Miniaturization Design for SRM
Satisfying the Requirements of N-T
characteristic Focusing Magnetic Saturation
Takahiro Kumagai 1*, Jun-ichi Itoh 1, Keisuke Kusaka 1, and Daisuke Sato 2
Department of Electrical, Electronics and Information Engineering, Nagaoka University of Technology, Japan
2
Nagaoka Motor Development Co., Ltd., Japan

Abstract—This paper proposes a miniaturization design
method for a switched reluctance motor from therequirement of N-T characteristic. In particular, a specificparameter related to magnetic saturation is defined in order
to arbitrarily determine how much to use the magnetic
properties of the iron core material. Typical motor
dimensions are obtained by setting the parameter related tomagnetic saturation, maximum electrical frequency,
maximum current density, maximum copper loss, conductor
slot fill factor, magnetic properties of material, and input
voltage. The designed motor is analyzed by a Finite Element
Method in order to validate the proposed method.
Index Terms—Switched reluctance motor (SRM), motor
design, N-T characteristic, magnetic saturation

I. INTRODUCTION
High-efficiency and high-power density motor havebeen studied actively [1-2]. With the improvement ofmotor design and magnetic material, high power densityand efficiency of switched reluctance motor (SRM) are
reported to be competitive to those of magnet motor [3].
In order to determine the typical motor diameters suchas stack thickness, stator outer diameter, and rotor outer
diameter, a traditional design method based on output
equation of output density and output coefficient is widelyused [4]. In this method, first, the output density and theoutput coefficient are determined, next, the typical motor
diameters are back-calculated from the required torque andoutput. The determination of these coefficients relies ondesigner’s experience or past design example due to its
dependencies on many factors such as the type andstructure of the motor, the kind of the iron core material,
and winding current. However, it is difficult for SRM’sdesign to rely on the designer’s experience and the past
design example because (ⅰ) magnetic saturation is used
actively unlike typical motor and (ⅱ) new magnetic
material is being developed. Inappropriate determinationof these coefficients leads to not only low efficiency andlow power density but also design rework due to thedissatisfaction of the required N-T characteristics.
This paper proposes the design method to satisfy therequired N-T characteristic and to realize the miniaturized
motor volume considering the magnetic properties. In
particular, the specific parameter related to the magnetic
saturation is defined in order to arbitrarily determine howmuch to use the magnetic properties of the iron material.

II. PROSED DESIGN METHOD
Fig. 1 shows a N-T characteristic. The N-T
characteristic is determined by the rated output Pout,maximum rotation speed max, and base speed base. The
maximum torque Tmax(=Pout/base) and the torque atmaximum speed Tmax(=Pout/max) are uniquelydetermined from Pout, base, max. Generally, a rectangularcurrent waveform drive is employed at lower base,whereas a single pulse drive is employed at higher base.
Fig.2 shows the SRM shape. In this paper, number ofphases m, number of stator poles Nr and rotor poles Ns,
stack thickness Lh, stator outer diameter Ds, rotor outer
diameter Dr, and number of turns N are designed to satisfy
the required N-T characteristic and to realize theminiaturized motor volume. The airgap length lg isdetermined by structural restrictions. In addition, thestator’s and rotor’s pole width ts, tr are calculated by thesepole arc angle s, r determined by the Corda triangle [4],whereas stator yoke thickness ys, rotor slot depth dr, andshaft diameter rsh are determined in order to avoid extreme
magnetic saturation [4]. Note that the magnetic flux
flowing on the teeth’s side is ignored for simplification.
(A)

Tmax

Constant output
characteristic●

Torque

1

(B)
●

Tmax

base

max

Rotation speed
Fig. 1 Output N-T characteristicsys
Lh:thickness

ds

lg
dr
tr

ts

Ds/2

rsh
Dr/2

Fig. 2 Dimensions of SRM

TABLE ICOMBINATIONS OF PHASE-NUMBERS AND POLE-NUMBERS

Start
Pout, max, based,
Vdc, sat, fe_max,
Jrms_max, Wc_max, ks, Bsat
m, Ns, Nr are selected
for min. motor volume

