Formulation electroniques

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS
Int. J. Numer. Model. 2002; 15:23–57 (DOI: 10.1002/jnm.430)
Equivalent circuit representation for integral formulations
of electromagnetic problems
H. Baudrand1,
* and D. Bajon2
1
ENSEEIHT 2 rue Camichel 31071 Toulouse, France
2
SupAero, 10 Av. Edouard Belin; 31055 Toulouse, France
SUMMARY
The purpose of this study is to investigate the resources offered by electromagnetic equivalent circuits as a
tool for the systematic derivation of integral equations and also, to propose consistent models for active
sources. Copyright # 2002 John Wiley & Sons, Ltd.
1. INTRODUCTION
Equivalent circuit models have been introduced many times in the development of integral
formulations as attempts to transpose field problems into network problems. Usually, in these
approaches, the building of the equivalent circuit is directed by the aim to emphasize on a
particular point of the formulation. No general convention arise from these attempts. For
instance, a multiport can represent either the junction between two waveguides or the numerical
procedure of the moment method [1].
However the systematic use of operators and the use of J ¼ H ^ n instead of H gives a
straightforward representation of the relations between fields on both sides of the surfaces where
the conditions to be solved are impressed. From this, the following study develops a coherent
representation of electromagnetic problems which allows a systematic derivation process of
integral equations.
The definition of an admittance operator on one side of a surface S is firstly given, along with
the equivalent circuit representing it. As in the vector circuits defined by References [2] or [3],
this circuit representation is in many ways comparable to conventional electric circuits except
that voltages and currents are related to electric and magnetic fields. Since scalar voltage and
current are linked by scalar impedance, it seems natural to link two-dimensional functions such
as electromagnetic fields, with operators.
The boundary conditions through a surface are also represented in terms of equivalent circuit.
Actually, the continuity conditions on the tangential magnetic fields through a surface take the
form of Kirchoff’s law of currents.
Copyright # 2002 John Wiley & Sons, Ltd.
*Correspondence to: H. Baudrand, ENSEEIHT 2 rue Camichel, 31071 Toulouse, France.

Nevertheless, the introduction of operators instead of conventional admittance is not the only
difference between conventional circuits and the equivalent circuits presented herein. A new type
of sources, the ‘virtual sources’, is introduced.
In integral formulations, the unknown functions are chosen in such a way that they partially
fulfil the continuity conditions and, the formulation consists in prescribing the remaining
conditions. These functions are usually defined on only one part of the domain on which the
boundary conditions of the problem are specified. These unknown functions are inserted in the
equivalent circuit through these so-called virtual sources.
In the circuit itself, these sources obey the same laws as their conventional counterpart. Their
virtual nature comes from their own specification: to cancel on one sub-domain while their dual
cancels on their definition sub-domain when the electromagnetic solution is reached. Thus, the
‘virtual sources’ do not exchange energy with the circuit since they should be determined in such
a way that, as in a Wheastone’s bridge, they act as a short or open circuit. The virtual sources
and their applications are described in detail in the third part of this study.
The last part of this study is devoted to real sources as required for global simulation of
circuits. Global simulation attempts to include active components in electromagnetic simulation
and more often resorts to the finite difference in the time domain method (FDTD) or to the
finite elements method (FEM) [4,5]. In this paper, an integral method approach to global
simulation is undertaken with coherent definition for active sources in order to overcome the
apparent inconsistency between localized definition for voltages and currents in the active
devices and electromagnetic response of the surrounding passive components.
Finally, the steps of the systematic procedure are detailed: statement of the equivalent circuit,
solving of the circuit to express the dual of the virtual and real sources in order to derive the final
equation of the problem which translates the properties of the virtual sources. Applying
Galerkin’s procedure provides the final numerical form of the problem.
2. ADMITTANCE OPERATOR
The admittance operator is strongly connected to the Green’s operator introduced in integral
formulations but with one fundamental difference: the admittance operator considered herein is
related to a half-space instead of the whole space around the distribution of the sources.
2.1. Definition of the admittance operator
Let S be a closed surface bounding the space in which the electromagnetic fields of the problem
are defined and S a surface supporting S meaning that the bounds of S are also limits for S as
seen in Figure 1.
At each point M of the surface S, E is the tangential electric field and J is defined as
J ¼ H ^ n ð1Þ
n being the normal to the surface S in outgoing direction, pointing into the region where the
fields are defined.
The introduction of n in the definition of J both orients the surface S and associates a real
vector to the tangential magnetic field instead of a pseudo-vector. The key point in this
definition is that the bounds of the surface S coincide with the limits of S.
H. BAUDRAND AND D. BAJON
24
Copyright # 2002 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2002; 15:23–57

The medium being a linear one, the integration of Maxwell’s equations yields a linear
relationship between E and J (Figure 2).
J ¼ #
Y
YE ð2Þ
2.2. Representation by an equivalent scheme
From now on, the surface S will conveniently be represented by two terminals in an electrical
network: the voltage across the terminals is related to the electric field while the outgoing branch
current represents J. The current flow has the same orientation as n: an incoming current in the
admittance operator.
The impedance operator is defined by
E ¼ #
Z
ZJ ð3Þ
2.3. Relation with Green’s functions
The Green’s functions link the fields to a given source distribution, electric or magnetic current
densities.
Using Green’s functions, the admittance operator introduced in (2) is obtained as follows. Let
us consider S in Figure 1 as a magnetic surface covered with a sheet of electric current density jS.
A direct or numerical evaluation gives the tangential electric field E as
E ¼ #
G
GjS ð4Þ
J
E
S
M
n
Figure 1. Representation of the admittance operator.
Y
Ŷ
E
J
E
Y
J ˆ
=
S
Figure 2. Equivalence scheme of an admittance operator.
EQUIVALENT CIRCUIT REPRESENTATION 25
Copyright # 2002 John Wiley Sons, Ltd. Int. J. Numer. Model. 2002; 15:23–57

On the magnetic field, the boundary conditions on S are
H1T H2T ¼ jS ^ n ð5Þ
Due to the magnetic nature of S, the tangential magnetic field H2T vanishes on S then (5)
becomes
HT ¼ jS ^ n ð6Þ
Cross-multiplying each term of (5) with n gives
HT ^ n ¼ ðjS ^ nÞ ^ n ¼ jS ð7Þ
In this case, J is then equal to jS and the impedance operator in (3) is the Green’s operator
itself
#
Z
Z ¼ #
G
G ð8Þ
In Figure 3, the #
Y
Y operator links together the two tangential vector fields defined on S and
acts, on the side pointed by n, as a terminating impedance for the surface S.
Due to the cross-product in definition (1) of J, E and J are of the same mathematical vectorial
nature while H is a pseudo-vector.
Real vectors being mostly preferred, the alternative choice consisting of introducing M ¼
n ^ E appears less convenient, as it conduces to handle magnetic voltages and magnetic
currents. According to the choice of magnetic equivalent circuits, all the equivalent circuits
derived in the following would have to be replaced by their dual. For the sake of completeness,
this point is illustrated and discussed later on in Section 8.2.
2.4. Fundamental property of #
Y
Y in a lossless medium
A natural definition for the inner product for vectorial functions with two components defined
on a surface S (i.e. tangential components) is
hf jgi ¼
Z
S
f *t
g dS ð9Þ
Y
Ŷ
E
J
s
j
J = _
js
Figure 3. Representation of a Green’s function.
26

