Design Methods of Transversally Laminated Synchronous ... · (the ratio of the d-axis inductance...

164 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 1, NO. 2, JUNE 2017

Abstract—Transversally laminated synchronous reluctance

machine (SynRM) are usually designed with multiple-layer flux

barriers to achieve high electromagnetic performance. This paper

summarizes three design methods to optimize the machine.

Related implementation procedures are detailed. Besides,

advantages and disadvantages of these methods are discussed.

Based on these conventional techniques, a comprehensive

optimization method is proposed, with which a prototype SynRM

is designed. The performances of this prototype are discussed to

verify the optimal design method.

Index Terms—Design method, flux barrier, synchronous

reluctance machine, torque ripple.

I. INTRODUCTION

YNCHRONOUS reluctance machine (SynRM) has attracted

much attention recently, thanks to the advantages of low

cost, robust structure and high torque density [1-3]. Compared

with induction machines (IM), the rotor of SynRM is simpler to

manufacture and produces less losses. In [4], testing with the

same speed and load torque, the temperature rise of an IM is

much higher than a SynRM. As for permanent magnet

synchronous machine (PMSM), the cost of SynRM is much

lower, and, problems such as the magnets demagnetization and

the performance variation with temperature rise do not exist. In

[5], a SynRM is optimized with the same volume of a ferrite

PMSM, and their electromagnetic torque is similar. Besides,

torque ripple of the SynRM is usually lower than that of the

switched reluctance machine (SRM), therefore, producing less

vibration and noise [6].

Define d-axis as the main flux path, namely the smaller

reluctance direction, and q-axis as the larger reluctance

direction. The SynRM is working based on the rotor saliency

(the ratio of the d-axis inductance and the q-axis inductance),

and the rotor design has been the main issue. Generally

speaking, the SynRM has three rotor topologies, namely the

physically salient pole (SP) structure, the transversally

laminated anisotropy (TLA) structure, and the axially

laminated anisotropy (ALA) structure [7]. The SP rotor in Fig.

1(a) has been perceived to be the simplest to manufacture. The

saliency is gained by cutting q-axis magnetic lamination, but

the pole arc of d-axis is shrunk meanwhile. Therefore, saliency

Shun Cai, Jian-Xin Shen, He Hao, and Meng-Jia Jin are with the Department

of Electrical Engineering, Zhejiang University, Hangzhou, China(e-mail:

[email protected], [email protected], [email protected], [email protected]).

of the SP rotor is the lowest among these three structures [8].

The saliency of the TLA rotor in Fig. 1(b) is produced by

punching several layers of air flux barrier in the q-axis. To

maintain rotor mechanical stress, ribs are preserved to connect

the adjacent iron segments, but the ribs cause flux leakage. It is

not difficult to manufacture the TLA rotor, and the

electromagnetic performances are better compared with the SP

rotor. The ALA rotor in Fig. 1 (c) is assembled by stacking

axial lamination layers and axial insulation layers alternatively

[9]. The saliency is usually considered to be the highest, for

example, in [10] the saliency ratio (i.e., Ld/Lq) of 10~20 is

achieved. However, there are mainly two problems about the

ALA rotor. First, the rotor structure is very complicated and the

cost of manufacturing and assembling is high. Second,

mechanically the rotor is not strong and can not operate at high

speed [11]. Therefore, the most promising structure for mass

production in industry application is the TLA rotor.

When the number of flux barrier layers for the TLA rotor is

small, the thickness of the iron segments is large and flux lines

in the q-axis can still pass along the iron segments. The flux

barriers can not separate the q-axis flux line adequately, and

thus the rotor saliency is decreased. It has been demonstrated

that the performance of SynRM with the TLA rotor can be

improved when adopting multiple-layer flux barriers [1], [11].

However, the parameters to be optimized become enormous

and are dependent on each other, making the optimization

procedure complex.

Now, finite element method (FEM) has been widely used to

optimize the electrical machines for accurate calculation and

accepTABLE time consumption [3]. If FEM is utilized to

optimize the motor parameters one by one, the final results may

be not optimal and sometimes even poor. If FEM is applied to

optimize the parameters globally, the parameters form a huge

matrix and the optimization time is not accepTABLE.

Therefore, how to optimize the SynRM both accurately and

rapidly is a key issue to investigate.

The purpose of this paper is to introduce various methods

and propose a comprehensive method to design the

transversally laminated SynRM. First of all, this paper

summarizes the conventional methods to optimize the SynRM.