Selection ofhighest Ns
within eq.(1)Select [m, Ns, Nr]with highest Ns from
Table.1 with in Eq.(1)Not coveredmax, Tmax=1.5
Dr is optimized
for min. motor volume

m Ns Nr  [4]Example of application [5]3 6 4 Highsupercharger, micro-machine
4 8 6 Highservo
3 12 8 High
pump, off-road vehicle
4 16 12 Low
traction, UAV
3 18 12 Low
electric vehicle
Higher Lower
electric vehicle
: inductance ratio = unsaturated aligned inductance /unaligned inductance, UAV: unmanned aerial vehicle

Dr=0
0.2

Yes
No
Cal. Lh by Eq.(5)Interlinkage flux F [Wb]Ds>Ds_lim

Cal. Ds by Eq.(13)Cal. Vvol. by Eq.(16)

Dr=Dr+dNo

Vvol.min>Vvol.

Yes
Vvol.min=Vvol., Lh_Vmin=Lh
Dr_Vmin=Dr, Ds_Vmin=Ds

Isat

0.15

sat=1

0.1

for N-T requirement

Cal. N by Eq.(15)Cal. ITmax by Eq.(4)Tout_max=TmaxTmax
Cal. Imax by Eq.(3)(4)
Tout_max=maxTmax
Sim. Tmax_ave , Tmax_ave
with ITmax, Imax
Tmax_ave≈Tmax
Tmax_ave≈Tmax
YesEndsat=1.5

0.05
0 is updated
Tmax[n]Tmax[n-1]×
Tmax/Tmax_avemax[n]max[n-1]×Tmax/Tmax_ave
NoFig. 3 Proposed design flow of SRM

Fig.3 shows the miniaturization design algorithm
satisfying the required N-T characteristic. As a input
constants, there are eight variables; desired N-T
characteristic (operating point (A) and (B) in Fig.1), input
voltage Vdc, saturation level sat, maximum electricalfrequency fe_max, maximum current density Jrms_max,
maximum copper loss Wc_max, conductor slot fill factor ks,
and saturation magnetic density Bsat. The motor diameters
are derived from the following the input constants or the
optimization condition in each section.
Section Am, Ns, Nr: operating point (B) and fe_max
Section B
Lh: operating point (A) and sat
Section C
Ds: operating point (A), Jrms_max, Wc_max, ks
Section D
N: operating point (A) or (B), Vdc
Section E
Dr: minimization of motor volume
In addition, Tmax, max are the adjustment parameters
for satisfaction of the required N-T characteristic, andthese adjustment method are explained in section F.
A. Number of Phases and Poles m, Ns, Nr
Table 1 shows the typical combinations of the phasenumbers and the pole-numbers and the examples of theseapplications. Selection of higher Ns leads tominiaturization of motor volume thanks to reduction of thesat=2
Unaligned
position0 is adjusted

Aligned position

10

20
30
Current i [A]4050

Fig.4 Relationship between current amplitude and magnetization
characteristic

stator yoke thickness and the coil-end length. Therefore,
the combination with highest Ns is selected within therange where electrical frequency at max does not exceed
fe_max; the range is expressed by (1).
Nr 

2 fe _ maxmax.......................................................... (1)Note that the selection of higher Ns also leads to increaseof the magnetic flux flowing on the side of the teeth. This
results in high unaligned inductance, i.e. low inductanceratio . In the proposed method, optimum design
considering its magnetic flux cannot be conducted because
it is ignored for simplification. According to Ref.[4], the is decreased from four-phase 16-12 type and three-phase18-12 type. Therefore, the three-phase 18-12 type is thelimit for the proposed design method ignoring themagnetic flux flowing on the side of the teeth. Since theSRM with relatively low Ns is used for low and medium
power application as shown in Table 1, the proposed
design’s target will be low and medium power application.
B. Stack Thickness Lh
Fig.4 shows the relationship between the current
amplitude and the magnetization characteristic. The
proposed method firstly defines the parameter sat as (2).
I sat  T max
I sat

where I sat Bsat lg
N 0........................... (2)where ITmax is the amplitude of the current reference at the
maximum torque and 0 is space permeability. The
magnetic flux density in the airgap reaches Bsat at thecurrent of Isat, i.e. Isat is saturation start point. Therefore,sat is the parameter to determine how much to use themagnetic properties from the saturation start point.
The instantaneous maximum torque under magnetic
linear region, i.e. i ≤ Isat, is expressed as (3).

