Nonstandard electrostatic problems for strips

Nonstandard electrostatic problems for strips Eugene J. Danicki, Yuriy Tasinkevych Polish Academy of Science, IPPT, Warsaw, Poland Abstract explain the ‘wave-scattering’ terminology: radiation cond., incident ...

1

Introduction

In classical electrostatics, the boundary value problem is formulated for electric field or its potential, governed by the Laplace equation and the boundary conditions on the surface of conducting body. The solution provides the electric field in the space around the body and the electric induction (the electric charge density) distribution on the body surface [Moon]. In this paper, the conducting body is actually a system of in-plane perfectly conducting and infinitesimally thin strips of arbitrary width and spacing (see example in Fig. 1). Such systems found important applications in microelectronics, particularly in the surface acoustic wave (SAW) devices, where they help to redistribute the time-harmonic electric field on the surface of piezoelectric substrate. Fig. 1. A broad view, including 2nd and 3rd overtone, of the spatial spectrum of electric charge on the system of five strips alone (solid line) and placed beside a conducting half-plane (dash); the inset presents the field distribution on the strip plane in the later case.

Due to the piezoelectric effect, the field excites the harmonic Rayleigh wave on the substrate. The conversion of electric to acoustic signal is the most effective if the wave-number of Rayleigh wave falls within the spatial spectrum of the field distribution on the substrate. The temporal spectrum of the wave generated by the electric impulse applied to strips well resembles the spatial spectrum of the electric charge distribution on the strip system [Morgan] (the involved coefficient between both spectra is the velocity of Rayleigh wave). In the analysis of such structures (called interdigital transducers, IDTs), the most important is the solution to the charge spatial spectrum on the plane of strips rather, than the charge spatial distribution on strips (this is why we call the problem ‘nonstandard’). Although they are simply related by the Fourier spatial transform, the transformation direct evaluation poses a serious practical problem due to the square-root singularity of the charge distribution on strips. Note that there can be tens or even hundreds of strips in the analyzed structures, and any interesting spectral lines corresponding to the SAW wave-number depends strongly on all these singularities. Moreover, wide domain of the spectrum is interesting for applications, even spanned over several transducer overtones (fundamental harmonics). Naturally, the numerical results can only be obtained, and attempting to perform the Fourier transformation (practically - using the fast Fourier transform, FFT), one have to sample the singular field at discrete values of the spatial variable, what inevitably leads to significant numerical errors (even if the field is perfectly evaluated [Tasinkevych-PhD]). To overcome this difficulty, a new method is proposed in this paper that evaluates the charge spectrum directly, formally without an earlier evaluation of the charge spatial distribution with subsequent application of the FFT algorithm. The integrals of the field and induction are only necessary for evaluation of the strip potentials and charges in order to formulate the equations resulting from the circuit theory (the Kirchhoff’s laws), taking into account that some strips have given potentials, others can be isolated or interconnected. In fact, several template solutions for spatial spectrum is constructed for given strip systems, and the superposition of them is searched to satisfy the circuit equations. The paper is organized as follows. Next section contains the preliminary matter introducing the spatial spectrum of electric field on the plane of strips, and the planar harmonic Green’s function that, establishing certain relation between the field components, replaces the Laplace equation in the considered boundary-value problem. Section 3 presents details of efficient numerical evaluation of certain ‘generating’ [Danicki-US96] or ‘template functions,’ presenting a series of independent solutions to the charge distributions on the plane of strips expressed in spectral domain. The superposition of these par-

1

tial solutions is discussed in the subsequent section to satisfy the Kirchhoff’s laws for strips, what makes the final solution to the basic electrostatic problem. Section 5 presents a possible approach to the complementary problem of strips embedded in a preexisting electric field. It is called an ‘electrostatic scattering’ here, because this preexisting field is assumed to be a spatial harmonic function like in an electromagnetic scattering problem. The solution of this problem is constructed exploiting the above introduced generating functions, and again is obtained directly in the spectral domain, yielding a complete constructive theory of both cases important in the analysis of SAW devices: for the field generated by the given strip voltages (the case of generating IDTs), and for the field generated on the strip plane by the surface acoustic wave propagating on a piezoelectric substrate supporting strips. The induced strip charge describes the SAW detection by strips, and the spectrum of the induced field is responsible for the Bragg scattering of SAW by strips [Danicki-US04].

