ABSTRACTI. The introduction ofthe concept of "Admittance Operator" allows a systematic procedure for the computation of passive microwave circuits elements ...
Representation by Equivalent circuit of the Integral methods in Microwave passive elements.
H.BAUDRAND *
ABSTRACTI allows a systematic The introduction of the concept of "Admittance Operator" procedure for the computation of passive microwave circuits elements by integral method. First, an "equivalent circuit" of the electromagnetic problem has to be established. This circuit includes sources and elements which generally are self-adjoint operators. The following procedure consists in expressing the trial functions as l"virtual sources (voltage or currents) which do not consume energy. The efficiency of this procedure is described in the analysis of some structures which include several discontinuities with metallic elements or surface impedances. I - INTIRODUCTION
The integral methods are preferently used in cases where the devices present discontinuity surfaces S; with metal or dielectric separating different media where the Maxwell equations have analytical solutions (for instance fig.1). EO S3--U// ms,SE. E2.
/////E
S2 S4
Fig. I
Therefore the problem reduces in writing boundary conditions on each discontinuity surfaces. The Galerkin procedure applied with trial functions in spectral domain as in the transverse resonant method or real domain (Green's functions) gives homogeneous equations (calculation of propagation constant or resonant frequency) or inhomogeneous equations (discontinuity in waveguide or circuit excited by a source). In a few cases ([1] ...[4]) a representation of the problem is proposed in the form of multiport representing the interactions between the modes at the location of the discontinuities. These representations contribute entire to the calculation of the structure but do not permit the
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resolution of the given problem. The introduction of "admittance operator" ([5], [6], [7]) defined as one-port or two-port allows the definition of a "current denlsity" J analogous to the current line in classical TEM lines.
II-DEFINITION AND UTILISATION OF ADIMITTANCE OPERATOR Let a domain D, bounded by a surface S ( oriented with a normal vector n ) (fig 2a). The relation between the tangential field E and the current T TAin on S is linear and may be calculated by the dyadic Green's formula. This relation is represented with a one or two-port network (fig.2b). The operator Y presents the particularity that if the domain D contains only lossless media, by writing Y - jX, the operator X is a self-adjoint one
J1
J2~~~~~~~~~~~~~~~~~~~J
Fig.2e: One -port
J
Fig.2b:Two-port.
H1 -H1T =JsA^ n
J1
J2+Js=O
Fig. 2c The boundary conditions of magnetic field are translated in this represenltation by the classical Kirchoff law (fig.2). The sources and trial funlctions are represented in the equivalent circuit by voltage or current sources. Each type of discontinuity surface, does correspond to a two- port (some examples are given in fig.3), the device terminations correspond to active or passive one-port. The equations of the problem are obtained by writing that the trial functions are "'virtual"l (i.e. the complexe power which go through these sources is zero).This rule has been widely used in the problems of waveguides by the Spectral Domain Approach. In the example given in fig.3-1, the obstacle is such that in each point of the discontinuity either J = 0 or E = 0. With the usual conventions of the inner product (i.e. the 'integral of the product of the two functions 1360
or
their conjugate), this property is written < Et J
as :
>=0
where Et is the trial function of the electric field which is zero on a perfect metal. Let HI the indicator function of the isolator part of the obstacle. We have the implication: HI J = 0
0
(for all n)
provided that the trial functions Etn form a complete set of basis functions. In a same manner, in the case of fig 3-2, only two equations have to be solved, because of the choice of trial functions Jtn Etn HME2 0 O
and HM (E+ZsJl)
=0
(1)
=
0
(2)
gives with (1)
=0
The Galerkin method is later used to obtain a matrix representation of the problem. We can see that the equations in this representation are simply obtained by writing that the dual quantities of the trial functions (i.e. electric field if J is the trial function, and reciprocally) are zero in the considered interfaces. TYPE OF DICONTINUITY EQUATION 1) Thin perfect metal
obsLacle
J1+J2=-J E I -E2=E
HII E=O HI
J=OH
TRIAL FUNCTIONS
EQUIVALENT SCHEME
EL)
--J1
L =0 HM E=0I
J2-. l_
_
(Ot)E21 9TE
tEI
_~~~~~~
HI(E1-E2)=O
2)
Change of dimensions
of waveguide with surface impedance 75H
Et=E1I-E2
J _EI 2) KJI(E +ZsJ2)=O L=E H .t=E HM E2=-O t=
HI Jt=O
Li
lE
HM: Function of metal-(= ion metal, 0 elswere) HI: Function of isolator( HI= 1-HM)
Fig.3: Four-port equivalent circuit dicontinuities 1361
of two type of
III- APPLICATION: The classical network theory is applicable (Thevenin or Norton theorems for instance) to the development of the formulation. Another property is that this systematic procedure gives in most cases variational expressions for the unknown quantities (propagation constant or impedance of the circuit). To illustrate this method some results are given on the following devices -Waveguide discontinuities The double discontinuity between rectangular and circular waveguides (fig.4) is studied with a great precision with only four trial functions.
