24, NO. 6, NOVEMBER 1988. THE SLOT IMPEDANCE: EXPERIMENTAL VERIFICATION. S. Ratnajeevan H. Hoole, Thomas H. Walsh, George H. Stevens.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 24, NO. 6, NOVEMBER 1988
THE SLOT IMPEDANCE: EXPERIMENTAL VERIFICATION
S. Ratnajeevan H. Hoole, Thomas H. Walsh, George H. Stevens Department of Engineering Harvey Mudd College, Claremont, CA 91711
Abstract: Computational cost is high in eddy current finite element solutions because the magnetic vector potential A varies exponentially, resulting in fine meshes. The surface impedance boundary condition allows us to eliminate eddy current parts without external current sources. In slots, however, the’variation of A is very rapid even in the air,and modeling these parts in electric machinery is very costly. This paper introduces the slot impedance, which is very similar to the surface impedance and allows us to remove the slots using the slot impedance as a boundary condition.
Eq. 3 is known as the impedance boundary condition [2-61 and is used to eliminate these parts. It has been shown that this condition is valid even at comers [6,7]. In the analysis of slots, however, the comers interact, and flux decays exponentially down the slot (Fig. 2). Again the cost is prohibitive, particularly in electric machinery with numerous slots. Mayergoyz, Emad, and Sherif [ 8 ] have addressed this using analytical models of the slot. Here we use a transmission line model of the slot to give an impedance boundary condition such as eq. 3 to close the slot from the solution domain. The model is verified in the laboratory.
INTRODUCTION In eddy current problems in 2-dimensions, we usually solve the diffusion equation [ 11
-p-’V2A = JO - j o o A
(1)
for the complex vector potential A. In solving this by the finite element method, numerous nodes are required in conducting parts with (T # 0 to model the rapid variations, and, as a result, solution is expensive. When Jo = 0, and the depth of penetration, d=[w~l.o]-~’~, is low in relation to device dimensions, A vanishes within 2 or 3 multiples of d from the device surface so that the surface is effectively infinite in extent. This condition is nearly always true. For example, in copper or steel at power frequency, d is about 6 mm. Under 1dimensional behavior (Fig. 1)
A = A&-F; y= f i o p ~ ~ ] I n Et / Ht = Z,= y lo = dwp/o / B O ; &Van = yA
I
-
@=A
A=O Fig. 2: The Comer
(2)
(3)
THE TRANSMISSION LINE MODEL OF THE SLOT The transmission line model of a meter depth of the slot is presented in Fig. 3. The series reluctance elements carry the flux in, and the shunt elements allow leakage to the opposite face. The MMF drives the flux. Now, the reluctance is proportional to length and inverse area. Thus, if zy and 22 are, respectively, the series and shunt elements per meter in the x-direction over length Ax, we have z16x and z2/6x. To get zl,observe that the flux along a meter depth of the suface is the surface vector potential A = Wjo. z1
= MMF/mlength Fludm width
=
Hix lm = icu (Et / jo)xlm Z,
(4)
from eq. 3. The flux in Z2 /6x flows through the a i r 22 =
Fig. 1: Flux Penetration
MMFacross~aDg = Fludm width
0018-9464/88/1100-3156$01.0001988 IEEE
HXE Bxlm
= g k l
(5)
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Fig. 3: The Transmission Line Model
EXPERIMENTAL VERIFICATION: SLOT To verify the model, the arrangement of Fig. 4 was excited by a very low current of 50A to ensure linearity. The slot extends from the middle in two directions. For this long slot of 1 foot length, according to eq. 8, the magnitude of @ must fall by a factor of e in length (8/2)1/2(Cl/opo00)~/4cos(n/88). This distance ds is proportional to dg. Fig. 5 gives the log of gap length of the measured values of @ off the search coil at different distances, for gap lengths g = 6.35mm and 14.29mm. The coil was taped to a ruler and moved in to facilitate reading x, while its ends were connected to a digital voltmeter. The measured values of ds from the graph of Fig. 5 are 3.175cm and 4.824cm. But if we predict ds for g = 14.29mm from our theory, ds2 = d,l g2 / gl = 3.175 14.29/6.35 = 4.7625cm, an error of a mere 1.275%.
For deep slots, @2 = 0. For short slots of length L,
Since the slot is terminated at a surface impedance zlg. These expressions may now be used, just as the surface impedance, to terminate the slot [2-71:
For the long slot example,
r""l
-
SEARCH COIL
I I
IC) 1 I
I
Fig. 4: The Laboratory Slot Model
1
STEELPLATES
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4
0
I
2
4 6 Distance into slot (cm)
8
10
Figure 5: Variation of Logarithm of Flux Density with Depth in a Slot
CONCLUSIONS A transmission line model of the slot has been proposed and experimentally verified. f i e series elements of the model are derived from the surface impedance and the shunt elements from the properties of air. This model has been verified in the laboratory and allows us to define the slot impedance which may be used like the surface impedance to eliminate slots from the solution region.
[6] S. Ratnajeevan H. Hoole and C.J. Carpenter, “Surface Impedance Models for Comers and Slots,” ZEEE Trans., MAG-21, pp. 1841- 1843, 1985. [7] S. Ratnajeevan H. Hoole, Konrad Weeber, and N. Ratnasuneeran G. Hoole, “The Natural Finite Element Formulation of the Impedance Boiundary Condition in Shielding Structures,”J. of App. Phys., April, 1988. [8] I.D. Mayergoyz, F.P. Emad, and M. Sherif, “Electromagnetic Field Analysis of Unbalanced Regimes of Synchronous Machines,“ J . of App. Phys., 63, pp.3188- 3190, 1988 .
REFERENCES [ l ] M.V.K. Chari, “FInite Element Solutions of the Eddy Current Problem in Magnetic Structures,” IEEE Transactions, PAS-93, pp. 62-72, 1974. [2] S.R.H. Hoole, “Surface Impedance Behaviour of Iron at Comers and in Slots,” MSc. Dissertation, University of London, 1977. [3] T.H. Fawzi, M.T. Ahmed, and P.E. Burke, “On the Use of the Impedance Boundary Conditions in Eddy Current Problems,” (Invited), ZEEE Trans., MAG-21, pp.1834-1840, 1985. [4] I.D. Mayergoyz, F.M. Abdel-Kader, and F.P. Emad,”On Penetration of Electromagnetic Fields in Nonlinear Conducting Ferromagnetic Media,” J. of App. Phys., Vol. 55, pp. 618-629, 1984. [5] K.F. Ali, M.T. Ahmed, and P.E. Burke, “Surface Impedyce BEM Techniques for Nonlinear TM-Eddy Current Problems, J. of App. Phys., Vol. 61, pp. 3925-3927, 1987.
ACKNOWLEDGEMENTS: This project was supported in part by the Harvey Mudd College/Southem California Edison Center for Excellence in Electrical Systems.
