Electrostatic field of a system of aligned electrodes

Journal of Electrostatics, 26 (1991) 187-200 Elsevier

187

Electrostatic field of a system of aligned electrodes Dorel Homentcovschi Central Institute of Mathematics, Academic Street 14, Bucharest, Romania

(Received August 18, 1989; accepted in revised form May 25, 1991 )

Summary This paper provides an analytical expression for the electrostatic field of a system of aligned electrodes, the case of the piecewise homogeneous medium (two different dielectrics on the two sides of electrodes ) and that of one or two infinite electrodes are discussed too. The paper provides also the Maxwell capacitance matrix of the system. The problem is of a proper interest and can be also used as a model problem for analysing other steady electromagnetic systems.

1. Introduction

Two-dimensional boundary value problems for Laplace'sequation occur frequently in physics and engineering.In particularthey occur in many branches of microwave engineering. Though numerical methods for treatingboundary value problems become very popular,'the analyticalmethods keep their importance due to the precision of the resultsand simplicityand generalityof the given formulae. Beside these, analyticalresultscan be used as test problems for numerical methods and they can also be combined with some numerical techniques to eliminate the difficultiesof a purely numerical approach [I ]. This paper proposes an analyticalsolution for the problem of determining the electrostaticfieldcorresponding to a system of aligned electrodes.This problem arised in two differentdomains of microelectronics:the firstis the problem of determining the electricalparameters of a distributed resistive structure [2 ] and the second isconcerned with the determination of Maxwelrs capacitance matrix of a multiconductor coupled microstrip line [3,4].Both problems were solved by using a rather intricatemathematical tool including boundary value problems in complex plane and singularintegralequations. The solutionproposed in thispaper issimpler.It needs only some numerical quadrature and some algebra.Moreover, the method provides also the explicit form of the electrostaticfieldin the whole domain. In the cases of a single and a pair of stripsthe analyticalsolution of the problem isknown [5 ].This isthe point of departure of our considerations.In 0304-3886/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

188

Section 2 we deduced the form of the electrostatic field for a single and two electrodes and, further on, in Section 3 we generalised this form to an arbitrary number of strips. The determination of the Maxwell capacitance matrix involves the same type of hyperelliptic integrals like in papers [2,3 ]. The results are also generalised to the case of the piecewise homogeneous medium and also to the case of one or two electrodes extending to infinity. The importance of the results is intrinsic and the results can also be used in other domains by means of some suitable conformal mapping (as was the case in papers [ 2,3 ] ). In fact, the complex plane with some aligned electrodes can be considered now as a model problem for analysing other steady electromagnetic systems. 2. T h e c a s e s o f a s i n g l e a n d o f t w o a l i g n e d e l e c t r o d e s

2.1. Single electrode case

Let us start with considering the case of a single conducting strip, charged at the potential V~, placed in vacuum between points a~, b~ in the complex

z-plane. We choose the real axis as being in the electrode direction (Fig. 1 (a)). Let

F(z)= V(x~y)- iU(x,y)

(1)

be the complex electrostatic potential [6], V ( x , y ) stands for the potential thnction and U(x,y) denotes the flux function. The electrostatic field follows from the relation (2)

E,-iEvffi-F'(z)

By symmetry arguments the semi-axes (-or,at), (bt,~) will be flux lines, hence we have U ( x , O ) - U ~ for x e ( - o v , a ~ ) and U(x,0) = U2 for xe (b,ov).

'I

® X

~/ / l ~I r i l / / ~(IlI l / i

F / ~./ ;, / / / / D~

/ i i /~._ ~

" / / / / / / / . , ~ / 7 ~ ,, , T / / ,, T,( / / / / /

....... ~ "

-,~

'f//////1,f/////~

Ib

U - - ~ -

lc

Fig, I. The physical domain and the g and w planes for a single electrode case.

189

By two successive conformal mappings z=

bl-al Z-I b,+a, 2

2

'

Zfsin w

(3)

the domain in the upper half-plane, Im{z} > 0, is mapped into the half strip in Fig. l(c). The complex potential reads now simply

F(w)-

in

+ u, 2i I-V~

w+

(4)

and hence

dw dZ U , - U2 Ex-iE~.=-F' (w) dZ d z in

•

1.(bl-a,~

-1

(5)

We obtain finally k

(6)

Ex-iEyffix/ (z_al) (z_bl )

where k is a real constant. The square root function in (6) is defined in the complex z-plane with the cut [a,,b~] on the real axis and has real positive values on the semi-axis ( - oo,a, ). We have in fact

fx/[x-a,l'lx-b,I, Its/Ix-at

I

Ix-o~l,

for z ~ x < a t ,

for zffix+ i0, xe (a,,bl), for zffix-i0, xe (a,,b,),

(7)

for z ffix > bl.