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Basic solutions for planar systems

Let the harmonic potential on the plane y = 0, assumed independent of z, be exp(−jrx) exp(jωt),

(1)

where the time-dependence is not much important for electrostatics, but will help us to formulate the standard circuit equations for the strip currents rather than the strip charges; r is the arbitrary spatial spectral variable and ω is frequency of the field (the temporal dependence will be neglected in further ~ = −grad(ϕ) notations). On the basis of the Laplace equation for electric potential of electric field E ∆ϕ = 0,

(2)

one obtains that the y-dependence of the solution for ϕ vanishing with growing distance from the plane y = 0 (in other words, satisfying the equivalent ‘radiation condition’ for electromagnetic waves) is exp(−|ry|). Assuming the vacuum space of dielectric constant o , the following equations result for the field on the plane y = 0+ (just above the plane) E(r) = jr, D(r) = rSr o ,

(3)

where Sν = 1 for ν ≥ 0 and −1 otherwise is adopted (rSr = |r|; r is assumed real), and to shorten further notations, we applied E = E1 and D = D2 . It is convenient to introduce the planar harmonic Green’s function here defined by G(r) = E(r)/D(r) = jo Sr ,

(4)

for the field components on the plane y = 0: the tangential field E1 and normal induction D2 , satisfying the radiation conditions (vanishing at |y| → ∞; the other, complementary class of the field satisfying the opposite, that is growing at infinity, will be considered in the scattering problem later). This Green’s function will replace the Laplace equation in all the analysis that follows for the fields on the plane y = 0 but, in order to simplify further notations, we apply o = 1. It is known [Moon] that the charged halfplane x < 0 induces the following field on the plane y = 0+ D(0) (x) = ..., x < 0; D(0) (r) = ... E (0) (x) = ..., x > 0, E (0) (r) = ...

(5)

in the spatial, and spectral (at the right) representations. It is convenient to introduce the complex field defined by Φ = D − jE, (6) in both spatial and spectral domains that yields for the halfplane

D

(0)

Φ(0) (x) = ... (x) = Re{Φ(0) (x)},

=..., ˆ E (0) (x) = Im{Φ(0) (x)},

(7)

for any x (with correctly chosen value of the square-root functions); = ˆ notes the correspondence between the spatial and spectral representations. The above follows the well known electrostatic theorem that 2

real and imaginary parts of any harmonic function represent a solution of certain electrostatic problem. The boundary conditions for the considered problem here are E(x) = 0 on strips and D(x) = 0 outside strips, appended by the ‘radiation condition’ that the field vanishes at |y| → ∞. This idea is exploited further below. Concerning the spectral representation, it is very important to note that it has semi-finite support, the feature of great importance for further numerical analysis. Namely, it results from Eq (4) that Φ(r) = D(r) − jE(r) = (1 + Sr )D(r) that is zero for r < 0, for fields satisfying the radiation condition, Eq. (4). The definition (6) has been chosen to obtain the convenient support r ≥ 0. For given Φ(r), the the representation of D and E in spectral domain can be inferred from the above as   j 1 Φ(r), r ≥ 0, Φ(r), r ≥ 0, and E(r) = jS D(r) = (8) D(r) = r Φ∗ (−r), r < 0, −Φ∗ (−r), r < 0, 2 2 what can be easily checked by substitution to Eq. (6) and (4). Moreover, functions [D(r − p), E(r − p)] are a pair of another harmonic function [D(x) − jE9x)] exp(−jpx) satisfying Eq. (2), but not necessarily Eq. (4). Now consider a strip of width 2a. The field spatial distribution [Moon] is described by 2 2 −1/2 (0) (0) Φ(1) (x; a) = =Φ h (a − x ) i h(x − a)Φ (x + a),i∗ Φ(1) (r; a) = Φ(0) (r) exp(−jra) ∗ Φ(0) (r) exp(−jra)

(9)

(the superscript ∗ means the complex conjugation), respectively for spatial and spectral domains, as resulting from the known convolution theorem for the Fourier transforms. Taking into account the semifinite support of Φ(r), the convolution is explicitly Z r (1) Φ (r; a) = ...dξ = J0 (ra), r > 0, (10) 0

where J − 0 is the Bessel function. Repeating the above, one easily obtains similar expressions for the system of two strips: Φ(2) (r) = [Φ(1) (r; a1 )e−jrb1 ] ∗ [jΦ(1) (r; a2 )e−jrb2 ],

