Table Of Content~IEEE TRAN SACTI 0 NS ON
MICROWAVE THEORY
AND TECHNIQUES
A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
AUGUST 1995 VOLUME 43 NUMBER 8 IETMAB (ISSN 0018-9480)
PAPERS
Variational Propagation Constant Expressions for Lossy lnhomogeneous Anisotropic Waveguides ...................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Liu and K. J. Webb 1765
Inverse Scattering Method for One-Dimensional Inhomogeneous Lossy Medium by Using a Microwave Networldng
Technique ...................................................................................... T. 1. Cui and C. H. Liang 1773
Ridge Waveguide Polarizer with Finite and Stepped-Thickness Septum .................. J. Bornemann and V. A. Labay 1782
Radiation Loss of Y-Junctions in Rib Waveguide .................................... L. Cascio, T. Rozzi, and L. Zappelli 1788
Experimental 6-GHz Frozen Wave Generator with Fiber-Optic Feed ........................ J. B. Tha.xter and R. E. .Bell 1798
Modeling of Active Antenna Array Coupling Effects-A Load Variation Method ........... S. Sancheti and V. F. Fusco 1805
New Multilayer Planar Transmission Lines for Microwave and Millimeter-Wave_Integrated Circuits ................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P.-C. Hsu and C. Nguyen 1809
Chirosolitons: Unique Spatial. Solitons in Chiral Media ....................................... K. Hayata and M. Koshiba 1814
Active Antenna Array Behavior ............................................ D. E. J. Humphrey, V. F. Fusco, and S. Drew 1819
High-Frequem:y Reciprocity-Based Circuit Model for the Incidence of Electromagnetic Wa:ves on General Waveguide
Structures ..................... : ........................................................... D. De Zutter and F. Olyslager 1826
Analysis of Compact E-Plane Diplexers in Rectangular Waveguide .............................. A. Morini and T. Rozz,i 1834
Design of a Multicoupled Loop-Gap Resonator Used for Pulsed Electron Paramagnetic Resonance Measurements .....
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Sakamoto, H. Hirata, and M. Ono 1840
A New Boundary Integral Approach to the Determination of the Resonant Modes of Arbitrarily Shaped Cavities ..... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Arcioni, M. Bressan, and L. Perregrini 1848
Gaussian Beam Optical Systems with High Gain or High Loss Media ................ A. A. Tovar and L. W. Casperson 1857
Extraction Techniques for FET Switch Modeling ............. A. Ehoud, L. P. Dunleavy, S. C. Lazar, and R. E. Branson 1863
Control of Mode-Switching in an Active Antenna Using MESFET ...... M. Minegishi, J. Lin, T. ltoh, and S. Kawasaki 1869
Simplified Mode-Matching Techniques for the Analysis of Coaxial-CavitycCoupled Radial E-Plane Power Dividers ...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. E. Bialkowski, J. Bornemann, V. P. Waris, and P. W Davis 1875
Accurate Quasi-TEM Spectral Domain Analysis of Single and Multiple Coupled Microstrip Lines of Arbitrary
Metallization Thickness ......................................................................................... J.-T. Kuo 1881
An X-Band Acousto-Optic Variable Delay Line for Radar Target Simulation ............................................ .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. C. Zari, C. S. Anderson, and W. D. Caraway III 1889
(Continued on back cover)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 8, AUGUST 1995 1765
Variational Propagation Constant Expressions for
Lossy Inhomogeneous Anisotropic Waveguides
Yinshang Liu and Kevin J. Webb, Member, IEEE
Abstract- Based on reciprocal relationships for the adjoint components. In addition, the divergence-free condition must
operator, we derive a variational formulation for the propaga be enforced in the variational formulation. In this paper, a
tion constant satisfying the divergence-free condition in lossy
systematic procedure for the derivation of such a formula
inhomogeneous anisotropic waveguides whose media tensors have
tion is presented. Additionally, two variational formulations
all nine components. In addition, with some advantages over
previous representations, two variational formulations have been for reciprocal waveguides with the longitudinal part of the
derived for waveguides with the transverse part of the media medium tensor decoupled from the transverse part are derived.