2.2. The case of two aligned electrodes In the case of two aligned electrodes in Fig. 2(a) let us denote by V, and V~ the corresponding potentials of electrodes. Again, the segments of the real axis outside the strips are flux lines. In this case we consider a bilinear transform

Z-az+b

(8)

-cz+d and a Schwarz-Christoffel mapping Z

W--C'~

dZ

(9)

The first function maps the physical domain into the symmetrical domain with

190

';~// ,~ / / / / / / / ot

A

0

b~ [

//7,'///////b/~2a O~

I////////A//////i/

-1

®

|

x o2

.1

2c

Fig. 2. The physical plane and the Z and w planes for the case of the two electrodes.

respect to the OY axis in Fig. 2(b). The constant p > 1 follows from the condition of the two-ratios invariance,characteristicto bilineartransforms (1 + ~ ) 2 / ( 4 f l ) = [ ( a 2 - a l ) ( b 2 - b l ) ] / [ ( b 2 - a ~ ) ( a 2 - b l ) ]

(10)

Finally, the domain corresponding to the upper half-plane Ira{z} > 0 will be the rectangle in Fig. 1 (c). The complex potential reads now simply

F(w)_V2- VI w+ V2+ VI 2

(II)

2

The electrostaticfieldin the physica ~.plane is

Ex-iEy=

c' ~:i.i(Z2,fl~)

ad-bc (cz+d) 2

(12)

or, in the variable z,

k' Ex-iEy= ~/ (z-a~ ) (z-b~ ). x/~z-a~) (z-bz)

(13)

k' being again a real constant. The square root determinations in (13) are

defined similar to (7). It is to be noticed the similarity of the obtained expressions for the electrostatic field in the two particular cases mentioned in Sections 2.1 and 2.2. 3. The general case of n aligned electrodes

Let the n aligned electrodes [a~,bl],...,[a,,b,] be changed at V~, ...,11,potentials,respectively(Fig. 3).

191

yl ° 0

a~

bI

a ~t

bz

I

I

am

brn

,

I

t

an

bn

x

'j----r

Fig. 3. The physical domain in the general case.

Taking into account formulae (6) and ( 13 ) we look for the electrostatic field of the system of n electrodes in the form n

Z kjzJ-' •

jffil

E,-iE,= 1-](z)

(14)

where kl, ..., k,, are real constants to be determined and n

l-I( z ) = l-[

m=l

x/(Z-am)(z-bm)

(15)

The square root determination must be defined like in relation (7). In fact, the expression (14) for the electrostatic field is supported by some mathematical arguments concerning the general solution of the homogeneous Volterra boundary value problem [6]. The function defined by relation (14) is homomorphic outside the strips and has the proper behaviour of the electrostatic field at the electrode ends. On the real axis we have n

Z k xJ-' E,(x,O)

Ey(x,O)=O

J=~ -

for xe (-oo,at)

(16)

III(x)l'

Vn

"Z kj J-'

E,(x,_O) =0,

E~(x,_+O)= + (-I) -,~

II-I(x)l

-

forxe(am,bm), m=l,...,n (17)

n

Z kJxj-' jffil N(x,0I= ( - 1 ) " i--l-I(x) I '

E~(x,O)=O

forxe

(bm,am+~), m = l ,

...,n - 1

(18) ?l

Ex(x,O)=-t) .i---, iH(x)l ,

E~(x,O)=O

forxe(b.,oo)

(19)

In fact, these relations express the very fact that the electrode segments are

192

potential lines and the gaps are flux lines for the function defined by formula

(14). As concerns the behaviour at infinity we have

Ex-iEy=k"+o(z -2) Z

(20)

Let us now apply the Gauss law of electrical flux to a cylinder X with generatrices orthogonal to the z-plane and having a unity height. Because on the lateral surface nx=dy/ds, ny= -dx/ds we get

" " da--Im{r Q-fJeE.n fe(E. -iEy) (dx+idy )}

(21)

=Ira

X

Here Q is the total charge inside the surface and iVis the profile curve of the cylindrical surface. In case we consider the curve iVas being the circle Izl--R including inside all the strips the residue theorem yields Qf2~eo'k,

(22)

