Approximate analytical solution of Helmholtz's equation in an inhomogeneous conducting region using hybrid synthesis method

ZESZY TY N A U K O W E P O LITEC H N IK I ŚLĄ SKIEJ Seria: E L EK TR Y K A z. 173

2000 N r kol. 1471

W ojciech B U R LIK O W SK I, K rzysztof K LUSZCZY Ń SK I Zakład M echatroniki

APPROXIMATE ANALYTICAL SOLUTION OF HELMHOLTZ'S EQUATION IN AN INHOMOGENEOUS CONDUCTING REGION USING HYBRID SYNTHESIS METHOD

S u m m a ry . The paper presents the analysis o f the electrom agnetic field in a basic m odel o f an induction m otor. The m odel contains an inhom ogeneous conducting region.

Field distributions are obtained using a hybrid synthesis m ethod. In order to take into account field inhom ogeneous problem s the m ethod em ploys special integral approach. It is based on analytical form ulae describing the eddy current distribution in an equivalent hom ogenous region. Basic features o f the m ethod are discussed. A ttention is focused on the error estim ation. Com parison w ith the results obtained when using FEM is provided.

PRZYBLIŻONE ROZWIĄZANIE ANALITYCZNE RÓWNANIA HELMHOLTZA W NIEJEDNORODNYM OBSZARZE PRZEWODZĄCYM WYKORZYSTUJĄCE HYBRYDOWĄ METODĘ SYNTEZY POLA ELEKTROMAGNETYCZNEGO

S treszczen ie. A rtykuł przedstaw ia analizę pola elektrom agnetycznego w uproszczonym m odelu silnika indukcyjnego. Model ten zaw iera niejednorodną warstwę przew odzącą. R ozkład pola w yznaczono w ykorzystując hybrydow ą m etodę syntezy. W m etodzie tej w ykorzystano specjalne podejście całkowe, aby uwzględnić niejednorodność analizow anego obszaru. Jest ono oparte na w yrażeniach analitycznych opisujących rozkład prądów w irow ych w zastępczym obszarze jednorodnym . Opisano podstaw ow e w łasności m etody. U w aga została skupiona na analizie błędu metody.

Załączono porów nanie uzyskanych w yników z rezultatam i otrzym anym i przy w ykorzystaniu m etody elem entów skończonych (MES).

1. IN T R O D U C T IO N

P urely analytical solutions describing the distribution o f the electrom agnetic field are restricted to the analysis o f hom ogenous problem s [l]-[3 ]. This also applies to the steady-state analysis o f eddy currents described in the presented paper. This lim itation can be easily overcom e by the use o f num erical m ethods [l]-[3 ],[5],[6], H ow ever, internal errors encountered in num erical solutions o f the field equations cause serious problem s in certain applications [1], [2]. O ne o f these applications is the evaluation o f the electrom agnetic force/torque [5], [6]. It led to the developm ent o f various hybrid field m ethods [4], [5], They incorporate advantages o f both analytical and num erical m ethods. D ifferent approach is presented in [6] w here the com parison betw een the results obtained w ith different m ethods is considered. Proposed hybrid synthesis m ethod o f solution o f the H elm holtz’s equation, in the opinion o f the authors, is supplem entary to the solution o f the P oisson’s equation presented by R ogow ski ([1], p p .82). It applies both to the m odel structure and the em ploym ent o f the F ourier’s series analysis in the m athem atical form ulation o f the m ethod. The hybrid synthesis m ethod w as em ployed in the evaluation o f parasitic torque in an induction m otor [4]. The paper presents results o f researches aim ed at the estim ation o f the error introduced by the method.

2. D E SC R IPT IO N OF TH E M ODEL

The inhom ogeneous basic m odel o f an induction squirrel cage m otor analysed in the paper is show n in Fig. la. It is a part o f an infinitely long region o f w idth g and length 2 r (r- pole pitch). This region is bounded by tw o parallel plane surfaces o f infinite perm eability (//=«>).

T hese are the stator and rotor cores, respectively. In the low er o f these, there lies a series o f conducting rectangular bars {yc *Q, P =P0) o f height h. The current sheet o f the stator (s) is placed on th e surface o f th e stator core [1], [4].