Implement procedures are stated, and both the advantages and

disadvantages of these method are compared. Then, a

comprehensive optimization method is proposed to design the

SynRM parameters preliminarily. This method is applied to

optimize a 5kW prototype SynRM. The design procedure is

detailed and the performances of the prototype are discussed to

Design Methods of Transversally Laminated

Synchronous Reluctance Machines

Shun Cai, Jianxin Shen, Senior Member, IEEE, He Hao, and Mengjia Jin

S

CAI et al. : DESIGN METHODS OF TRANSVERSALLY LAMINATED SYNCHRONOUS RELUCTANCE MACHINES 165

validate this method.

(a)

(b)

(c)

Insulation Layer

Lamination Layer

Fig. 1. Three different rotor structures of SynRM. (a) SP Rotor. (b) TLA Rotor.

(c) ALA Rotor.

II. CONVENTIONAL DESIGN METHODS

Much research has been done to improve the calculation

speed and achieve high performances [3], [12-29]. In this

section, the conventional methods to optimize SynRM are

classified as three types.

A. Combination of FEM with Optimization Algorithm

One of the solutions to solve this problem is to combine the

FEM with various optimization algorithms [12-19]. By

utilizing a certain algorithm to search the potentially optimal

parameters according to the proceeding calculation results, the

parameter matrix to be solved can be simplified and the

calculation efficiency can be improved significantly.

In the first step, a parametric model is established, and the

parameters to be optimized are listed. The initial parameters are

given to calculate in the beginning. Take a rotor with two

U-shape (2U) flux barriers [14] as an example to analyze, the

rotor section is shown in Fig. 2. There are 8 parameters in the

rotor, and obviously, they are dependent on each other.

In the second step, the constraint conditions of the

parameters are set, including the parameter range conditions,

the geometry constraint conditions, and the electromagnetic

constraint conditions. The parameter range conditions include

the variation range of each parameter to be calculated, and these

1h2h4h

3h

4t3t

2t1t

Flux Barriers

q axis

d axisIron Segments

Fig. 2. Parametric modelling of rotor with 2U flux barriers.

determine the solving time. If the variation range is too large, it

may take a long time to converge. If the variation range is too

small, the final result may be not optimal. So it is important to

set a reasonable variation range. The geometry constraint

conditions mean the possible combinations of the geometric

structure. For the 2U flux barriers rotor of Fig. 2, the total

thickness of the two layers of flux barriers and three layers of

iron segments along the q-axis is equal to the difference

between the rotor outer radius and inner radius. Therefore, the

summary of h1~h4 is less than the difference of rotor outer

radius and inner radius. Similar constraint condition is also

suiTABLE for the t1~t4. When the above conditions are

satisfied, each combination of parameters can be corresponding

to a rotor structure. The rotor can be optimized but the

optimization time is very long. To reduce the optimization

matrix, the electromagnetic constraint conditions can be

attached. For the 2U flux barriers rotor of Fig. 2, the main flux

lines pass through the segments along the d-axis. Although the

leakage flux of the q-axis may shift the flux lines, the flux

linkage of the d-axis is much larger than that of the q-axis. To

distribute the flux density uniformly, the radial thickness of the

second iron segment (h3) should be similar to the tangential

thickness (t3). Similar constraint condition is also suiTABLE

for the other layers of flux barriers. By limiting the difference

and the ratio, the matrix to be optimized can be reduced

significantly.

In the third step, the objective functions, maximum iterations,

and the minimum convergence limit are set. The conventional

objective to design a machine is to ensure the torque density

and power density. Since the SynRM has inherent drawbacks of

poor power factor, and large torque ripple, the objective

function should be set with comprehensive weights of these

performances. The maximum iteration and minimum

convergence limit determine the calculating time and accuracy,

and ought to be coordinated properly.

In the last step, the computer calculates and outputs the

optimal parameter design. Firstly, the objective function of the

initial model is solved. Then the computer searches among the

parameter matrix to be optimized within the constraint

conditions, and calculates the objective of the new parameters

corresponding prototype. The objective function is compared

with the proceeding result until the maximum iteration or the


minimum convergence limit is reached. In the subsequent

calculation, the computer can utilize various optimization

algorithms to search potentially optimal parameters according

to the results calculated before, such as the genetic algorithm

[12-14], response surface methodology [15], [16], sequential

unconstrained minimization technique [17], Taguchi method

[18], particle swarm optimization technique [19] and so on. At

present, most commercial softwares have embedded these

optimization algorithms, and are capable to take multi-core

parallel calculating to reduce the calculation time.