(11)

where a1,2 and b1,2 are the corresponding widths and displacements of strips having the shifted spatial field representation (9). The coefficient j is introduced in the second term to obtain D(x) = Re{Φ(x)} and E(x) = Im{Φ(x)}, conveniently analogous to Eq. (7). For three (N = 3) strips and more, we have h i Φ(N ) (r) = [Φ(N −1) (r)] ∗ jΦ(1) (r; aN ) exp(−jrbN ) , (12) Φ(N ) (x) ∼ [(x − x1 )(x − x2 )...(x − x2N )]−1/2 , for spectral and spatial representations, respectively, where xi are the strip edges. If there is a conducting halfplane, the convolution with Φ(0) in spectral domain is necessary, and the corresponding multiplication by Φ(0) (x) shifted by x0 in spatial domain; this introduce another term (x − x0 ) in the above formula (Appendix A). As can be easily checked, the above discussed field vanishes fast with |x| → ∞, indicating the multipole character of the charge distribution on strips. Thus, it cannot form a basis of complete representation of the arbitrary field generated by the system of strips. The system can, for instance, possess a net charge different from zero, inducing the electric field vanishing at infinity like 1/x. The harmonic function representing such cases and exhibiting the square-root singularity at the strip edges is the above derived function Φ(x) multiplied by a polynomial function of degree not exceeding N − 1. Hence, N independent template solutions result, all satisfying the boundary and radiation conditions (4) Φ(N,0) (x) = Φ(N ) (x), Φ(N,1) (x) = xΦ(N ) (x), · · · , Φ(N,n) (x) = xn Φ(N ) (x), · · · ,

(13)

which yield the corresponding field E (n) and induction D(n) on the plane y = 0. Applying the known Fourier transform theorem that multiplication by x in spatial domain corresponds to differentiation in spectral domain, one formally obtains the above functions in the spectral domain Φ(N,0) (r) = Φ(N ) (r), Φ(N,1) (r) =

d (N ) dn Φ (r), · · · , Φ(N,n) (r) = n Φ(N ) (r), n < N, dr dr 3

(14)

yielding the field spectral representations E (n) (r) and D(n) (r), according to Eq. (8) (in Sec. 3, these functions will be redefined for the numerical advantage). The superposition of these N independent functions suffices for representation of arbitrary field that can be generated by the system of N strips subjected to N circuit constraints: given strip voltages or charges, or interconnections. The next section presents a convenient method for evaluation of the above spectral ‘generating’ or ‘template functions’ for practical use. Their spatial counterparts can be computed with help of FFT, if necessary. In practical systems, both the source terminals are connected to another strips in the system, rendering the system electric neutrality (the currents flowing to and out of the system are in perfect balance as flowing through the same source). This means that the distant field excited by the charge distribution on strips behaves like the field generated by an electric dipole, at most, vanishing faster than 1/x. Hence, the field spectral representation must vanish at r = 0. It is evident that the function (N,N −1) ˆ (N,N −1) (r), Φ(N,N −1) (r = 0) 6= 0,  Φ(N,N −1) (x)=Φ 1 Φ (r), r ≥ 0, D(x) = F −1 {D} ∼ 1/|x| at |x| → ∞ D(r) = Φ(N,N −1)∗ (−r), r < 0, 2

(15)

must be excluded from the set of functions representing the charge distributions on strips, because this, and only this function represents field vanishing like 1/x at infinity, what is evident from Eq. (12). The other functions represent field vanishing like x−k , k = 2, ..., N , and their spectral representations behave like rn , n = 1, ..., N − 1 at r → 0, according to the limit theorem for the Fourier transforms. The above shows that at our disposal is a set of N − 1 template functions which superposition, with certain coefficients, must satisfy the circuit equations. Let each strip be connected to either of the source terminals. It is clear that the conditions on the strip voltage differences can be set only, not on the absolute voltages of strips. This shows that there are N − 1 circuit equations in this example, and similarly for any cases, proving that there is a complete system of equations for evaluation of all superposition coefficients mentioned earlier. To formulate the equations however, we need to evaluate ¯ i (or, conveniently for application, the currents Ji = jω Q ¯ i flowing to strips), voltages Vi and charges Q for fields represented by each ‘generating function.’ Details on the formulation and solution of the circuit equations are presented in Sec. 4. Below we consider the induction D(0) and field E (0) evaluated from Eq. (8) using Φ(N,0) instead of Φ: D(r) = Φ(N ) (r)/2, r ≥ 0 or Φ(N )∗ (−r)/2, r < 0, yielding results marked by the upper index (0); the analysis for other functions, involving multiplication by xn in Eq. (13) and indexed by (n), is similar. According to the definition of the electric potential E = −dϕ/dx, the strip voltages can be obtained by integration of E (0) (x), what in spectral domain, corresponds to the division by r of the E (0) (with accuracy to an unimportant constant due to the voltage difference being only involved in the circuit equations). As concerns the strip charge, the function Q(0) is first defined being the integral of D(0) (x). Using the Green’s function (4), one obtains: Z ∞ 1 |r|−1 D(0) (r)e−jrx dr, V (x) = F {−jE(r)/r} = 2π Z −∞ ∞ j Q(0) (x) = F −1 {jD(r)/r} = r−1 D(0) (r)e−jrx dr. 2π −∞ (0)

