An analytical Network for Switched Reluctance
Machines with Highly Saturated Regions
Q. Yu, C. Laudensack, D.Gerling
Universiteat der Bundeswehr Muenchen, Germany
Qiang.Yu@unibw.de
Abstract- In this paper an analytical network for flux linkage
and flux density distribution of a switched reluctance machine
(SRM) is developed based on the magnetic equivalent circuit
(MEC). Due to the positional dependent reluctance in the air
gap, all paths are separately modeled. Instead of using empirical
equations, the effective air gap reluctance is analyzed from
machine geometry. Due to high local saturation in the teeth, the
machine is discretized into adjustable elements. In addition, the
slot leakage and flux distribution in the end part are taken.
Finally, the simulation is discussed and compared with PC-SRD
and finite element method (FEM).
Keywords- SRM, magnetic curves, high saturation, magnetic
path, machine discretization.
I.
INTRODUCTION
The MEC method is often used in electrical machines [1].
The magnetic curves are prerequisite for analytical design of
SRMs. However, the precise calculation of such curves poses
great challenge by complex air gap path and extreme high
local saturation [2]. Very often the air gap reluctance is
analytically solved by empirical estimations and saturation
distribution is analyzed by FEM [3-4].
Although it is impossible to get exactly the flux linkage, it
is demonstrated in this paper how to simulate the machine by
making use of its special characteristics. First, different
phases are assumed to be independent. Next, the model is
divided into 3 types according to the degree of overlapping,
namely the unaligned and non-overlapping positions (Type
1), the partial overlapping positions (Type 2), and the total
overlapping including aligned positions (Type 3). Type 3 is
very simple as almost flux goes through the air gap vertically
and not discussed here.
II.
THE CALCULATING PRINCIPLE
Consider Path i at aligned position. From stator teeth it
goes through the air gap vertically, and then split in half to
each side of rotor yoke (Fig.1).
The network is obtained by investigating one quarter of the
machine. For the air gap, stator and rotor teeth, the effective
width is divided in half so that the value of reluctance is
doubled. There are two of such elements connected in series,
so the coefficient is 4 times. The inductivity is
Li =
( 4 NTP )
2
4 R AG + 4 RST + 4 RRT + RSY + RRY
(1)
where NTP is the number of turns per pole.
However, not all paths can be modeled by (1) because the
air gap reluctance is time dependent. Consider the circuit
nearby the unaligned position (Fig.2), similarly there are 2
loops in parallel. Each loop revolves around one quarter of
the machine and goes through the air gap twice. The two air
gap reluctances are represented by RAG1 and RAG2. They
depend on rotor position and are not necessarily the same.
Fig. 2. The Magnetic path and network of the second kind
The equivalent network in Fig.2 can be further divided into
2 circuits that their inductivities are connected in parallel,
according to the difference of the air gap reluctances.
Meanwhile, the reluctance of yoke and teeth of each sub-path
is estimated as half of the original (Fig.3).
The model is developed under a 3-phase 12/8 SRM and the
path is shown in Fig.1. The rotor position θ = 0° is defined as
aligned position while θ = 22.5° as unaligned position.
The machine is divided and each part is lumped by a
reluctance block, namely the air gap RAG, stator teeth RST,
rotor teeth RRT, rotor yoke RRY, and stator yoke RSY.
Fig. 3. The equivalent network of the second kind
So each path is converted and characterized by the air gap
that it goes through only once. With symmetry the inductivity
of Path i is alternatively written as
Fig. 1. The Magnetic path and network of the first kind
Li =
( 4 NTP )
2
2 R AG + 2 RST + 2 RRT + 0.5 RSY + 0.5 RRY
( 2)
Given rotor position, there are plenty of paths going
through the air gap with different reluctances, and each
should be modelled by (1) or (2). In SRMs, only the flux
tubes vertically going through the air gap, here called the
main path, tend to pass the overlapping area of teeth and then
splits in the yoke as the first kind while the rest paths are from
(2). The separate modelling for different paths is simple, but
decisive.
III.