The surface S in Figure 1 being bounded by purely reactive impedance conditions and
containing only lossless media, the flow of the Poynting vector has a nil real part
P ¼
Z
S
ðE* ^ HÞ n dS ¼
Z
S
ðE* ðH ^ nÞÞ dS ¼ hEJi ð10Þ
ReðPÞ ¼ ReðhEjJiÞ ¼ ReðhEj #
Y
YEiÞ ¼ 0 ð11Þ
Being independent of the electric field E, this property ensures that
#
Y
Y ¼ j #
X
X ð12Þ
The #
X
X operator is a hermitian operator defined, for any j and c as
hjj #
Y
Yci ¼ hj #
X
Xjci ð13Þ
The media being reciprocal, #
X
X is a symmetrical operator.
2.5. Two-port operators
The generalization of the admittance operator to the space bounded by two surfaces, S1 and S2,
in Figure 4 is straightforward from the relationships between E1, E2 J1 and J2, which have the
form
J1
J2

¼
#
Y
Y11
#
Y
Y12
#
Y
Y21
#
Y
Y22

E1
E2

ð14Þ
2.6. Example of an infinite waveguide
As an example for the derivation of the admittance operator, let us consider a rectangular
waveguide excited in such a way that only TEn0 modes propagate as in Figure 5. Knowing the
electric field in the cross-section Eyðx; z ¼ 0Þ in z ¼ 0, the calculation of the magnetic field, i.e. J,
follows from the expansion of the electric field in terms of the set of the electric fields of the TEn0
modes as
E ¼
X
n
EnfnðxÞ ð15Þ
22
21
12
11
ˆ
ˆ
ˆ
ˆ
Y
Y
Y
Y
1
E
1
J
2
J
2
E
s1
s2
1
E
2
n
2
J
2
E
1
J
1
n
Figure 4. Two-port admittance operator.

with fn, the electric field of the nth mode given by
fnðxÞ ¼
ffiffiffi
2
a
r
sin
np
a
ð16Þ
In this particular case, the inner product given in Equation (9) takes the form
hf jgi ¼
Z a
0
f * g dx ð17Þ
From the orthogonality of the functions fn, the components of the electric field in Equation
(15) are given by
En ¼ h fnjEi ð18Þ
which give for J:
J ¼
X
n
YMnEn fn with YMn ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðn2p2=a2Þ k2
0
q
jom0
ð19Þ
YMn being the modal admittance of the nth mode and k2
0 ¼ o2
m0e0.
Substituting (18) into (19) finally yields the expression of J as
J ¼
X
n
fnðxÞYMnh fnjEðxÞi ð20Þ
which can be rewritten as
J ¼
Z a
0
X
n
fnðxÞYMnfnðx0
ÞEðx0
Þ dx0
ð21Þ
or, using Dirac notation [13],
jJi ¼
X
n
j fniYMnh fnEi ð22Þ
Thus the admittance operator #
Y
Y is given by
#
Y
Y ¼
X
n
j fniYMnh fnj ð23Þ
εo, o
y
z
E
0
a x
Figure 5. Admittance operator in an infinite waveguide.
28

The functions fn for TE and TM modes being orthogonal functions, expression (23) is the
spectral expansion of #
Y
Y, and YMn can be seen as the circuit admittance for the nth mode. Thus,
expression (23) can be directly applied providing that TE and TM modes are supported by the
medium or the waveguide considered inside S, Figure 1. Introducing a biorthogonal set of basis
functions, expression (23) is extended to more general cases where the set of functions fn does
not fulﬁl the normation conditions, for example when vanishing modes or hybrid modes are
present.
As shown in Reference [10], the biorthogonal set of basis functions is such that
hjnj fmi ¼ dmn ð24Þ
where dmn is the Kroneker delta symbol and the spectral expansion of #
Y
Y takes the form
#
Y
Y ¼
X
n
jjniYMnhfnj ð25Þ
This expression is now suitable whatever the region in which electromagnetic ﬁelds are
considered.
3. BOUNDARY CONDITIONS THROUGH A SURFACE
The introduction of J allows the elaboration of an equivalent circuit for the boundary
conditions through a surface S. Following Figure 6, one can write
E1 ¼ E2
ðH1T H2T Þ ¼ ð jS ^ nÞ ð26Þ
Rewriting the second relation in Equation (26) in terms of J1, J2 gives
ðH1T H2T Þ ^ n ¼ ðjS ^ nÞ ^ n ¼ jS ð27Þ
With J1 ¼ H1T ^ n and J2 ¼ H2T ^ n, Equation (27) gives
J1 J2 ¼ jS ð28Þ
It is now straightforward that (28) may be summarized with the equivalent scheme in
Figure 7 which expresses the conditions E1 ¼ E2 and J1 ¼ J2 jS.
s
n
1
H
1
E
2
H
2
E
(1)
(2)
Figure 6. Boundary conditions through a surface.

Henceforth, the fields are two-dimensional functions denoted without arrows, unlike the
previous vector representations of the fields. Similarly, two terminals are displayed in Figure 7
showing the two sides of the surface S. The orientations of J1 and J2 follow the orientation of n
according to the general definition (1) which preserves the right orientation of jS.
From these conventions, a general surface with a surface impedance such that E1 ¼ E2 ¼
ZSðJ1 þ J2Þ is represented, as shown in Figure 8.
4. RESTRICTION OF THE DOMAIN OF THE OPERATOR: VIRTUAL SOURCES
Planar circuits such as coplanar wave-guides (CPW), microstrip structures or irises in
waveguides, consist of printed surfaces between stacks of homogeneous media. From the
formulation point of view, these printed surfaces shall be considered henceforth as discontinuity
surfaces.
In the particular case of discontinuity surfaces with uniform surface impedance, the integral
formulation is directly obtained by inspection of the equivalent scheme in Figure 9.
On S, the electric field verifies
ð #
Y
Y1 þ #
Y
Y2 þ #
Y
YSÞE ¼ 0 ð29Þ
This condition (29) expresses the existence of the fields in the equivalent circuit like a
resonance condition. The surface admittance operator #
Y
YS has to be understood in a general
1
E
1
J
2
J
2
E
s
j
Figure 7. Equivalent scheme of the boundary conditions.
1
E
1
J 2
J
2
E
s
j
Zs
Figure 8. Representation of a surface impedance boundary condition.
30

