Design of Flat Torque Switched Reluctance Motor Considering Asymmetric Bridge Converter Using Response Surface Modeling

8/4/2019 Design of Flat Torque Switched Reluctance Motor Considering Asymmetric Bridge Converter Using Response Surfac

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Proceedings of the 2008 International Conference on Electrical Machines Paper ID 1219

978-1-4244-1736-0/08/$25.00 2008 IEEE 1

Design of Flat Torque Switched Reluctance Motorconsidering Asymmetric Bridge Converter using

Response Surface ModelingJae-Hak Choi1, Yon-Do Chun1, Pil-Wan Han1, Dae-Hyun Koo1, Do-Hyun Kang1, Ju Lee2

1 Industry Applications Research Division, Korea Electrotechnology Research Institute, Changwon, Korea2 Department of Electrical Engineering, Hanyang University, Seoul, Korea

E-mail : [email protected]

Abstract-This paper presents an optimum design for obtainingflat torque of switched reluctance motor, which has highfluctuating torque due to its inherent salient structure. In order toreduce the fluctuating torque ripple causing noise and vibration,an optimization design technique has been introduced and

investigated to find geometric and electric variables by means ofcombining finite element analysis considering driving circuits andresponse surface modeling.

I. INTRODUCTION

Switched Reluctance Motor (SRM) has a lot of advantages

such as simple and rugged motor construction, high reliability,

and low cost [1]. However, SRM has some problems that limit

its applications because of its inherent structure. One of the

major problems is the fluctuating torque ripple that causes

undesirable acoustic noise and high vibration. The flat torque

depends essentially on geometric shape parameters and electric

circuit parameters, which have been adopted as two-

dimensional design variables. As shown in Fig. 1(a), thegeometric design variables are relative to the salient pole arc

such as stator pole arc s , and rotor pole arc r . The electric

design variables are relative to turn-on and turn-off angle,

which is decided by switch Qa, Qb and Qc in Fig. 1(b) [2].

(a) Configuration of initial model: switched reluctance motor

(b) Asymmetric bridge converterFig. 1. Cross section of switched reluctance motor and Drive Circuit

Gradient-based nonlinear optimization methods are

inefficient in this application where expensive function

evaluations are required, and in this application where

objective and constraint functions are noisy due to modeling

and cumulative numerical inaccuracy since gradient evaluationresults cannot be reliable. Moreover, it is difficult to be

integrated with analysis software, and they cannot be employed

when only experimental analysis results are available. In this

research an effective optimization method based on a response

surface modeling has been used to overcome aforementioned

difficulties. The optimum design, which minimizes fluctuating

torque ripple could be obtained from this work and has been

verified by experiment and analysis.

II. MODEL AND DESIGN VARIABLE

A. Model Discretion

Fig. 1(a) shows construction of 6/4 SRM with stator pole arc30 and rotor pole arc 30. The stator consists of three phases

and six salient poles with concentrated winding, and the rotor

consists of four salient poles. The drive circuit in Fig. 1(b)

consists of a single-phase diode bridge rectifier that converts

the input AC into DC and an asymmetric bridge converter that

supplies power to the SRM. The specifications of the

manufactured SRM are as follows. The outside diameter,

lamination length, air-gap length are 80.4mm, 80mm, and

0.4mm, respectively

B. Finite Element Analysis Tool

In SRM with no magnetic saturation, the instantaneous

torque is expressed by d/)(dLi/)i,(T =2

21 . Theelectromagnetic torque is proportional to the derivative of the

inductance,L, which is a function of rotor position, , and the

square of winding currents affected by inductance of windings.

Although this mathematical equation is often quoted for SRM,

it is not sufficient for accurate prediction of torque, because the

magnetic saturation effect can not be considered [1]. The finite

element method is essential for the precise calculation of the

nonlinear magnetic saturated torque [3].

Also, the switching conditions and freewheeling diodes of

the motor drive circuit have to be considered in finite element

analysis. To provide continuous torque, the drive circuit must


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Proceedings of the 2008 International Conference on Electrical Machines

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be commutated that the switches S1 and S2 of phase-A turn off

and the switches S3 and S4 of phase B turn on. When S1 and S2

of phase-A turn off, the current flows through the freewheeling

diodes D1 and D2. The current is restored to DC link capacitor

or flows through other phases. The current flow of this case is

shown in Fig. 2 as a dotted line. When switches of phase B

turn on, the current flows as the solid line.