There are 4 paths altogether and Path 1 goes vertically
through the air gap (Fig.5). With rotor moving towards left,
the point D at end of the teeth passes W, the crossing point of
OC and DE, and WD(θ) becomes negative.
When teeth are approaching, Path 2 becomes very small.
The boundary of Path 2 and 3 tagged with X is estimated by
the shortest way principle. Path 3 and 4 are not symmetric
with curvature. The width of Path 4 can be expressed with
non-overlapping length on the rotor tooth.
THE AIR GAP RELUCTANCE
The air gap reluctance is decisive as permeability of air is
very small against even highly saturated steel. The reluctance
is calculated based on geometry only. For each type, the
following principles apply.
(1) Each path is recognized by straight lines or arcs.
(2) The shortest way principle: the flux chooses the way
in the air gap through the minimum length, through
flux leakage perpetuates.
(3) The geometrical curvature is taken into account.
A. Type 1
IV.
DISCRETIZING THE MACHINE
Because of local saturation, it is not sufficient to present
each part of the machine by a single reluctance. As part of the
network, each part is further discretized.
A. Type 1
In Fig.6, the stator teeth are divided into 3 areas, namely
top, mid and end according to the degree of saturation. The
mid and end are sorted by slot leakage tagged with LK, and
the top subjects to local saturation, depending on rotor
position. Similarly, the rotor is divided.
Any path that going through each area is expressed by a
reluctance block, except for Path 1 and 5 at the top of teeth.
The circuits of totally 7 paths are also shown. Only Path 1 and
5 go through top, mid and end of both rotor and stator teeth.
Next, the size of each block is initialized without using
Ampere’s Law. For top reluctance of each path, the width and
length are initially set as the width of air gap reluctance.
Meanwhile, the lengths of end reluctances are defined as the
same as that of the leakage, so lengths of the mid are known.
Fig. 4. Modeling of flux paths in the air gap for Type 1
This type is defined from unaligned position to where teeth
begin to overlap. All the air gap flux can be collectively
represented by 7 paths (Fig.4). As rotor moves counterclockwise, Path 1 becomes the shortest whereas Path 5, 6 and
7 are gradually losing their potential.
The lengths OD and OF are equal to outer radius of rotor
teeth. The points A, I, L are not necessarily on the same line.
AI is vertical to flux Path 1. AL is in parallel with side of
teeth DN whereas DJ is in parallel with AI. C and F are
respectively mid point of the width of teeth AB and DE.
Giving rotor position, all the geometries of the air gap
reluctances are reduced to the widths of Path 1 and 5 only.
B. Type 2
Fig. 6. The network for Type 1
Fig. 5. Modeling of flux paths in the air gap for Type 2
For rotor teeth, the flux path is a little complicated as it
does not go vertically down. It is assumed that the flux enters
rotor teeth at an angle of π / 4 measured from horizontal axis,
and the actual reluctances in the mid for Path 3 and 4 can be
seen as an isosceles triangle linked with width of its air gap
reluctance. The width of mid, end and yoke reluctances are
initially evenly divided.
B. Type 2
Fig. 7. The network for Type 2
In Type 2, 4 paths are modelled. With increasing degree of
overlapping, the flux in rotor teeth begins to go vertically
down. In analogy, the reluctance at top of teeth in Path 1 is
modeled by a triangle, indicating degree of saturation.
The size of each reluctance is initialized in a similar way.
Note that when teeth are at the beginning of overlapping, the
size of rotor top reluctance in Path 4 may become very large
and even exceed the height of teeth because it comes from the
size of air gap expressed by (DE-AY) in Fig.5. In this case the
length is set lower than width as this reluctance is not
dimensional sensitive.
When teeth are far from overlapping and so very large the
air gap length, the top tends to be unsaturated. AA1 and DD1
are initially estimated as 1/15 of the width of the tooth.
However, the saturated region may expand as teeth are
approaching, depending on phase current. The area can be
iteratively updated through AA1 and DD1 by prescribing a
flux density, but at the cost of computing time. The expected
convergence varies with current and position, and is difficult
to obtain without resort to finite element analysis.