sense. In the case in which a uniform surface admittance is considered, it just reduces to a scalar
weighted identity operator.
Unfortunately, in the case of perfect conductors, the conditions impressed to the tangential
fields are piecewise conditions and electrical representation of these conditions is not available
with conventional circuit elements. YS or ZS are not defined in the whole surface and are infinite
or nil.
On the other hand, standard resolution process makes intensive use of unknown distribution
of sources to formulate the problems and takes great advantage of the restricted definition
domain of these unknowns which concerns only one part of the printed surface: the metallic part
in the case of the distribution of electric current in microstrip-like structures, the dielectric part
in the case of slot- or coplanar-like structures.
The test or weighting functions used to attain the numerical representation of the
integral equation are therefore defined on one or other of these sub-domains and the final
representation of the integral equation performs the restriction of the operator on their
definition sub-domain.
The restriction to a sub-domain is introduced in the equivalent circuits through virtual
sources. Directly inspired by the standard treatment of microstrip and finline problems with
perfect conductors, virtual sources are now defined below.
Two kinds of virtual sources are introduced: superficial electric current sources, i.e. J defined
in (1), and tangential electric field sources, E. They are defined on a sub-domain, D, of S and
they vanish elsewhere.
The definition of these virtual sources arises from their fundamental property:
The dual function of a virtual source cancels on the definition sub-domain of the virtual source.
A current-type virtual source is such that the tangential electric field vanishes on its definition
domain while the superficial current density J is zero on the definition domain of an electric field
virtual source.
The representations and the properties of the virtual sources are summarized in Figure 10 in
their definition domain D and in the complementary sub-domain CD. Whatever the nature of D,
CD is the complementary domain.
The decisive point is now to translate the boundary conditions on the whole surface, meaning
in each point of the surface S, by introducing a unique symbol attached to the virtual sources.
Afterwards, the integral equation of the problem will arise by applying the above fundamental
property of the virtual sources.
s
1
Ŷ
2
Ŷ
Ys
Ys E 1
Ŷ
2
Ŷ
Figure 9. Representation of a surface impedance in a cavity.

5. INTRODUCTION OF VIRTUAL SOURCE IN HOMOGENEOUS PROBLEMS
Let us assume that the surface S involves arbitrary materials such as surface impedance, perfect
conductors, thick unperfect conductors, etc., so that in each point of this surface, the boundary
conditions take the form of a two-port relationship. On each side of the surface the surrounding
media define two admittance operators, #
Y
Y1 and #
Y
Y2, respectively.
The building of the electromagnetic equivalent circuit consists in the construction of a
network including virtual sources defined in one sub-domain D and connected in such a way
that, following the rules given in Figures 10 and 11, in each point of S the equivalent circuit
expresses the boundary conditions corresponding to D and to CD.
Let us assume that the boundary conditions are given by a two-port Q1 on D and a two-port
Q2 in CD. The general two-port with sources is symbolized by Q in Figure 11.
In Figure 10, the virtual sources act like toggled switches in ‘on’ or ‘off’ position according to
the nature of the source, J or E, and the sub-domain concerned. In homogeneous problems, real
sources are not present. The parameter sought, propagation constant or resonance frequency, is
issued from the existence condition of electromagnetic fields in the structure. In consistence with
the definition of the virtual sources, this existence condition coincides with the property stated in
the above section (Section 3), which prescribes the cancellation of the dual of the virtual source
to reach the solution.
Thus, deriving the final integral equation requires extraction of the terminals from the overall
circuit, where the virtual sources are connected, as in Figure 12. Setting J0
and E0
as the dual
functions of J and E, the problem is
J0
E0

¼
#
H
H11
#
H
H12
#
H
H21
#
H
H22

E
J

ð30Þ
Standard network analysis supplies the expression of the operators #
H
Hij in (30) providing that
T or Pi equivalent representations of the boundary condition two-ports Q1 and Q2 are
E E=0
CD D
J=0
J
J=0
CD
E=0
D
Figure 10. Properties of virtual sources in D and CD.
32

introduced in the representation of #
H
H in Figure 12 and that commutativity rules are carefully
applied to these operators. In particular, some of these operators may be singular ones and the
formulation is directed by the systematic avoidance of their inversion.
The general condensed form of Equations (30) is
j0
¼ #
H
Hj with j ¼
E
J

and j0
¼
E0
J0

ð31Þ
the block partitioning of #
H
H being given by Equation (30). The next two steps towards the
solution are:
* to carry out the fundamental principle which cancels the dual function on the definition
sub-domain of the virtual sources,
* the application of Galerkin’s procedure.
These two steps are now detailed. Let gp be a set of basis functions defined on D to expand the
combination of fields associated to the virtual source as
j ¼ Spxpgp ð32Þ
The components xp in (32) are the unknowns of the problem and are given by
hgqjj0
i ¼ 0 for all q ð33Þ
Q1
Q2
Q
E J
Figure 11. Rules for boundary conditions specified in two domains.
Q
E J
J
1
Ŷ 2
Ŷ E Ĥ
J’
E’
Figure 12. Extraction of the virtual source terminals.

meaning that
X
p
hgqj #
H
Hgpi ¼ 0 ð34Þ
Finally, the numerical form of the problem expresses that the non-trivial solution is retained,
namely
det½ #
H
H ¼ 0 with ½ #
H
H q;p ¼ hgqj #
H
Hgpi ð35Þ
½ #
H
H being the matrix representation of the operator #
H
H.
6. EXAMPLES OF HOMOGENEOUS PROBLEMS
To illustrate this approach, let us start by considering the two simple examples of the
determination of the mode wavenumbers of a microstrip line and of a finline (Figure 13).
The sub-domains D retained to perform the resolution will, respectively, be the metal for the
microstrip line and the dielectric domain in the slot for the finline.
The equivalent circuits are given in Figure 14.
In the metallic domain D of the microstrip line, the dual of J cancels since the surface is a
perfect electric wall. For the finline, the metallic sub-domain is CD, E therefore must cancel.
In the dielectric domain, E and J are continuous, J cancels in Figure 14(a) as a virtual source
in CD and as the dual of the virtual source E in Figure 14(b).
1
Ŷ
2
Ŷ
1
Ŷ
2
Ŷ
S1
S2
S1
S2
D D
Figure 13. Microstrip line and finline with perfect conductors.
1
Ŷ
2
Ŷ
J’
S1
S2
(b)
1
Ŷ
2
Ŷ
E’
S1
S2
(a)
J E
Figure 14. Equivalent circuits for the microstrip line and the finline.
34

The dispersion equations of the two lines are therefore derived as
det½ð #
Y
Y1 þ #
Y
Y2Þ1
¼ 0 for the microstrip line
det½ð #
Y
Y1 þ #
Y
Y2Þ ¼ 0 for the finline
ð36Þ
arising from the application of the fundamental principle:
E0
¼ ð #
Y
Y1 þ #
Y
Y2Þ1
J ¼ 0 for the microstrip line
J0
¼ ð #
Y
Y1 þ #
Y
Y2ÞE ¼ 0 for the finline
ð37Þ
In reference [7], lossy coplanar lines constitute another instructive example. Planar circuits
with non-perfect metals with non-inﬁnite conductivity and non-zero thickness or with
supraconductor depositions involve two-port-like boundary conditions. The T-representation
of this two-port [7] is given in Figure 15 in the case of multilayered metalization [16].
In the case of coplanar structures, it is convenient to choose the resolution sub-domain D as
the dielectric domain in the slots. The global representation of the interface is then obtained via
the insertion of two virtual sources in Figure 16.
Z2
Z1 Z3
Figure 15. Two-port representation of boundary conditions impressed by multilayered non-perfect metals.
Z2
Z1 Z1
E
J
Z2
Z1 Z1
Z2
Z1 Z1
ou D
ou CD
Figure 16. Equivalent circuit of the interface of a lossy planar line.

