S-Domain Modeling of Rectangular Waveguidewith Dielectric Inset (3D case)

Marco BressanV Giuseppe Conciauro* and Wissam Eyssa**Dept. of Electronics, University of Pavia,

Pavia, Italy, Email: giuseppe.conciaurogunipv.it

AbstractWe introduce a new method for the determination ofthe Generalized Admittance Matrix of a rectangularwaveguide section loaded with a dielectric inset ofarbitrary shape. A State Space - Integral Equationapproach is applied which leads to the pole expansionof the admittance matrix in the s - domain.

1. IntroductionAs illustrated in other works [1] [2], the State Space-Integral Equation (SS-IE) Method is quite efficient forthe wideband modeling of passive waveguides and quasi-planar integrated circuits. The dimension of the state-space model obtained by this approach is much smallerthan that obtained in other ways (e.g. FEM-based algo-rithms [4]). This means that model order reduction is notmandatory. Furthermore, the reduced computing time andthe limited memory requirements make this method wellsuited for inclusion in efficient CAD tools.In [3] we introduced the application of the SS-IE methodto the determination of the General Admittance Matrix(GAM) of a rectangular waveguide section including ahomogeneous cylindric dielectric inset, in a simple 2Dcase. The extension to the 3D case, which is not trivial,is outlined in this paper.

2. TheoryFig.la represents a waveguide section including a di-electric inset of arbitrary shape and relative permittivityc. Let us consider the first N waveguide modes thatinteract significantly with the inset and excite them by themodal voltages v (v = 1, 2; n = 1, ..., N) at the portsS(1) and S(2) of the waveguide. The index n identifiesa well-determined mode of the rectangular waveguide(n <-* TE,O or TM,O).The field in the air region (Ea, Ha) can be considered asthe superposition of the effects of the field distributionon the ports due to the voltages v(>) and of the equivalentelectric and magnetic currents densities

J = x H' M = Eax n

defined over the air/dielectric interface or (Fig. lb). Theeffect of these currents, combined with that of the volt-ages, creates a null field in the dielectric region. Dueto the continuity of the tangential components of the

S (1)

II

{v(l !}V(2)1

an

b

S (2,

nuill -

field d -Mn

Fig. 1. Rectangular waveguide with a dielectric inset of arbitraryshape (a), schematic view of the structure with equivalent sources forthe determination of the field in the air region (b) and in the dielectricregion (c).

fields across or, the correct field (Ed, Hd) in the dielectricregion (Fig.1c) is generated by the sources -J and -M,with the ports closed by conducting planes. Also in thiscase the field in the complementary region is zero. Forthis reason, to study the field in the air or in the dielectricregion we consider the waveguide totally filled with airor dielectric, respectively. In the air region we have

2 N

ZaZVn)En)+Escv=l n=l2 N

Ha =V(Z)H') + Hscv=l n=l

(2)

(3)

The first term in (2) and (3) represents the incident field,whereas the second term is the scattered field generatedby the equivalent sources. The expressions of En>) andHn>) are known from waveguide theory.The modal current for the m-th mode at S(L) is

m;-; ( Yatm ~~~~~~(-)l-l h/> - dd (4)

where hm is the magnetic modal vector (normalization:f h2 dxdy = 1). Introducing (3) in (4) and using thereciprocity theorem we also have

2 N

~~(~ 1 S S V~] Hi-j) dxdyiim ninn1

+ (E?n , J)-(Hm(/' M) (5)

where fJ, gf=fg dcr. Equation (5) evidences thatthe crucial point to obtain the current-voltage relationship

TABLE 1

Cl,= en(n)- ()VZ~(V .en) Cn(l')

n(>) {-( in nn (&")sn <+ Z (V hn) 8nv)

fin(l> (((v) Sn) + (I1-Knc coth KnC) Cn))= z(V h)j2( (v) C(Y) _ ccothKnc (v))I(v i)(_l)vfincn)

OM2 2/27c(wFr)1(knrc) [ ( 1)j"

