Modeling and Control of Hybrid Stepping Motors

5/28/2018 Modeling and Control of Hybrid Stepping Motors

Modeling and Control of Hybrid Stepping MotorsKiyonobu Mizutani' Shigeo Hayashi* Nobuyuki Ma tsuiSum itomo Heavy Industries , Ltd.Research Development Cen terPower Transmission Con trols Gro up

6-1 Asahi-cho, Ohbu, Aichi, 474 Japan .Depar tment of Electrical and Com puter EngineeringNagoya Inst i tute of TechnologyGokiso, Showa, Nagoya, 466 Japan .

AbstractStepping motors have been widely used in industry and con-sumers products . However, unlike the conventional ac motors.stepping motors have not yet a clear equivalent circuit. Thismakes it difficult to achieve the precise torque and speed con-trols. B ecause the hybrid stepping m otor is promising as a low-speed, high-torque servo motor, it is expected to achieve a clearequivalent circuit.

Based on the s ta tor and rotor teeth arrang ements , th e perme-ance distr ibution with respect to th e rotor posit ion is obtained.The magnetic equivalent model is introduced by using the ob-ta ined permeance distr ibution. The voltage equation is , then,derived from the magnetic equivalent model. Th e d-q axis isdefined for the permeance distribution ; Th e orientation of tlied-axis is chosen on the center of rotor teeth that corresponds tothe peak of permeance. Based on this d-q axis, the dc motormodel of the three phase hybrid s tepping motor is derived Th econtrol strategies of torque and speed can be derived from thedc motor model. The effectiveness of the proposed modeling hasbeen verified by experiments.

1 IntroductionStepping motors ar e used not only as a conventional rotat-ing machines but as actuators for positioning in industries andconsumers products . Th e structure of their core and winding ar-rangements have various configurations and corresponding con-trol me thods are proposed and practiced.[11[2]However, unlike theother ac motors such as induction motors and blushless dc mo-tors , the s tepping motor has not yet a clear equivalent electriccircuit.[3] Among various types of st eppin g motors. th e hybridstepping motor is promising as a low-speed, high torque servo-motor and it is expected to obtain a clear equivalent circuit forprecise torque and speed control.In the paper, the modeling and control s tra tegies of bipolar-excited three phase hybrid s tepping m otor are presented. First ofall, the permeance d istr ibution with respect t o the rotor positionis calculated throug h a simplified flux inodel of th e given s tato rand rotor teeth arrangem ent. Based on the calculated perme-

ance distribution, the equivalent magnetic circuit of the motoris obta ined, from which the voltage equation is derived. Th evoltage equation is similar to that of the conventional blushlessdc motor. Th e d-q axis should be defined to obtain a dc m otormodel. Here, d-axis is chosen on th e center of rotor tee th th atis a peak point of permeance. As a result, a dc motor model ofthe three phase hybrid s tepping motor is obtained. T he control289-7803-1462-x/93 03.0001993IEEE

strategies of torque and speed ca n be easily derived from the d-qmodel of t he motor.Since the original equivalent magnetic model is closely relatedto the s ta tor and rotor teeth arrangement, the equivalent mo-tor param eters of th e f inal dc motor can be estimated under anappropria te driving condit ion during th e design stage. The ef-fectiveness of the proposed modeling is verified by comparisonsbetween th e experimental and the calculated control character-istics durin g the design stag e of th e motor.

2 Permeance ModelThe configuration of the bipolar-excited three phase steppingmotor is shown in Fig.1, where the concentric windings are lo-cated on the s ta tor core and the poles of s ta tor and rotor have anumber of teeth .

a bs ta to r rotor

/ stsator

Figure 1: Configuration of three phase s tepping motor.

Consider a permeance between th e stator an d rotor poles uti-der t lie assumption t hat a. reh tive permeability of the iron core isinfinite and th at th e lea.kage flux i n the direction of the plane per-pendicular to t he paper can be neglected. Since the s ta tor wind-ing is a concentric winding located on the sta.tor pole. the stat orflux linkage in on e phase of the s ta tor winding (a-b in Fig .1) isexamined. As previously explained, since the poles have a nuin-ber of teeth , the b asic exaniina.tion of th e flux linkage should begiven for one set of teeth com bination. Fig.2 shows the sta torand ro tor t ee th ar rangemen t to examine the flux ~a . t h . [ ~ ]

n the fig ure, ideal flux pa.th is P I . but oth er flux pa.ths P2 - Psshould be considered in the actual m otor because of th e existenceof leakage flux. Here, five kinds of flux paths having a. length