As detailed in reference [7], the virtual sources no longer represent the electric field in the slots
as in the example in Figure 14, but a combination of fields which, vanishing for instance in the
metal sub-domain, allow the advantages of the restriction of the sub-domain to be extended to
the non-perfect metal problem.
Figure 17 shows the attenuation of a coplanar line in membrane technology versus the
operating frequency, for a thickness of metalization around 2:6 mm; calculation and
measurement are in good concordance [16].
The influence of the thickness of metalization on the characteristic impedance of a coplanar
line in membrane technology is shown in Figure 18.
Non-local interactions as in MIM capacitors can also be introduced in the equivalent circuit
representation of boundary conditions. The MIM capacitor is introduced in the formulation by
a set of boundary conditions expressing the propagation of the fields between the two metallic
plates [17].
In all the cases mentioned above, the calculation of the operators #
Y
Y1 and #
Y
Y2 is based on a
spectral expansion whose comprehensive derivation is given in References [6,7] and in
Reference [13].
7. INHOMOGENEOUS PROBLEMS: INTRODUCTION OF REAL SOURCES
In practice, planar circuits include either active devices or probe excitations as in planar
antennae. For optimal design purpose, it is of major importance to account for this excitation,
Figure 17. Attenuation against frequency of a coplanar line in membrane technology. Comparison with
measurements in Reference [16].
36

in order to compute and predict the input impedance presented by the overall passive parts to
the active devices. In the modeling of planar circuits, active devices will appear as ‘planar
sources’ while probe excitations will appear as ‘sources of modes’. As shown below, it is actually
possible to reduce a ‘source of modes’ to a ‘planar source’ by inserting a coupling two-port
which performs the mode transformation between the guided mode of the source of modes and
the field distribution on the planar circuit [18,19].
The main interest of planar sources lies in the capability to link non-linear localized elements
to their electromagnetic environment. One or two active sources are substituted to a non-linear
element depending whether this active element is a one or a two-port. In the case of a transistor,
for instance, two coupled sources are displayed. On the one hand, these sources allow the
calculation of the passive circuits surrounding the non-linear element and on the other hand
they enable the non-linear characteristics of the non-linear devices to be introduced. This double
property allows the global simulation of a complete active circuit to be undertaken. This point
was first reported as the ‘Compression method’ [20] and carried out in the frame of the moment
method.
At first glance, a major difficulty seems to arise: the characteristics of the passive surroun-
ding elements are calculated in terms of electromagnetic fields E and H while the charac-
teristics of the localized active devices are given in terms of current and voltage, I and V,
respectively. The first requirement for the sources is therefore to support the double definition,
(I; V) and (E; H). This major condition considerably restricts the choices for defining planar
sources.
Another question concerns the geometrical shape of the active devices. Is it necessary to
modelize the intimate complexity of the active devices? Is the interdigital structure of a transistor
or the shape of the metal contact of a Gunn diode significant in practice?
Usually, an equivalent homogenous medium has been adopted [4] in place of the active device
or, when attempting to perform a detailed description of the element in a global simulation, only
an extremely succinct passive environment has been considered [5].
40
50
60
70
80
2 4 6 8 10 12 14 16 18 20
Metalisation thickness in µm
Characteristic
Impedance
Figure 18. Influence of the thickness of the metalization on the characteristic impedance of a coplanar line
in membrane technology.

In any case, the concept of dimensionless localized active elements which should solve the two
previous questions, because of the natural definition of V and I on a localized element and the
inexistence of an internal structure for a dimensionless device, has been used in the moment
method [21], in FDTD [22] and in the finite element method [23].
The localization of the active elements in these three approaches results from the way of
computation. For instance, the finite element method easily suggests the replacement of one
edge by a resistor or by one integration of the electric field. However, for the evaluation of the
dual function, which enables the determination of the impedance values (from a known
potential, the flow of the magnetic field provides the current), a cell with finite dimension has to
be considered.
7.1. Difficulties related to localized sources
In a slot annular antenna loaded with active devices [8,9], real sources have been introduced to
predict the system’s performance. It has been observed that the input impedances of the real
sources had values depending on their lateral dimensions and were diverging Figure 19(a) when
the spatial extension of the annular sector in Figure 19(b) was tending towards zero.
This behaviour can be related to the divergency obtained in the calculation of a self-
inductance as the diameter of the wire tends towards zero.
To illustrate this point, let us now consider an infinitely narrow current source, with
infinitesimal extension, displayed in the middle of the cross-section of a rectangular waveguide
as in Figure 20.
The current density in the Oy-direction is given by
J ¼ JyðxÞ ¼ I0dðxÞdðzÞ ð38Þ
where dðxÞ is the Dirac distribution. The waveguide is infinite in both z 0 and z50 directions
and the equivalent circuit of this structure is shown in Figure 21.
The admittance operator #
Y
Y is that of an infinite waveguide supporting TEn0 modes, with n an
odd integer, whose spectral development is given by
#
Y
Y ¼
X
n
j fniYMnhfnj ð39Þ
with YMn ¼ gn=jom, fn ¼
ffiffiffiffiffiffiffiffi
2=a
p
cos np=a and gn ¼
ðn2p2=a2Þ k2
0
q
.
Figure 19. (a) Imaginary part of the input impedance of the active planar source in
(b) against the spatial extension. From Reference [8].
38

The electric field excited by this source is given by
E ¼
1
2
#
Y
Y
1
J ¼
1
2
X
nodd
fnð0Þ #
Y
Y
1
MnhfnjJi ð40Þ
With Jðz ¼ 0Þ ¼ J0dðxÞ, expression (40) becomes
E ¼
jom
a
I0
X
nodd
1
ðn2p2=a2Þ k2
0
q ð41Þ
The series in (41) diverges as lnn when n tends to infinity n ! 1.
7.2. Towards a definition of planar sources
From the above considerations, two requirements should be satisfied to define a planar source:
* The source should have a finite size.
* A non-ambiguous definition of the currents and the voltages should be available.
Fulfilling these conditions suggests defining a planar source as bounded by two opposite
electric walls, between which the voltage drop can be evaluated, and closed by magnetic walls as
in Figure 22 in order to ensure a convenient current calculation.
-a/2
0
J
y
x
+ a/2
Figure 20. Current source in an infinite waveguide.
Ŷ Ŷ
J
Figure 21. Equivalent circuit of the structure in Figure 20.