A _diag(An) diag(A ) 1 B _ diag(Bn) diag(Bn)[diag(An ) diag(An) [diag(Bn ) diag(Bn) J

A~ {JlKn cothfKncAn -=l

{ coth cnc- n c csch2KncBn i coth Knc

(( ) = C

In

(2_1 z & 2) =Z

n - 0I -snc coth snc

B = 2Kn sinh Kncn - _csch^;nc

lin

8(v) _ sinh K,4(v) c(v) _ cosh k,n4(")Sn sinh snc n Kn sinh sncThe upper (lower) definition holds when n corresponds to a TE (TM) mode.

for our structure is the determination of the equivalentcurrents, as functions of the applied voltages.

As in other works (e.g., [1] [5]), in order to obtain thesystem equations in the form of state-space equations,we use an hybrid representation in which the fields,generated in the rectangular cavity by J and M, are givenas the sum of a quasi-static part (Boundary Integrals)plus a high frequency correction expressed as a truncatedResonant Mode Expansion (RME). Also EJ() and Hn>)can be put in a similar form, in which the quasi-staticpart is given in closed form as described in the Appendixof [1]. We have

E 9- u _12 nr(k2+nn - n kn (k2 + .2) Fnr (6)

3 0(-)Hn1(s(S,710n (7)-n

ST10 kr(k r + 5S2)

(S vEo,_to, ,LLO/IEo

where 5nr, Rnr and knr are the electric and magneticeigenvectors and the corresponding wavenumber of them-th mode of the rectangular cavity (m ii, r) (nor-malization: f SnrdV f -H2rdV = 1). The explicitexpressions of UWj), W ), and Oyr) are given inTab. 1. By using the representation of the incident field(6),(7) and the BI-RME representation of the cavity fieldssustained by J and M, we can express the tangentialfields at a generic point Ir c cr as

ON THE AIR-SIDE

Et = _' TaM Sr1GAJ anrknrQEnr)t

n,r

+ : I: Vn ,nt + 2v=l n=l

_S GF_'0

v(i>nt +

KS

1 SMTHta = _ S M +Tj -

S '10

i2 N

° v=lI n=lI

(8)

VI- anr(0nr)tT/o n,(r

sWn>t _2xf (9)

ON THE DIELECTRIC-SIDE

Et' =- SeJ + TaM +s'1oGAJ

+Zbn,rknr(Srir)t + 2n, r

d 1t ST1O TJ+ s6GFM

'o

(10)

(1 1)+ S1 bnr(%(nr)t 2'10 n,r

where anr and bnr are the mode amplitudes in the air-and in the dielectric-filled cavity

s'1oknr(r nr, J)anr

~2s (knr M)

k2nr (k2r + S2)

. 1: 0(,)v()v=1,2

(12)

bn ._ST1oknr(Enr, J)- 26 (nrr M) (13)brir knr (knir + S2,E)and the other symbols represent the integral operators

SeJ V j ge(jr, Ir1) V' Jf(fe) d7/'

GAfJ:= GAJ 5Enr (Snr (r) )tn,r nr

GAfJ j (G -(r -r) J(r')) /c'TJ j (v x Go(i,r') J(r')) cdc'

smM:= V7 j gm(? r ) V' M(r') d/'

G M j (GO rXir )e

M ) dc

TaM j (v x GO(i, r') M()) cr'

In the expressions above ge and gm are the Green'sfunctions (GFs) for electrostatic and magentostatic scalar

potentials in the rectangular cavity, Go and Go arethe quasi-static dyadic GFs for electric and magneticvector potentials in the 'Coulomb gauge'. The GFs arecalculated as image series, the convergence of which isaccelerated by the Ewald technique [6]. The correct rep-resentation of the field discontinuities across the currentsheets is ensured by taking into account the singularityof the GFs ge, g-, Go and Go. Hence, the accuracy ofthe representation of the discontinuities is not affected bytruncating the infinite modal series in equations (8)-(1 1)and retaining the first cavity modes such that

{ Co/b in (8-9)knr <(,Wmax (14)