According to the analysis above, the diagram of the

combination of FEM with optimization algorithms can be

described in Fig. 3.

As can be seen from Fig. 3, most of the work are undertaken

by the computer. The tasks for the designer are the parametric

modelling, and setting the objective functions, initial

parameters and constraint conditions. The professional

requirements for the designers are relatively low, and the

optimal design can be achieved as long as the conditions are set

reasonably.

The drawbacks of this method are the huge calculation task

for the computer and the long solving time. For the 2U flux

barriers rotor of Fig. 2, the number of parameters to be

optimized increases by 4 when the flux barrier increases by one

layer. At the same time, the dimension of the parameter matrix

increases, and the calculation time boosts. Therefore, with the

increases of flux barrier layers, the calculation efficiency

significantly drops, and it becomes much harder to achieve the

optimal design.

B. Analytical Method

Since the FEM is time-consuming, much research has been

done to seek for the replacement of FEM. In [24] and [30], an

analytical method is proposed based on the magnetic circuit

model to evaluate the SynRM with single- and double-layer

flux barriers, respectively. In [23], an analytical model is raised

based on the synthesis of d-axis and q-axis MMF effect. In [29],

a detailed model is discussed to solve the SynRM with

asymmetric flux barriers. The analytical calculation is much

faster compared with FEM. In [20], a saturating lumped

parameter model is presented to take the saturation into account.

In [21], a nonlinear reluctance network model is established to

evaluated the SynRM with a complex rotor structure. FEM is

applied in the beginning to divide the reluctance network

properly. Then the machine can be calculated and optimized

with the fast reluctance network method. Since the saturation

and iron losses are considered, the results can be satisfactory

even with high load [22].

In this part, an analytical method based on the magnetic

circuit model is introduced to evaluate the SynRM with

multiple-layer flux barriers. The calculation speed is very

advantageous, and the result coincide with the FEM when the

saturation is not severe [28].

The rotor topology discussed in this part is shown in Fig. 4.

Each flux barrier has an even thickness and so is with the iron

segments, in order to keep the flux density uniformly

distributed and to avoid local saturation. The number of flux

barrier layers is nb, the thickness of the kth layer flux barrier is

tbk, the circumferential length of the kth layer flux barrier is lbk,

and the position of the kth layer flux barrier end is bk. The flux

barrier from the outermost to the shaft is called as the 1st layer

flux barrier, the 2nd layer flux barrier, …, and the nbth layer

flux barrier.

Parameters to

Be Optimized

Reaching Maximum

Iterations or Minimum

Convergence Limits?

FEM

Calculation

Computer

Solving

Yes

No

Design

Requirements

Optimization

Finshed

Prototype with

Certain Parameters

Solving Objective

Functions

Setting Constraint

Conditions

Setting Initial

Parameters

Parametric

Modelling

Setting Objective

Functions

Searching

Parameters with

Algorithms

Fig. 3. Diagram of combination of FEM with optimization algorithm.

If the saturation in both the stator core and rotor core is

neglected, the magnetic circuit of one pole can be presented as

in Fig. 5, where Rbk is the reluctance of the kth flux barrier, Rgk

is the reluctance of the air-gap adjacent to the kth layer iron

segment, bk is the flux going through the kth layer flux barrier,

gk is the flux through the air-gap adjacent to the kth layer iron

segment, fsk is the average MMF of the stator adjacent to the kth

iron segment, and frk is the MMF of the kth rotor iron segment.

Take the kth layer flux barrier as an example, the flux

conservation can be expressed as

( 1)bk b k gk (1)

where bk and gk can be expressed as

sk rkgk

gk

f f

R

(2)

( 1)rk r k

bk

bk

f f

R

(3)

Define the angle between two flux barrier ends as

1k k k (4)

Then Rbk and Rgk can be expressed as

0

bkbk

bk ef

tR

l l (5)

0 22

ef

gkg

k ef

RD

l

(6)

where 0 is the permeability of vacuum, ef is the effective

air-gap length, lef is the effective axial length and Dg is the

diameter of air gap (the average of the rotor outer diameter and

stator inner diameter).

The three phase winding MMF in the stator frame can be

expressed as


0, 1, 2,

6 1

3( ) cos( )

ip m w

s s s

i

N I kf p t

p

(7)

bkl

bkt

Iron Segments

Flux Barriers

bk

1st

kth

nbth

d-axis

q-axis

Fig. 4. Rotor parameters in one pole.