As Path 1 is not as decisive as it is in other types, usually
the size is estimated as constant, but less accurate when
extremely high current is used. Here the size AA1 and DD1 is
assumed to change proportionally from unaligned position as
1/15, to commence of overlapping as 1/5 of width of teeth.
The widths of mid, end reluctances of teeth and the yoke
are also managed. A convergence is prescribed, based on the
fact that the flux density of any part (the mid, for example) is
circumferentially identical, as is shown in Fig.8.
Type 2 is modeled in a similar way. The highly saturated
region is expressed by an isosceles triangle with one side
equal to the length of overlapping. However when the aligned
position is approaching, the flux density on top of teeth makes
less outstanding and height of the region is not necessarily
equal to the length. Finally the triangle region reduces to a
block.
VI.
OTHER FACTORS
A. The slot leakage
V.
MANAGING THE SIZE OF ELEMENTS
The size of elements indicating the geometry of path is
subject to effective air gap length and degree of saturation.
Here the iterative updating method is used and each type is
discussed.
For Type 1, all reluctances in the air gap and top of teeth
can be regulated by the size of Path 1 in the air gap. The flux
distribution at the edge of teeth is shown in Fig.8. Path 1
chooses the shortest length centered by AD into rotor tooth.
Fig. 9. The flux leakage paths
Fig. 8. Detailed paths at highly saturated region for Type 1 (left); Regulating
the size of mid, end reluctances (right)
The sizes of the highly saturated region and the air gap are
geometrically related. It is assumed that the flux within the
triangle travels at an angle of π / 4 measured from horizontal
axis. The highly saturated region can be collectively seen as
an isosceles triangle and the width of air gap reluctance
depends only on the side AA1 or DD1.
The main flux leakage happens at the root of stator teeth.
The flux flow in the slot is shown in Fig.9, and there are in
general 2 categories. Path 1 revolves around bottom while
Path 2 passes across the slot. As the circumferential distance
between teeth is very large, Path 2 is in this case negligible.
Path 1 can be lumped by a reluctance block and the flux
goes through in parallel with that in the stator teeth. The
effective length is the path in the air and estimated as an arc
with radius of 1/4 height of stator teeth hst as
lmag _ lk = (1 / 4 )i hst i(π / 2 )
(3)
B. The machine end part
Type 1 is taken as an example and the flow chart is shown
in Fig.11. The program is characterized by 2 loops and each is
regulated by convergence from a prescribed value. The inner
loop donates to material permeability from B-H curve while
the outer to consistency of flux density in end, mid and yoke
part of the machine by adjusting the dimensions for each
reluctance.
Tab.1 Local flux density at highly saturated region under 140A
I = Finite Element Method
II= Analytical Method
Angle
22.5°
17.5°
12.5°
7.5°
2.5°
0°
Fig. 10. Flux lines at the end part, upper: aligned position, middle: partial
overlapping position, lower: non-overlapping position [5].
The paths at the end part of each type are quite similar.
They are characterized by vertical lines through the air gap
and smooth arcs elsewhere (Fig.10). The main difference
between aligned and partial overlapping is the height of the
path measured from X direction, which depends on the degree
of overlapping. At non-overlapping angles, the equivalent
length in parallel with the air gap lmin no longer equals to the
air gap length δ, but relates with position.
VII.
ST Mid
I
0.90
1.05
1.38
1.72
1.84
1.94
II
0.92
1.10
1.43
1.61
1.69
1.79
ST_max
I
0.72
1.55
1.98
1.70
1.52
1.61
II
0.74
1.69
1.87
1.53
1.34
-
RT Mid
I
0.41
0.76
1.18
1.35
1.45
1.62
II
0.36
0.88
1.22
1.46
1.54
1.63
RT__max
I
0.64
1.44
1.88
1.58
1.52
1.45
II
0.72
1.59
1.86
1.43
1.34
-
The machine is first simulated with constant single phase
current. The flux density at different parts is obtained and
compared with FEM. Tab.1 shows that the density by both
methods fit each other well. The flux density at the highly
saturated area (Stator Teeth and Rotor Teeth) is tagged with
ST_max and RT_max.
