Integral equation formulation and solution for extended vertical current sheets in multilayered planar structures


  • M. Vrancken,

    1. Katholieke Universiteit Leuven, Department of Electrical Engineering: Electronics, Systems, Automation and Technology, Division Telecommunications and Microwaves, Leuven, Belgium
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  • G. A. E. Vandenbosch

    1. Katholieke Universiteit Leuven, Department of Electrical Engineering: Electronics, Systems, Automation and Technology, Division Telecommunications and Microwaves, Leuven, Belgium
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[1] The integral analysis of multilayered planar structures owes its high efficiency to the excellent matching of the Spectral Domain Approach (SDA) to the planar nature of the problem. This highly efficient approach is strongly encroached upon when arbitrary three-dimensional (3-D) surfaces in a multilayered environment have to be analyzed. However, for 3-D structures that consist of horizontal and strictly vertical conducting surfaces, the analytical possibilities of the SDA can be exploited further. The integral equation formulation is adapted to the geometry by expressing it in a hybrid dyadic-mixed potential form. The solution is improved with a combined spectral-spatial domain approach, where part of the reaction integrals are done in closed form in the spectral domain. This modified method avoids most theoretical and numerical problems, while the majority of practical 3-D problems in stratified media can still be handled.

1. Introduction

[2] The full wave analysis of electromagnetic radiation and interference phenomena in planar antennas, circuits and interconnections in multilayered media is most efficiently done by an integral equation formulation combined with a Spectral Domain Approach (SDA) [Itoh, 1980]. The incorporation of the complicated multilayered environment in an analytical manner provides more efficient and stable solutions as compared to local differential equation based methods like Finite Difference Time Domain (FDTD) [Taflove, 1997] or Finite EleMents (FEM) [Volakis et al., 1997] that rely more on purely computational power to resolve the relevant physical phenomena. As the geometry of the structure becomes more complicated, and deviates more and more from the purely planar case, these advantages fade away, and local methods may become competitive. This occurred first with the need to model probe feeds [Zheng and Michalski, 1990] and short circuits in patch antennas and via's and air bridges in integrated circuits [Tsai et al., 1996]. More serious problems threaten the integral equation approach when arbitrary 3-D surfaces have be analyzed. For such a case, currents are present in a continuous manner at all positions and can have arbitrary orientation. The theoretical formulation of a Mixed Potential Integral Equation (MPIE) becomes cumbersome since one is confronted with multiple scalar potentials [Erteza and Park, 1969] or a dyadic vector potential kernel [Michalski and Zheng, 1990]. These problems translate into severe numerical problems such as the need to evaluate higher order Sommerfeld integrals and an angular azimuth dependence for the components of the vector potential. Furthermore, a large number of Green's functions need to be computed and stored such that reaction integrals can be evaluated by numerical interpolation [Gay-Balmaz and Mosig, 1996]. The aforementioned problems are avoided when we focus on structures as the one depicted in Figure 1. This is a 3-D structure subject to the limitation that the conducting surfaces are either horizontal, parallel to the layers, or vertical, normal to the stratification of the medium. This is the only geometrical limitation, since the vertical current sheet can have arbitrary dimension, can cut through dielectric interfaces, and can connect or intersect the horizontal conductors. This kind of geometry allows to refurbish the integral equation approach by exploiting the analytical possibilities of the SDA to the utmost. The formulation of the electric field is adapted to the geometry in a hybrid dyadic-mixed potential form which avoids the problems of a full 3-D formulation. This leads to a blending of the previously separately used Electric Field Integral Equation (EFIE) [Faché et al., 1992] and the Mixed Potential Integral Equation (MPIE) [Mosig, 1988]. The EFIE relates fields directly to currents via a dyadic Green's function, while in an MPIE charges are introduced, such that scalar and vector potential Green's functions are distinguished. We will use the EFIE for vertical field and current components and the MPIE for the horizontal or transverse components (see section 2). The discretisation of the integral equation into a set of linear equations is done with a method of moments [Ney, 1985], which requires evaluation of reaction integrals [Rumsey, 1954] which can be performed in the space domain [Sercu et al., 1993] or the spectral domain [Horng et al., 1992]. The MPIE formulation in the space domain is most popular since it avoids the 1/R3 spatial singularity of the dyadic formulation by a rather physical method: derivatives of the Green's functions are transfered to the expansion and test functions such that charges are obtained. The same problem is overcome in the EFIE spectral domain approach in a more numerical manner, by evaluating the reaction integrals in the spectral domain. The extra integrations improve the slow spectral decay of the Green's functions. Again, our solution of the integral equation combines both approaches with a mixed spectral-spatial domain approach (see section 3) where the evaluation of the reaction integrals is spread out over both domains. The (z, z′) dependent part of the reaction integrals can be done analytically (see section 3.4) in the spectral domain, prior to the numerical inverse Fourier transform. The remaining integrals are done in the space domain, where the MPIE retains its advantages. The proposed methods were implemented and applied to a number of problems shown in section 4 which show that most 3-D problems in multilayered media can actually be handled.