1sf

2sf

bsnf

1g

2g

bgn

1gR

2gR

bgnR

1bR2bRbbnR

1rf2rfbrnf1b2bbbn

Fig. 5. Magnetic circuit in one pole.

where Np is the number of coil turns per phase, Im is the

amplitude of stator current, p is the number of pole pairs, is

the order of spatial harmonics in stator current, kw is the

winding coefficient of the th harmonic, s is the position in the

stator frame, is the electrical angular speed and is the phase

angle of current.

By solving equation (1)~(7), the rotor MMF in each iron

segments is shown as (8), and the coefficients are listed as

(9)~(13).

0, 1, 2,

26 1

(cos )( )

iw

rk g k

i

kf D

p

(8)

( 1)2

pmt

(9)

1

, , 1 ,

1

( )sin( ) sin( )b

b

n

k k i k i bi k n bn

i

m m p m p

(10)

1

, ,

1

,1

,

; if

; if

b

b

n q

i j l q j

q i l i

i jn q

l q j

q j l i

b a b i j

m

a b i j

(11)

1 1

1

1

1 [ (( ) ) ]i i i

g bik j i

k jjef bi

aD t

al

(12)

, ;2

ig bi

i j k

k jef bi

D tb a i j

l

(13)

Since the stator MMF and rotor MMF have been expressed

in (7) and (8), the output torque can be calculated according to

[31], and expressed as (14).

2

0

0

( )( )

2

g ef s rr r r

ef r

pD l dfT f d

d

(14)

The advantages of analytical method are the fast speed. The

dimension of the coefficient matrix of (9)~(11) is nb×nb, which

is much smaller compared with the huge node matrix number of

the FEM. So the analytical method can be used to set the initial

parameters and make preparation for the later FEM precise

calculation.

However, the iron core saturation and stator slot opening

effect are neglected in the analytical model. Since the saturation

may reduce the inductance and induce the d-q axis inductance

variation with rotor position, the average torque and torque

ripple are deteriorated under saturation. The SynRM

performance is strongly sensitive to the saturation, therefore,

the analytical method can not provide very accurate result.

C. Combination of FEM with Optimization Algorithm

As has been stated, the method to combine the FEM with

optimization algorithm is very time-consuming whereas the

analytical method can not guarantee the accuracy. So some

researches have been focused on a compromised method to

achieve high performance with high optimization speed. Some

constraint conditions are attached to reduce the number of

parameters to be optimized, decouple the parameters, and

optimize them either one by one or globally [13].

One widely used constraint method is to regard the rotor flux

barriers as uniform slots on the rotor, and therefore the angles

between every two adjacent flux barrier ends are the same [32],

as shown in Fig. 6. The parameters to be optimized become

independent on each other and it is much easier to illustrate the

effect individually. There are generally two types of constraint

conditions as following.

One type is to propose a parameter called as the number of

pseudo barrier ends [31-33], as shown in Fig. 6(a). The number

of flux barrier layer is defined as nb, the angle of adjacent flux

barrier ends is in electrical rad. Then the pseudo flux barrier

ends in one pole-pair nr can be expressed as (15). Note that,

although there is no real flux barrier end beside the q-axis, the

rotor is still divided as several pseudo uniformly distributed

ends. Therefore, nr>2× (2nb-1). The pole arc angle of the first

real flux barrier 1 can be expressed as (16). The consequence

in Fig. 6(a) is nb=3, nr=18, otherwise the relationship is not

fixed as .

2

rnp

(15)

1

1

4 2

rb

nn

(16)

The other refinement technique is to define by the angles

between the adjacent flux barrier ends [27], as shown in Fig.

6(b). The angle between adjacent flux barrier ends is m, the

controlled angle of the first pseudo flux barrier is , and the


parameters satisfy the equation (17).

1

( )2 2

b mnp

(17)

(b) Defined by the angle of

adjacent flux barrier ends

(nb=3)

1

2m

1

2m

m

m

m

m

m

m

Iron Segments

Flux Barriers

d axis

q axis

1

2

1

2

1

Iron Segments

Flux Barriers

d axis

q axis

(a) Defined by the number of

pseudo barrier ends

(nb=3,nr=18)

Fig. 6. Uniformly distributed flux barrier rotor.

The former constraint condition defines the pseudo flux

barrier ends in one pole-pair (nr) as an integer. Therefore the

pole arc angle of the first real flux barrier 1 must be the

multiplier of the angle of adjacent flux barrier ends , and the

constraint conditions are more strict. When the nr is extended to

the decimal, the two constraint conditions, in Fig. 6 (a) and (b)

are actually the same.

