диафрагмированные волноводные фильтры / 29b9be85-40b0-45d2-baf4-697fb88d1f6e

European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2000 Barcelona, 11-14 September 2000c ECCOMAS

NEW IDEAS FOR FREQUENCY/TIME DOMAIN MODELING OF

PASSIVE STRUCTURES FOR MMICS

P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia 1

Centro Interuniversitario per la Ricerca sulle Microonde e Antenne (CIRMA) Dipartimento di Elettronica, Universita` di Pavia

Via A. Ferrata 1, 27100 Pavia, Italy

e-mail: conciauro@ele.unipv.it, web page: http://ipvmw5.unipv.it/

Key words: MMIC, Planar Circuits, Numerical Modeling, BIRME Method

Abstract. This paper outlines a new philosophy for the numerical modeling of shielded microstrip structures printed on multi-layered lossy substrates. An algorithm is described which directly yields the mathematical model of the structure starting from field equations. The model, obtained in the form of the pole expansion of the admittance matrix, represents the structure like a lumped-element network. Any type of frequencyor time-response can be deduced from this representation in negligible times, with high frequencyor time-resolution. Some comparisons with results obtained by using the popular Spectral Domain Approach show that the procedure presented in this paper permits a dramatic time saving.

1TMR fellow at the Dipartimento di Elettronica, Pavia; presently at the THD, Darmstadt.

This work was supported by E.C. under TMR programme contract no. ERBFMRXCT960050 C.N.R. and by C.N.R. under contract no. 99.00037.PF48.

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

1 INTRODUCTION

Many “full wave” methods are currently available for the frequency-domain or time-domain analysis of passive structures used in Monolithic Microwave Integrated Circuits (MMICs). The computer codes based on these methods yield many samples of some frequencyor timeresponses. The required number of samples can be very large, especially in the wideband analysis of a structure having a rapidly varying frequency response or—equivalently—in the calculation of a slowly decaying time response. A new field calculation has to be carried out for obtaining a new sample, so that a long computing time can be necessary.

In some papers concerning the numerical modeling of waveguide components [1]-[6] and 2D planar circuits [7], some of the present authors showed that a dramatic reduction in computing time can be obtained by using the Boundary Integral - Resonant Mode (BI-RME) method. This method differs radically from other more usual ones, because it makes use of the field analysis to determine the mathematical model of a microwave component rather than the numerical values of some response at discrete frequencies or times. The model, consisting in the pole expansion of the Y- or Z-parameters, can be used for calculating in negligible times any type of frequencyor time-response to a very high degree of resolution.

This paper aims to demonstrate the feasibility and the advantage of the BI-RME approach in the modeling of shielded microstrip structures for MMICs (Fig. 1). A layered substrate is considered, with one ore more slightly conducting layers characterized by the complex permittivity:

For simplicity, a delta-gap voltage excitation is considered [8]. To show the feasibility of the method without introducing unessential complications, a zero-thickness and lossless metallization is also assumed.

y

Figure 1: Shielded microstrip circuit printed on an interface (z = 0) of a layered medium: side view (left), top view (right). The region includes the metallization (shadowed area) and the delta-gaps (segments t1; t2; : : :)

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

2 THEORY

The shadowed area shown in Fig. 1 represents the metallization, including the gaps t1; t2; : : : ; tK where the exciting voltages v1; v2; : : : ; vK are applied. The excitation gives rise to a surface cur-

rent ~ distributed in the metallization, and to a set of gap-currents 1 2 K . The positive

J i ; i ; : : : ; i

direction of the gap-voltages and currents is defined by the normals ~n1; ~n2; : : : ; ~nK . We have:

Z

~ ¢

ik = J ~nkdtk (2)

tk

Over the surface the tangential electric field must satisfy the boundary condition:

k=1

where ±k denotes a delta-function supported by the segment tk. According to (3), the tangential electric field must vanish everywhere on , with the exception of the gaps, where it corresponds to the values of the applied voltages.

The electric field can be represented by the modal series

where: the primed and double-primed quantities refer to the modes TEnp and TMnp of the rectangular waveguide of dimensions a and b, respectively (Fig. 1); vectors ~enp are the (normalized) electric vectors of the waveguide modes; coefficients Znp are the modal input wave impedances of the layered waveguide, closed by the plane conductors at z = z0 and z = zN , as seen in

where ¬np = 4 if np 6= 0 and ¬np = 2 otherwise. The impedances Znp correspond to the one-dimensional Impedance Green's Functions (IGF) involved in the modal representation of the field in a layered medium [10] [11], calculated with the observation and the source points at the plane of the metallization. The values of Znp can be obtained by using the general closedform expression of the IGF given in [11], specialized to the case of a rectangular box with permittivities of the type (1).