Figure 1.

A 3-D structure in a multilayered medium where only horizontal and vertical conducting surfaces are present.

2. Hybrid Dyadic-Mixed Potential Formulation

[3] The electromagnetic field in an arbitrarily layered medium is most conveniently obtained by performing a Fourier transform of the transverse spatial (x, y) coordinates to the (kx, ky) wavenumber domain. The Fourier transform of a spatial function G(x, y, z) is written as equation image(kx, ky, z) and is defined in Appendix A. The problem can thus be reduced to a standard transmission line problem for the TE and TM parts of the field [Itoh, 1980]. Each layer k of the medium has properties ϵk, μk and corresponds to a section k of the transmission line that is determined by a propagation constant

equation image

and the characteristic impedances

equation image

where kρ = equation image is the transverse spectral wave number. We assume that a current source is present in layer j at position z′, and that we need the electric field at an observation position z in layer i as indicated in Figure 2. All required Green's functions can be expressed with the voltage and current G = I, V on the transmission line caused by voltage and current sources S = I, V (indicated in superscript). Together with the T = TE, TM separation (indicated in super-superscript), eight possible functions can appear

equation image

where each function equation imageijST(kρ, z, z′) depends on the spectral wavenumber kρ and the source z′ and observation z positions. These functions now occur in a hybrid dyadic-mixed potential field formulation as given below. The expressions will be given for the moment in the spectral domain. The inverse Fourier transform will be discussed in section 3 when we take a look at the mixed spectral-spatial domain approach. For relating transverse (t = x, y) or horizontal field components to transverse current components, we retain the usual mixed potential formulation, which is expressed in the spectral domain as

equation image

For expressing the vertical or z components of the field as a function of the vertical component of the current source, we could use a mixed potential form, but there is no need to do so. Spatial 1/R3 singularities will be reduced in section 3.4 by analytical integrations, so we will use the simplest dyadic relation

equation image

For the cross-coupling terms, relating transverse to vertical components, we will use an intermediate formulation:

equation image
equation image

The vertical part is directly expressed with a (dyadic) Green's function, while for the transverse part of the formulation, the spatial derivatives will be interpreted and used (as in a mixed potential form) to define charges.

Figure 2.

The equivalent transmission line representation for the field in a stratified medium.

3. Mixed Spectral-Spatial Domain Approach

[4] The electric field integral equation is discretized using a method of moments [Ney, 1985]. The current is expanded using standard rooftop functions [Rao et al., 1982] The elements of the method of moments coupling matrix are obtained as reaction integrals [Rumsey, 1954]

equation image

where the integrations extend over the surfaces S, S′. The fields equation imagei are related to the currents equation imagej by Green's functions, which for a layered medium are obtained via the SDA approach with a numerical inverse Fourier transform.

3.1. MPIE-Space Domain Approach

[5] For the electromagnetic coupling between horizontal currents, the normal MPIE space domain approach is used. The field in the space domain is obtained from equation (4) as

equation image

where GttA and Gϕ are called the vector and scalar potential kernels and are obtained by an inverse Fourier transform from the expressions in equation (4). Inserting this into equation (8) gives fully

equation image

The order of integration and inverse Fourier transform is explicitly indicated to compare with the expressions given below. Note that the double derivative present in equation (4) was transferred to the expansion and test functions to obtain the charges Q = ∇t.equation imaget/(−jω). The remaining spatial singular behavior is of the order 1/R = 1/equation image with ρ = equation image and Δ = ∣zz′∣.

