explicitly shown.

The use of (B-1) and (B-10) is straightforward and very useful, because it allows

us, with a simple substitution of gfor g, to account, albeit approximately, for the effects

of the stator slots on magnetizing inductance calculations, as well as flux linkage due

to permanent magnets.

For machines with both stator and rotor slots, the concept of Carters coefficient

can still be used; however, in this case

=g gc cs r (B-11)

where

cw w

wgc w

g

rrs rt

rts rs

= +

+ +

41

4

ln

(B-12)

and where wrsand wrtare the width of the rotor slot and rotor tooth where it meets the

air gap.

Before concluding, it should be noted that (B-10) and (B-12) are based on a geom-

etry in which tooth tips do not exist or are neglected. In cases where this is not appli-

cable, the same methods can be used to find an alternate expression for Carters

coefficient.

carters coefficient

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