It is generally believed that there are three main parameters

for the SynRM, viz, the number of flux barrier layers, the

position of flux barrier ends, and the thickness of flux barriers

[1], [11], [34]. The number of flux barrier layers is defined by

nb, the position of flux barrier ends is defined by the pseudo flux

barrier ends in one pole-pair nr or the angle between adjacent

flux barrier ends (m), the thickness of flux barriers is defined

by the insulation ratio [1], [27], as shown in (18).

ins

Widthof fluxbarrierk

Widthof ironsegment (18)

According to the parameters defined above, the SynRM can

be optimized with these three main decoupled parameters. Note

that, the insulation ratio is corresponding to the the real flux

barrier. For the pseudo flux barrier in Fig. 6(a), the insulation

ratio kins is zero.

In [32], the influences of these parameters are analyzed

theoretically in accordance with a magnetic circuit model. In

[26] and [25], the parameters are optimized for the maximum

torque density and minimum torque ripple, respectively, based

on ideal sinusoidal-wave MMF. In [2], the design principle of

nr to achieve low torque ripple is proposed. In [35], the design

principle of nr is illustrated further with FEM. In [36], the

optimization rule of nr is concluded with the combination of the

number of flux barrier layers and the stator winding

configuration. In [37] and [38], the effect of the three main

parameters are investigated on the basis of mass finite element

calculation.

As can be seen from above analysis, the parameters of the

original SynRM model are dependent on each other and the

matrix to be optimized is huge. With the constraint conditions,

the number of parameters to be optimized decreases

significantly. It becomes much simpler to investigate the effect

of each parameter and more efficient to design the SynRM.

Nevertheless, the theoretical constraint conditions cannot

take the nonlinear problem into consideration. Therefore, the

optimized prototype with this method may be not optimal.

Besides, with the increase of constraint conditions, the

calculation speed can be improved at the sacrifice of

comprehensive performance.

III. A COMPREHENSIVE OPTIMIZATION METHOD

On the basis of the three conventional design methods, this

section proposes a comprehensive optimization method. This

method can help predict the performance rapidly, and set the

initial value of the parameters. The influences of each

parameter is illustrated to achieve fast design procedure.

When the air-gap length is relatively small and the diameter

of air-gap (namely the average of the stator inner diameter and

the rotor outer diameter), (19) is satisfied and the MMF drop on

the air-gap can be neglected compared with that on the rotor

flux barriers.

2

g bk

ef bk

D l

t (19)

Since only the fundamental of the stator MMF rotates at the

same speed as the rotor and produces steady MMF drop on the

rotor flux barriers, only the fundamental of stator MMF is

considered hereafter when analyzing the rotor sTABLE MMF.

For the constraint conditions in Fig. 6(a), the rotor MMF and

the fundamental of stator MMF are shown in Fig. 7, where i is

the angle between the stator current vector and the d-axis. Note

that the consequence in Fig. 7 is when the current vector aligns

with the q-axis. Otherwise the fundamental of stator MMF may

shift whereas the rotor MMF is always symmetric along the

d-axis and q-axis.

The rotor stair MMF can be obtained from the fundamental

stator MMF, and the electromagnetic torque can be calculated

in (14). The average torque is shown in (20), where kr is only

dependent on the rotor parameter and called as rotor coefficient,

as shown in (21).

2

0 19sin 2

g ef p m

ave r

ef

D l N I kT k

p

(20)

1

1rf

brnf

2rf

r

f

rf

0 ( )2

s if

2 p

0

.

.

.

q axis d axis

Fig. 7. Ideal MMF distribution in rotor coordinate.


2

2

( 4 2)2 sin

2

( 4 2)

2( 1)sin sin

( 2 )1 cos

2sin

r br

rr

r b

br

r br rb

r

r

n nn

nk

n n

nn

n nn nn

n

n

(21)

As can be seen from (20), the ideal average torque is

proportional to 1/p. However, when the number of pole-pairs is

small, the yoke thickness is large, thus the slot area to contain

the stator windings is reduced. Besides, a small number of

pole-pairs means relatively long end-winding, and both the

volume and weight are increased. So the number of pole-pairs

for the SynRM is usually selected as 2 or 3.

Define split ratio as the ratio of the stator inner diameter to

the outer diameter. As can be seen from (20), the average torque

is proportional to the square of the current Ampere-turns.

Therefore, the SynRM needs to have sufficient slot space to

contain the stator winding, thus the split ratio is usually smaller

than that of the PMSM [39]. However, if the split ratio is too

small, the rotor can not produce sufficient saliency. Besides, the

split ratio is also determined by the number of pole-pairs.