tensors decoupled from the longitudinal part. However, to obtain Applied in finite element methods, these formulations will
a variational formulation for a general lossy reciprocal problem
have some advantages over previously published formulations
the waveguide must be bi-directional. Each of the variational
in the literature since these new formulations are more general
expressions results in a standard generalized eigenvalue equation
with the propagation constant appearing explicitly as the desired or accurate.
eigenvalue. The stationarity of the formulations is shown. It For a general lossless (reciprocal or nonreciprocal) problem,
is also shown that for a general lossy nonreciprocal problem the variational functional always exists. For a general lossy
the variational functional exists only if the original and adjoint reciprocal problem, the variational functional exists only if the
waveguide are mutually bi-directional.
waveguide is bi-directional. For a general lossy nonreciprocal
problem, the variational functional exists only if the original
I. INTRODUCTION and adjoint waveguide are mutually bi-directional. (The orig
RECENTLY, finite element methods have been applied inal and adjoint waveguide are mutually bi-directional if for
extensively to waveguide problems [l]. When using finite each mode with the propagation constant ry in the original
element methods, it is desirable to use a variational functional. waveguide there exists a mode with propagation constant
A thorough study of the variational electromagnetic problems -ry for the adjoint waveguide. A waveguide is bi-directional
based on the reaction concept introduced by Rumsey [2], [3] if modes with propagation constant ry and -ry will always
and Harrington [4] has been presented by Chen [5]. Also, exist simultaneously for the same waveguide). An example
many different variational expressions for the propagation of a lossy nonreciprocal waveguide and its corresponding
constant have been derived for each specific problem [6]-[10]. variational formulation for the propagation constant has been
However, further investigation of variational expressions for proposed by Chen [5]. While the divergence-free condition
the propagation constant is still needed. For instance, Berk's is not enforced in Chen's formulation, the variational for
[3] and Kumagai's formulations [9], [10] for the propagation mulation given by our procedure will automatically satisfy
constant are restricted to lossless media. Spurious modes the divergence-free condition. We begin with a discussion of
[11] will occur in Rumsey's [2] formulations for the prop variational formulation issues, such as the choice of inner
agation constant, since the divergence-free condition is not product, variational variables (magnetic field or electric field
enforced in the variational expression. Davies [7] as well as or both electric and magnetic field; full vector, i.e., three
Chew's formulations [8] for the propagation constant only components of the field or only the transverse component
work for media with the longitudinal part of the medium of the field), the stationarity of variational formulations for
tensor decoupled from the transverse part. The Euler equation the propagation constant and the existence of the adjoint
for another of Davies' formulations [6] in terms of the full field. Next we derive the various variational expressions in
magnetic field vector does not satisfy the vector magnetic field the form of standard generalized eigenvalue equations where
equation, as pointed out by Hoffmann [12]. Each of the above the propagation constant appears explicitly as the desired
formulations has some failings. Hence, a new formulation for eigenvalue.
the propagation constant in general waveguides is needed.
For generality, the new formulation must be able to be used
II. VARIATIONAL FORMULATION ISSUES
in waveguides containing lossy inhomogeneous anisotropic
media where the tensor constitutive parameters have all nine Consider a differential equation
Manuscript received January 12, 1993; revised April 24, 1995. This work
was supported in part by the Semiconductor Research Corporation Contract M·P=O (1)
93-DJ-211.
The authors are with the School of Electrical Engineering, Purdue Univer
where M is a linear operator for describing the underlying
sity, West Lafayette, IN 47907-1285 USA.