This relation puts into evidence the physical meaning of the coefficient kn; it equals the total charge of the system devided by 2~o. It must be given as a data of the problem. In most cases we have Q-- 0, hence we consider, further on, the case k, vanishes, The constants k,, ..., k._ t must be determined by using the proper potentials of the strips. As we have Ex ffi - 0 V/Ox, relation (18) implies lira4.1

f Xj- t dx, ~ ki(1)" Vm-V,.+,=j= bm I II(x) I .-

I

n-1.

re=l,

(23)

""'

By using notations Qr¢l+ 1

Xj- l

Ai~=(-1)~

II-[(x)1dx, m=l, ...,n-l, j=l, ...,n - 1

bm

By. _ ( _ 1 ) ~ + , j" Gra

(24) xj - '

II-I(x)l

dx,

m---1,...,n, j f l , . . . , n - 1

relation (23) can be written in the form

193 n--1

kjAj,~=Vm-Vm+~,

m - l , ..., n - 1

(23')

i=1

These relations can be used to determine the constants kj (j = 1, ..., n - 1 ) when the strips potentials are given. In fact, we can write the solution in the form -.b

.~

k - [AT]-z'V

(25)

where f~T - [kl, ..., k,_~], ~T= [ V 1 - I72, ..., V , _ I - I;',] and [A T] is the transpose of the [A ] matrix. Let us denote by a~k the elements of the matrix [A T ] - ~. We have n--I

n--I

n

hj=mZ+ l "TmV= mffil Z

Z aTm-lVm

m----2

(26)

.-b

Hence we obtain the vector k in terms of the strip potentials

[All7

(27)

where ~ T _ [V1, ..., V,] is the potential vector and the matrix [A] is defined by relations d~ =a}'~,

dj,=-d}',_~, j = l , ..., n - I djm----QTm--Q~m_l, m = 2 , . . . , n - - 1 , j = l , . . . , n - 1

(28)

By introducing the values k~, ..., k,_~ in formula (14) we obtain the electrostatic field in the whole plane. 4. Determination of Maxwell's capacitance matrix of the structure

The Gauss law applied to a surface enclosing the strip [arab,, ] gives bm

Qm = ~ e { E y ( x , + O ) - E y ( x , - O ) }

dx (29)

am

o_1

bm

= 2 e o j ~ k j ( - 1 ) "~

I

xJ--1

dx,

am

Qm being the charge on the ruth strip. By using notation (24) we have n--1

Qmffi - 2~o ~'. kjBj,,,

rn = 1, ..., n

(30)

jffil Let ~T._

[QI, ..., Q,] be the charge vector. Relation (Y~-).becomes

Q = -2~o[BW]k

(30')

194 -b

In relation (30') we substitute the values given by formula (27) for the k to yield

Q = -2Eo[S T ] [A ]'~

(31)

This relation provides the Maxwell capacitance matrix of the system in the form [C] = - 2eo[B T ] [.4 ] (32)

5. Case of the piecewise homogeneous medium Let us now suppose the upper half-plane filled with a medium of ~z dielectric constant and the lower half-plane by a e2 dielectric constant medium. In this case on the Ox axis on the gaps (the space between electrodes) the following coupling relations must be satisfied

el'Ey(x,+O)=e2"Ey(x,-O),

E,,(x,+O)=E,,(x,-O)

(33)

Simply, by inspection, from relations ( 16 )- ( 19 ) it can be seen that all these relations are fulfilled. In fact, in the gaps E y - 0 while the E~(x,O) component has the same values in both media. Hence the above analysis concerning the determination of the electric field is also valid in the considered inhomogeneous case.

In the determination of the Maxwelrs capacitance matrix in relation (29) we must take into account that e is different on the two sides of the electrode

[ a,,,b,, ]. Hence relation (30) becomes t~-.. |

Q,,,f-(E1+E~)j~IkjBj,n,

mfl,...,n

(34)

Finally, in this case, the capacitance matrix of the system will be [C.o.hom ] = - (~ + e2)" [BT]" [A ]

(35)

By comparing relations (32) and (35) we obtain the relation el + E2

(36)

between the two capacitance matrices.

6. The cases the electrodes extend to infinity

6.1. A single electrode extends to infinity Let us suppose now that we have a system of n + 1 electrodes lying on segments [a],bl], ..., [a,,b,], [a~+l,Qo]. In this case we have

195 12

Z kj J-' •

j=l

E x - i E y = 1-i(1) (z)

(37)

where

For the new square root function we have x/am+~ z _ y x / l a , , + , - x l ,

--[+i~/[aa-~-xl,

forz=xa,,+,

(38)

...kn

The constants k~, are determined by relation similar to (27) where n is replaced by n + l and l'I (x) by 1-I(~)(x). In order to obtain the Maxwell capacitance matrix of the system we write the relation tl

Qr,= - 2~o ~ k,yB ym, (1)

m= 1, ..., n

(39)

j----1

Here •-~m (~ is given by the second relation (24) if we replace l-I ix ) by l-I (1, (x). We have ~__2¢o[B(~)]w. [A (~)].~r

(40)

The vector ~w__. [Ql, ..., Q, ] having only n components. In order to obtain the Maxwell capacitance matrix of the considered system we must add to the matrix [0(1)] ffi- 2~o[B (1)]T[A (1)]

(41)

a line which consists in the sum, with changed sign, of all the other lines.