This finally results in a configuration similar to the cross-section of a waveguide which would
support a fundamental TEM mode.
Let E0 be a source of electric field. The function E0 has an amplitude V and depends on
spatial variables through f0, as in expression (42)
E0 ¼ Vf0 ð42Þ
From the previous discussion, f0 is the quasi-static solution of the bidimensional problem with
boundary conditions between C; D; E and F in Figure 22.
Setting
f0 ¼ =j ð43Þ
the quasi-static problem has the following form:
=2
j ¼ 0 with
j ¼ jA on CD
j ¼ jB on EF
qj
qn
¼ 0 on CE and DF
8

:
ð44Þ
The voltage drop across AB is given by
V ¼ V
Z B
A
f0 dl ¼ ðjB jAÞV ð45Þ
thus ðjB jAÞ ¼ 1.
An interesting property of the function f0 is that the total current flow running between CD
and EF in Figure 22 is given by the scalar product hf0jJi, J being the current density defined in
(1). The expansion of J on an orthogonal set of basis functions has the following form:
J ¼ J0f0 þ
X
n0
Jnfn ð46Þ
in which the functions f0; f1; . . . ; fn, defined in the domain CDFE of Figure 22 are also the modal
expansion functions in the waveguide with cross-section CDFE, the functions f1; . . . ; fn referring
to the higher-order modes. Therefore, the amplitude of the current supplied by the real source is
given through
hf0Ji ¼ J0hf0j f0i ¼
Z
S
J0ðrjÞ2
dS ¼
Z
S
J0= ðj=jÞ dS ð47Þ
Electric wall
Magnetic wall
A
B
C
D
F
E
n
n
S
Figure 22. Model of a planar source.
40

The function j being the solution of problem (44), identity (47) is valid and
hf0jJi ¼ J0
Z
CDFE
j= n dl ð48Þ
Due to the magnetic walls, the integral in (48) vanishes between CE and DF and there
remains:
hf0jJi ¼ J0
Z
CD
j= n dl þ
Z
FE
j= n dl ð49Þ
hf0jJi ¼ J0 jB
Z
FE
f0 n dl þ jA
Z
CD
f0 n dl ð50Þ
The integrals both along EF and CD are opposite in sign because of the n orientation and of
Gauss’s theorem.
The total current flow corresponding to the distribution f0 is given by
R
FE J0f0 n dl, thus:
hf0jJi ¼ ðjB jAÞ
Z
EF
J0f0 n dl ¼ ðjB jAÞI0 ¼ I0 ð51Þ
The function f0 acts finally as the shape function which ensures reliable expressions for the voltages
and the currents.
7.3. Examples of shape functions
The shape functions are different for E-sources and J-sources. For J-sources, the definition and
the demonstration are quite similar to the previous ones:
J0 ¼ I0f 0
0 gives V0 ¼ h f 0
0jEi ð52Þ
Case of rectangular sources: In rectangular configuration, Figure 23 (a), f0 and f 0
0 are given by
f0 ¼
0
1
b
8

:
; f 0
0 ¼
0
1
a
8

:
ð53Þ
y
0
x
a
(a)
r2
r1
(b)
b
Figure 23. (a) Rectangular, (b) annular planar sources.

Case of annular sources: As shown in reference [8] the radial and angular components of the
shape functions in annular geometries Figure 23(b) are given by
f0 ¼
1
rlnr2
r1
0
8

:
; f 0
0 ¼
1
ry
0
8

:
ð54Þ
7.4. Equivalence between sources
From a mathematical point of view, real sources cannot be dimensionless. On the other hand,
the choice of the source is somewhat arbitrary, since it is most often impossible to describe the
active device in the global circuit accurately. This lack of standard reference giving an open
choice, the remaining question is how to attain the same result whatever the choice of the
source. This last requirement will in fact highlight more properties to be satisfied by the sources.
Let S be a planar source in a given environment. The impedance seen by the source is
determined from numerical computation. Another source, S0
, will be said, to be equivalent to S if
the distribution of the electromagnetic field in the overall circuit is kept unchanged. Obviously, the
computed input impedance seen by S0
will be different from that computed from S. As a matter
of fact, these two impedances are related by a coupling two-port as described below.
In the case of very small active devices with regard to the wavelength, the existence of
equivalent sources is straightforward for planar circuits. Consider for instance two sources, S1
and S2, connected to a microstrip line as in Figure 24.
The description of this source in a metallic box in terms of boxed-modes, as used in an
integral method performed in the spectral domain (for instance the spectral domain approach
[12,14,15]), generally gives good results in spite of the truncation of the spectral development. If
the upper terms of the spectral development does cover the source entirely, i.e. if the length of
their semi-period is greater than the largest dimension of the source, these terms have a
negligible effect on the computation. The difference between the two sources lies, in this case, on
the shape functions f01 and f02.
In general cases, the equivalence between two sources has to be verified. Changing one source
for another not only implies a change in the shape function but also a change in the internal
impedance of the source. In the case of a transistor for instance, substituting a source for
another results in a change in the non-linear relationships between the voltages and currents in
both the grid–source and the drain–source spaces. Furthermore, the reduction of the interdigital
structure of a transistor to a simple pair of coupled sources invokes a multi-scale approach [11]
and requires considerations beyond the scope of this paper. Nevertheless, in the next paragraph,
some insights into this point are given as concluding remarks.
S1 S2
f01 f02
Figure 24. Two equivalent sources in a microstrip line.
42

7.5. General approach to real sources
The justifications for applying a planar model for active components may be found in the
following considerations.
Let an active component (a Gunn diode for instance) be embedded in a cylinder S as in
Figure 25.
The electric field being expanded on an orthogonal set of basis functions ffng, the
characteristics of the active component may be described by a multiport having each port
excited by the corresponding spectral component of the electric field on the expansion function
fn.
This multiport simply results from the algebraic representation of the impedance or the
admittance operator. The potential oscillating behaviour is represented by harmonic sources on
each port in Figure 25.
Only quasi-static data being available for the active element, the ports corresponding to the
higher-order modes f1; . . . ; fn are loaded by an impedance matrix describing the close
electromagnetic environment of the active component, responsible for localized reactive energy
due, for example, to edge effects. The equivalent scheme of the active component is finally as in
Figure 26.
If this Z-matrix is independent of the externally connected circuits, the active component can
be described only from its response to the fundamental mode ff0g. On the other hand, if the
Eo
Eo
Jn
En
JO
S
S
fo
En
Figure 25. Active element and its multiport representation.
eo
fo
Active
element
e1
f1
en
fn
Z
Figure 26. Representation of an active element and its environment.