D 
N 2 0 Lh  r 
 2  i 2 ..................................... (3)lg

On the other hand, the instantaneous maximum torqueunder magnetic saturation region, i.e. i ≥ Isat, is expressedas (4).
Tout _ max 

Ns
1
D 
NBsat Lh  r  (i  I sat ) .............................. (4)
m
2
 2 

Fig.5 shows the comparison of the instantaneous
maximum torque between the theoretical value calculatedby (3) and (4) and the FEM results. As shown in Fig.5, thetheoretical value is the almost same as the FEM results.
The average torque is smaller than the instantaneous
maximum torque due to the torque ripple. In this paper,
the ratio of the instantaneous maximum torque to theaverage torque is defined as Tmax=Tmax_max/Tmax at
operating point (a). Assuming with ITmax ≥ Isat, the stack
thickness Lh is expressed as (5) from (2) and (4).
Lh 

2m0T maxTmax

1 ........................................... (5)N s Dr Bsat lg ( sat  )2
2

Selection of higher sat leads to the reduction of thestack thickness; therefore, the motor volume is reduced by
active use of the magnetic properties of the iron material.

6
Instantaneous maximum
torque Tout_max[Nm]

Ns 1
m 2

Error:7.5%

4
3

Error:6.8%

2
1

sat=10

10

Sslot _ J 

dT max  sat Bsat lg
Scoil NIT max dT max............ (8)ksks J rms _ max
ks J rms _ max
0

where Scoil is the conductor area (copper core wire area) in
the slot. By solving Sslot ≥ Sslot_J with (7) and (8), therequired slot depth ds can be derived (Omitted due to thepage limitation). Note that the minimum slot depth ds
which satisfies Sslot ≥ Sslot_J is defined as ds_J.
Next, the determination method to satisfy the maximum
copper loss Wc_max is explained. The copper loss isexpressed as (9) from (2) and (6).
Wc  N s

B l
 R Lcoil N
 L
I RMS 2  N s R coil ( sat sat g
ks Scoil N
ks Scoil0dT max )2 ...... (9)

40

50

20
Max. Error: 6.4%
in the exam. motor

10

Line: Eq.(6)Bot: FEM results

0
0

500

1000
1500
2000
Rotation speed [r/min.]

2500

3000

Fig. 6 Relationship between speed and current RMS value (dTmax=1/3)
Statorts
2

Slot spacefor one coil

s
2
2
Ns

s
2
Dr
 lg
2

ys

ds

(a) Stator slot of SRM
ds

Cross sectionof slot spacefor one coil

Lend



On the other hand, the required slot area per coil Sslot_J
is expressed as (8) from (2) and (6).

20
30
Current i [A]

Lh

Cross sectionof stator tooth  D
Sslot ds 2  (  sin  s )( r  lg )ds ..................... (7)2 Ns
Ns 2 2sat=2

Fig.5 Instantaneous maximum torque

1 TdT max
IT m ax 2 dt  IT max dT max .......................... (6)T 0where dTmax is the ratio of the electrical period to the pulse
width. As shown in Fig.6, the theoretical value is thealmost same as the FEM results.
Fig.7 shows the diagrams of the stator slot. Fig.7 (a) is
the stator slot, whereas Fig.7 (b) is a cross section of thestator tooth and winding.
First, the determination method satisfy the maximum
current density Jrms_max is explained. The slot area per coil(the shaded area in Fig.7(a)) is expressed as (7).

sat=1.530

C. Stator Outer Diameter Ds
Fig.6 shows the relationship between the speed and thecurrent RMS value. Assuming that the current waveformis the pulsed current with the current amplitude ITmax, thecurrent RMS value is expressed as (6).
I RMS Error:8.7%

Line: Eq.(3) and Eq.(4)Bot: FEM results

50Current RMS value IRMS [A]Tout _ max 

Cross sectionof winding
Lcoil Lend4ts(b) Cross section of stator tooth and winding of SRM
Fig.7 Diagrams of stator slot

where Lcoil is the average coil length per turn as shown in
Fig.7(b), is expressed as (10).
Lcoil  2 Lh  2ts   Lend ............................................. (10)Lend 