I I tI

F

Figure 2: Flux in a.ir gap

from g t o g + 2d are considered and each path corresponds topermeances PI - Ps. Then, the to ta l permeance between theteeth shown in Fig.2 can be given as;P = Pl + 2(Pz + P3 + P4) + Ps (1)

Here, each permean ce is given below.t - xPl = po

2 g + 2dx g 2 d - ; x xP4 = Po-log

where x is a sta tor-rotor displacement in m m. Wit h differentsta tor-rotor displacement, each permeance can be calculated forcorresponding flux path.

r,

Figure 3: Permeance varia t ion with respect to rotordisplacement angle

Fig.3 shows the permeance varia t ion with respect to the rotordisplacement angle, where 0 = 0 means the teeth axes of the

sta tor and rotor are in l ine . As shown in Fig.l, a pole has anumber of teeth and th e permeance of the pole can be obtainedas a parallel combination of permeances of each tooth. Assuminga. pole has n teeth. a resultant permeance of a pole Pg can berepresented a.s;

3 Actual Motor PermeanceAs stated above, a resultant permeance of a pole is obtainedunder the assumption that a relative perme ability of the iron coreis infinite. However, it is well known that an actual core mate-rial has a relative permeability of an order froni several hundredsto several thousan ds. Since a ratio of flux path length in theair gap and in the core is about 1:1000, magnetic re luctance ofthe gap is considered to be nearly equal to that of the iron coreand the a resultant permeance of a pole would be reduced fromthat obtained in Eq.(3) . Moreover, the magnetiz ing curve of theiron core generally has a nonlineari ty and this makes an accu-rate estimation of permeance difficult. To meet these problems,the experimental comparison between th e actual permeance and

Eq.(3) have been done for a number of prototypes of differentsize and capacity. From these experim ental comparisons. it isconcluded tha t E q.(3) should be modified by using coefficientsICl, K Z whose values are determined by motor constructions andcore materials.

P ( 0 ) = K I P g+ z


s t a to rI\ de S - - s ide NIcoilII b

s coiltor

Figure 5: Configurat ion of three phase hybrid s tepping m otormotor is chosen on the center of rotor teeth t ha t has a peak pointof permeance. Fig.4 shows a definition of d-axis of the stepping

5 Magnetic Circuit of Three PhaseHybrid Stepping MotorFig.5 shows the conf igurat ion of the prototype of th e threephase hybrid s tepping motor . Th e s tator pole has an concen-trated winding and poles of t he s t a to r and ro tor has a numberof teeth. An axial-magnet ized permanent m agnet is mounted onthe rotor and th e teeth of the rotor ar e displaced by a half pitchin rotor cores A a n d B. The rotor core A is first considered inorder to examine t he var iat ion of permeance.In Fig.5, for example, th e motor has a symmetr ic conf igurationand, therefore, th e permeance var iat ions of phase U , and U, withrespect to th e rotor displacement are th e same. Th e correspond-

ing permeance is represented as PLr In the s imilar manner , PVand PW are the corresponding permeances for phases VI, V2 andphases Wl, W2.Similarly, Pb, P; and Pty are defined for phasesU , V a n d W in the rotor core B. N ext , as s ume tha t the w ind ingsUl and U2 are in ser ies and generate the f lux of the sam e polar ityfor phase [ current . Then , magnetromotive forces of phase U ,V and W are represented as H ul , H u ~...,H w 2 and

Here, each magnetromotive force can be represented as a productof number of turns of the s ta tor winding and exci ted c urrent ofeach phase an d;

H~ = N,auH v = hTS i vHW = N , i w

Taking account of a symmetry of the motor construct ion, theequivalent magnetic circuit of the motor in Fig.5 can be rep-resented as in Fig.6, where H , is a magnetromotive force of apermanent magnet on the rotor.[6]

coreotor tcore A 1rotor

1 IFigure 6: Equivalent magnetic circuit of three phase hybrid

s tepping motor

6 Voltage EquationsThe vol tage equat ion of the three phase hybrid s tepping m otorcall be derived from a calculation of a flux linkage of each ph asebased on the equivalent magn etic circuit in Fig.6. From Fig.6,the flux linkages Xu, AV and X w for phases U , V a n d W aregiven below