Therefore, these kinds of conditions must be considered in

analysis. The time-stepped voltage source Finite Element

Method (FEM) is coded with the circuit equation considered

turn-on, turn-off switches and freewheeling diodes.

C. Significant Design Variables and Formulation

Inductance profile varies with the combination of stator and

rotor pole arcs, and influences the torque characteristics. Each

phase inductance profile shifts 30 in 6/4 SRM. For the flat

torque and maximum average torque, the pole arcs of the stator

and rotor have to be more than 30. If the pole arcs of stator

and rotor are smaller than 30, a large torque ripple will beperiodically generated. It is impossible to obtain flat torque

although the phase current flows ideally as shown in Fig 3(a).

Fig. 3(b) shows inductance profile of A-phase, switching

current of A-phase and torque characteristic of three phase, and

illustrate torque generation principles with pole arc

combination when s =30 and r 30. In order to generateflat torque, the flat current of each phase has to flow in rising-

inductance period (30), and can be possible by adjusting the

on and off. The stator pole arc is set to 30, because widening

the pole arc of rotor is better than widening the pole arc of

stator with respect of high slot fill factor. Torque ripple is

ideally able to be zero while increasing average torque.

Consequently, the pole arcs of rotor and stator, turn-on angle

and turn-off angle among electric and geometric variables are

selected as design variables for optimization. The object

function and design variables are represented by (1)

Object Function:

Minimize torque ripple, Tripp

Subject to:

Average torque, Tave 0.1Nm

Stator & rotor pole arcs s = 30, 30 r < 60,

Turn on angle, (60 s )/2 on + (60 r )/2

Turn off angle, 30off 30 + ( r s ). (1)

Fig. 2. Current flow in the drive circuit

(a)s &r < 30 (b) s =30 andr 30

Fig. 3. Torque generation principles

III. OPTIMIZATION PROCEDURE

The approximate optimization procedure is useful tool to

find optimum value about undefined relative equations

between objective and design variable. This method

approximates objective and constraint functions to quadratic

functions within the reasonable design space and sequentially

optimizes the approximate optimization problems in the

context of the design space adjustment strategy. Approximateoptimization problem is converged by agreement with the

actual function within an acceptable tolerance for error. The

approximate optimization based on a response surface

modeling has been applied to the optimum design of SRM.

Approximate optimization procedure consist of four parts;

Design of Experiments (DOE), Finite Element Method (FEM),

Response Surface modeling (RSM), Optimization. Firstly, The

analysis points through the DOE is well adapted to contain the

combinatorial exploration of numerous finite element

simulations required by the investigations on the effect of all

design variables of a given device [4]. Secondly, the response

values of analysis points are obtained through 2-D FEMcoupled with circuit equations of the converter. Thirdly, a

response surface and equation are estimated from the analyzed

response values by using RSM and regression analysis.

Fourthly, a feasible design region is chosen for an optimum

design. The conjugate gradient method in Microsoft Excel has

been used when deciding the best optimal model with

estimated quadratic regression equations.

A. Response Surface Modeling

The Central Composite Design (CCD) on the various DOE is

well adapted to contain the combinatorial explosion of

numerous finite element simulations required by the

investigations on the effect of all design variables of a given

device [4].RSM is a set of useful mathematical and statistical

technology. RSM statistically approximates the relationship

between the response value from performing FEA and design

variables. To make an approximate function, Least Square

Method and Variable Selection Method are used. To evaluate

the function, Analysis of Variance (ANOVA) is used. Among

many DOEs, CCD that is generally used for polynomial models

is used because maximum information can be obtained with the

number of minimum analysis times for the system.


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Fig. 4 shows the analysis points for CCD when there are two

factors for 2-Levels. The number of the CCD analsysis can be

calculated as follows.

c

k nkn ++= 22 (2)

where kis the number of design variable, 2k is the number of

experiment for the 2k factorial design, 2kis the number of axial

point, and nc is the number of replication for the center point.

RSM statistically approximates the relationship between the

response value, y, from performing FEM analysis and the

design variables with an error, and the equation can be

expressed in (3).

