ANALYTICAL CALCULATION OF LEAKAGE AND INDUCTIVITY
FOR MOTORS / ACTUATORS WITH CONCENTRATED WINDINGS
Dr.-Ing. Dieter Gerling
Philips Research Laboratories
Weißhausstraße 2
52066 Aachen, Germany
Fax: +49-241-6003-465; e-mail: gerling@pfa.research.philips.com
Abstract: There are many applications, where a
concentrated winding around an iron core is used, e.g. in
single-phase motors and transformers. For optimum
dimensioning of the iron core cross section it is essential to
know the magnetic field of the coil outside the core, because
the leakage of this part usually contributes to a large extend
to the flux density in the iron core.
In this paper, an exact analytical method to calculate the
field of the coil outside the iron core is presented, together
with some simplifications that are useful to determine the
respective flux density in the iron core.
x
coil
A
iron
y
B
M
z
M
B
A
section BB:
section AA:
1. INTRODUCTION
In the fields of electrical machinery and transformers it
is well-known to wind a coil around an iron core. A widely
spread example is the single-phase two-pole permanent
magnet motor / actuator: As a motor it is found in many
household devices and pumps, as an actuator it is used as a
direct drive of the throttle plate for internal combustion
engines in the automotive industry, see figure 1.
coil
iron
magnet
Fig. 1: Single-phase permanent magnet actuator.
The principle construction of the electromagnetic device
(neglecting mounting holes and some detailed curvature) is
the same for all the applications, see figure 2:
magnet
lstator
Fig. 2: Geometry of the single-phase two-pole permanent magnet
motor / actuator and definition of the coordinate system.
Mostly, rather sophisticated design procedures for
calculating e.g. torque and power consumption exist, but
often these procedures have one major drawback: As they do
not calculate the leakage flux of that part of the coil which is
outside the iron, the flux density level of the iron (i.e. the
saturation level) and the inductivity are not computed
correctly. As the coil is wound around the iron core (see
section BB in figure 2) the leakage is rather large. Therefore
it is decisive for the quality of the electromagnetic design to
incorporate the leakage field of the coil: Neglecting the
leakage gives the wrong inductivity, the actuator / motor
saturates at low current and therefore the output torque is too
low.
Up to now this problem can only be solved with a 3dimensional calculation using the Finite Element Method
(FEM). Two-dimensional FEM-calculations give wrong
results as well, because then the large coil overhang in axial
direction of the magnet is neglected. As 3-dimensional FEMcalculations are rather costly in terms of calculation time as
well as pre- and post-processing, this method can not be
included into a design procedure.
Because of this an analytical calculation of the leakage
field of such a coil wound around an iron core will be
presented in the following. To avoid an iterative calculation
of the flux density in the iron core, the relative permeability
of the iron is assumed being infinite ( µ r, Fe → ∞ ). This
interesting half plane y ≥ 0 this approximated field problem
is equivalent to that one we get with the help of the method
of reflection, see figure 3.
coil (N*I)
x
iron
h stator
means that the presented method is only valid if saturation is
avoided (a condition which usually is desired for the
discussed electromagnetic devices).
The assumption of infinite relative permeability of the
stator iron has the following consequences:
• The flux density level of the iron core calculated by this
procedure has to be checked against the material data: If
the calculated value is above the saturation level of the
material, some input data have to be changed to remain
below the saturation level of the material.
• A single value of the coil inductivity will be given. As
the inductivity decreases with increasing saturation, the
coil inductivity is constant below the saturation value.
z
y
a)
x
iron
coil (N*I)
h coil
z
y
2. FIELD CALCULATION
In figure 2 the Cartesian Coordinate system is defined. In
the following we will restrict the calculation of the magnetic
field to values y ≥ 0 in the x-y-plane.
Additional contributions to the flux density in the iron
core come from the part of the coil in the x-z-plane. As the
method of calculation is in principle the same, this will not
be described in detail. With this, the total three-dimensional
field problem is divided into several two-dimensional ones.