transverse dimensions of the component are small enough, the planar character of the source is
justified.
These considerations support the previous choice for the definition of the sources. The J0-
source will be preferred to the E0-source if the conservation of current flow is naturally valid
inside the active element as in high input impedance devices. In this case, the current has only
one component If0 while the electric field may have several components.
The modeling of the active device is most often the concern of a quasi-electrostatic approach
(most of the semiconductor-based devices belong to this category) which at first gives the
potential then the electric field and at finally the conservative current. As an example, a
comparison between J0 and E0-sources is given in the next section (Section 8.1). The connections
in Figure 25 being valid, a scaling change allows one source to be related to another.
Let us consider as an example, the simplified active device whose top view is given in
Figure 27(a).
An electric field source is connected to a transistor via a microstrip line in AB and CD, which
is in turn connected to another, wider microstrip line in A0
B0
and C0
D0
. Let S1 be an active
element.
As in Figure 25, only the fundamental mode is assumed to be connected to the outer medium
while the higher-order modes are loaded by the electromagnetic environment. Then, in ABCD,
the E1-source is efficiently represented by its amplitude on the fundamental mode f1 connected
to the surrounding environment through the coupling two-port Q1 in Figure 27(b).
Thevenin’s theorem then provides a representation of E1 and Q1 by an equivalent electric field
source E2 with internal impedance z2 in Figure 27(b).
(b)
A’B’C’D
’
Q1 Q2
E1 =Vf1
(s2) (s3 )
Q2
ABCD A’ ’ B’ ’C’ ’D’ ’
z2
E2
z3
E3
(s1)
S2
S3
A’
C’ D’
B’ S1
C’’ D’’
A’’ B’’
C
A
D
B
(a)
Figure 27. A planar source (a) and its possible representations (b).
44

Applying the same approach to the upper scale (A00
B00
C00
D00
) yields another equivalent electric
field source E3 with internal impedance z3 in Figure 27(b). This recursive approach is a localized
one and, the higher-order harmonics still being insensitive to the environment, the sources
S1; S2 and S3 are equivalent sources since substituting one for the other does not disturb the
overall electromagnetic conditions in the global circuit.
Such an approach ensures a great saving in computation time in the global analysis of the
circuits, since in the different scales (i.e. the connections between S1 and S2 and between S2 and
S3) the details of the circuit are analysed elsewhere. The analysis of these connection two-ports,
Q2 for instance, reduces to that of a step between two rectangular waveguides containing one
iris. The first waveguide is with AB–CD as electric walls and AC–BD as magnetic walls and the
second one with A0
B0
–C0
D0
as electric walls and A0
C0
–B0
D0
as magnetic walls. The thin iris is
placed between the two guides and consists of a metallic part AA00
B00
B and CC00
D00
D and a
magnetic part A0
A00
C00
C0
and B0
B00
DD00
D0
. This magnetic part expresses that the superficial
current density is zero on these surfaces.
Concretely, to perform the global analysis of circuits including active devices, the compromise
to make is to find equivalent sources, which fulfil the equivalence requirements and are as large
as possible.
7.6. Mode sources
In uniaxial devices such as filters and tapers, the source is the fundamental mode of the structure
given by the amplitude of the electric field Ve0 or of the J-field, Ij0 [13]. e0 and j0 are the
bidimensional functions describing the transverse distribution of the fields.
Case of E-sources: All the higher-order modes reflected at the first discontinuity of the
structure being attenuated at the location of the source, they see an infinite waveguide or, in
other words, an impedance operator #
Z
ZM, while the fundamental mode sees a short circuit. Thus,
the equivalent circuit of the source is as in Figure 28:
The total electric field on the source is given by
E ¼ Ve0 #
Z
ZMJ ð55Þ
The interpretation of this expression is as follows:
* for the fundamental mode, E ¼ Ve0,
* for the higher-order modes, E ¼ #
Z
ZMJ,
Y
E
J
Ijo
M
Ẑ
Veo
Y
E
J
M
Ŷ
(a) (b)
Figure 28. Equivalent circuit for (a) E-type and (b) J-type sources.

where the minus sign comes from the orientation given to J. Relation (55) describes the
behaviour of an infinite waveguide.
The expression of the magnetic field being
J ¼ Ij0 þ
X
n0
In:jn ð56Þ
the amplitude of J on the fundamental mode is given by
I ¼ he0jJi ð57Þ
with the normation relations
he0jj0i ¼ 1; he0jjni ¼ 0 ð58Þ
This expression is the same as for the planar sources. In the particular case where TE and TM
modes are supported, the choice of a unique function to express E and J seems more convenient.
Actually, letting f0 be the transverse electric field for the fundamental mode and setting
E ¼ Vf0 ð59Þ
J is given by
J ¼ If0 þ
X
n0
Infn ð60Þ
and
I ¼ hf0jJi ð61Þ
with the normation relations
h f0j f0i ¼ 1; h f0j fni ¼ 0 ð62Þ
In these cases,
#
Z
ZM ¼
X
n0
j fniZMnhfnj ð63Þ
The impedance computed with the normation condition he0jj0i ¼ 1 is closely related to the
reflexion coefficient G which, in a general way, is defined as
G ¼
1 y
1 þ y
with y ¼
I
V
ð64Þ
which gives in the TE–TM case:
G ¼
1 y
1 þ y
with y ¼
1
YM0
I
V
ð65Þ
YM0 being the admittance of the fundamental mode.
Case of J-sources: This case is quite similar to the previous one as seen in the equivalent circuit
given in Figure 28(b). One should note that #
Y
YM in Figure 28(b) is not the inverse of the operator
#
Z
ZM in Figure 28(a) since neither involves the component on the fundamental mode in its
spectral development. The operator #
Z
ZM is the inverse of #
Y
YM only in the manifold of the higher-
order modes. Inspection of Figure 28(b) gives the general expression of J as
J ¼ Ij0 #
Y
YME ð66Þ
46

The amplitude of E on the fundamental mode is now given by
V ¼ hj0jEi ð67Þ
Case of incident waves: The waves are given by
A ¼
1
2
ffiffiffiffiffiffiffiffiffi
ZM0
p ðE þ ZM0
JÞ
B ¼
1
2
ZM0
p ðE ZM0
JÞ
with E ¼ Vf0 and J ¼ If0 ð68Þ
for TE–TM waves. Like E and J, the waves A and B are bidimensional functions and have the
general form A ¼ A0f0, A0 being the amplitude of the wave such that
A0f0 ¼
1
2
ZM0
p ðE þ ZM0JÞ or E ¼ 2
ZM0
p
A0f0 ZM0J ð69Þ
Thus, an equivalent circuit is given for the fundamental mode in Figure 29(a), for the higher-
order modes in Figure 29(b) and for all the modes in Figure 29(c), #
Z
Z
0
being given by
#
Z
Z
0
¼
X
1
n¼0
j fniZMnh fnj ð70Þ
For hybrid modes, the functions e0 and j0 are different. One of them, e0 for instance, has to be
chosen for the definition of the waves since A or B will otherwise never cancel. The two
equivalent circuits corresponding, respectively, to these two choices are given in Figure 30.
Figure 30(a) gives
2A0e0 #
Z
Z
0
J ¼ E with #
Z
Z
0
M ¼
X
n¼0
jeni
1
henj jni
henj ð71Þ
while in Figure 30(b), the expression of the #
Y
Y
0
M operator is
#
Y
Y
0
M ¼
X
n¼0
j jni
1
hjnjeni
hjnj ð72Þ
Y
E
J
Mo
Z
ˆ
(a)
o
o f
A
ZM0
2
2
Y
E
J
M
Ẑ
(b)
Y
E
J
Mo
Z '
ˆ
(c)
o
o f
A
ZM0
2
2
Figure 29. Equivalent circuit for a source of modes: (a) for the fundamental mode;
(b) for the higher-order modes; (c) for all the modes.