Generally speaking, a small number of pole-pairs means a

thicker yoke and a smaller split ratio [36].

As for the winding configuration, the average torque is

proportional to the square of fundamental winding coefficient,

as shown in (20), therefore, the winding configuration ought to

exhibit high fundamental coefficient. Besides, the

electromagnetic torque originates from the interaction of the

stator MMF and rotor MMF, as shown in (14). So when the

harmonic order of the stator MMF coincides with the rotor

MMF, certain harmonic torque ripple is produced. As can be

seen from Fig. 7, when the rotor flux barrier is symmetric, all

the harmonic orders of the rotor MMF are odd. So, the low odd

harmonics in the stator MMF should be avoided.

The main harmonic order for the stair rotor MMF in Fig. 7 is

k1×nr1, where nr1 is the odd number that is nearest to nr, and k1

is an integer. The main odd harmonic order for the stator MMF

is k2×Ns/p±1, where Ns is the number of stator slots and k2 is

the integer that ensures the harmonic order is odd. When the

winding configuration is chosen, selection of nr should avoid

the main harmonic order of stator MMF to achieve low torque

ripple.

Relationship of the rotor coefficient kr with nb and nr is

shown in Fig. 8 according to (21). With the increase of flux

barrier layer number, the rotor coefficient boosts. But when the

number of flux barriers is nb>4, the rotor coefficient hardly

increases with the number of flux barriers. Considered the

manufacturing and processing technique, the number of flux

barriers layer should not be too large. And the selection of nb

should also be combined with that of the nr to achieve a large kr.

With the above analysis, the number of pole-pairs, split ratio,

winding configuration, number of flux barrier layers, and flux

barrier distribution can all be determined preliminarily. Then,

the initial model of the SynRM can be established, and the

parameters can be further optimized with FEM.

The comprehensive optimization procedures are described in

Fig. 9. The preliminary design is largely based on the designer's

experience. The computer calculation includes the optimization

of stator core and rotor flux barriers, and the calculating time

can be reduced significantly. The computation can make up for

the limitation of the designer's experience.

The advantage lays on the high efficiency to design the

SynRM. The parameters are decoupled and can be optimized

one by one. The optimization procedures do not take long time,

and, generally high performance of the SynRM can be

achieved.

However, the optimization result relies on professional

knowledge of the designer and require abundant experience.

Besides, since the constraint conditions are proposed based on

the ideal model, the final prototype may be not optimal.

IV. DESIGN EXAMPLE

In this section, a prototype is optimized with the

comprehensive method proposed above. The design procedures

are detailed according to the diagram shown in Fig. 9. The

performances of this machine are discussed to verify this design

method.

A. Motor Specification

The specification and design requirements of the prototype

SynRM are listed in TABLE I. To satisfy the power density, the

average torque should be maximized. To reduce the vibration

and noise during operation, the torque ripple should be

minimized.

B. Design Procedure

To improve the rotor saliency and reduce the end-winding

length, the number of pole-pairs is selected as 3. Besides, the

split ratio of SynRM is smaller than that of PMSM, and the split

ratio is selected as 0.55 preliminarily when the number of

pole-pairs is 3.

Since the number of pole-pairs is small, modular

concentrated winding of which the numbers of slots and poles

satisfying Ns=2p±1 or Ns=2p±2 do not exist. To improve the

fundamental winding coefficient, the conventional

concentrated winding of which the numbers of slots and poles

satisfying Ns/2p=3/2 is not considered. Therefore, distributed

winding is rather common for the SynRM. For integer-slot

distributed winding, the order of the main harmonic k2×

Ns/p±1 is odd when k2 is an integer. For fractional-slot

distributed winding, the main odd harmonic order is generally

larger. For example, for the Ns/2p=9/2 configuration, the main

harmonic k2×Ns/p±1 is odd only when k2 is an even number.

Therefore, 6-poles 27-slots configuration is selected for the

prototype.

The main odd harmonic order in the stator MMF is 18k±1,

where k is an integer. So the selection of nr should avoid the

multiplier of 18. In accordance with Fig. 8 to gain high rotor

coefficient, the rotor main parameters are selected as nb=5,

nr=29.


After the basic design, the initial model of the prototype

SynRM is established. The other parameters such as the stator

tooth thickness, stator yoke thickness and rotor barrier

thickness can be optimized with FEM. Besides, the mechanical

strength should be enhanced to guarantee the tensile strength at

the maximum speed. The optimized prototype is shown in Fig.