Separation of the flux density in several contributions and
adding these up is allowed because of the linearity of the
problem (the assumption of infinite relative permeability of
the stator iron was made, see above).
b coil
b)
x
(X, Y)
coil (2*N*I)
ζ
z
y
2.1. Derivation of an equivalent problem
To come to a solution we are regarding the following 2dimensional problem first: instead of considering the outside
radii of the stacked iron, a lamination with rectangular
outside contour is assumed. In addition, the case of a coil
facing a finite iron will be approximated by the case of a coil
facing an infinite iron half plane1, see figure 3. In the
2*b coil
c)
Fig. 3:
a) 2-dimensional field problem (finite iron);
b) equivalent 2-dimensional field problem (infinite iron
half plane);
c) method of reflection.
1
The problems of finite and infinite iron are equivalent, because
they can be transformed into each other by using the conformal
mapping (especially the "Schwarz-Christoffel mapping", see [1]).
This equivalence will not be used in the following, the infinite iron
half plane will be taken as an approximation.
The Cartesian Coordinate system is defined as shown in
figure 2, because then the interesting field distribution is
represented by the first quadrant ( x ≥ 0 , y ≥ 0 ). This makes
the graphical representation easier when the software tool
"Mathcad 7.0" [2] is used for the calculation of the field
distribution.
The field problem described above can now be solved by
using the law
G G
H ⋅ dl = N ⋅ I .
(1)
∫
To get an impression which degree of simplification is
permissible, the following three calculations will be
performed:
• concentrating the coil to a single point in the x-y-plane
at x = y = 0 (i.e. "0-dimensional current distribution");
• concentrating the coil to a line segment ( y = 0 ,
1
1
− hcoil ≤ x ≤ hcoil );
2
2
• regarding the real 2-dimensional coil ( −bcoil ≤ y ≤ bcoil ,
1
1
− hcoil ≤ x ≤ hcoil ).
2
2
In the last two calculations the total current ( 2 ⋅ NI ) has
to be distributed equally on the line segment or area, so that a
surface current density or a current density, respectively, has
to be considered. The solution of equ. (1) will then be gained
by integration (over line segment or area, respectively). In
the following, the positive direction of the current flow is
assumed to be in the positive direction of the z-axis.
If the coil is concentrated to a single point, the solution
of the field problem is very easy: The field lines are
concentric circles around the coil, the field strength decreases
inversely proportional to the distance from the coil. As we
are calculating for air in the whole plane (see figure 3c) the
G
G
flux density becomes B = µ 0 H .
2.2. Calculation for the 0-dimensional
current distribution
For a specific point ( X , Y ) of the x-y-plane the x- and ycomponents of the flux density distribution become (the
index "0" stands for the 0-dimensional current distribution):
µ
− sin(ς )
B0, x ( X , Y ) = 0 2 ⋅ NI
2π
(2a)
X 2 +Y 2 ,
B 0, y ( X , Y ) =
µ0
2 ⋅ NI
2π
cos(ς )
X 2 +Y 2 .
(2b)
( X , Y ) ≠ (0,0)
For ( X , Y ) = (0,0) the field strength is not defined as the
G
value for dl in equ. (1) becomes zero. The angle ς can be
calculated from ς = arctan(Y / X ) .
2.3. Calculation for the 1-dimensional
current distribution
If the coil is concentrated to a line segment ( y = 0 ,
1
1
− hcoil ≤ x ≤ hcoil ), the x- and y-components of the flux
2
2
density distribution for a specific point ( X , Y ) can be
calculated as follows (the index "1" stands for the 1dimensional current distribution):
µ 2 ⋅ NI
B1, x ( X , Y ) = 0
2π hcoil
( X − x, Y ) ≠ (0,0)
B1, y ( X , Y ) =
µ 0 2 ⋅ NI
2π hcoil
( X − x, Y ) ≠ (0,0)
1
hcoil
2
∫
1
− hcoil
2
1
hcoil
2
∫
1
− hcoil
2
− sin(ς )
( X − x) 2 + Y 2
cos(ς )
( X − x) 2 + Y 2
dx
dx
,
(3a)
.
(3b)
 Y 
The angle ς can be calculated from ς = arctan
.