3.2. Dyadic-Mixed Spectral-Spatial Approach

[6] Using equation (5), we could similarly compute

equation image

but the spatial singularity that has to be integrated is 1/equation image. To avoid this problem, the (z, z′) dependent part of the surface integrals can be brought “inside” of the inverse Fourier transform as

equation image

where l, l′ are curves in the X, Y plane as indicated in Figure 3. The (z, z′) integrals can be done in closed form as will be demonstrated in section 3.4. These extra integrations will reduce the spatial singular behavior to a ln(Δ + equation image) behavior. The resulting sequence of operations is again given explicitly in equation (11) and pictured in Figure 3. Notice that because of the dyadic formulation, no charges appear in this part of the formulation.

Figure 3.

The combined spectral and spatial domain approach for the evaluation of the reaction integrals for two vertical currents. Closed form integration in the spectral domain facilitates the numerical inverse Fourier transform and reduces the singularity in the spatial domain.

3.3. Cross-Coupling Terms

[7] We now evaluate the electromagnetic coupling from a vertical to a horizontal current. From equation (6), we obtain the spatial domain expression

equation image

The spatial singular behavior of the occurring Green's function is now reduced by combining the techniques of the previous two cases: the spatial derivatives are transferred to the observation current, and part of the reaction integral is done in the spectral domain. When inserted in equation (8), this gives

equation image

Notice that only the current is required for the vertical current and only the charge for the horizontal current. The sequence of operations is depicted in Figure 4. The reciprocal coupling is obtained starting from equation (7) with a similar approach as

equation image

We will now show how to perform the (z, z′) integrals that are required in the above formulations in closed form. An analytical method is possible thanks to the equivalent transmission line representation of the Green's functions. We will also indicate how the integrations reduce the spatial singular behavior as claimed above.

Figure 4.

The combined spectral and spatial domain approach for the evaluation of the reaction integrals for the coupling of a vertical current to a horizontal current.

3.4. Analytical Integration

[8] The Green's functions in a multilayered medium show translational invariance only in the transverse (x, y) coordinates, so we can use ∇t = −∇′t. For derivatives involving (z, z′) coordinates, we cannot simply interchange source and observation coordinates, since the layers of the medium are “in the way”. However, equivalent relations can be found based on the equivalent transmission line representation. For derivatives involving the observation z coordinate, we have the usual transmission line equations

equation image
equation image

(without considering the delta function contributions when z = z′). For derivatives to the source z′ variable, we obtain the reciprocal relations

equation image
equation image

where instead of the current or voltage itself, the source type changes.

[9] With the above derivative relations, it becomes possible to evaluate the convolution of any Green's function with the basis functions that expand the current in the z direction analytically via a partial integration. This obviates the numerical interpolation procedure that is often used to evaluate the reaction integrals [Gay-Balmaz and Mosig, 1996].

[10] When we need to compute the coupling between two horizontal currents, but now both flowing on a vertical sheet, we can use equation (9) and as before the (z, z′) part of the surface integrals can be moved “inside” of the inverse Fourier transform. Although the currents are expanded with rooftop functions, in this case, their (z, z′) dependence is just a pulse function. For a horizontal source current we obtain, using equation (18) with G = V and T = TE, TM

equation image

and when it is an observation current, we can use equation (16) to obtain

equation image

When the expansion and test functions overlap, care has to be exercised when z = z′ by considering the discontinuities that occur at the source position on the transmission line of Figure 2 or by introducing explicitly the delta-functions in equations (16), (17), (18), (19). For vertical currents, we need to evaluate equation (11) with linear expansion functions for Jz as

equation image

with G = I, T = TM and for the integration with a linear test function, we get

equation image

The above formulas have to be used together to obtain a full Galerkin formulation. In general, the analytical integrations can change the type of the Green's function and introduces factors 1/γi2 or 1/γj2. An overview of the possible asymptotic behavior of the spectral Green's functions is given in Appendix B. Together with the fact that for large spectral wave numbers, we have

equation image

we can verify that the decay in the spectral domain is always improved and this corresponds to a less singular behavior in the spatial domain.