6.2. Tw~ electrodes approach infinity Let the aligned strips be ( - av,bo], [al,bl], ..., [an,b,], [a,+l,av]. We suppose the first and last electrode have the same potential V, + 1. We have n

Z kjzi-1

Ex-iEy=i~(2)(z)

(42)

where

,_d-I

(43)

We consider the square root determination

~ +_ix/Ix-bol , for z=x+_iO, x bo.

(44)

196

All the developments in Sections 6.1 hold also if we replace 1-[<~) (z) by l'I <2)(z). We obtain thus the constants kl, ..., kn and the Maxwell capacitance matrix of the structure. 7. Application to the a n a l y s i s of i n t e r d i g i t a l t r a n s d u c e r s

The circuit model for interdigital transducers considered in [7 ] accounts for the spatial distribution of the driving electric field in transducers with arbitrary metallization ratios and polarity sequences. It gives a description of transducer operation at both fundamental and higher harmonic frequences. In order to use the circuit model it is necessary to find a suitable scalar function of the electric field to use as a source term for acoustic waves. It is assumed that this source term is the normal electric field component, appropriately normalized. The analysis given in [7] is based on the assumption that the electric field under a given electrode depends only on the dimensions and polarities of that electrode and its nearest and next nearest neighbors (local environment) and also that the electrode widths and spacings vary slowly from electrode to electrode. The problem is solved by numerical integration of the integral equation which gives the surface charge density on the electrodes. The mathematical model given in Fig. 3 corresponds exactly to the problem considered in [7 ] where the upper half-plane is filled by air and the lower one by the piezoelectric substrate. The normal component of the electric field at the electrode surface is given by relation tt~l

Z__

Ey(x,O)-(-1)m#li-[(x) I , x (am,bm), m--1,.,.,n

(45)

and the constants kj are determined by solving the linear system (23'). In relation (42) it is assumed that the total charge is zero. The parameters of the equivalent circuit of the interdigital transducer are determined in terms of capacitance C associated with the change and voltage on one electrode and of the spectral excitation function. The determination of the capacitance matrix was discussed above. The spectral excitation function is defined as

F.,(#)=K

)}

(46)

Here ~ denotes the Fourier transform with respect to the argument fl, K is a normalization quantity and 0m(x) is the characteristic function of the segment (am,b,,) [equals 1 for x¢ (a,,,bm) and zero otherwise ]. To obtain the spectral excitation function we put f ( x ) - E ,(x,O

Ix-a..I . Ix-b.,I

(47)

197 and consider an expansion of this function into a series of modified Chebyshev polynomials.

[(x)=~+ ~ftcos(l arccos2X- (am+bm)~

(48)

In fact, this relation can be written in the form

f(bm2am COSt"{"bm2ara)m~'Jt'l~l~l

lt, ~E(O,x)

(49)

and consists of a Fourier series. The coefficients ft can be easily computed by using a FFT algorithm. Further on we have

•. ~

fcosl[-/ arcc°s-2x-(bm+am)~ --ttrt j ..

_

_

_

_

Um

x/Ix-aml lX-bml

}

Om(X)

(50)

-exp{ipbm2am}(i)t'J,(flbm2am) By uAng relations (47)-(49) we get

Fm(')=Tcexp{i];~bm'{'am]~'~'L~'Jc~,~(i)t, °~ iJt(flbm2am)]

(51 )

which gives the expansion for the spectral excitation function in terms of cylindrical Bessel functions. Relation (51) completes the considered model circuit for interdigital transducers for aa~ electrode and any environment. 8. Discussion about the application of the study of the resistive distributed structures

The above analysis can also be applied to other steady states of the electromagnetic field.Let us consider, as an example, a distributedresistivestructure which consists of a thin film area of resistivematerial which has on its boundary two -- or several -- conductor terminals (electrodes) separated by insulated regions. In this case U is the stream function, the dielectricconstant is replaced by the square resistance parameter R D and the charges on electrodes by current flowing through them [2]. Using an appropriate conformal transformation maps the physical structure into the upper half-plane, Ira{z}> 0. Since the perpendicular component of the electricfieldat the electrode line in gaps is zero we can cut the upper half-plane from the lower one, the solution remaining unchanged. (In fact, this is equivalent to putting ~ = R[]=,= oc in the

