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 r ,air = 1 µ r , Fe → ∞ ), 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:
w = ln (ζ ) ,
w = u + jv
(1)
ζ = ξ + jψ
The conformal mapping between the ζ -plane and the z -plane can be obtained by using the SchwarzChristoffel-mapping (see e.g. [3]); again, the distance between “C-B” and “D-E” is assumed to be zero:
1
1
dz
= A (ζ + 1) 2 ζ −1 (ζ − 1) 2 ,
dζ
z = x + jy
(2)
1
 ζ 2 −1 2
= A 2 
 ζ 
The integration of equ. (2) gives (see e.g. [3]):
1


z = Aζ 2 −1 2 − arccos  1  + B

where A and B are constants.
With equ (1) we get: ζ = e w ; and in total:

ζ 
(3)
1


z = A e2w −1 2 − arccos  1w  + B
e




(4)
y-coordinate
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:
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 r ,air = 1 µ r , Fe → ∞ 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 r ,air = 1 µ r , Fe → ∞ ) are assumed. This conformal mapping delivers:
ζ = ew
(
= eu+ jv
= eu ⋅ e jv
= eu ⋅ cos ( v ) + j sin ( 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 µr,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:
z (u → ∞ ) = Aew + B
(6)
which is the same principle dependency like in (5).
The flux density for y = 0, x > 0 will be calculated from
G G
G
G
v∫ Hdl = Θ, B = µ0 H
(7)
Assuming that the closed line in (7) is a field line, we get:
v∫ Hdl = Θ, B = µ0 H ⇒
B = µ0 Θ
leff
(8)
The effective length leff can be calculated using the following assumptions:
•
Because of µ r ,air = 1 µ r , Fe → ∞ the magnetic resistance of the iron path can be neglected, i.e.
the iron path does not contribute to the effective length leff .
•
For x = 0 the effective length leff is equal to the gap width δ .
•
For x > 0 the effective length leff can be calculated from the above analytical result (half circle),
enlarged by the gap width δ (for the analytical solution a gap width of δ = 0 has been assumed).
In total we obtain:
leff = π x + δ
(9)
and further:
B ( x ) = µ0
Θ
(10)
π x +δ
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:
Bnorm ( x ) =
B ( x)
= δ = 1π x
B ( x = 0) π x + δ 1 +
δ
For large values of x (i.e. x δ ), equ. (11) can be reduced to:
(11)
Bnorm ( x δ ) = δ = δ ⋅ 1
πx π x
(12)
The following figure shows the results of the different calculation methods (in all calculations the same
gap width δ 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 δ 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:
Bδ = µ0 ⋅ n ⋅ I ,
δ
(13)
where n is the number of windings of the magnetising coil, I is the current in the coil and δ 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 µr,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)