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

It is noted that, due to the trigonometric form of ~enp0 and ~enp00 , equation (4) represents the components of the field in the form of Fourier series, like in the Spectral Domain Approach (SDA) [9]. Using the values of the impedances at a given frequency, and introducing (4) into (3), an equation is obtained that determines the spectral components of the current density at that frequency. By means of the Method of the Moments (Galerkin version) we could transform this equation into an algebraic system, equivalent to the one obtained through the SDA. Then, like in this approach, we could determine the spectral components of the current frequency- by-frequency, and could obtain a collection of samples of the current/voltage relationship at discrete frequencies.

The BI-RME method used in this work introduces a substantial change to the SDA. The nodal point of the BI-RME method is the use of approximate expressions that represent the impedances as rational functions of the frequency. As we shall see, using these expressions, the MoM leads to determine a mathematical formula relating the admittance matrix to the frequency.

The approximate representation of the impedances is obtained starting from the following formulas, deduced from the singularity expansion of the IGF discussed in [11]:

All the quantities appearing in these equations can be calculated by applying the general formulas given in [11]. Here we limit ourselves to discuss the equations.

The pole pairs placed at !npq and °!npq§ are related to the resonating frequencies of the mode TEnpq (primed symbols) or TMnpq (double-primed symbols) of the layered box. These poles are slightly displaced upward from the real axis, due to the losses in the

medium (Fig. 2).

Further imaginary poles (jªnp1; jªnp2; : : : ; jªnpH ) are present in TM-mode impedances, whose number depends on the number of conductive layers. In the practical case of slightly conducting layers these poles are very close to the origin of the !-plane.

A pole at the origin of the ! plane can also exist in TM mode impedances, provided the metallization is placed at the interface between insulating layers (the coefficient xnp0 in

(7) differs from zero only in this case).

The quantities Rnp00 , Snp0 , Snp00 (and the poles near the origin, in the case of TM modes),

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

Figure 2: Pole pattern of Znp00

determine the low-frequency behavior of the impedances. They are defined as

The quantities in parenthesis are regular near to the origin and their values can be calculated numerically, by using the closed-form expressions of the impedances given in [11]. Coefficients Rnp00 , Snp0 , Snp00 are real-positive, due to the passivity of the structure and because the real and imaginary parts of the impedances are even and odd functions of !, respectively. It is also noted that the frequency-independent term is absent in (6), because the modal impedances of TE modes are zero at zero frequency.

The extraction of the low-frequency terms in (6) and (7) accelerates the convergence of the infinite pole expansions given in [11]. After the extraction, thanks to the improved convergence, the expansions can be truncated by retaining the only singular terms included in some circle C around the origin (the symbol ßC in (6) and (7) denotes the truncated series). The radius of C must be larger than the upper limit (!max) of the frequency band of interest, in such a way as to include in the circle the resonances at frequencies below !max and a sufficient number of higher resonances. Thus the radius is fixed to some value!max, where is an accuracy factor larger than 1 (Fig. 2).

An useful simplification can be made in (7), since we are not interested in high accuracy at very low frequencies. In fact we can take into account the effect of the singularities near the origin by replacing them with a single pole jªnp:

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

where:

where we introduced the “amplitudes” of the resonant modes of the box:

Equation (9) is the “BI-RME representation of the field”, because it includes Boundary Integrals of the type (5) (in the quasi-static contribution represented by the first two series), and two Resonant Mode Expansions (the last two series, related to the TE and TM resonant modes of the box).

Resonating frequencies of the box become larger and larger with increasing the mode order.

As a consequence, with increasing n and/or p, the poles !np0 1; !np0 2; : : : and !np00 1; !np00 2; : : : move at increasing distances from the origin, up to the point that all of them go outside C. For this

reason the RMEs include only a finite number of terms, corresponding to all the triads (npq) such that !npq0 2 C or !npq00 2 C. It is convenient to convert the three-index notation into a single-index one, for instance, by ordering the poles included in the RMEs into ascending sequences. Thus, from now on we shall replace, for instance, the symbols !npq0 , znpq0 , a0npq with !i0, zi0, a0i. The ordering identifies the triad (npq) corresponding to the index i. We shall denote by P and Q the upper limit of i, in the cases of TE and TM resonant modes, respectively; in other words, P and Q are the numbers of the resonant modes considered in the RMEs.

By substituting (9) into (3) we obtain the basic equation to be solved. The solution is obtained by Galerkin's method. The current is approximated by the formula

where f~umg and fw~mg are two sets of suitable two-dimensional vector basis-functions defined on and the weights cm, dm are unknown. All basis functions have zero normal component at

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

the edges of the metallization. Functions ~um are solenoidal (rT ¢ ~um = 0), in order to permit an adequate representation of the solenoidal currents in the zero-frequency limit. Since the subspace of solenoidal vectors is spanned by f~umg, vectors w~m and all their possible combinations must be non-solenoidal. This implies that the functions rT ¢ w~m must be linearly independent.

From (11) and (5) we obtain the spectral amplitudes of the current in terms of the spectral amplitudes (u~mnp; w~mnp) of the basis functions2:

The superscript T denotes transposition.