Sslot  D
ds  (  sin  s )( r  lg ) .............. (11)
ds
2 Ns
Ns
 2 2

D

D
 
Ds  2  r  lg  ms ( r  lg )sin  s   d s  .................. (13)
2
 2
2


D. Number of Turns N
The determination method to avoid higher the backelectromotive force (EMF) Vemf at base speed than the
input voltage Vdc is explained. The method of determining
from the maximum speed is omitted due to the page
limitation. The back EMF Vemf at base is expressed as (14).
Vemf 

Fsat
N
D 
base  s NBsat Lh  r  base ....................... (14)
m
m
 2 

By solving Vemf ≤ Vdc with (14), the number of turns N
is expressed as (15).
NmVdc
 N ( A)D ................................... (15)
N s Bsat Lh  r  base 2 

Note that the maximum number of turns N which
satisfies Vemf ≤ Vdc is defined as N(A). In order to reduce themaximum current amplitude for inverter, the maximum
number of turns N(A) which satisfies Vemf ≤ Vdc is selected.
E. Rotor Outer Diameter Dr
By regarding the motor volume Vvol. as the cylinder’svolume with the stator diameter and the motor’s axial
length, the motor volume Vvol. is expressed as (16).
Vvol . (T max , Dr ) 
4

Ds 2 Lall 4

Ds 2 ( Lh  2Lend ) ................ (16)Where Lall is the axial length of the motor including thecoil-end length. The parameters Ds, Lh, Lend in (16) are thefunction of Dr as explained in the previous sections;however, they are also the function of Tmax which is theadjustment parameter. Therefore, first, the motor
dimensions with initial value of Tmax are determined bysearching for Dr in order to minimize (16), next, the Tmax
is adjusted by FEM. These processes are repeated untilachievement of the desired torque.
F. Parameter Adjustment with Simulation
As shown in Fig.3, Tmax and max are adjusted bymultiplying the previous value of Tamx and max by theratio of the desired torque to the average torque in thesimulation. This flow is repeated until the simulated
average torque is the almost same as the desired torque.
This process is achieved with only few repeats because
Tmax and max can be accurately adjusted with the use ofpreviously adjusted Tmax and max. Note that the current
reference is back-calculated from (2) or (3), whereas the

TABLE II
DESIGN REQUIREMENTS AND CONSTRAINTS.
Design requirements
Output power
Pout
0.75kW
Maximum speed
max
5000r/min (Tmax=1.43Nm)
Base speed
base
3000r/min (Tmax=2.39Nm)
Input voltage
Vdc
48V
Saturation levelsat
1.6
Max. electrical frequency
fe_max2kHz
Max. current density
Jrms_max
10A/mm2
Max. copper loss
Wc_max75W (Wc_max/Pout=0.1)
Space factor
ks
50%
Sat. magnetic flux density
Bsat
1.64T (35H300)
Design constraints
Airgaplg
0.25mmStator pole arc
10deg.
s
Rotor pole arc
10deg.
r

0.45

Motor volume Vvol. [L]

By solving Wc ≤ Wc_max from (9) to (11), the required
slot depth ds can be derived (Omitted due to the page
limitation). Note that the minimum slot depth ds which
satisfies Wc ≤ Wc_max is defined as ds_C.
In order to reduce the motor volume, the minimum slot
depth ds which satisfies both Sslot ≥ Sslot_J and Wc ≤ Wc_max
is selected; therefore the slot depth ds is expressed as (12).
ds  max(ds _ J , ds _ C ) ................................................. (12)On the other hand, the stator outer diameter Ds is thesum of the rotor outer diameter, the air gap length, the slot
depth, and the stator yoke thickness, is expressed as (13)Coil-end: consider
Lend is Eq.(16)0.4

Dr_Vmin=60.2mm.
(Lall/Ds=2.4)

0.35
Dr_Vmin=71.8mm.
(Lall/Ds=3.0)

0.3

Coil-end: ignore
Lend=0

0.25
20

40
60
80
Rotor diameter Dr [mm]100Fig. 8 Relationship between rotor diameter Dr and motor volume Vvol.