IEEE Log Number 9412696. physical problem and P is the unknown field quantity. The
0018-9480/95$04.00 © 1995 IEEE
1766 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 8, AUGUST 1995
corresponding functional, F, can be represented as [5], [6] formulation in (2) is to find pa. The choice of inner
product will determine the adjoint operator Ma and the
(2)
corresponding adjoint solution pa, and hence the vari
In (2), pa denotes the solutions with the corresponding ational functional F. The real inner product is defined
adjoint operator Ma, i.e., pa satisfies Ma · pa = 0. We have as
J
=
6F (6PalM. P) + (PalM. 6P) (Jig) = f · g dS (7)
= (6PalM. P) +(Ma. pal6P).
whereas the complex inner product is defined as
Hence, the first variation of the functional F vanishes
j
when M · P = 0 and Ma · pa = 0. In (2), (Pa IM · P)
(Jig)= f* · gdS (8)
represents the inner product between pa and M · P. In
electromagnetic problems, M is given by Maxwell's equations
with f* being the complex conjugate of f, and f and
and P represents the electromagnetic field. The formulation for
g being two arbitrary vectors. The integral is two
the functional in (2) has several drawbacks.
dimensional, since two-dimensional waveguide prob
First, direct use of the functional in (2) does not yield a
lems are considered. Since the operator for Maxwell's
desirable eigenvalue matrix equation. If we write Maxwell's
equations is hermitian in a lossless system, we would
equations in the form of M · P = 0, we have for the vector
usually choose a complex inner product in the lossless
magnetic field source-free wave equation
case. Therefore, the operator will be self-adjoint if we
\7 x t -1 · \7 x H - w2/f · H = 0 (3) choose a complex inner product. Hence, in a lossless
system the adjoint solution pa will be related to P by
with H being the magnetic field vector, E the permittivity
pa = P. Equation (4) becomes
µ
tensor of the medium and the permeability tensor of the
medium. With the differential operator in (3), the functional F =(HIV x t -1 · \7 x H - w2/L · H). (9)
F = (Hal\7 x t -1 · \7 x H - w2µ-1 · H) (4) Many useful formulations may be deduced from (9), for
example, the classical result given by Berk [3]
will be variational about the fields that satisfy (3). Direct use
.
j
of this functional in the solution of a waveguide problem t -
\7 x H · 1 · \7 x H* dr!
gives either a matrix eigenvalue equation with frequency as
w2 = J (10)
the eigenvalue [13], or a quadratic eigenvalue equation with
H·lf·H*dr!
propagation constant as the eigenvalue [14]. The latter could
be seen by assuming that the field has ei(wt--yz) dependence,
where the integration can be two-dimensional or three
with 'Y being the propagation constant. With this assumption,
dimensional, depending on the specific problem. The
the functional in (4) has the form
complex inner product is also directly related to the
F = (Hal(Y't - h i) x t-1 energy concept in a lossless system.
·(Y't - hi) x H -w2/f·H) (5) On the other hand, a real inner product should be used
in the lossy case. The reason is that phase information
where \7 = Y't - hi and Y't = x(8/8x) + y(8/8y). in the propagation constant will be Jost in a complex
This functional will yield a quadratic eigenvalue equation
inner product formulation. Here, the primary concern
with 'Y being the eigenvalue. For computational simplicity,
is anisotropic inhomogeneous lossy media. Therefore, a
it is desirable that the formulation should yield the standard
real inner product is implied if not noted otherwise.
generalized eigenvalue equation form
2) Choice of variational variables-magnetic field or elec
tric field: When the permeability is homogeneous, the
(6)
divergence-free condition \7 · B = 0 can be reduced to
with >. = 'Y or 'Y2 and A and B being either differential or \7 . H = 0. Hence, the normal component of magnetic
matrix operators. field will be continuous at all points over the domain.
Second, the functional F from (2) introduces another set of In addition, Ampere's law in a source-free waveguide
unknowns, pa.
will result in the continuity of the tangential component
Third, spurious modes will occur in using the functional of the magnetic field at all points over the domain.
given in (4) [I I]. This is because the vector that satisfies (3) Thus, it is convenient to use the magnetic field to set
does not automatically satisfy the divergence-free condition up a variational functional when the permeability is
/LH.