−1

(16)

¯ = 2Q because it is the integral of the induction difference on both sides The charge distribution is Q of the plane y = 0, and D2 (y − 0) = −D2 (y + 0) = −D. Naturally, V (x) is constant on strips because E(x) = 0 there, and Q(x) is constant between strips because D(x) = 0 there. Thus we apply for the ith strip voltage Vi and current Ji (0)

Vi

(0)

= V (0) (ξi ), Ji

= 2jω[Q(0) (ζi ) − Q(0) (ζi−1 )], i = 1, ..., N,

(17)

where ξi is a point within the ith strip domain (best - at its center), and ζj can be at center of spacing between two neighboring strips, j +1 and jth; they can be easily evaluated from the strip edges xi , except ζ0 and ζN which value can be applied at certain distances in front of the first strip and after the last one; both values are equal because Q(N ) (x → ±∞) are equal. Note that all the above evaluations involve only the spectral representation of D(0) (r). The other advantage of this approach is that we may not bother with the square-root sign of Eq. (12) to evaluate the correct Vi , Ji .

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3

The template functions

YT: write a text basing on ”Strip electrostatics,” your paper on J.Tech.Phys. (but another examples) and dissertation, appended by detailed descriptions of advanced numerical subjects concerning convolution. Define Φ(N,n) ... ATTENTION: CHECK or complete all the above formula!!!! Then start with this: As seen from the above discussion, the ‘generating functions’ Φ(r) are crucial for the evaluation of the spatial spectrum of induction D(r). This requires careful numerical ...

4

Evaluation of the charge spectrum

YT: write a text including detailed descriptions of advanced numerical subjects concerning the condition factor etc. Plus examples: with/without error, for 5 strips and 5+halfplane (Appendix A) ...

5

The scattering problem

This section strongly relies on the earlier paper [Dan-UFFC] presenting the ‘electrostatic scattering’ problem for periodic strips (or groups of strips) with period Λ = 2π/K. There are also defined the ‘generating functions’ which, due to the system periodicity, are actually the discrete spectral functions represented by the Fourier series Fn . They are obtained by convolutions like in Eq. (9) but involving Legendre polynomials Pn (·) instead of Bessel function J0 (ra). On the strength of the asymptotic expansion [Magnus-Oberhettinger-Tricomi] Pn (cos θ) = J0 ([2n + 1] sin[θ/2]) + O(sin2 θ/2), θ ∼ aK → 0,

(18)

we infer that the difference between the functions obtained in both cases vanish if r = nK, K → 0. We are going to exploit this approximation for K small but finite, in developing analogous solution to the scattering problem using the earlier evaluated function D(r), Eq. (15). As presented in Sec. 3, this function is evaluated at discrete values of the spectral variable r = iK, K = ∆r. Thus, it is the discrete series in the numerical analysis, and actually representing, on the basis of the theory of FFT, the periodic function in spatial domain with certain large period 2π/K. This authorize us to introduce notations Di = D(iK), Ei = jSi Di ; D−i = Di∗ . (19) In the periodic case, only one series Fn is evaluated, and all other which correspond to D(n) for different n ∈ (0, N −1), are obtained by shifting the index of Fn by certain values to obtain series Fn−m . Moreover, if m ≤ N/2, the ‘generating functions’ Fn−m satisfy the radiation conditions and thus are accounted for in the field representation in order to satisfy the corresponding circuit equations, including the system electric neutrality. This is because F|k|≤N/2 = 0. In the case of aperiodic systems, the functions D(n) have this analogous property that, behaving like rn−N , have small values over broader domain r ≈ 0. As will be shown below, the series Di introduced in Eq. (19), with D0 6= 0, will play important role in the solution of the considered scattering problem. Note that the pair of functions with shifted index Dn−m and En−m places the values D0 , E0 at the spectral line r = mK, what corresponds to multiplication by exp(−jmKx) in spatial domain. Naturally, such multiplied functions still satisfy the boundary conditions on the plane of strips: E(x) = 0 on strips and D(x) = 0 outside strips. Moreover the corresponding function Φ evaluated from Eq. (6) is also a harmonic function. This property will be exploited below with this important remark that D and E evaluated from Eq. (8) using the shifted Dn−m Di = Di−m and Ei = Ei−m = jSn−m Dn−m

