ISEF 2005 - XII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Baiona, Spain, September 15-17, 2005 CALCULATION OF THE MAGNETIC FIELD IN THE END WINDING REGION OF UNSATURATED ELECTRICAL MACHINES WITH SMALL AIR-GAP Dieter Gerling, Andreas Schramm Institut für Elektrische Antriebstechnik (IEA) Professur für Antriebstechnik und Automation (EAA) Universität der Bundeswehr München Fakultät für Elektrotechnik und Informationstechnik (EIT) Werner-Heisenberg-Weg 39 D - 85577 Neubiberg, GERMANY Phone: +49-89-6004-3712, Fax: +49-89-6004-3718 E-mail: andreas.schramm@unibw-muenchen.de, dieter.gerling@unibw-muenchen.de Abstract - This paper presents a simple, but nevertheless quite precise analytical model for calculating the magnetic field distribution in the end winding region of radial flux machines. The model is compared to numerical calculations using FEM and to measurements on several different machine topologies. The accordance of the analytical calculations to numerical simulations and measurements is very satisfying. 1. Introduction The magnetic field in the end winding region of a radial flux electrical machine is composed of two parts: the field produced by axial leakage of the air-gap field, the field excited by the end windings. In the following, only the first part of the field in the end winding region will be calculated. The second part is examined in detail in [1] and [2]. Both parts can be superposed to get the total field. This superposition generally is valid, because in the end winding region linearity is guaranteed as the field is dispersing in air. In this paper, radial flux machines will be analyzed. Nevertheless, it is quite obvious how to apply this analysis to axial flux machines. 2. The Field Produced by Axial Leakage of the Air-Gap Field To calculate the field in the end winding region of an electrical machine that is produced by the axial leakage of the air-gap field, the following model is investigated: A two-dimensional cross-section (cut along the axis of the machine) is regarded at. The extension of stator and rotor perpendicular to the machine axis is assumed being large against the air-gap width. The axial extension of the end winding region is characterized by x > 0. With such a model, the field in the end winding region that is produced by the axial leakage of the air-gap field can be described for conventional machines with a constant and small air-gap (like induction machines and synchronous machines). In addition, reluctance machines are covered at least in the aligned position.
3. Conformal Mapping Assuming that the relative permeability of iron is much higher than of air ( µ rair, = 1 µ rfe, ), the field problem can be solved by conformal mapping using the Schwarz-Christoffel-mapping, see e.g. [3,4]. Fig. 2: Coordinate systems for applying the conformal mapping. Regarding the w -plane, the field between the two infinite extended fixed magnetic potentials v = 0 and v = π is known (please compare with the field of an infinite plate capacitor): the field lines are represented by u= const., the lines of equal potential are represented by v= const.. The conformal mapping between the w -plane and the ζ -plane is known (see e.g. [3]), if the distance between C and D is assumed to be zero: ( ζ ) w= ln, w= u+ jv ζ = ξ + jψ (1) The conformal mapping between the ζ -plane and the z -plane can be obtained by using the Schwarz- Christoffel-mapping (see e.g. [3]); again, the distance between C-B and D-E is assumed to be zero: 1 1 dz ( ) 2 1 = A ζ + 1 ζ ( ζ 1 ) 2, z = x+ jy dζ 1 2 2 ζ 1 = A 2 ζ (2) The integration of equ. (2) gives (see e.g. [3]): where A and B are constants. w With equ (1) we get: ζ = e ; and in total: 1 2 2 ζ 1 arccos 1 ζ z= A + B (3)
1 2 w 1 2 arccos 1 ew z= A e + B (4) The function given in equ. (4) can be evaluated, e.g. using the software package MathCad 2001. The result, which gives a qualitative impression of the field lines, is shown in figure 2: y-coordinate x-coordinate Fig. 2: Evaluation of the analytical formula equ. (4). As a simplification of the above conformal mapping, just the conformal mapping between the w -plane and the ζ -plane (which should be turned by 90 ) can be taken: If µ rair, = 1 µ rfe, holds true, the influence of any part with ψ < 0 has no relevance (because the distance between C and D is assumed to be zero). This means that even the model in the ζ -plane represents the field problem, if ideal boundary conditions (especially µ rair, = 1 µ rfe, ) are assumed. This conformal mapping delivers: w u+ jv u jv ζ = e = e = e e ( cos( ) sin( )) = + eu v j v (5) From equ. (5) can be deduced that the field lines ( u= const., 0 v π in the w -plane) are half circles in the ζ -plane. This result corresponds very well with the result shown in figure 2. To check the results obtained from analytical formulae against numerical FEM-calculations, a system like shown in figure 4 will be analyzed: The energized coil generates a magnetic flux in the iron core with some leakage flux at the gap. The interesting part of this field calculation is x > 0. The result of such an FEM-calculation is given in the following figure 3b). Fig 3: a) Model for the FEM-calculations; b) results for µ r,fe =1000.