06/_90 in (10-11)

where WOmax is the maximum frequency of interest and( is an accuracy parameter (typically ( 2 3). Inequations (8)-(11), J,M and the mode amplitudes arethe unknowns.We use the following approximations of the currents

PM

MJI zcpjip iJp=l

1 Pj.- dpv1

where {f Ji} and {vJ'} are appropriate two-dimensionalvector basis functions. To meet the physical constraintat s = 0, it is important that these basis functions canrepresent solenoidal currents, exactly. We indicate withQM < PM and Qj < Pj the number of independentfunctions in the sets {V. i7 } and {V J }, by which werepresent the magnetic and electric charges on or.As in [3], the continuity of the tangential components ofthe field is enforced by using the Galerkin's method (Jpand J' are used as test functions for electric and magneticfield equations, respectively), giving rise to two matrixequations

Smc+ s2GFc+ Td +s2Haa +s2Hdb = (V + s2W) v (16)d A-SeC -_GAd+ Tc+s2EaKaa+EdKdb =Uv (17)

where the tilde denotes the transpose, v = u$> } is thevector of voltages, a = {anr}, b = {bnr}, C {Cp},d {dp} and the other matrices are defined as

2KE4,SmnJ) -±K~,e§Sij: 2 Ui, uj Si = 1+e (Vi, SVj

GF: (1(+)V, GFj) 2 vi Taj)

Emi: 8m i) Hiui

n(>i) := (Un>), v)'K : diag{ki}

V(ni) : Vn ) Vi) W(ni): (CVn ) ) V)

GAj 2 (vi, Govj [EaKa Ea] [EdKd Ed]It is stressed that all matrices are real and independent

(15)

of s. In equations (16) and (17) matrices E, H and Kare marked by the letters a and d in order to distinguishthe different number of resonant modes considered in theair-filled and in the dielectric-filled cavity, respectively.Further two matrix equations correspond to the defini-tions of the mode amplitudes given in (12) and (13)

Ka2(Ka2 + s2I) a + KaE3 d + s2Ha c = s2E v (18)Kd2(Kd2 + ES2I) b + KdEd d + ES2Hd c 0 (19)

where ij : rm O') where i P-*(n, r), j P-*(m, v)and d is the Kronecker symbol. In order to obtain thesystem equations in the standard form, we note that Seis a symmetric semidefinite positive matrix; it can be putin the form

S = QA 1Q (20)

where A is a positive definite diagonal matrix of di-mension equal to the rank of Se (Qj). Further equationis obtained by introducing an additional set of statevariables q defined by

s2Aq =-Qd (21)

Finally, the equation relating the modal currents on portsto the voltages and to the state variables is obtained bysubstituting (6), (7), (15) into (5)

i= ((A + s2B)v -s2a -(V + s2W)c -Ud)'10 (22)

where matrices A and B are defined in Tab. 1.Equations from (16) to (22) represent the system in thes-domain and can be put in matrix form as follows

(M + s2N)x = (L + s2L')v

i=1 ((A + s2B)v (L + 2Lf)x)

where x = {variables and

M= 0EaKa

/Ka2O

N= Ha

O9 O

(23)

(24)

a, b, c, d,q} is the vector of the state

0Kd40

EdK d

0

0cKd2cHd00

00sm

T0

HacHd

GF00

KaEaKdEd

TGAQ

00000

o001Q0

0000A/

(010

L= lvilul

101

101

Matrices M and N are symmetric of order Ma + Md +PM + PJ +Qj, where Ma and Md are the number of the

S S (2)

0c

Fig. 2. A waveguide section loaded by a cylindric dielectric inset.

resonant modes considered in the air-filled and dielectric-filled cavities, respectively.The GAM, which is naturally symmetric, is given by

Y='0 (A+sn(LB- ')M sN) 1L5L)

STI

The inverse of matrix (M + s2N) can be calculated inthe form of a pole expansions (for instance, see [4]), bydetermining the eigensolutions of the homogeneous prob-lem (M+A2N)y = 0. Without going into mathematicaldetails, the rank of matrix N is Ma + Md + PM + QJand this is the number of the eigenvalues. Some ofthe eigensolutions correspond to resonant fields that can

exist in the recatngular cavity with the dielectric inset;the remaining eigensolutions are spurious and refer tothe structure obtained exchanging air with dielectric. Asexplained in [3], the spurious solutions do not couplewith the ports and, hence, do not give rise to actualpoles of the GAM. Therefore, the elimination of spuriouseigensolution is not necessary.