control parameters of on-angle and off-angle are decided
according to Ref.[4].
III. DESIGN EXAMPLE OF 0.75KW SRM
Table 2 shows the design requirements and constants ofthe design example. In the design example, Pout is 750W,
max is 5000r/min, and base is 3000r/min. Therefore, Tmax
is 2.39Nm and Tmax is 1.43Nm. Assuming that 35H300 is
employed for the motor core; Bsat is 1.64T from the valueof the magnetic flux density at 5000A/m in the data sheet.
A. Number of Phases and Stator- Rotor- PolesIn the design example, fe_max is 2kHz. This is because
the sampling frequency is assumed to be 20kHz; and thevalue with a margin of 1/10 is set as fe_max. By calculating
the Eq.(1), the condition of Nr ≤ 24 should be satisfied.
Therefore, three-phase 18-12 type SRM which is the limitof the proposed method is selected according to Table 1.
B. Miniaturization design
Fig.8 shows the relationship between the rotor diameter
Dr and the motor volume Vvol.. As shown in Fig.8, there isa point to minimize Vvol. by changing Dr. This minimum
point is decided by two factors as following. (ⅰ) Lh can be
reduced by increasing Dr from Eq.(5) because the torquedepends on the rotor size, i.e. the air gap area. (ⅱ) Ds isinevitably increased by increasing Dr from Eq.(13)because the stator diameter is outside of the rotor diameter.
Therefore, there is a trade-off relationship in two factors

0Error: 0.19%

30

Tmax=1.43Nm

1

Current i[A]

40

4

3.0

5

Fig. 9 Number of trials and average torque
TABLE IIIDESIGN REQUIREMENTS AND CONSTRAINTS.
Main parameters
Number of phasesm3
Number of stator poles
Ns
18
Number of rotor poles
Nr
12
Rotor diameterDr
60.2mmStator diameter
Ds
101.4mmThickness of motor coreLh
33.7mmOverall axial lengthLall
41.9mmNumber of turns
N
92turns (f=0.69mm)
Control parameters
Current amplitude at
ITmax
34.1A
operating point (A) in Fig.1
Current amplitude at
28.4A
Imax
operating point (B) in Fig.1
Lall=41.9mm
Lh=33.7mm

iw

iv

iu

iu

IRMS=18.4A(Jrms=8.3A/mm2)(Wc=65.2W)

10

0
4.0

2
3
Number of trials [times]iw

20

Error: 0.96%
Torque T[Nm](B)

Tmax=2.39Nm

Tout

2.0
Tave.=2.39Nm(Error:0.19%)

1.0
0.0
0

5

10

15
20
25
30
35
Mechanical angle [deg.]4045(a) Operating point (A)40
Current i[A]

(A)

iv

iu

30

iw

20

iu

IRMS=15.4A(Jrms=6.9A/mm2)(Wc=45.9W)

10
0
4.0

Torque T[Nm]Average torque [Nm]

2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1

Tave.=1.43Nm(Error:0.96%)

3.0
Tout

2.0
1.0
0.0

0

5

10

15
20
25
30
35
Mechanical angle [deg.]4045(b) Operating point (B)D s=
101.4mmFig.11 Simulation results of currents and torques at operating points (A)and (B)

Dr=60.2mm
coil-end
core

Fig. 10 Cross section of designed SRM(ⅰ) and (ⅱ). The point where (ⅰ) and (ⅱ) are balanced is theminimum point of motor volume Vvol.. In addition, the ratioof Lall/Ds which minimizes Vvol. is smaller in considerationof coil-end. This is because the large ratio of Lall/Ds, i.e.
flat shape, leads to increase of the axial length of the motor
due to the coil-end.
C. Parameter adjustment with FEM
Fig.9 shows the average torque when Tmax and max
are adjusted according to the algorithm explained in thesubsection 2-F. The initial value of Tmax and max is 1.5.
The larger number of trials makes the average torque thedesired torque and finally converges. In this case, whenfour trials are conducted, the error rate is 1% or less.
D. Validation of designed motor
Table 3 shows the designed motor diameters, whereas