By testing both sides of (3) with the u- and w-functions we obtain a set of M0 + M00 linear algebraic equations relating the weights and the mode amplitudes to the applied voltages. Another set of 2(P + Q) equations relating the weights and the mode amplitudes is obtained by substituting (12) and (13) into (10). The equation obtained by testing with the u-functions are used to express the c-variables as functions of the others. Then, the c-variables are eliminated from the remaining equations and the following approximation is made3:

vectors.

3Due to the smallness of ªnp the approximation (15) is inaccurate at very low frequencies only.

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

TABLE I - MATRIX DEFINITIONS

we obtain a linear system relating the unknown variables dm, b` and the mode amplitudes to the gap voltages. Equations (16) are included in the system, after expressing the b-variables as functions of the d-variables (see (26) in the Appendix). These manipulations aim to make all the coefficients of the system to depend linearly on the frequency, in such a way as to write the system in the form of a matrix equation of the type:

In this equation v is the vector of the gap voltages and x is the vector of the unknowns, defined as

x = (d1; : : : ; dM00; a01; : : : ; a0P ; a01; : : : ; a0P ; a001; : : : ; a00Q; a001; : : : ; a00Q; b1; : : : ; bM00)T

and L, M, N are frequency-independent matrices (see Table I)4.

Furthermore, through the elimination of the c-variables from (14) also the gap currents can be written as functions of v and x:

Therefore, substituting the solution of (17) we obtain the following equation, which formally gives the admittance matrix of our circuit:

4L and M are N £ N matrices, with N = 2(M00 + P + Q).

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

The particular form of the matrix (L°j!M) is the fundamental result of the BI-RME approach. In fact, it can be shown that, thanks to this form, the inverse matrix can be obtained as5:

where n and xn are the eigenvalues and the eigenvectors obtained as solutions of the generalized eigenvalue problem

with the normalization xTMx = 1. Equation (20) explicitly gives the frequency dependence of the inverse in the form of a pole expansion, the poles being obtained as the eigenvalues of (21)6.

It is noted, however, that the direct use of (20) is not convenient, because it requires the calculation of all the eigensolutions (in practical cases their number can be of the order of hundreds), most of which represent poles very far away from the frequency range of interest. To avoid this problem, it is possible to use the alternative formula7

where the weight of the singular terms decreases as fast as (j!= nj)2. Thus, it is possible to ignore all poles located outside the circle C defined above (we are interested in frequencies below !max) and we can limit ourselves to determine only the eigenvalues located inside the circle 8. Substituting (22) into (19) we obtain the following expression, valid up to !max:

Equation (23) can be rewritten in the standard form of a pole expansion:

5The mathematical details cannot be reported here. Equation (20) and the existence of the inverse matrices in Table I, shall be discussed in a forthcoming journal paper.

6It can be shown that the eigenvalues and the corresponding eigenvectors are present in complex conjugate pairs, and that all eigenvalues have negative real part.

7Equation (22) is obtained by extracting from the summation the first two terms of its power expansion around

!= 0 and by re-expressing these terms by the corresponding terms obtained directly by power-expanding (L ° j!M)°1.

8No particular computational burdening derives from the calculation of the terms involving L°1, since this matrix must be calculated in any case to solve the eigenvalue equation (21).

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P. Arcioni, M. Bressan, G. Conciauro, A.R. Olea-Garcia

3 RESPONSE IN FREQUENCYAND TIME-DOMAIN

The mathematical model (24) is an approximation for the exact pole expansion of the Y- matrix. In the approximation the infinite sequence of singular terms (typical of distributedelement structures) is replaced by a finite one (typical of lumped-element networks). The expression Y0 + j!S1 takes into account the effect of the infinity of poles placed outside of the circle C.

Due to the low-frequency approximations (8), (15), (26) some inaccuracy can be expected in the range of very low frequencies. However, the width of this range decreases with decreasing the conductivity of the medium, so that, in practical applications, the inaccuracies are confined in an uniportant portion of the band of interest.

Equation (23) can be used for calculating very rapidly the values of Y-parameters at many frequencies in the band of interest. The rapidity of calculation permits to use a very small frequency step, in such a way as to have a resolution appropriate to reproduce irregular curves (filter responses, sharp resonance peaks, etc.). Any kind of circuit parameters (e.g., scattering parameters) can be deduced in negligible times from these values.

The time-domain current response to any kind of voltage excitation can be calculated by convolution integrals, using as kernel the transform of (24), i.e.:

C

Also in this case there is no problem in calculating a very large number of samples of y(t), with a small sampling interval and for a long time. These results can be used for time-domain simulation of complete MMICs including the modeled structure, possibly together with nonlinear devices.

4 RESULTS

In this section we present some results of the BI-RME method and compare them with results obtained by a commercial code based on the SDA (EMSightTM included in “Microwave Office” by Applied Wave Research, inc.). In all our calculations the u- and w-functions were suitable combinations of “rectangular roof-tops”, giving rise to field patterns of the type shown in Fig. 3. The strip-type solenoidal basis functions (Fig. 3c) serve to take into account the d.c.

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