 X −x
2.4. Calculation for the 2-dimensional
current distribution
Considering the coil in its real 2-dimensional expansion
1
1
( −bcoil ≤ y ≤ bcoil , − hcoil ≤ x ≤ hcoil ), the x- and y2
2
components of the flux density distribution for a specific
point ( X , Y ) can be calculated as follows (the index "2"
stands for the 2-dimensional current distribution):
( X , Y ) ≠ (0,0)
µ
2 ⋅ NI
B 2, x ( X , Y ) = 0
2π hcoil ⋅ 2 ⋅ bcoil
1
hcoil
2
bcoil
∫ ∫
1
−b
− hcoil coil
2
− sin(ς )
( X − x) 2 + (Y − y ) 2
dx dy , ( X − x, Y − y ) ≠ (0,0) ,
(4a)
µ
2 ⋅ NI
B 2, y ( X , Y ) = 0
2π hcoil ⋅ 2 ⋅ bcoil
1
hcoil
bcoil
2
∫ ∫
1
−b
− hcoil coil
2
cos(ς )
( X − x) 2 + (Y − y ) 2
Y − y
The angle ς can be calculated from ς = arctan
.
 X − x
dx dy , ( X − x, Y − y ) ≠ (0,0) .
y-coordinate in mm
25
2.5. Numerical evaluation
hcoil = 33 ⋅ 10 −3 m and bcoil = 9 ⋅ 10 −3 m . There is no field
plot for the 0-dimensional current distribution (i.e. current
concentrated in the center of the coordinate system), because
the result is obvious: We get concentric circles around
x = y = 0 with e.g. B0 = 40mT for a radius of 10mm .
y-coordinate in mm
0.017
15
0.017
0.022
10
0.022
15
0.02
0.018
0.016
0.022
0.024
10
0.022
0.02
0.018
0.024
0.018
5
0.01
0.022
0.022
0.016 0.02
0.014 0.018 0.024 0.024 0.02
0
0
5
10
15
20
25
x-coordinate in mm
vector is always perpendicular to the iron surface if
µ r , Fe → ∞ is true). Because of this, the calculation time can
0.028
be decreased by far. From the following figure 6 we can
deduce, that even for calculating the flux density on the iron
surface we have to consider the 2-dimensional current
distribution to obtain the correct result.
5
0.034 0.028 0.022
0.034
0
5
0.018
It is obvious from these calculations, that only the
calculation for the 2-dimensional current distribution gives
the right result. For evaluating the flux density in the iron
core of the coil and the inductivity of the coil, it is sufficient
to know the flux density on the iron surface, i.e. the flux
density for y = 0 . In addition, only the y-component has to
be calculated: The x-component is always equal to zero,
because µ r , Fe → ∞ has been assumed (the flux density
0.017
0
0.016
Fig. 5: Distribution of the magnetic flux density (in T) for
the 2-dimensional current distribution.
25
20
0.014
0.016
20
To solve equ. (2a,b) to (4a,b) the software package
"Mathcad 7.0" [2] has been used, where the integrals in equ.
(3a,b) and (4a,b) have been computed numerically. In the
following some results of the calculation with Mathcad are
given. Figures 4 and 5 show the flux density distribution for
the first quadrant ( x ≥ 0 , y ≥ 0 ) for 1- and 2-dimensional
current distribution. The x-axis (please refer to figure 2) is
shown horizontally, the y-axis is shown vertically.
The calculation has been performed for NI = 1000 A ,
(4b)
10
15
20
25
x-coordinate in mm
Fig. 4: Distribution of the magnetic flux density (in T) for
the 1-dimensional current distribution.
flux density in T
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
50
40
30
20
10
0
10
20
30
40
50
x-coordinate in mm
Fig. 6: Flux density versus x-coordinate for y = 0 and for 0-, 1- and 2-dimensional current distribution
(dashed, dotted and solid lines, respectively).
3. PRACTICAL RESULTS
To further reduce the complexity and the calculation
time of the software program, the following simplifications
will be introduced:
1
• For x ≤ hstator (please refer to figure 3 for the
2
definition of hstator ) we take the calculation performed
for the 2-dimensional current distribution (even if the
real field distribution will be deformed a little bit by the
finite expansion of the stator iron);
1
• for x > hstator we assume the current to be
2
concentrated into one point at ( x, y ) = (0,0) .