Fig.10 shows the cross section of the designed motor. The
designed motor has m=3, Ns=18, Nr=12, Ds=101.4mm,
Dr=60.2mm, Lh=33.7mm, N=92turns.
Fig.11 shows the simulation results at operating point(A) and (B) in Fig.1. It is confirmed that the torque withan error rate of 1% or less is obtained at each point. Notethat the stator diameter is decided from the constraint of
maximum copper loss. The current density is 8.3A/mm2
for the design with 10A/mm2, whereas the copper loss is65.2W for the design with 75W. These difference is due tothe error with Eq.(6) as explained at Fig.6.
E. Effect of magnetic saturation
Fig.12 shows the current paths at operating point (A)and point (B) in cases of sat=1.2, 1.6, 2.0. As shown in
Fig.12, the selection of larger sat leads to the use of deeper
magnetic saturation. Therefore, sat is the specificparameter in order to arbitrarily determine how much touse the magnetic properties of the iron material.
Fig.13 shows the relationship between the magnetic
saturation level sat and the motor volume. Note that theInterlinkage flux F[Wb]0.18

TABLE ⅣDESIGN PARAMETERS.
2605SA1(B50=1.53T)(A)

ITmax36.1A(sat1.2)

0.12
(B)

0.06

Isat30.1A

0
0

0.4

0.8
1.2
1.6
Standardized current i/Isat [-]2

Interlinkage flux F[Wb](a) magnetic saturation level sat=1.2
0.18

(A)

ITmax34.0A(sat1.6)

0.12
(B)

0.06

Isat21.3A

0
0

0.4

0.8
1.2
1.6
Standardized current i/Isat [-]2

Interlinkage flux F[Wb](b) magnetic saturation level sat=1.6
0.18

(A)

0.12

(B)

ITmax35.6A(sat2.0)

Isat17.8A
0

0.4

0.8
1.2
1.6
Standardized current i/Isat [-]2(c) magnetic saturation level sat=2.0

Motor volume Vvol. [L]

Fig.12 Current paths at operating point (A) and point (B) under different
magnetic saturation level
0.44
0.42
0.4

0.38

around sat=1.6

0.36

0.32
1.2

1.4
1.6
1.8
2
2.2
Magnetic saturation leval sat [-]Number of turns

N

Motor volume

Vvol.

Dr
Ds
Lh
Lall

1.8
62.6mm106.2mm
33.8mm
42.4mm
94turns(f=0.72mm)

1.6
59.2mm101.5mm
30.4mm
38.6mm
98turns(f=0.68mm)

0.376L0.312Lthe required N-T characteristic and to realize miniaturized
motor volume considering the magnetic properties.
Table 4 shows the design parameters in case of
employing an amorphous alloy (2605SA1) and high-grade
silicon steel (35JNE300) into a motor core. The volume of
the motor of 35JNE300 which has higher Bsat is about 17%
smaller than that of 2605SA1. From the above results, it isconfirmed that it is possible to optimally design the motor
according to the magnetic properties of the magnetic
material without output equation relying on designer’s
experience or past design example.

This paper proposeed a miniaturization design method
for a SRM from the requirement of N-T characteristic. In
particular, sat was defined in order to arbitrarily
determine how much to use the magnetic properties of theiron core material. The typical motor dimensions are
obtained by setting the parameter related to magnetic
saturation, maximum electrical frequency, maximum
current density, maximum copper loss, conductor slot fill
factor, magnetic properties of material, and input voltage.
The designed motor was analyzed by a FEM in order tovalidate the proposed method.
REFERENCES

0.34
1

sat

IV. CONCLUSIONS

0.06
0Magnetic saturation level
Rotor diameter
Stator diameter
Thickness of motor core
Overall axial length

35JNE300(B50=1.74T)

2.4

2.6

Fig.13 Relationship between magnetic saturation level sat and motor
volume.

motor volume is minimized with the proposed method ateach sat. As shown in Fig.13, in case of lower sat than
1.6, the motor volume is reduced thanks to active use of
the magnetic properties of the iron material. On the other
hand, in case of larger sat than 1.6, the motor volume isincreased. This is because the selection of larger sat leads
to larger ratio of Lall/Ds for the reduction of the current
density and the copper loss. This results in increase of theaxial length due to the coil-end. Therefore, in case ofexcessively large sat, the volume reduction effect is lower
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