The steady state distribution o f the electrom agnetic field in the m odel is described by the H elm holtz’s equation. It is form ulated in the reference fram e fixed in the rotor. Resulting two- dim ensional (2D ) field equations are form ulated in term s o f the com plex m agnetic field vector potential A = [0,0, A ] . F or a the v-th field space harm onic they have the follow ing general form:

V 2d ( x , y ) = p p 0/ A ( x , y ) ^

w here A is com plex m agnitude o f the vector potential, p = j c o is the operator o f tim e differentiation, co is relative angular frequency o f the v-th m agnetic field space harm onic in the rotor reference fram e, pQ ,y a re m aterial param eters.

To analyse the field, stator and rotor cores are represented by the appropriate boundary conditions applied on their bounds [1]. The rem aining m odel region is divided into two subregions having different m aterial properties:

A pproxim ate analytical solution. 41

• air-gap region (G), ( y=0,

• electrically inhom ogeneous region (C), w here conductivity is described by the follow ing expression :

y ( x ) = (1 - z ( x ) ) r (2)

c

w here z(x) is an auxiliary function (Fig. lb ), yc is the conductivity o f the bar.

In this inhom ogeneous m odel analytical solution o f the field equations could not be obtained. It is due to the spatial dependence o f the conductivity in the region (C) [1], Therefore, to obtain field distribution for the inhom ogeneous m odel, we have to resort to a num erical approxim ation.

In order to obtain analytical solution the hybrid synthesis m ethod is applied [4], The m ethod is based on the analytical solution o f H elm holtz’s equation in the hom ogenous conducting region [1], [3],

T he proposed approach introduces two model changes, w hich are crucial to the problem analysis:

• geom etry alteration, w hich affects the conductivity o f the conducting region, changing the inhom ogeneous region (C) into an equivalent hom ogeneous one (H),

• insertion o f an additional com pensating w inding in the region (H), [4].

R esulting hom ogenous m achine m odel is shown in F ig .lc.

S tator core

(G^ V S tator winding (s),J_v ( x) i

E 3 m ' V A

a)

z(x)

br

Rotor core, T ^ \C o n d u c tin g b a r^

b)

(G j Com pensating winding ( + ) , / ł ) (x,v) ,

C)

Fig. 1. a) Inhom ogeneous m achine m odel, b) auxiliary function z(x), c) equivalent hom ogeneous m achine model Rys. 1 .a) N iejednorodny m o d el m aszyny, b ) funkcja pom ocnicza z(x), c) zastępczy jed n o ro d n y m odel m aszyny

The hom ogenous m odel is excited by tw o sources:

• ¿ (v \ x ) ejm‘ - the stator w inding current sheet o f the v-th space harm onic, being the driving one in the m odel,

¿ * \ x ) = a v cos(w a / r ) + b v sin (w a:/r) ^ w here a v , b vaie know n current coefficients o f the tw o-phase stator w inding,

• J <+)(x,v) eJwt - the com pensating w inding current sheet, placed on an arbitrary depth d in region (H),

,(*). , v ,<+>, , (4)

I ( x , v ) = L J x ( x , v )

A e A

w here A = {A] , A,... A j^ } is a set o f N space harm onics o f the com pensating w inding related to the v-th stator space harm onic,

■ ¿ ^ \ x ,v ) = a A(v)cos(2.7Dt/r) + ¿ /l(v)sin(A ® c/r)

w here a ,i(v), bx(v) are u n k n o w n current coefficients for each A-th space harm onic o f the two phase com pensating w inding.

The m ain feature o f the com pensating w inding can be described in the follow ing way:

reduction o f the influence o f the current flo w in g in region (H) outside the bar subregion (I.e.

for z ( x ) - l ) on the o verall fi e l d distribution.

The above current sheets constitute set o f boundary conditions. The resulting field problem s described by (1) are solved separately for:

• the stator v-th space harm onic,

• each A-th space harm onic o f the com pensating winding.