3. Validation

We applied this method to analyze a waveguide sectionloaded by a cylindric inset of height smaller than b(Fig.2). In Fig.3 we present the results obtained by thismethod, compared to the results obtained by the codeHFSS. The simulation is relative to a waveguide sectionof dimensions 22.86 x 10.16 x 20 mm; the inset iscentered on the base of the waveguide, its radius is6 mm, its height is 8 mm and E = 3. The basis functions{vp} used to approximate the electric current J are theones described in [1]. The basis functions {fJ'} used torepresent M are of the same type, but modified to fitthe different boundary conditions of the magnetic currentat the contact between or and the waveguide wall. Inthis simulation we used: Ma = 93 Md = 196; PM =

Pj 130; QM = Qj = 64. The total computing timewas 30 sec (code not yet optimized). Fig.3 shows a

perfect agreement with the results of HFSS. A list of thefrequencies relative to the eigensolutions is presented inTab.2. Only the frequencies denoted with * correspond tonon spurious modes. It is evident (Fig.3) that the spuriousresonances have no effect on the response of the circuitin the band of interest.

HFSSSs-]I-

-8 9 10 [Ghz] 1 3

Fig. 3. Scattering parameters for the fundamental mode TElo. Inthis band only two resonances are not spurious and the correspondingfrequencies are indicated by the dashed lines.

References[1] F. Mira, M. Bressan, G.Conciauro, B.Gimeno, V.Boria, "Fast

S-Domain Modeling of Rectangular Waveguides with Radially-Symmetric Metal Insets" IEEE Trans. Microwave Theory & Tech.,vol. 53, no. 4, pp. 2397-2402, April 2005.

[2] G. Conciauro, P. Arcioni, M. Bressan, "State-space Integral-equation method for the S-domain modeling of planar circuitson semiconducting substrates", IEEE Trans. Microwave Theory &Tech., vol. 51, no. 12, pp. 2315 - 2326, Dec. 2003.

[3] M. Bressan, Giuseppe Conciauro, Wissam Eyssa, "Integral-Equation Method for the S-Domain Modeling of RectangularWaveguide with Dielectric Insets (2D case)", IMS 2006, SanFrancisco, California, June 11-16, pp 1049-1052.

[4] A.C. Cangellaris, L. Zhao "Model Order Reduction Techniquesfor Electromagnetic Macromodelling Based on Finite Methods"Int. Jour. Numerical Modeling, vol. 13, no. 2/3, pp. 181-197,March/June 2000.

[5] P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, L. Perregrini,"Frequency/Time Domain Modeling of 3D Waveguide Structuresby a BI-RME Approach" Int. Jour. Numerical Modeling, vol. 15,no. 3, pp. 3-21, March 2002.

[6] P. P. Ewald, "Die berechnung optisher und electrostatischer gitter-potentiale", Ann. Phys., vol. 64, pp. 253-287, 1921.

TABLE 2

FREQUENCIES (GHZ) CORRESPONDING TO EIGENVALUES

CALCULATED IN THE BAND 8 . 13 GHZ.

1 7.237 7 10.824 13 12.6242 7.253 8 11.313 14 12.7183 9.650 9 11.595 15 12.819*4 10.589 10 11.815* 16 12.9415 10.755 11 12.0216 10.763 12 12.104The * denotes non spurious resonances.

0-

[dB] ,

-10 ,S1l1

02O

30-o HFSS

SS-IE

4010

-[

[Ghz] 1 3

1e

ISI-11