Xu =

AV =

x w =

29


-2(Pu + P b ) ( P w + P L ) H u-2(Pv + P L ) ( P w + P b ) H V+ { P w ( P t , + PL + PL)-P;Y(Pu + Pv + Pw)}H,] (8)

( 9 )where

C P = P u + P v + P w + P; + P; + PLFrom the as s umpt ions tha t Pu , PV and Pnr are the s inusoidalfunctions of the rotor displacement as shown in Fig.3 and aregiven by Eq. (3) , Pu, PVand Pw are given below.

pu = P + pacaseP v = + P a c o s ( 8 5 ( 1 0 ){ Pw = P + P o c o s ( 8+ 5

{ P& = P Pocos(8 + g xBy replacing 8 in Eq.(lO) with B x , Pt,, P; and P;Y are obtainedas shown below.

PL = P P o c o s 8Pt. = P Pocos(8 2 (11)Rearranging E qs(6)-( 8) , the fol lowing results are obtained.

8 - . 4 -A = iPN:Zu PN:iv PN:iw

A V = -gPN:Zu + -PN:iv P N : i w3 3

+Po COS t ,H, (12)4 - . 8 -23

A w = - - P N f Z u P N : i v + ?PN:Zw2

pa os(e .)N,H, (13)4 - . 4 - 8 -3 3

+Po COS(^ + - . )N,H, (14)

Rated volt. 28 ( V )Rated curr . 18 A )R ated ou tpu t 250(W)Weight 10.0 ( K g )

where

Rate d torque 2.45 (kgf in)Rated speed 100 ( rpm)No. of pole 222Dimension 140 x 160 ( inm)

L , = M , = $PN:PONSH , = M,I,From discussions above, th e vol tage equat ion of the three ph asehybrid stepping motor is given as follows : (R, : Stator windingresistance)

[ = [ 0 R, 0 1 2v w 0 0 R, 2= [ R , R, 0 ] [ u

0 0 R , 2,

R, o o z

Here, the perman ent m agnet on th e rotor is represented as M,I,,where M , i s t he mutua l induc tance and I , is the equivalent mag-net izing current . And a leakage indu ctan ce of t he s t a to r w ind ingis neglected. This equat ion is the sa me as th at of the conven-t ional brushless dc moto r . However , i t is noted here tha t themotor character is t ics can be es t imated by this equat ion with-ou t exper iment s , becaus e the permeance func t ion , the numberof winding turns a nd th e magnetrom otive force of the perm a-nent "net ar e known values durin g the design stage .

.l_, .- . . .... --.

7 Validity of Modeling

The specification of the prototype is given in Ta.ble1. Fig7shows the speed emf waveform of the m otor. From the figure.a. negligible small distortion is observed. This nieans that anassumption of sinusoidal permeance variation is valid.

( a ) Motor s peed 100 rpm (5V /d iv)

(b) Motor speed 25 r p m ( 1 V / d i v )

Figure 7: Speed emf waveformFrom Eq.( 15), the s tator winding inductance, emf constantand torque constant are given as

8 -L , = M - PN: , PohrsHm= M T I m- 3respectively. A s a validity of modeling, a comparison betweendesign value and m easured value is effective. Th e measurem entof torque con stant is performed by the fol lowing manner . Fig.8shows the m easurement of torque constant .Stator windings are connected as shown in the f igure and s t i ff -ness character is t ics are measured under the constant dc exci ta-tion for var ious rotor angle. By this metho d, winding currents

292

-I I II.______- _ I ______


Torque pick upAngle sensor

w

p: number of poles pareU : rotational speed

HandleV

?---

Figure 8: Measurement of torque constant

W ,- 1IJs + DI >

[ I I 1

I I0 10 20 30 [AI

Figure 9: Cur rent vs. torque characteris t ics

are Zu = I , 2 v = 2w = - ; I and the corresponding torqueexpression is obtained as,(17)32DC= --M,.I,IsinO

by substituting winding currents into Eq.(16). Fig.9 shows theresult of TDc( = 27r) vs. dc exciting curre nt. Torque constantis obtained from the gradient of the characteristics.Table2 shows the comparison between the designed and mea-sured values of th e test mo tor. Th e validity of modeling is wellproved by the fact tha t th e measured torq ue consta nt, emf con-stan t and in ductanc e show good agreements with the correspond-ing design values.