The general solutions can be w ritten in the follow ing form:

• for the air-gap region (G)

.(') / \ /-Ah . , , Y ny fyl'i (6)

An = c o s( m x ) + G 2„ S i n ( m 0 j p 3„ e + G Ane ) t n

• for the hom ogeneous conducting region (H)

An = Z \lL i„ cos(nx) + H 2„ s m (n x )\H 3ne + H Ane ) n

w here G ^ , . , H (^ , . are constants adjusted to fit the particular boundary conditions resulting from the continuity o f

• the norm al com ponent o f the flux density B ,

• the tangential com ponent o f the field strength H ,

for stator (n= w z/r, i=s) or com pensating w inding (n=An/x, i=+) respectively, o n = j w v + " 2 ■

In order to evaluate y (+)( i, v) , the follow ing constraint equation is form ulated [4]:

,(+>( x ,v ) + z (x ) f ] \ x , y ) d y = ^g 0. ^{(8 )} g - h

A pproxim ate analytical solution. 43

dx = 0

Fig.2. S im plified graphical interpretation o f the constraint equation (8) Rys.2. U proszczona graficzna interpretacja rów nania w ięzów (8)

It results from the im plem entation o f the A m pere's law to eddy currents j(x,y) induced in the hom ogenous conducting region H by both w indings (Fig.2):

v x , » . , (9)

¿ ( x ,y ) = ¿ v ( x , y ) + ¿ (x , y ) ,

U) (,) (10)

l v (x,y) = - p r 4 W ) ,

c

l * \ x ,y ) = -p r E 4 +W ) = E / +W ) .

c ~

X X

A pplication o f the F ourier’s analysis to (8) leads to the algebraic system o f equations [4], The unknow ns are 2 N current coefficients gj_ ( v), bj_(v) o f the com pensating w inding. The system m atrix is full w hich is typical for the integral field m ethods (e.g. BEM). Then the final field distribution is obtained by superposition o f solutions (6), (7) for both windings.

3. E V A LU A TIO N OF TH E ERR OR

T he constraint (6) is satisfied only in an average (integral) sense. It results in error in the obtained field distribution.

The follow ing three approaches are feasible to im prove the accuracy o f the proposed method:

• increase o f the num ber o f harm onics N in the analysis [4],

• changing the position o f the com pensating w inding (analysed in the paper),

• increase o f the num ber o f the w inding current sheets.

The first one is related to the basic properties o f the Fourier series describing auxiliary function z(x) (8). The third o f the above m ethods is proposed as an extension to the basic idea described in the paper. It will be based on the division o f the region (C) into subregions along its height and application o f the proposed m ethod to each o f these.

In the paper only the second m ethod is described. Two cases are distinguished, based on the value o f th e skin depth 5 = ~J2 /(coy/u)[2,3]. The optim um position o f the com pensating w inding is:

• for the resistance-lim ited case ( S » h ) => d= 0.5 h - in that case the eddy current density is alm ost constant along the height o f the region (H). Therefore to com pensate the magnetic field generated by the current flow ing outside bar subregion (I.e. for z(x )= l), the com pensating w inding should be placed near the m iddle o f the height o f the region (H),

• for the inductance-lim ited case ( 5 « h ) => d= 0 - in that case the eddy current is concentrated near the surface o f the region (H). To com pensate the m agnetic field generated by the current flow ing outside bar subregion the com pensating w inding should be placed on the boundary o f the region (H) and the air gap (G).

A bove cases are based on the properties o f the m agnetic field generated by the currents distributed in th e hom ogenous space. H ow ever, as there are tw o infinitely perm eability surfaces on both sides o f th e analysed region, infinite num ber o f im ages influences the distribution o f the field [1]. Therefore in practice the solutions for d= 0 and d= 0.5 h should be thought as a m easure o f the error lim it for the actual field distribution.

4. R ESU LTS

The calculations w ere perform ed for the follow ing data: yc =58e6 [S/m], r=0.1[m]r, rr-T/4, b r/rr= 0 .5 , v= l, N=100, O)=2n50[rad/s], g=5.5e-3 [m], h=0.9g. R esulting model o f h a lf o f the bar region show n in Fig. 3.

Fig.3. H a lf o f the b a r region o f the analysed m achine m odel R ys.3. P o ło w a obszaru p ręta w analizow anym m odelu m aszyny

The com parisons w ere draw n betw een the results for:

• the hom ogenous m achine m odel (Fig.2c), obtained using the m ethod proposed in this paper.

Tw o values o f d w ere chosen: d=0, d=0.5h,

• the inhom ogeneous m achine m odel (Fig.2a) in w hich the electrom agnetic field is evaluated w ith the help o f the finite elem ent m ethod (FEM , A N SY S® ). A sm all part o f the finite elem ent m esh is show n in Fig.4.

