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APPENDIX B CARTER’S COEFFICIENT



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 g with 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 cs is the stator Carter ’s 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 w corresponding to one-half of a stator slot width plus one-half of a stator tooth width. Thus w=



1 1 wss + wst 2 2



(B-2)



where wss is the stator slot width and wst is 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 w may be expressed as Φ=



μ0 l (wss + wst ) 2g



(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 (Φ2) will have to travel further. Our goal will be to establish a value g′ such 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.



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627



CARTER’S COEFFICIENT



Φ=



μ0 l (wss + wst ) 2g ′



(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, Φ = Φ1 + Φ 2



(B-5)



The first term is readily expressed as Φ1 =



μ0 wst l 2g



(B-6)



The second term is more involved. At a position z (see Fig. B-1), the distance from the rotor to the stator along the indicate path is g + πz/2. Thus, the field intensity along this path may be estimated as H=



 g + πz / 2



(B-7)



The flux Φ2 may be expressed as wss / 2



Φ2 =



∫



(B-8)



Bldz



z=0



Substitution of (B-7) into (B-8) and noting that the fields are in air yields Φ2 =



2 μ0 l ⎛ π wss ⎞ ln ⎜ 1 + ⎝ π 4 g ⎟⎠



(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



wst



wss w



z



stator tooth



g f1 f 2 rotor



Figure B-1. Carter’s coefficient.



628



CARTER’S COEFFICIENT



cs =



wss + wst 4 g ⎛ π wss ⎞ ln 1 + wst + π ⎜⎝ 4 g ⎟⎠



(B-10)



Observe that g, g′, and cs can 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 g′ for 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 Carter ’s coefficient can still be used; however, in this case g ′ = gcs cr



(B-11)



where cr =



wrs + wrt 4 gcs ⎛ π wrs ⎞ ln ⎜ 1 + wrt + ⎝ π 4 g ⎟⎠



(B-12)



and where wrs and wrt are 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 geometry in which tooth tips do not exist or are neglected. In cases where this is not applicable, the same methods can be used to find an alternate expression for Carter ’s coefficient.