8 Control of Three Phase HB

Figure 10: Block diagram of d-q model of three phasehybrid s tepping motor

following voltage equation results.[']

where

Fig.10 shows the block diagram representation of Eq.(18).T he figure shows the coupling term between d axis and q axiscomp onents exists. Since the coupling term makes th e controlcomp licated, de-coupling is achieved by a. modificatio n of appliedvoltage shown belowv d = U; w,L,iqv = U, f W, ( l i E + L a i d ) (19)

where vi and U, are the reference voltages and v d and u are theactua l de-coupling applied voltage. By combining E qs( 18) , (19),the de-coupled block diagram representation of the three phasehybrid stepping motor is shown as in Fig.11.

vd -1 1 1 zI R , + s L , >T

R , L , Js + DFigure 11: De-coupled block diagram representation ofthree pha.se hybrid step ping moto r

9 Experimental ControlCharacteristicsStepping Motor Fig.12 shows the basic speed c ontrol system of the three pha.seBy applying a well known d-q transformation to Eq.(16), the hybrid stepping motor.

293


Table 3: OutlineInvert erCarrier freq.DC voltageDead t imeCurr. detectorPosition sensoiControlControl period

100

. xperimental systemIGB T inverter10 kHz48 I 2 p e cNNC- 1 0 GA

2000 p / r EncoderTMS32010100 f i sec

w; 100: z =

I I50

Figure 12: Speed control system

~ j 40-

In designing the speed regulator, the general assumption ismade th at the transfer function of th e current minor loop is 1and the frictional coefficient D = 0 ( J = 0.51 x Kgm)The outl ine of th e controller is given in Table3. Fig.13 showsthe s tep response of speed for two different speed reference an dFig.14 shows the s teady st a te speed a nd torque characteris t ics .It is noted here that the speed below 40 rpm in Fig.14 could notmeasured directly. Since the DSP in the controller can calculatethe speed in t he form of position difference in one sampling , thecalculated result was used as a speed below 40 rpm. In this case,a resolution of speed w as 1.5 rpm .

100 1

w; 20

LO 40 60[%load]

Figure 14: Steady st at e speed and torq ue chara.cteristics

of the voltage equation is closely related to the motor pole coil-figuration. Therefore, the motor performa nce can be estimate dfrom the voltage equation during th e design stage of th e motor.The control model of the niotor can be easily obtained by ap-plying a well-known d-q transform ation. Th e validity of proposedmodeling is well proved by t he comparison between th e designedand measured motor par amete rs and the control characterist ics .The authors wish to thank Mr. Makoto Nakamura for his assis-tance in experiments.References

PI

131

141

Figure 13: S tep response of speed 151

I6110 ConclusionsStart ing from a simple flux model in the a ir gap, the mag-netic equivalent mod el of the three phase hybrid s tepping motor

is deduced. Th e voltage equation is obtained from the magneticequivalent model. Since th e procedure origina tes from th e geo-metrical configuration of stator and rotor poles, the parameters

[71

294

P.J. Lawrenson et al. Variab le speed switched reluctan ce mo-to rs. P roc . IEE, ~01 .127, t .B , No4, pp.253-265, October,1980R.Kishnan et al. Design procedure for switched reluctancemotors. Conf. Record, IEEE-IAS Annual Meeting, Col-orado, pp.858-863, Oct obe r, 1986P.P.Acarnley et al. Detection of rotor position in step-ping and switched motors by monitoring of current wave-forms. IEEE, Trans. Industrial Electronics, vol.IE-32, No.3,pp.215-222, Augu st, 1985S.Ueno, Y.Takeda, S.Ishikawa, T.Hira sa. T hru st Char acter -is tics of V R T ype A ctuator Considering Inductor Teeth . Na-tional Convention Record, IEE Japan, Industry App. pp.761-766( 1988)(in Japanese )N.Matsui, H.Yam ai, H.Dohmeki Tor que Control of the Hy-brid Stepping Motor. Proc. of the Incremental Motion Con-trol. System s an d Devices pp.239-247(1987)Takura Application Guide of Stepping Motors. Sogo Den-shi Publication, Co. (1987)(i11 apanese)N.Matsui, K.Kameda, T.Takeshita DSP-B ased Software De-Coupling Current C ontrol of Brushless Motor. ~01.107,No.2,pp.215-222 Trans. of IEE Japan (1987)(in Japanese)

Modeling and Control of Hybrid Stepping Motors

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