+= ),,,( 21 kxxxfy (3)

where u=f(x1, x2, , xk) is the true response function that has

kdesign variables, and denotes the random error that includes

measurement error on the response and is inherent in theprocess or system.

For most of the response surfaces, the functions for the

approximations are polynomials because of its simplicity,

though the functions are not limited to the polynomials. The

response surface is described as follow.

ji

k

j

k

i

k

ij

ijjjj

k

j

jj xxxxy =

= +==

+++=1

1

1 1

2

1

0 (4)

where represents regression coefficients, x is the design

variable, and kis the number of variables.

B. Approximate Optimization Process

Fig. 5 describes the optimization procedure of the SPSRM indetail. The computational procedure is as follows:

Step 0. Set the initial design and the design space. The

initial design space is assumed 50% ~ 100% of the whole

design space that includes the initial design.

Step 1. Select CCD (2k+2k+nc) sampling points within the

design space. Calculate the sampling points set by Finite

Element Analysis (FEA).

Step 2. Approximate the objective and constraints functions

to quadratic polynomial functions by RSM and ANOVA.

Step 3. Find an approximate optimum using the

approximate objective and constraint functions.

Step 4. Evaluate actual objective and constraints at theapproximate optimum value by real FEA.

Step 5. Check convergence at the approximate optimum

using actual objective and constraints function values. If the

approximate optimization problem is converged, then

terminate the optimization. Otherwise adjust the design

space.

Step 6. Select CCD (2k+2k+nc) design points within the

new design space. The new sampling points set consists of

the previous approximate optimum points and newly

selected design points. Go to Step 2 again.

Fig. 4. Central Composite Design for 2-Levels 2 Factors

Fig. 5. Approximate Optimization procedures

IV. RESULTS COMPARISON AND DISCUSSION

A. Geometric Design Variable Optimization: Rotor Pole Arc

Fig. 6 shows regression analysis results of the rotor pole arcs

on conditions that the stator pole arc is set to 30 as explained

in section II-C, and that the turn-on and turn-off angle is set to

0 and 37.5 to raise phase current as shown in Fig. 3(b). The

quadratic equation of the torque ripple and average torque are

respectively estimated in (5) and (6). Table I show the results

compared with the initial model as shown in Fig. 1.

2)(18.0)(8.158.391 rrripT += (5)

)(0058.04006.0 raveT = (6)

(a) Torque ripple (b) average torqueFig. 6. Regression analysis of torque characteristics with rotor pole arc

s=30

r=44


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B. Electric Design Variable Optimization: Switching Angles

Table II and Table III show the optimum results of the

switching variables on the initial shape model as shown in Fig.

1 and the optimum shape model as shown in Fig. 6,

respectively.

Fig. 7 shows the feasible region of turn-on and turn-off onthe optimum shape. The gray color area indicates the feasible

region and includes optimum value.

Fig. 8 shows the inductance profile and current waveform of

one phase of motor. The optimum turn-on and turn-off of the

initial shape are -6.0 and 45.0 as shown in table II.

TABLE IGEOMETRIC VARIABLES DESIGN RESULTS OF ROTOR POLE ARC

Rotor Pole ArcOptimization

InitialPoint

(FEM)

(1)Approx.Optimum

(RSM)

(2)RealOptimum

(FEM)

ConvergenceError

(1) vs. (2)

Torque ripple (Trip) 73.8% 44.6% 43.5% 2.4%

Torque average (Tave) 0.225Nm 0.144Nm 0.143Nm 0.7%

Rotor pole arc (r) 30.0 43.96 44.0 -at the same switching angle (turn-on angle: 0, turn-off angle: 37.5)

TABLE IIELECTRIC VARIABLES 1STITERATION DESIGN RESULTS OF INITIAL SHAPE

Turn-on and offat initial model

InitialPoint

(FEM)

(1)Approx.Optimum

(RSM)

(2)RealOptimum

(FEM)

ConvergenceError

(1) vs. (2)



Turn-on angle (on) 0.0 -6.0 -6.0 -

Turn-on angle (off) 37.5 45.0 45.0 -

at the same initial shape (stator pole arc: 30, rotor pole arc: 30)

TABLE IIIELECTRIC VARIABLES 3RDITERATION DESIGN RESULTS OF OPTIMAL SHAPE

Turn-on and -offat optimum model

ShapeOptimum

(FEM)

(1)Approx.Optimum

(RSM)

(2)RealOptimum

(FEM)

ConvergenceError

(1) vs. (2)



Turn-on angle (on) 0.0 -0.1 -0.1 -

Turn-on angle (off) 37.5 35.0 35.0 -

at the same Optimum shape (stator pole arc: 30, rotor pole arc: 44)

Fig. 7. 3rd iteration feasible design region of turn-on and off on optimum shape

The negative torque is generated because the phase current

flowed in the falling inductance period. However, the

switching angles of the optimum shape are -0.1 and 35.0 as

shown in table III. The negative torque is not generated

because the phase current was off before the falling inductance

period.