\7 · B, where B = In the formulations presented in this homogeneous and the permittivity is inhomogeneous. On
paper, the divergence-free condition will be incorporated into the other hand, the electric field should be used as the
the variational functional to avoid the occurrence of spurious variational variable when the permittivity is homoge
modes. Variational formulation issues will be discussed in the neous and the permeability is inhomogeneous. Since the
following: former case is common, we will give the derivation of
1) Choice of inner product-real inner product or complex the functional in terms of magnetic field. The functional
inner product: The first step to simplify the variational in terms of electric field can be found by duality. When
LIU AND WEBB: VARlATIONAL PROPAGATION FOR ANISOTROPIC WAVEGUIDES 1767
the longitudinal part of the medium tensor is coupled must be satisfied. Since F = 0 when A· X .x - >.B · X .x = 0
to the transverse part, the longitudinal component of is satisfied, we have the result shown in (15). To show
either magnetic or electric field has to be represented by that F in (16) is variationally stable about the true field,
both the transverse magnetic and transverse electric field. from (16), we have
Therefore, the variational functional must be represented
8F = (8XflA · X.x - >.B · X.x)
in terms of both electric and magnetic field for this case.
3) Choice of variational variables-full vector or only the + (Aa. Xf - >.Ba. Xf l8X.x). (19)
transverse component: The functional in terms of only
Hence
transverse field has fewer unknowns in the resulting
matrix eigenvalue equation, but results in a more compli oF = 0 (20)
cated expression and involves additional differentiation
whenever (6), (17), and (18) are satisfied. Therefore,
due to the incorporation of the divergence-free condition.
F is variationally stable about the true field. To show
In this paper we will give both three-vector and two
that the expression in (15) is variationally stable, with
vector variational expression forms.
>. = (C / D), the following relationship is useful
4) Variational stationarity of propagation constant T A
variational functional F, in terms of the field, does c
8>. = 8
not necessarily yield a variational expression for the D
propagation constant /. For example, the functional C+oC C
defined by D+oD D
F = (A . <l>-y - /<l>-y IA . <I>-y - /<l>-y) (11) CD+oCD-DC-CoD (21)
(D + 8D)D
will be stable about the true field solution of A · <I>-y -
Hence
/<l>-y = 0. Since the first variation of the functional F
oCD - CoD
can be written as
o>. = (D + 8D)D . (22)
oF = (o(A · <I>-y - 1<I>-y)IA · <I>-y -1<I>-y)
Keeping only first order terms, we have
+ ((A· <I>-y - 1<I>-y)l8(A · <I>-y -1<I>-y) (12)
8>.D = oC - >.oD. (23)
we have 8F = 0 when A· <I>-y -1<I>'Y = 0. Since F = 0
when A · <l>-y - /<l>-y = 0, we have Using (15) and (23) we have
(13) o>.(Xf IB · X.x) = o(XflA · X.x) - >.o(Xf IB · X.x)
Hence, = (8XflA · X.x - >.B · X.x)
2 (A . <l>-y IA . <I>-y) - (XflA · 8X.x - >.B · 8X.x)
I - (14)
- (<I>-yl<I>-y) . = oF. (24)
However, 81 = 0 only when Aa · <l>-y - /<l>-y = 0, Since in general (Xf IB · X.x) "I- 0 [15], [16], we have
i.e., 1 is variational only when A is self adjoint. From
the above example, the / expression derived from a (25)
variational formulation for the field is not necessarily
whenever ( 6), (17), and ( 18) is satisfied. Therefore, the
variationally stable. Thus it is important to show directly
eigenvalue (propagation constant) is variationally stable
the variational stability of /.
about the true field solutions.
We want to show that the following expression for
5) Existence of the adjoint field: Although the functional in
>., derived from the functional in (2) and eigenvalue
(16) always yields a variational expression for the eigen
equation in (6), will always be variationally stable
value >., the variational expression in (16) is meaningful
>. = (Xf IA· X.x) only when the adjoint field solution exists, as given by
(15)
(Xf IB · X.x) (17). The adjoint field equation, (17), actually is not an
eigenvalue equation since it is restricted by (18), i.e.,
where X.x satisfies A · X.x - >.B · X.x = 0 with A and
>.a = >.. We can still consider ( 18) as an eigenvalue
B being either differential or matrix operators and >. the
equation and solve for its eigenvalue. If A and B in (6)
eigenvalue. The functional F in (2) can be written as
are not self-adjoint, in general >. may not be a eigenvalue
of (17). If>. is not a eigenvalue of (17), the equation
with (26)
(17) has only a trivial solution Xf. Hence, (15) and (16) will
not be a variational expression in this case.