X _Jolt$F t ~ ~ ~ ~ ~VlE Fig.4: Thick trans,ji ;tion between two waveguides. verifying the edge conditions. In the equivalent network, the sources JO and J'O are the amplitudes of the transverse magnetic fields of the fundamental mode. The calculation of Eo and E'o is required for the impedance matrix of the discontinuity in the fundamental mode. -Radiating patch In the case of the radiating patch excited by a coaxial line (Fig.5) one shows that by using the electric field as trial function on the surface S, one avoids taking some current sources on the feed line as required in previous studies [8]. L 5.~~~~~~~~SSG m
)\)
81.5
[ A]
°~~~~~~~~~.SW OL1. %
4.0
Fig.5
30 50 Fig.6: Dispersion curve of fund. mode. 15
The method is also being applied to more complex problems : -The coupling between coplanar line and fin-line with finite 1362
thickness on anisotropic substrate (Fig.6) . In this case, it can be matrices of smaller size can be used compared to usual noticed that studies [9]. -The superconducting microstrip line with a thickness greater than the London length. This line is an example of line which presents a surface impedance and furthermore, it illustrates the constraint which appears in the choice of sources in order to obtain a solvable formulation of the problem. Fig.7 shows the equivalent network and fig.8 shows the variation of effective dielectric constant calculated by this method versus the London length.
Fig.7: Superconductor microstrip. 14
V.3-
c
12'
C:
0 U
10-
C) .-a
w
Fig.8: London's length XL(enA) IV- CONCLUSION: We have shown with some examples a systematic procedure to solve the circuit elements by the integral method. This procedure needs the resolution of a multiport network ixn which classical impedances are replaced by linear operators. It is applicable to many other types of problems such as, diffraction , biffurcation etc... The main advantage is that , when the discontuity surfaces are chosen , the procedure gives automatically a variational formulation for the results. 1363
Acknowledgements: The examples mentioned in this paper are provided respectively by P. COUFFIGNAL, H. AUBERT, and P. MEISSE. (Institut National Polytechnique de Toulouse, and Centre National d'Etudes Spatiales Toulouse).
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electromagnetiques"t.
Colloque O.H.D. Rennes Sept. 1989 [ 6] C. VASSALO "Dicontinuit6s abruptes dans les guides optiques" Colloque O.H.D. Rennes Sept.1989 [ 7] H. BAUDRAND "Formalismes variationnels en e6lectromagnetisme. Application a la modelisation en microondes". Colloque O.H.D. Rennes Sept. 1989 [8] N.G. ALEXOPOULOS "Recent research advances in microstip antennas and circuits a7 UCLA"I Proceeding of J.I.N.A. Nice 8-10 Nov.1988. pp.209-211. [9] J. BORNEMAN, F. ARNDT "Calculating the characteristic impedance of fin-lines by transverse resonance method". IEEE Trans on MTT Vol. MTT-34 Nil Jan 86 pp.85-92. 1364