The product #
Z
Z
0
M
#
Y
Y
0
M is equal to
#
Z
Z
0
M
#
Y
Y
0
M ¼
X
n¼0
jeni
1
h jnjeni
h jnj ð73Þ
Let E ¼
P
n¼0 Vnen then,
#
Z
Z
0
M
#
Y
Y
0
ME ¼
X
n¼0
enVn ¼ E ð74Þ
showing that #
Z
Z
0
M
#
Y
Y
0
M is the identity operator.
8. EQUIVALENT CIRCUIT OF DEVICES INCLUDING REAL AND
VIRTUAL SOURCES
Let us consider a boxed planar circuit as in Figure 31. The box characteristics are such that the
admittance operators are #
Y
Y1 and #
Y
Y2. The treatment of surface admittances being quite similar
to their treatment in the case of homogenous problems, let us assume that the surface S of the
circuit contains only metal ðMÞ, dielectric ðDÞ and sources ðSiÞ.
Y
E
J
(a)
Y
E
J
M
'
Ŷ
(b)
M
'
Ẑ
2A0e0 2A0e0
Figure 30. Equivalent circuit of sources of incident waves.
S1
M D
S
1
Ŷ
2
Ŷ
Figure 31. A general planar circuit.
48

Two sources, a virtual and a real one being involved in the formulation process, four
possibilities can be considered as summarized in the table of Figure 32 since the test functions
(virtual sources) and the real sources can be either of current or electric field type.
Detailed below as an example is the case where the virtual source is of electric field type and
the real one is of current type, (line 2, column 1 in Figure 32).
The property applied in Figure 33 is such that if one considers that a current source is
‘radiating’ on a dielectric surface, and an electric field source (i.e. a magnetic current source) is
radiating on a metallic body, the domain of the virtual sources covers the domain of the real
source if the latter is of dual type of the virtual source, as in boxes two and three in the table of
Figure 32. In the first box of the table, the current source is not on M and, in the fourth box, the
electric source is not in D.
The equivalent circuit of the structure studied in Figure 31 is completed by addition of both
the terminating operators #
Y
Y1 and #
Y
Y2. With the choice in Figure 33, one obtains the equivalent
circuit given in Figure 34.
Hereafter, it will be seen to be convenient to arrange the equations of the problem represented
by the equivalent circuit in a matrix form such that:
* the member on the left-hand side contains the real and virtual sources,
* the member on the right-hand side is made up of the dual functions.
E0
J

¼
0 1
1 #
Y
Y1 þ #
Y
Y2

If0
E
ð75Þ
I fo V fo
I fo
V fo
J
E I fo
V fo
E
E
J
Real
Source
Virtual
Source
Domain M M+S1
Domain D+ S1 D
(1) (2)
(4)
(3)
Figure 32. Equivalent circuits of the interface with the domain of virtual sources.

Let gp be a complete set of test functions for E
E ¼
X
p
xpgp ð76Þ
xp are the unknowns. Galerkin’s procedure applied to (75) acts as follows:
the ﬁrst line is projected on f0:
h f0jE0i ¼ V ¼
X
p
h f0jgpixp ð77Þ
the remaining lines are projected, respectively, on g1; g2; . . . ; gp, giving
hgpjJi ¼ 0 for all p ð78Þ
since the dual function cancels on the domain of the virtual sources as shown in Figure 33.
Thus,
hgpjJi ¼ 0 ¼ hgpj f0iI þ
X
q
hgpjð #
Y
Y1 þ #
Y
Y2Þgqixq ð79Þ
I fo
E
I=0 I fo I=0
Dielectric Source Metal
Interface S
Figure 33. Equivalent circuits of the interface S with regard to the domain of the virtual and real sources.
I fo
1
Ŷ
2
Ŷ
J
S1 S2
Eo
E
Figure 34. Equivalent circuit of a planar structure.
50

and setting
hgpjð #
Y
Y1 þ #
Y
Y2Þgqi ¼ ½ #
Y
Y1 þ #
Y
Y2 p;q and X ¼
x1
xp

and A ¼
hg1j f0i
hgpj f0i

ð80Þ
Equations (75) take the condensed form
V ¼ At
X
0 ¼ AI þ ½ #
Y
Y1 þ #
Y
Y2 X
ð81Þ
hence,
V ¼ At
½ #
Y
Y1 þ #
Y
Y2
1
AI ð82Þ
As an application, the equivalence between current and field sources is now focused as
mentioned in Section 7.5 above.
8.1. Equivalence between current sources and electric field sources
The circuit in Figure 35 simply consists of an open-ended microstrip line fed as in Figure 31:
Let fgpg be a complete set of functions defined on the metal for the case of a source of
current-type. The set fg0
pg of functions defined on both the metal and the source domain used in
the case of field-type sources will be defined later on. The two cases correspond respectively to
boxes one and two in the table in Figure 32.
Current-source: In order to ensure the continuity of the currents, the x-component of the
function f0 is extended on the metal sub-domain as represented in Figure 36.
This point is of prior importance for the convergency of the computation of the series
expansion. This ‘extension’ does not indeed modify the evaluation of the current in the final
result since the electric field on this sub-domain is zero (the source acts as a short circuit on the
metal).
J
1
Ŷ
2
Ŷ
(a)
Jo
(b)
1
Ŷ
2
Ŷ
Jo
Eo
E
x
M
D
0 S
Figure 35. Planar circuit with its equivalent circuit for two types of sources:
(a) current-source; (b) field-source.