and where
The functional in (16) works only when the eigenvalue
(18) (6) and its adjoint (17) have identical sets of eigenvalues,
1768 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 8, AUGUST 1995
i.e., (18) is satisfied. This is the reason why Chen's with
technique [5], although intended for solving general
nonself-adjoint problems, in general works best for self L = [Ln L12] (32)
L21 Lz2
adjoint operators in eigenvalue type problems, such
= 1...., A 1 A " - 1 _
as those for wave propagation in lossless waveguides. L ll = WEtt - - V t X Z - Z X v t - WEtz - Etz (33)
W µzz Ezz
In lossless systems, the operator is self-adjoint using
complex inner product, therefore ( 18) is always true. L 12 = - "v t X - 1 ZA X µ- zt - E-ztZA X -1 "v t (34)
For a lossy nonreciprocal system, if the original and µzz Ezz
adjoint waveguide, where the medium tensor is trans L 21 = - "v t X -1 ZA X E-zt - µ- tzZA X - 1 "v t (35)
posed to that of the original waveguide, are mutually Ezz µzz
- 1 , 1 , ...., _ 1_
bi-directional, ( 18) can also be satisfied. We will give an Lz2 =wJitt - - Vt X z - z X Vt -Wµtz - µtz (36)
example later in this paper. In the next section, we derive W I] Ezz µzz
the variational expression for the propagation constant r = [ o Jz x (37)
for general waveguides. z jz x I 0
and I being identity operator. H z and E z are related toe and
Ill. DERIVATION OF THE VARIATIONAL h by
EXPRESSION FOR THE PROPAGATION CONSTANT
jwµzzH z = -jwjj,zt · h +Vt· (z x e) (38)
In the discussion of two-dimensional waveguide problems,
it is assumed that the current source inside the waveguide is jWEzzEz = - jWEzt · e - Vt· (z X h). (39)
zero and the boundary is either pee (perfect electric conductor)
Premultiplying (31) with r z results in
or pmc (perfect magnetic conductor) or at infinity. Again,
the field dependence is assumed to be e1(wt-v), with z the A · <I>,, - ry<I>,, = 0 (40)
longitudinal direction. If the vector magnetic field is separated
with A = r z · L. Note that r z · r z = I. In lossless problems,
into transverse and longitudinal parts, we have
(31) is self adjoint with the choice of a complex inner product.
(27) Therefore, we have
where h = Hxi + Hyf) is the transverse component. In the
same way, we have
as the variational expression for 'Y in lossless problems with
(28)
the choice of a complex inner product. In lossy nonreciprocal
problems, we have
The media tensor is defined in matrix form as
['Eu
= Etz ]
E = - (29)
Ezt Ezz
as the expression for ry, which is variational only if the
where Ett is a transverse dyadic, ftz and fzt are vectors and
original waveguide and the adjoint waveguide are mutually
Ezz is a scalar. The definition for Ji is similar. In reciprocal
bi-directional. In lossy reciprocal problems, we have
problems, these tensors are symmetric.
The following criteria have been adopted for the selection (<I>_,,IL<I>,,)
'Y =
of the operator representing Maxwell's equations used in the (<I>_,,1r z<I>,,)
variational functional:
as the expression for ry, which is variational only if the
1) the operator is represented in terms of only transverse
waveguide is bi-directional. The reason that <I>_,, is the adjoint
fields,
solution for lossy reciprocal problems is explained in the
2) the operator is self-adjoint in lossless system,
following paragraph.
3) the operator will lead to a standard general eigenvalue
The permittivity tensor and the permeability tensor of
equation with the propagation constant appearing explic
the media are symmetric in lossy reciprocal systems. For
itly as the desired eigenvalue,
Maxwell's equations, as represented in (31) and (6), A is self
4) the operator can be used in lossy, inhomogeneous, adjoint and Ba = - B. Here we have A = L, X;., = <I>,,,
anisotropic reciprocal or nonreciprocal problems.