(20)

fail to satisfy Eq. (4) (Ei 6= jSi Di , this can be easily check by inspection), and thus fail to satisfy the radiation condition (the corresponding field does not vanish at |y| → ∞). Note however that this failure takes place only due to the spectral lines 0 ≤ i ≤ m; all other lines satisfy well the Eq. (4). It may be instructive to check that Laplace equation (2), that sounds E,1 + D,2 = 0 for the field exp(−j(i − m)Kx ± |i − m|Ky), is satisfied by each of its harmonic components (choosing the proper sing in the exponential function). To proceed further, another class of field is introduced here - the incident harmonic field of wavenumber rI = IK and complex amplitudes DI and E I . This field does not vanish at infinity; being the incident 5

wave it rather grows with |y| → ∞ and thus is the functions of type exp(|ry|). As opposite to the scattered field, Eq. (4), the incident field obeys: E I = −G(IK)D = −jSI DI

(21)

(for o = 0). To simplify the analysis, I > 0 is applied in this paper. The field on the strip plane y = 0 is a sum of the scattered (marked below by upper index s) and ‘incident’ spatial waves; it must satisfy the boundary conditions on this plane. Noticing that the field {Di−m , Ei−m } does it, we attempt to express the surface field by the combination (the summation convention applied over repeated indices) ¯i = Eis + E I δiI = jαm Si−m Di−m , E ¯ i = Dis + DI δiI = αm Di−m , D

(22)

where δ is the Kronecker delta and αm are unknown coefficients. One needs only to add the requirement that the scattered field obeys Eq. (4), that is Eis = jSi Dis ; explicitly αm (1 − Si Si−m )Di−m = 2DI δiI .

(23)

It may be checked by inspection that the solution to this infinite system of equation (for i in infinite limits) can be solved with αm , 1 ≤ m ≤ I + 1, for the assumed I > 0. Indeed, for any i > I, the term in bracket turns to zero satisfying the homogeneous equation (δI,i>I = 0), and similarly for any i ≤ 0, provided that m takes values in the above limits. For i = I we have αI+1 = 1/D1∗ Other αm can be evaluated in recursive manner, starting with equation i = I down to i = 1. The resulting field distribution may include a net charge, D0 6= 0. To assure the system electric ¯ i with coefficient α0 chosen to obtain neutrality, one needs to add α0 Di to the evaluated D ¯ 0 + α0 D0 = 0. D

(24)

This completes the solution of the scattering problem; the resulting surface field in the spectral representation Di = αm Di−m , Ei = jαm Si−m Di−m ; 0 ≤ m ≤ I + 1, (25) satisfies the boundary conditions at the strip plane; Fig. 5 presents the computed example. Moreover, the induced strip voltages and charges can be evaluated in the already presented manner, Eq.(16) in the corresponding discrete form (r = iK). These should be treated like originated from the negative external voltage source. If, for example ith strip is assumed grounded but get certain potential due to the incident field, than a combination of ‘template functions’ D(n) must be added with proper coefficients evaluated as presented in the earlier sections. Certain comment is needed, however, concerning the evaluation of αI+1 and others. For K → 0, the involved D−1 → D(r → 0) seems correct. Only applying the discrete r results in somewhat odd value D(−K) instead. Numerical experimentations show that we may apply Di−m+1 instead of Di−m or Si−m−1 instead of Si−m in Eq. (23) with good result. Fig. 5. Scattering by 5 strips: a) the induction D(x) and b) its scattered part Ds only, showing the ‘specular reflected’ field (equal −DI ) outside the system of strips. c) The evaluated surface potential V and Q(x) shows the induced strip voltage and charge; the difference of values of Q on both sides of the structure equals zero indicating the structure electric neutrality.

FOR INSPECTION and VERIFICATION!!! + Fig.: scatt. for 5 strips.

ED contribution finished, 5/06/2004 6