The finite elem ent m odel w as form ulated using the follow ing assum ptions (Fig.4):

• it covers one pole pitch o f the m achine,

A pproxim ate analytical solution. 45

• additional layers o f elem ents, representing rotor and stator cores, w ith relative perm eability /ur =10000, are placed along the active region (F ig .la ),

• appropriate boundary conditions are applied on the borders o f the model to account for the sym m etry o f the field (A (r Q )= -A (r j) and high perm eability o f the m achine cores (N eum ann's d A /d y = 0 , r = r s u T r ) [1],[3],

• current is injected into the nodes on the stator inner surface according to the formula:

/„, = cos(w am It) + j sin(vOTcm It) ( 1 2 )

w here m = l,2 ,3 ,...,M + l is the num bering o f nodes, M =800 is the num ber o f elem ents along the pole, x m - position o f the m -th node.

The stator w inding current coefficients a v , b v in the analytical field model were adjusted to ensure the sam e m agnetom otive force M M F as in the case o f the finite elem ent model. Results obtained along h a lf o f a pole pitch ( z/2) are show n in F ig.5.

Rotor - p / v i”\ \ Statoi core F 0 \ X — U ) core

i i i i i i i i i i i i n i i n i i i r \ (

=

)

Fig.4. F inite elem en t m odel o f th e analysed m achine with the description o f the boundaries and model dim ensions

Rys.4. M odel analizow anej m aszyny w ykorzystujący m etodę elem entów skończonych z opisem brzegów i w ym iarów m odelu

1.0

o.o

0.0

a)

2.5

□stance, x (cm)

5.0

□stance, x (cm) C)

b)

□stanœ, x (cm)

d)

□stance, x (cm)

Fig.5. a) A uxiliary function z(x). Spatial distribution o f the real com ponent of: b) the com pensating winding current, c) the norm al com ponent o f the m agnetic flux density on the surface o f the stator (y=0), d) the tangential co m p o n en t o f the m agnetic flux density on the surface o f the hom ogenous conducting region

(y=(g-h)‘)

Rys.5. a) F u n k cja p o m o cn icza z(x). Rozkład przestrzenny części rzeczyw istej: b) prądu uzw ojenia kom pensa­

cyjnego, c) składow ej norm alnej w ek to ra indukcji m agnetycznej n a pow ierzchni stojana (y=0), d) składow ej stycznej w ektora indukcji m agnetycznej na pow ierzchni jednorodnej w arstw y przew odzącej

(y=(g-h)')

5. C O N C LU SIO N S

C om parison has been m ade betw een the results obtained using proposed hybrid synthesis m ethod and the F E m ethod. A s the value o f the skin depth for the analysed m odel is 5 & 0.9e- 3 [m ], and therefore 5 « h / 5, neither o f the tw o conditions m entioned before (for resistance- lim ited and inductance-lim ited case, respectively) is fulfilled.

F rom Fig. 5c,d it is apparent that the FE solution in the air gap for an inhom ogeneous m odel is w ithin the lim its established by the field distributions obtained using the proposed m ethod for (d= 0, d= 0.5h). D iscrepancy on the edges o f the m odel, near x= 0 and x=t (not included) m ay result from the FEM internal errors [1]. Presented results prove correctness o f the proposed approach.

A pproxim ate analytical solution.. 47

The m ost outstanding advantage o f the proposed m ethod is the com putational time required to obtain solution. The analytical evaluation using M ATLAB environm ent takes only about 1/300 o f the com putational tim e required to obtain the reasonable solution using FEM (A N SY S® ) for the m odel considered. It is especially im portant w hen num erous field solutions are necessary, as it is the case in the evaluation o f the parasitic torques in induction squirrel cage m achines [4], [6], W e are working on the im plem entation o f the R ogow ski’s method as the error estim ator [1].

REFEREN CES

1. K .J.B inns and P .J.Law renson and C.W .Trow bridge, “The analytical and numerical solutions o f electric and m agnetic fields” John W iley, N ew York, 1992,

2. A .K raw czyk and J.A.Tegopoulos, “N um erical m odelling o f eddy currents” Clarendon Press, O xford, 1993,

3. J.K .Sykulski, K .Paw luk, R.Sikora, R.L.Stoll, J.Turow ski, K.Zakrzew ski "Computational m agnetics", C hapm an & Hall, London 1995,