The test functions gp, defined on the metal, have cosine functions as the x-component.
Setting #
Z
Z ¼ ½ #
Y
Y1 þ #
Y
Y2
1
, the equivalent circuit in Figure 35 gives the two relations:
E0
E

¼
#
Z
Z #
Z
Z
#
Z
Z #
Z
Z

J0
J

ð83Þ
Galerkin’s procedure gives
V0 ¼ h f0j #
Z
Zf0iI0 þ At
X
0 ¼ AI0 þ ½ #
Z
Z X
ð84Þ
with
J ¼
X
p
xpgp; A ¼
hg1j f0i
hgpj f0i

; X ¼
x1
xp

and ½ #
Z
Z p;q ¼ hgpj #
Z
Zg;qi ð85Þ
Thus the input impedance of the line seen from the source is, in this case, given by
Zs ¼ h f0j #
Z
Zf0i At
½ #
Z
Z 1
A ð86Þ
Electric field sources: The function f 0
0 associated to the electric field source is constant on the
source, the test functions g0
p are defined on the whole domain M [ S. The equivalent circuit in
Figure 35(b) gives:
J0
E

¼
0 1
1 #
Z
Z
0

E0
J

ð87Þ
and the admittance seen from the source is now deduced as
Ys ¼ A
0
t
½ #
Z
Z
0 1
A0
ð88Þ
gp
x
f0
0 s
M
Figure 36. The f0-function and the test functions gp for the case in Figure 35(a).
52

with
A0
¼
jhg0
1j f 0
0i
hg0
pj f 0
0i

and ½ #
Z
Z
0
p;q ¼ hg0
pj #
Z
Zg0
qi ð89Þ
The two results (86) and (88) seem quite different. However, in (88) let us choose the test
functions as
g0
1 ¼ f0 and g0
pþ1 ¼ gp ð90Þ
Since g0
p cancels with p 1 on the domain of the source, A0
in (89) becomes
A0
¼
h f0j f 0
0i
0
0

ð91Þ
and the ½ #
Z
Z
0
matrix takes the form
½ #
Z
Z
0
¼
h f0j #
Z
Zf0i A0
A ½ #
Z
Z

ð92Þ
A direct derivation gives
A
0
t
½ #
Z
Z
0 1
A0
¼
jh f 0
0j f0ij2
h f0j #
Z
Zf0i At½ #
Z
Z 1
A
ð93Þ
As shown before in Section 7.3, the shape functions f 0
0 and f0 are such that
h f 0
0j f0i ¼ 1 ð94Þ
Thus, providing the right choice for g0
p, in particular g0
1 which is constant on the source domain,
ensures that the two results are the same.
The extension of the source domain being small enough with regard to the circuit domain, the
numerical evaluation converges as long as the higher-order test function is such that its half-
period is approximately of the same dimension as the source. With any other choice of source
for the calculation, the calculated impedances will depend on this source; a coupling two-port
will be required to establish the equivalence.
8.2. Dual definition for fields and currents
Instead of the electric pair E and J, the magnetic pair H and n ^ E was the other alternative
choice to give a representation of electromagnetic field problems. In equivalent magnetic-like
circuits, in which the voltage is associated to the magnetic field and the current to the electric
field, one would have to introduce n ^ E as the counterpart of J defined in (1).
In that case, the impedance operator of a half-bounded space in Figure 37 is equal to the
admittance operator previously introduced in (1) since
H ¼ #
Z
ZmJM with JM ¼ n ^ E ð95Þ

The equivalent circuit corresponding to a surface S with a surface impedance ZS is then such
as in Figure 38
since
H1 H2 ¼ ysE ^ n ¼ ysJM ð96Þ
With these conventions, the approach is quite similar to the previous one. The equivalent
circuits are simply the dual to those obtained above. Thus, for instance, the circuit in Figure 34
becomes that of Figure 39.
Nevertheless, some reasons can be found to prefer the choice made throughout this paper. It
actually seems more convenient to handle electric currents rather than magnetic ones since they
are physically present in the whole structure, in particular when dealing with planar circuits
since, except in the excitation of the structure, a magnetic current is most often coupled to
another one with the opposite sign. Besides, the definition of the reflexion coefficient directly
refers to the electric field since G ¼ 1 on an electric wall, because the reflected and incident
electric fields are opposite. Moreover, on the one hand, the behaviour of active devices is mostly
controlled by the electric field and, on the other hand, structures supporting TEM propagation
mode are essentially relevant to electrostatic approximation. For these reasons it has been found
preferable not to introduce the p=2 rotation on the electric field prescribed in (95).
8.3. Systematic procedure for planar circuit analysis
The approach developed herein allows equivalent circuits to be built for deriving integral
equations from given boundary and continuity conditions. Let us summarize all the steps of this
approach.
H
JM
M
Ẑ
Figure 37. Impedance operator with H and JM.
Y
JM
H2
H1
ys
Figure 38. A surface impedance ZS and its equivalent circuit.
54

Given j, a set of virtual sources; j0 a set of real sources and c and c0 their respective dual
functions, the equivalent circuit is built as in Figure 40.
Let f0 be the shape function of the real sources and fgg a set of test functions.
j0 ¼ f0a
h f0c0i ¼ b
ð97Þ
with a ¼ I or V, b ¼ V or I. a and b represent the amplitudes of the real sources. a being
known, the objective is to determine the dual quantities b and the characteristic matrix of the
problem such that b ¼ Ma.
Let g be a set of test functions, j is given by
j ¼ x g ð98Þ
In Equation (98) the dot in x g stands for the summation of all the products with the test
functions.
Since hg ci ¼ 0, one ﬁnds
b
0

h f0
#
H
H11 f0i h f0
#
H
H12gi
hg #
H
H21 f0i h g #
H
H22gi

a
x

ð99Þ
1
ˆ
m
Z
Ho
2
ˆ
m
Z
M
J
Figure 39. Equivalent circuit of Figure 34 in terms of JM and H.
ϕ
ϕ
ϕ0
ϕ0
ψ0
ψ
Ĥ
=
Figure 40. General equivalent circuit and its matrix representation.

The second line of the sub-matrix finally gives
h f0
#
H
H11f0ia þ hg #
H
H22gix ¼ 0 ð100Þ
and the amplitudes b are given by
b ¼ ½h f0
#
H
H11f0i hf0
#
H
H12gihg #
H
H22gi1
hg #
H
H21f0i a ð101Þ
with testing functions g chosen among classical ones, roof-top, entire domain functions, finite
elements, etc., as abundantly described elsewhere.
9. CONCLUSION
This paper is aimed at giving an overview of the potential applications of the equivalent circuit
approach to derive integral formulations for electromagnetic boundary problems.
In this overview of the applications of electromagnetic equivalent circuits to the integral
formulation process, it has been shown how virtual sources enable one the benefit of the
restriction of the definition sub-domain of the test functions, more than in the case of perfect
conductors boundary conditions.
From this point of view a great deal of situations, not exhaustively enumerated here such as,
air bridge boundary conditions, multilayered circuits and active antennae, reduce to the same
analysis scheme.
Furthermore, the definition of equivalence classes of sources enables us to rigorously define
the behaviour of active devices and to include them in electromagnetic analysis. In a circuit with
high integration levels and increased compacity, this approach allows us to analyse
electromagnetic interactions between active devices, unlike resorting to a component library
in which each component is defined independently of its electromagnetic environment. Until
now, the passage between (V; I) characteristics and the (E; H) electromagnetic description of
passive circuits was subjected to the definition of a characteristic impedance, well defined only in
TEM lines. From this study this passage has been widened to any bidimensional circuit without
resorting to any TEM interconnect.
More generally a new look at the design of circuits seems to emerge. Indeed, circuits are no
longer a set of cascaded elements but a set of bidimensionally interacting elements.
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