10.

Fig. 8. Relationship of rotor coefficient kr with nb and nr.

Number of

pole-pairs

（2~3）

Initial Split Ratio

(0.5~0.6)

Winding Configuration

(Large Fundamental Coefficient

and High Harmonic Order)

Stator Core

Optimization

Selection of nr

(Avoiding the Harmonics

of Stator MMF)

Selection of nb

(Combination with nr to

Produce Large kr)

Rotor Flux Barrier

Optimization

Computer FEM OptimizationDesign by Experience

Optimization

Finished

Fig. 9. Diagram of the comprehensive optimization method.

TABLE I MOTOR SPECIFICATION AND DESIGN REQUIREMENT

Rated Power 5kW

Rated Speed 3000rpm

Maximum Speed 5000rpm

Winding Connection Y-Connected

Rated Line-Line Voltage 380V

Rated Current Density 5.5A/mm2

Stator Outer Diameter 181mm

Shaft Diameter 55mm

Core Axial Length 75mm

Slot Filling Factor 65%

C. Performance Discussion

The flux density distribution under the rated operation

condition is shown in Fig. 11, and the flux density variation

with rotor position are detailed in Fig. 12.

A+

A-

A-

A+

A-

A-A+

A+

A+

A-

A+

A-

A+

A+A+

A-

A-A-

B+

B+

B+

B+B+

B+

B+

B+B+

B-

B-B-

B- B-

B-

B-

B-

B-

C-

C+

C+C+

C+

C+C+

C+

C+C+

C-

C-

C-C-

C-

C-

C-C-

Fig. 10. Cross section of optimized prototype.

The maximum flux density in the stator tooth, stator yoke,

and rotor iron segment is about 1.6T, close to the knee point of

the lamination BH-Curve. It can help ensure the material

utilization but meanwhile avoid saturation. The radial flux

density in the stator yoke is nearly zero, whereas the tangential

flux density almost only contains the fundamental. The

tangential flux density of stator tooth is nearly zero while the

rial flux density almost only contains the fundamental. The

tangential flux density outweighs the radial flux density in both

the rotor iron segment and rotor bridge, besides, the tangential

flux density contains abundant 9th order harmonic.

The excitation field of SynRM is established by the stator

armature current. When the rotor iron segment is aligned with

the stator tooth, the reluctance is small and the flux lines pass

through that layer of iron segment. When the rotor iron segment

is aligned with the stator slot, the reluctance increases and the

flux lines pass through the adjacent layer of iron segment.

Therefore, the stator slots influence the flux density distribution

in the rotor and presents the tooth harmonic in the iron segment.

The numbers of pole-pairs and stator slots are 3 and 27. So the

main harmonic order in the rotor iron segment is 27/3=9th.

The SynRM has its rotor flux bridges saturated in order to

reduce the inductance of q-axis and produce the rotor saliency.

But the saturation point in the rotor flux bridge is not fixed. The

saturation point may shift because of the stator slots, and thus,

the 9th order harmonic in the flux bridge increases.

The d-axis inductance and q-axis inductance variations with

the d-axis current and q-axis current are shown in Fig. 13,

where the current is normalized by the rated current. The q-axis

magnetic circuit is saturated to reduce the q-axis linkage flux.

When the current amplitude is small, the rotor flux bridge is not

saturated and the q-axis inductance is relatively large. With the

increase of current, the rotor flux bridges become saturated.


Stator Yoke

B (T)

0.00.20.40.60.81.01.21.41.61.82.02.22.4

Stator Tooth

Rotor

Flux Bridge

Rotor

Iron Segment

Fig. 11. Flux density distribution under rated operation condition.

0 60 120 180 240 300 360-2

-1

0

1

2

Rotor Position (Ele-deg)

Flu

x D

ensi

ty (

T) Stator Yoke

Radial

Tangential

0 60 120 180 240 300 360-2

-1

0

1

2


Flu

x D

ensi

ty (

T) Stator Tooth

0 60 120 180 240 300 360-2

-1

0

1

2


Flu

x D

ensi

ty (

T) Rotor Iron Segment

0 60 120 180 240 300 360-3-2-10123


Flu

x D

ensi

ty (

T) Rotor Flux Bridge

Fig. 12. Stator and rotor core flux density variation with rotor position under

rated operation condition.

The q-axis inductance is then reduced and hardly varies with

the current. Comparatively, the d-axis magnetic circuit is less

saturated and the inductance varies with the current amplitude.