From a qualitative comparison can be obtained that the field distribution is the same like in the analytical investigations: The field lines are half circles in the air half plane, and they are perpendicular to the iron surface. 4. Detailed Comparison of the Different Calculation Methods To compare the different calculation methods, the magnetic flux density for y= 0, x> 0 will be analyzed. From the analytical solutions, only the simplified method (equ. (5)) will be regarded: For large values of u equ. (4) becomes: which is the same principle dependency like in (5). The flux density for y= 0, x> 0 will be calculated from zu w ( ) = Ae + B (6) Hdl =Θ, B =µ 0H (7) Assuming that the closed line in (7) is a field line, we get: Hdl =Θ, B = µ H B = µ Θ (8) l 0 0 eff The effective length l eff can be calculated using the following assumptions: Because of µ rair, = 1 µ rfe, the magnetic resistance of the iron path can be neglected, i.e. the iron path does not contribute to the effective length l eff. For x = 0 the effective length l eff is equal to the gap width. For x > 0 the effective length l eff can be calculated from the above analytical result (half circle), enlarged by the gap width (for the analytical solution a gap width of = 0 has been assumed). In total we obtain: leff = π x+ (9) and further: ( ) = 0 Bx µ Θ π x + (10) To make the comparison between analytical and numerical calculations much more clear, a normalized description will be used: the flux density will be normalized to the flux density value at x = 0. Consequently, we get: B norm ( x) ( ) Bx = = = 1 Bx ( = 0) πx + 1+ π x (11) For large values of x (i.e. x ), equ. (11) can be reduced to:
Bnorm ( x ) = 1 πx = π (12) x The following figure shows the results of the different calculation methods (in all calculations the same gap width has been used); all values are normalized to 1 for y= 0, x= 0. To make the following graph generally applicable, even the x-coordinate is normalized. It is quite obvious, that a normalization to the gap width is useful. As normalized values are compared, this comparison is valid for any circumferential position of an electrical machine. 5. Comparison to Measurements In the following the above derived analytical field calculation will be compared to measurements. These measurements are taken at two different motors: Unsaturated reluctance motor in aligned position: In circumferential direction the field is measured in the middle of one tooth (to avoid field disturbances coming from the tooth edges) and normalized to the value just at the iron surface. Unsaturated induction motor with blocked rotor and non-rotating magnetic field (the overlapping 3-phase winding of this motor is energized by DC-currents). The measurements of the magnetic flux density are made with a Teslameter using a Hall-Effect probe. As, in case of the reluctance machine, the air-gap is too narrow to measure the flux density within, the flux density in the air-gap is calculated as follows: B ni = µ 0, (13) where n is the number of windings of the magnetising coil, I is the current in the coil and is the radial length of the air-gap. It can be obtained from the above figures that the analytical description of the field in the end winding region of electrical machines, which has been derived in chapters 2 and 3 of this paper, corresponds quite well to the measurements. It has to be emphasized that these measurements have been taken on different motor topologies (reluctance motor and induction motor) with even different winding design (concentrated winding and overlapping winding). Fig. 4: Total flux density (normalized) in the end winding region ( y 0, x 0 a) analytical calculation according to equ. (11), b) numerical (FEM) calculation with µ r,fe =1000, c) measured (unsaturated induction motor), d) measured (unsaturated SRM, aligned position). = > ):
The good correspondence clearly visible in figure 4 comes from the fact that the air-gap leakage by far is the most important part of the total field in the end winding region, and this air-gap leakage is independent from motor topology and winding design. 6. Conclusions The mathematically derived analytical model of the magnetic field distribution in the end winding region presented in this paper on the one hand matches with the results of the numerical calculations using FEM, and on the other hand agrees with the measurements performed on a reluctance machine with concentrated windings and on an induction machine with a symmetrical three phase winding, as well. Altogether, from the end of the core to infinity the general gradients of the curves resemble. The presented analytical model is simple and can be evaluated very easily and fast, as the contribution of the end winding itself on the magnetic field distribution is being neglected. This can be done without too much inaccuracy, as this contribution is rather small compared to the field emerging from the air-gap as long as the air-gap is small. References [1] Schramm, A.; Gerling, D.: Analytical Calculation of the End Winding Leakage Inductance Based on the Solution of Neumann Integrals, published on ISIE 2005 [2] Gerling, D.; Schramm, A.: Calculation of the magnetic field in the end winding region caused by axial leakage of the air gap field and the end winding leakage inductance of an unsaturated reluctance machine at arbitrary rotor position., Technical Report No. 5-2004, University of the Federal Armed Forces Germany, Munich, 2004 [3] Simonyi, K.: Theoretische Elektrotechnik, VEB Verlag der Wissenschaften, 1980 (in German) [4] Lehner, G.: Elektromagnetische Feldtheorie, Springer-Verlag, 1994 (in German)