3.5. Inverse Fourier Transform

[11] After the analytical integration, the numerical inverse Fourier transform is performed. The decay of the Green's function for large kρ is further improved by extracting the quasi-static behavior in closed form. The factors 1/γi2, 1/γj2 introduced by the analytical integrations do not produce any extra singularities when kρ = ki, kj, which can be verified by expanding the expression in series. The physical pole singularities (surface wave) and branch point singularities (space wave) are also extracted [Demuynck et al., 1998]. The purely numerical work involves a real axis integration [Katehi and Alexopoulos, 1983] of a smooth and rapidly decaying spectral function. The extracted parts can be added again in the spatial domain as simple formulas by analytical inverse Fourier transforms.

4. Analysis of 3-D Problems in Stratified Media

4.1. A Problem With a Large Vertical Current Sheet

[12] Figure 5 shows a vertical plate of dimensions w = 10cm, h = 6.25cm cutting through a dielectric layer of thickness d1 = 2.5cm and ϵr1 = 5.1 in the surrounding air with ϵr0 = 1.0. The excitations are two dipoles positioned at d2 = 2cm in the air-dielectric interface as indicated in Figure 5. The frequency is 1.875 GHz. The resulting current distribution is shown as a vector plot in Figure 6, where we used 16 × 10 square segments on the plate. Figure 7 displays the required behavior of the magnitude of Jz over the air-dielectric interface. For clearness, we used 10 segments along Y as before, but 20 segments along the Z direction. Of course, the current profile will depend strongly on the position of the excitation.

Figure 5.

An extended vertical plate cutting through a dielectric layer and illuminated by two horizontal dipoles.

Figure 6.

Vector plot of the resulting current distribution on the plate of Figure 5.

Figure 7.

Magnitude of Jz on the plate of Figure 5.

4.2. Short Circuited Microstrip Patch Antenna

[13] Figure 8 shows a patch antenna where one side is short circuited with the ground plane by a vertical plate. The structure was proposed and measured by [Gay-Balmaz and Mosig, 1996]. The dielectric layer has ϵr1 = 2.33 and a thickness t1 = 1.57 mm. The patch has dimensions Wp = 30 mm, Lp = 42 mm. The probe feed is positioned at Wf = 9 mm, Lf = 4.2 mm. Figure 9 compares the measured results for the input impedance by [Gay-Balmaz and Mosig, 1996] with our computations in the frequency range 2.60–2.95 GHz in 36 points on a Smith chart.

Figure 8.

A probe fed short circuited microstrip patch antenna.

Figure 9.

Input impedance of the antenna of Figure 8 from 2.60–2.95 GHz, 36 points.

4.3. Packaged MMIC

[14] Another area of application is the analysis of packaging effects. The behavior of (Monolithic) Microwave Integrated Circuits (MMIC's) packaged in a metallic box as depicted in Figure 10 can deviate from the original unpackaged behavior due to box resonances and proximity effects of the side walls [Faraji-Dana and Chow, 1995]. The sharp resonance effects can be combatted by introducing absorbing materials, such as the lossy layer attached to the cover of the box in Figure 10. Nevertheless, there usually remain frequency shifts and other perturbations of the circuit response. For the circuit in Figure 10, the shunt stub with w = 1.4 mm, l = 1.9 mm originally has a stop band around 11 GHz, and this behavior is not much altered when the circuit is positioned at xc = 7.5 mm, yc = 17 mm. The example is taken from [Burke and Jackson, 1990]. The box has dimensions a = 15 mm, b = 24 mm, c = 12.7 mm and covers the circuit on a substrate of t1 = 1.27 mm, ϵr1 = 10.5(1 − j0.0023). The absorbing layer has t3 = 1.27 mm and ϵr3 = 21(1 − j0.02), μr3 = 1.1(1 − j1.4). Figure 11 compares the results of [Burke and Jackson, 1990] with our own computations for the amplitude of S21 from 9.0–12.0 GHz in 49 points. When the circuit is moved to the middle of the box at yc = 12 mm, the excitation of a box resonance around 10.8 GHz significantly alters the overall behavior as can be seen in Figure 11. The results of [Burke and Jackson, 1990] were computed with a “boxed” Green's function, where the effect of the side walls is incorporated in the formulation, such that the current on the box does not have to be computed. In our analysis, the current on the vertical walls of the box was completely expanded using rooftop functions, while the top cover is a part of the layer structure. The analysis of [Burke and Jackson, 1990] is of course more efficient for this particular structure, but is limited to strictly rectangular completely closed boxes, while our more general approach allows to model irregularly shaped boxes and possible openings in the sidewalls.