The flux linkage at unaligned and aligned and positions are
compared with PC-SRD. It is shown in Fig.12 that both
methods fit each other well and some divergence occurs
under large current.
RESULTS AND DISCUSSION
With geometry and material input, the main program is
characterized by recognizing different model types from a
rotor position.
Fig. 12. Flux linkage at aligned and unaligned positions
Next, the magnetic curves at all positions are calculated.
There are altogether 23 curves with positional difference of 1
degree (Fig.13). Starting from unaligned position, the rotor
angle from 22.5° to 16.5° is simulated by Type 1 while 0.5°
belongs to Type 3. In general the flux linkages at different
types connect to each other well.
The FEM in 3 dimensions is used to compare. High
accuracy is achieved by such a simple analytical network.
VIII.
Fig. 11. Flow chart for Type 1
CONCLUSION
An analytical network is developed to calculate magnetic
curves and flux density for SRM, which is characterized by
Fig. 13. The magnetic curves, left: analytical, right: FEM
time dependent air gap and high saturation. The model is
divided into 3 types and different flux paths are modeled
separately. The air gap reluctance is obtained from machine
geometry and empirical equations are avoided. The machine
is discretized according to the degree of saturation and all
elements are adjustable. The sizing oriented method is simple
and time saving, and high accuracy is achieved.
Institute for Electrical Drives and Actuators, Universitaet der
Bundeswehr Muenchen, Germany, 2004.
BIOGRAPHIES
Qiang Yu was born in China, 1982. He graduated from the Northwestern
Polytechnical University in Xi’an, China in 2008 and is now with the
Institutes of Electrical Drives, Universitaet der Bundeswehr, Muenchen,
Germany, as a Ph.D. student. His technical interest includes field analysis and
switched reluctance machines.
REFERENCES
[1]
[2]
[3]
[4]
[5]
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A. Arthur, “Analytical Computing the Flux Linked by a Switched
Reluctance Motor Phase When the Stator and Rotor Poles Overlap”,
IEEE Trans. on Magnetics, Vol. 36, 2000, Page (s): 1996-2003.
G.J. Li, X. Ojeda, S. Hlioui, et al, “Comparative Study of Switched
Reluctance Motors Performances for Two Current Distributions and
Excitation Modes”, IEEE Conf. Industrial Electronics, IECON, Porto,
Portugal, Nov. 2009, Page(s): 4047-4052.
R. Krishnan, “Switched Reluctance Motor Drives”, tutorial book, CRC
Press, ISBN: 0-8493-0838-0, 2001, Page(s): 30-77.
D. Gerling, A. Schramm“Calculation of the Magnetic Field in the End
Winding Region Caused on Axial Leakage of the Air Gap Field and the
End Winding Leakage Inductance of an Unsaturated Reluctance
Machine at Arbitrary Rotor Position”, Technical Report, TB 05-2004,
Christian Laudensack was born in Germany, 1981. He graduated in 2007 at
the University of Federal Defense, Munich, in aerospace engineering. His
employment experience included the German Federal Armed Force, the IAB
GmbH, Ottobrunn, the Systemzentrum für Luftfahrzeugtechnik, Erding, and
the Institute for Electrical Drives and Actuators at Universitaet der
Bundeswehr, Muenchen, Germany. His special fields of interest include
switched reluctance machines.
Dieter Gerling, born in 1961, got his diploma and Ph.D. degrees in Electrical
Engineering from the Technical University of Aachen, Germany in 1986 and
1992, respectively. From 1986 to 1999 he was with Philips Research
Laboratories in Aachen, Germany as Research Scientist and later as Senior
Scientist. In 1999 Dr. Gerling joined Robert Bosch GmbH in Bühl, Germany
as Director. Since 2001 he is Full Professor and Head of the Institute of
Electrical Drives at Universitaet der Bundeswehr, Muenchen, Germany.