Fig. 9 and Fig. 10 show the waveform of one phase terminal

voltage and three phase current on initial model and optimum

model, respectively. The currents are measured as ratio of

4.2(A/div) to 3(V/div) in Fig. 9 and as ratio of 2.8(A/div) to

(2V/div) in Fig. 10. Both of the motor voltage and current

waveforms generally regard as important electric parameters

for a prediction of motor performance. The flat-top current

doesnt flow at initial model in Fig. 9, otherwise the flat-top

current flows at optimum model in Fig. 10. Here it can be

known the flat-top current waveform is important to make flat

torque waveform, because the motor torque is proportional to

the square of winding currents. If the rising and falling ofphase current is fast or slow according to the switching angle,

the flat torque can not be obtained.

(a) Initial shape (b) Optimum shapeFig. 8. Inductance profile and current of initial and optimum motor

(a) simulation

(b) experiment (20V/div, 4.2A/div)Fig. 9. The voltage and current of initial model


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Fig. 11 shows the energy conversion loop of initial model

and optimal models. Estimation for the average torque could be

illuminated with areas on the energy conversion loop. It can be

known that the initial model has an advantage with respect of

the high average torque, because the energy conversion loop of

initial shape is much wider than that of optimum model.

Fig. 12 shows the analyzed torque waveform. The torque

ripple is reduced from 73.8% to 43.5% by optimizing the

geometric pole arcs, and is reduced from 43.5% to 38.5% by

optimizing the electric switching angles. The torque ripple is

reduced by about two times than the initial one. The average

torque was satisfied of constraint condition.

(a) simulation

(b) experiment (20V/div, 2.8A/div)Fig. 10. The voltage and current of optimum model

Fig. 11. Energy conversion loop

Fig. 12. Instantaneous torque waveform

V. CONCLUSIONS

Firstly, to minimize torque ripple, the geometric rotor pole

arc has been optimized from initial 30 to optimum 44 by

using the approximate optimization. Secondly, the optimum

combination of electric turn-on and turn-off angle can also be

obtained for each initial and optimum shape. The torque rippleis reduced by about two times than the initial one. It can be

also known that there is the trade-off between a torque ripple

and an average torque. This paper measured the phase currents

and the terminal voltage of both models and then shows that

the analysis method considering the drive circuit is suitable for

SRM optimization. The results prove that the optimization

procedure is efficient in this application where expensive

function evaluations are required and in this application where

objective and constraint functions are noisy due to modeling

and cumulative numerical inaccuracy. The optimization

introduced in this article may be also used effectively for

various electric machines.

REFERENCES

[1] T. J. E. Miller, Switched Reluctance Motors and their control, OxfordUniversity Press, 1993, pp. 53-70

[2] Jae-Hak Choi, Youn-hyun Kim and Ju Lee, "Geometric design of polearcs considering electric parameters in switched reluctance motor,"International Journal of Applied Electromagnetics and Mechanics, vol.19, no.1-4, pp. 275-279, 2004.

[3] Jae-Hak Choi, Tae-Heoung Kim, Yong-Su Kim, Seung-Jun Lee, Youn-Hyun Kim, and Ju Lee, Finite Element Analysis of Switched ReluctanceMotor Considering Asymmetric Bridge Converter and DC Link VoltageRipple,IEEE Transactions on Magnetics, vol.41, no.5, 1640-1643, May2005.

[4] Box, G. E. P. and Willson, K. B., On the Experiment Attainment ofOptimum Conditions,Journal of the Royal Statistical Society, Series B.,13, pp.1~14, 1951.

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Design of Flat Torque Switched Reluctance Motor Considering Asymmetric Bridge Converter Using Response Surface Modeling