4. W .B urlikow ski and K .K luszczynski, “ Determ ination o f the parasitic torques in an induction squirrel cage m achine em ploying analytical field m ethod: analysis and validation” Proceedings o f International Conference on Electrical M achines IC E M ’98, Istanbul, Turkey, Septem ber 1998, 1300-1305,

5. T .S .L ow and C.Bi and Z .J.L iu “A hybrid technique for electrom agnetic torque and force analysis o f electric m achines” CO M PEL V o l.16 No.3 (1997) pp. 191 -205,

6. S. J. Salon and D .W .B urow and C. Slavik and R.Bushm an and M .J.DeBortoli,

“C om parison o f pulsating torques in induction m otors by analytical and finite element m ethods” IEEE Trans. M agn vol.31 No.3 M ay 1995 pp.2056-2059.

Recenzent: Dr hab. inż. M arian Łukaniszyn Profesor Politechniki Opolskiej

W płynęło do R edakcji dnia 15 kw ietnia 2000 r.

Streszczenie

R ozw iązania analityczne opisujące rozkład pola elektrom agnetycznego ograniczają się do analizy problem ów w obszarach jednorodnych [1]. Dotyczy to rów nież przedstawionej w artykule analizy stanów ustalonych przy zasilaniu sinusoidalnym [2,3]. O graniczenia pow yższe m ożna pokonać w ykorzystując dyskretne algorytm y num eryczne [1,2,3,5,6]. Ich zastosow anie w analizie sił oraz m om entów elektrom agnetycznych wiąże się jednak z licznym i problem am i natury obliczeniow ej [5,6]. D oprow adziło to do rozw oju wielu

rodzajów tzw. m etod hybrydow ych [4,5,6], W ykorzystują one zalety obydw óch w spom nia­

nych uprzednio podejść: analitycznego oraz num erycznego.

A nalizow any w artykule niejednorodny m odel m aszyny elektrycznej przedstaw iono na r y s .la . W m odelu tym nie je s t m ożliw e w yznaczenie analitycznego rozw iązania układu rów nań po la [1]. P roblem ten rozw iązano w ykorzystując h y brydow ą m etodę syntezy pola [4],

O trzym any n a jej podstaw ie zastępczy m odel jednorodny przedstaw iono na ry s.lc.

W stosunku do uprzednio przedstaw ionej w ersji m etody [4] w prow adzono następujące zm iany:

• grubość w arstw y przew odzącej h je st dow olna,

• uzw ojenie kom pensacyjne (+) m oże być usytuow ane na dowolnej głębokości d (rys.lc).

Istn ieją trzy sposoby popraw y dokładności m etody:

• zw iększenie liczby harm onicznych pola N [4],

• zm iana położenia uzw ojenia kom pensacyjnego d,

• zm iana liczby w arstw prądow ych tw orzących uzw ojenie kom pensacyjne.

P ierw sza z pow yższych m etod w ynika z w łasności szeregu Fouriera opisującego funkcję pom ocniczą z(x) w yk o rzy stan ą w opisanym algorytm ie. Z kolei trzecia proponow ana metoda popraw y dokładności je s t rozw inięciem proponow anego algorytm u.

W artykule uw zględniono jedynie d ru g ą z przedstaw ionych m etod. O pierając się na definicji głębokości w nikania 5 w yróżniono dw a skrajne przypadki:

• słabego efektu naskórkow ości ( S » h ) => dla którego przyjęto d= 0.5 h,

• silnego efektu naskórkow ości ( S « h ) => dla którego przyjęto d=0.

R ozw iązania dla pow yższych dw óch przypadków stanow ią m iarę błędu ograniczając zakres rzeczyw istego rozw iązania. Popraw ność przedstaw ionej analizy uproszczonej zweryfikow ano w ykorzystując m etodę elem entów skończonych (M ES) na przykładzie modelu przedstaw ionego n a rys.3. O trzym ane w yniki przedstaw iono na rys.5.

P od staw o w ą zaletą przedstaw ionej m etody w stosunku do M ES stanowi czas obliczeń.

W przypadku przedstaw ionego m odelu stosunek czasów obliczeń w ynosił 1/300. Jest to szczególnie istotne w przypadku m odeli w ym agających w ielokrotnych obliczeń. Przykładem je s t w yznaczanie m om entów pasożytniczych w m aszynach indukcyjnych klatkow ych [4],

A ktualnie prow adzone są prace nad w ykorzystaniem m etody Rogow skiego w roli estym atora błędu [1].