The saturation decreases the d-axis inductance as well as the

rotor saliency. Under the rated operation condition, the d-axis

inductance and q-axis inductance are 19.5mH and 6.2mH,

respectively, hence, the rotor saliency is 3.1.

The torque waveform with the maximum torque per ampere

(MTPA) control is shown in Fig. 14. The average torque and

torque ripple are listed in TABLE II, where the torque ripple is

defined as the peak-to-peak value of the electromagnetic torque.

The ratio of torque ripple to average torque at rated working

point is about 2%. The fluctuating torque is thus small, and the

prototype can operate steadily.

00.2

0.40.6

0.81

00.2

0.40.6

0.815

10152025

id

iq

Ld (

mH

)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

5

10

id

iq

Lq (

mH

)

Fig. 13. d-axis inductance and q-axis inductance variation with d-axis current

and q-axis current.

0 60 120 180 240 300 3600

5

10

15

20

I=0.2Im

I=Im


To

rqu

e (N

.m)

Fig. 14. Electromagnetic torque with MTPA control.

TABLE II TORQUE PERFORMANCE UNDER MTPA CONTROL WITH DIFFERENT CURRENT

AMPLITUDE

Current Amplitude Average Torque Torque Ripple

0.2 0.8Nm 0.08Nm

0.4 3.5Nm 0.28Nm

0.6 7.9Nm 0.34Nm

0.8 12.7Nm 0.31Nm

1 17,2Nm 0.36Nm

V. CONCLUSION

SynRM has been attractive alternative in some traditional


AC drive applications. Especially the TLA rotor structure, with

the combination of excellent electromagnetic performance as

well as simple manufacturing process, is perceived to be the

most promising in mass industrial production. The

electromagnetic performance of the TLA SynRM is high under

the condition that the rotor has multiple-layer flux barriers.

However, the parameters to be optimized become enormous

and are dependent on each other when the number of flux

barriers is large, making the optimization procedure rather

complex.

This paper summarized three conventional methods to

optimize the SynRM. The first method is to combine FEM with

the optimization algorithm. This method can achieve optimal

design on the sacrifice of large calculating time. The second

method is to utilize the fast analytical method to replace the

precise FEM. This method can improve the calculating speed

significantly, whereas the calculation result is not accurate with

the core saturation. The last method is to attach constraint

conditions to refine the parameter relationship. The constraint

condition can help reduce the number of parameters and save

the calculation time. However, the final prototype may be not

optimal under these constraint conditions.

Based on the conventional techniques, this paper proposed a

comprehensive optimization method to design the SynRM. The

initial parameters are selected according to the ideal model.

Then FEM is applied to make precise calculation and

optimization. Since the parameters are decoupled and

preliminarily optimized, it is easy and fast to obtain high

performance of the motor.

A prototype is designed with this comprehensive method for

the high torque density and low torque ripple. The

implementation procedures are detailed to demonstrate the

method. Finally, the machine performances are discussed to

ensure that the prototype can operate steadily.

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Shun Cai received the B.S. and M.S.

degree in electrical engineering from

Zhejiang University, China in 2014 and

2017, respectively. He is currently

pursuing the PHD degree at the

University of Sheffield, UK. His

research interests include synchronous

machine topologies and optimization.

Jian-Xin Shen received the B.S. and M.S.

degrees from the Xi’an Jiaotong University,

China in 1991 and 1994, respectively, and

the PhD degree from the Zhejiang

University, China in 1997. He was with the

Nanyang Technological University,

Singapore (1997-1999), the University of

Sheffield, UK (1999-2002), and IMRA

Europe SAS, UK Research Centre, UK

(2002-2004). Since 2004 he has been a Professor of Electrical

Engineering at the Zhejiang University. He has published more

than 220 technical papers, including 5 prize papers, and he

holds over 30 patents. He is an IET Fellow and an IEEE Senior

Member. His main research interests include design, control

and applications of electrical machines and drives, and

renewable energies.

He Hao received the B.S. and PhD degrees

from the Zhejiang University, China in

2008 and in 2013, respectively. He was an

Electrical Engineer at Hangzhou Wahaha

Group Co., Ltd (2013-2014). Since 2014

he has been a Post-doctor at the Zhejiang

University. His main research interests

include design, control and applications of

electrical machines and drives.

Mengjia Jin received the B.S. and PhD

degree from Zhejiang University, China, in

2001 and 2006 respectively. Now he is

working with Electrical Engineering

Department of Zhejiang University as an

associate professor. His works focus on

electrical machine design and drives.

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