Figure 10.

Studying the effect of the metallic package and absorbing materials on the circuit behavior of the MMIC.

Figure 11.

Amplitude of S21 for the boxed circuit of Figure 10 as computed in 49 points in the frequency range 9.0–12.0 GHz. The response is influenced by the location yc in the box.

4.4. Microstrip Rectangular Spiral Inductor With an Air Bridge

[15] In the following example, we investigate the influence of the finite thickness of the upper piece of the air bridge of a microstrip rectangular spiral inductor. The main dimensions accompanying Figure 12 are ϵr = 9.8, d = 635 μm, ws = 625.0 μm, s = 312.5 μm, w = 312.5 μm, h = 312.5 μm. The upper piece of the air bridge will be modeled as a flat strip with zero thickness t = 000.0 μm or as a rectangular box with the same thickness as the vertical studs of the air bridge t = 312.5 μm. The structure was originally introduced by [Rittweger and Wolff, 1990]. Measurements were performed on a structure where the air bridge was actually a wire with circular cross section of diameter 317.5 μm. A Finite Difference Time Domain (FDTD) analysis where the upper piece was assumed infinitely thin was performed. An improved analysis with an MPIE-SDA integral equation technique were the finite thickness was taken into account was performed by [Bunger and Arndt, 1997]. These results have been taken as reference curves in the following graphs. In Figure 13 we compare these reference results with our own computations for the amplitude of S11 and S21 for t = 000.0 μm on the left graphs and for t = 312.5 μm on the right graphs. Computations were performed from 2.0–20.0 GHz in 145 frequency points.

Figure 12.

Microstrip rectangular spiral inductor with an air bridge.

Figure 13.

Amplitude of S11 and S21 for the microstrip spiral inductor of Figure 12 as computed in 145 points in the frequency range 2.0–20.0 GHz. The computed results by [Bunger and Arndt, 1997] were taken as reference result on all graphs. The influence of the thickness of the upper piece of the air bridge is compared for t = 000.0 μm (left graphs) and t = 312.5 μm (right graphs).

5. Conclusions and Ongoing Work

[16] For 3-D structures in layered media that consist of horizontal and strictly vertical conducting surfaces, the electromagnetic field is formulated in an adapted form that has characteristics of both a dyadic and a mixed potential form. In this way, theoretical and numerical problems of traditional MPIE formulations are avoided. The chosen geometry also allows to exploit the analytical possibilities of the Spectral Domain Approach further. Part of the reaction integrals can be done in closed form in the spectral domain, prior to the numerical inverse Fourier transform. This leads to a combined spectral and space domain approach for the evaluation of the reaction integrals. Examples and numerical results demonstrate the applicability of the method to real 3-D problems in multilayered media and the results achieved so far.

[17] Ongoing work includes the incorporation of magnetic sheet currents and coplanar transmission lines. For every day use, additional effort should be spent on improving computational efficiency, for example by exploiting symmetry to reduce the number of reaction integrals that has to be evaluated.

Appendix A:: Definition of the Fourier Transform

[18] With the use of

equation image

the Fourier transform pair is defined as

equation image
equation image

which can be more easily evaluated by using polar coordinates in the spatial and spectral domain as

equation image

such that the inverse Fourier transform reduces to a one-dimensional Fourier-Bessel transform

equation image

where Jo is the Bessel function of the first kind and order 0.

Appendix B:: Overview Asymptotic/Singular Behavior

[19] The following Fourier transform pairs give the correspondence between the asymptotic behavior for large kρ in the spectral domain and the singular behavior for small transverse distances ρ in the spatial domain. The distance ∣zz′∣ is written as Δ.

equation image
equation image
equation image
equation image