The second assumption is motivated by the fact that for
1
y = 0 , x > h stator the flux density calculated for the 02
dimensional current distribution is a good approximation for
the flux density calculated for the 2-dimensional current
distribution. For the example calculated above we get a
relative error of about 7% ( hstator = 49 ⋅10 −3 m ).
With this simplification a numerical evaluation of the
actuator shown in figure 1 has been carried out. It could be
shown that the flux density in the iron core coming from the
coil leakage is of the same magnitude like the flux density
coming from the desired flux path (i.e. the flux path crossing
the rotary magnet). This means that for an orientation of the
rotary magnet like shown in figure 2 the flux density of the
iron core is doubled just because of the leakage. Evaluating
the flux density in the iron core with maximum contribution
of the magnet (i.e. a magnet position perpendicular to what is
shown in figure 2) results in about 25% of the flux in the iron
core coming from leakage.
The inductivity can be calculated from the derivative of
the flux linkage with respect to the current:
L=
∂Ψ
.
∂I
(5)
The flux linkage can be obtained from:
Ψ = NΦ ,
∫
Φ = BdA .
A
(6)
As we assumed that no saturation occurs in the stator
iron, the flux density generated by the permanent magnet has
no influence on the derivative in equ. (5).2 Because of the
assumed linearity, equ. (5) and (6) can be simplified to
N ⋅B⋅ A
. In addition, we will calculate the inductivity
L=
I
just from the flux density in the iron core and the iron core
cross section:
L=
N ⋅ Bcore ⋅ Acore
I
(7)
The neglection of the coil area when calculating the
inductivity is motivated by the fact that the flux density is at
least a factor of 50 smaller than in the iron core and that not
all field lines are coupled to all turns of the coil (the cross
sections of iron core and coil are of the same order of
magnitude). In table 1 the inductivity values obtained by
different methods are summarized:
Table 1: Inductivity values obtained by different methods.
L / mH
difference
(from meas.)
analytical
calculation
w/o
leakage
6.5
-58.6%
analytical
calculation
with
leakage
14.6
-7.0%
3D FEM
calculation
[3]
measurement
13.6
-13.4%
15.7
-
From table 1 can be deduced that the analytical
calculation considering the leakage comes quite close to the
measured value. The same holds true for the 3-dimensional
FEM calculation. It is obvious that the calculation without
taking into account the leakage leads to wrong results.
2
If saturation occurs, the change of the flux linkage with a change
of the current is smaller than for no saturation. This means that the
inductivity is smaller for the case of remarkable saturation.
4. CONCLUSIONS
In this paper an analytical quasi 3-dimensional
calculation of leakage and inductivity for motors / actuators
with concentrated windings is presented.
To get access to an analytical solution, linearity is
assumed and the 3-dimensional problem is divided into
several 2-dimensional ones. It becomes obvious from this
paper that calculating the exact 2-dimensional field
distribution in each plane is essential for the knowledge of
the leakage flux, which itself is decisive for the flux density
level in the iron core as well as for the inductivity.
The presented method leads to results that are of similar
quality like results obtained from 3-dimensional FEM
calculations, but with the great advantage of dramatically
reduced computing time. This advantage becomes even more
pronounced, when small changes are made to the design: For
FEM calculations the whole model has to be set up again,
whereas for the analytical calculation just some input
parameters have to be adjusted. Therefore, the presented
method is qualified to be embedded into the design
procedure of motors / actuators with concentrated windings
(see fig. 1 and [4]).
References
[1]
[2]
[3]
[4]
Simonyi, K.: Theoretische Elektrotechnik, VEB Verlag der
Wissenschaften, Berlin, 1980 (in german)
Mathcad, Version 7.0, May 1997, MathSoft Inc.
MSC/ARIES, Version 7., April 1996, MSC/EMAS, Version
3., July 1994, The MacNeal-Schwendler Corporation
Gerling, D.: "Design of a Single-Phase Reluctance Actuator
with Large Linear Stroke", 8th European Conference on Power
Electronics and Applications (EPE), 1999, Lausanne,
Switzerland (to be published)