198

solution for the piecewise homogeneous medium). The electrokinetic field is given by relation (14) where kn = 0 and the constants kl, ..., kn_ 1 being determined by relation (25). The admittance matrix of the multiterminal resistive distributed structure is half the above determined Maxwell capacitance matrix. The obtained formulae enable us to draw the potential and stream lines (by integrating numerically the corresponding differential equation) and also to study the termic effects in distributed resistive structures. Some applications of this kind were given in [8]. 9. Conclusions

The analytical expression for the electrostaticfieldof a system of aligned electrodes, given in this paper, simplifiesconsiderably the analysis of m a n y electrostaticproblems which interestsmicroelectronics. For a given system of electrodeswe must firstlycompute some hyperelliptic integrals. Next, for a given electrodes loading we solve a linear system and finallythe electricfieldfollows by evaluating some elementary complex functions.The numerical steps involved in solvingthe problem are very simple and there exist very good routines for solving each of them. By conformal mappings the considered model problem can be used to analyse also other electromagnetic systems. In fact,the solutions given in papers [2-4] are based on this idea. Appendix. The uniqueness theorem In determining the electrostatic field of the considered system we stated that the strip potentials are given. We can also have the case the charges Qmare given as well ss~he case we know some potentials and some charges. In order to put into evidence the necessary conditions for determining an unique solution of the problem we give a uniqueness theorem. We start with the identity

div(cVgrad V)~(grad V)2+~V.AV

(52)

where the dielectricconstant e is assumed constant. By integrating relation (52) on a circleof radius R (includinginside all strips),with the center at the origin,and by using the Gauss theorem we obtain e I:l~R

cV~-~n ds IzlfR bm

f

+ ~ V~, e{Ey(x,+O)-Eylx,-O)}dx m~=l

¢ Qm

(53)

199

As the totalcharge of the system vanishes we have

V=O(1),

OoV=o(R-2) for R.oo

(54)

and consequently the inte~al on the circle [z i= R vanishes as R ~ or. W e obtain

VmQm=Z (vm- Vn)Q

c(E2+E2) dxdy = m----1

(55)

m-----1

By using this relation we can state the uniqueness theorem: The electrostatic field of the given system is completely determined if we give for every electrode the potential or the charge. In the case the charges are n--I given for n--1 electrodes we have Qn= - Y~m=l Q~ and hence no other boundary condition is needed for the nth electrode. This theorem provides that the system (23') has a unique solution. Let now all the electrodes have the same potential V~ = ~ =--. = V. = Vo. In this case the relation (55) gives Ex- iEy =-0. Therefore tl

~Cij--0.

j_-

(56)

By symmetry we also obtain

~ Cijffi0

(57)

iffil

This relation can be used to determine the last line of the Maxwell capacitance matrix.

References 1 R. Levy, Conform transformation combined with numerical techniques, with applications to coupled bar problem, IEEE Trans. Microwave Theory Technique, MTT-28 (4) (1980) 369375. 2 D. Homentcovschi, A. Manolescu, A.M. Manolescu and C. Burileanu, A general approach to analysis of distributiveresistivestructures,IEEE Trans. Electron Devices, ED-25 (1978) 787794. 3 D. Homentcovschi, A. Manolescu, A.M. Manolescu and L. Kreindler, An analytical solution to the coupled stripline-likemicristrip line problem, IEEE Trans. Microwave Theory Technique, 36 (6 ) (1988) 1002-1007. 4 D. Homentcovschi, A cylindrical multiconductor stripline-likemicrostrip transmission line, accepted for publication in IEEE Trans. Microwave Theory Technique. 5 E. Durand, Electrostatique,t.I,II,Masson, Paris, 1966. 6 D. Homentcovschi, Complex variable functions and applications in science and technique, Ed. tehnic~, Bucharest, 1986 (in Romanian).

200 7 W.R. Smith and W.F. Pedler, Fundamental and harmonic frequency circuit - - model analysis of interdigital transducers with arbitrary metalization ratios and polarity sequences, IEEE Trans. Microwave Theory Technique, MTT-23 (11 ) (1975) 853-864. 8 D. Homentcovschi, A. Manolescu and A.M. Manolescu, Problems concerning the laplacian field in bidimensional domains with applications in microelectronics, Microelectronica, 15, Ed. Acad. RSR, (1987) (in Romanian).