,\ = ry, and B = r z· L is self adjoint with the choice of
Let a real inner product when the medium tensor is symmetric
[15]. We also haver~= -rz with the choice of a real inner
(30) product. The set of eigenfunctions for (6) is identical to the
set of eigenfunctions for (17) with X~ = X_;., and ,\a = ,\.
Maxwell's equations in operator form [ 15] satisfying the above Hence, we have
criteria can be written as
,\ = (X-;.,IA · X;.,)
(41)
(31) (X-;.,IB · X;.,)
LIU AND WEBB: VARIATIONAL PROPAGATION FOR ANISOTROPIC WAVEGUIDES 1769
with the choice of a real inner product in lossy reciprocal we have
systems.
E_-y =E-y,x(x, -y, - z).i - E-y,y(x, -y, - z)fj
IV. ADJOINT SOLUTIONS FOR LOSSY PROBLEMS - E-y,z(x, -y, -z)z
In the following, the expression for the adjoint field for H_'Y =H-y,x(x, -y, - z).i - H-y,y(x, -y, - z)fj
lossy systems will be given. The eigenmodes in the adjoint
- H-y,z(x, -y, - z)z. (46)
waveguide and the eigenmodes in the original waveguide
satisfy the orthogonality relationship [15]
Next, considering the waveguide with inversion symmetry, we
have
E_'Y =-E-y,x(-x, -y, - z).i-E-y,y(- x, -y, - z)fj
where N-yf3 is some constant. Assume an normalized set such - E (-x -y -z)z
,, z ' '
that Nf3f3 = 1. From the orthogonality relationship in (42), H_'Y =H-y,x(-x, -y, - z).i+H-y,y(-x, -y, - z)fj
we have
+ H-y,z(-x, -y, -z)z. (47)
L
l<T>-y) = (<T>-yl<T>fJ)II'z<T>~). (43)
Hence, ( 41) can be written as
{3
From (43), we have (48)
with <T>_-y given by (44)-(47). The formulation in (48) has the
with
following properties:
(44) 1) It contains second order derivatives in space. Higher
order interpolation functions, which are second order
with as long as D is nonsingular. In general, relationship differentiable, are commonly used in many problems
(4 4) holds even for nonreciprocal (lossy, anisotropic and such as waveguides with convex polygon shapes [18]
inhomogeneous) waveguides. However, (44) is not useful or and surface modes in microstrip [19]. Even with these
necessary in actual numerical calculations. If the adjoint field
higher order elements, lack of continuity in the first order
does exist, we can represent the original field as well as the derivative between elements may be problematic in the
adjoint field as a linear combination of basis functions with
case of second order derivatives in the functional. The
unknown coefficients. By taking the variations with respect to
second order derivatives can be reduced to first order
the adjoint field, the resulting matrix equation will contain only
derivatives using integration by parts, as shown later in
the unknown coefficient of the original field. This approach is
this section.
variational only if the adjoint solution exists. The existence
2) The formulation is represented by only transverse fields.
of the adjoint solution relies on the matrix A in (44) being
The longitudinal components will be given by Faraday's
nonsingular, which is not always known a priori.
and Ampere's law, represented in (38) and (39), re
For reciprocal problems, it has been shown [ 17] that the
spectively. Hence, the divergence-free condition will be
relation between X-y and X_-y can also be found if the wave
satisfied since B or D will be represented as the curl
guide possesses one of the following symmetries: reflection
of a vector.
symmetry in a plane perpendicular to the waveguide axis;
3) The permittivity and permeability tensors can have all
180 degree rotation symmetry about an axis perpendicular
nine components in the formulation. In reciprocal sys
to the waveguide axis; inversion symmetry in a point on
tems, the material tensors must be symmetric. There
the waveguide axis. With the medium tensor in the form of
are no other restrictions on the elements of the material
(29) and Etz = E;t, iitz = µ;t, the field with a propagation
tensors in (48) as long as the waveguide is bidirectional.
constant - "( can be found by the transformation relations of
The material tensors may be complex, functions of
the field under a symmetrical operation. First, considering the
frequency, or functions of position. Hence, (48) can
waveguide with reflection symmetry, we have
be applied in lossy, inhomogeneous, anisotropic recip
rocal problems. Note that if the symmetric conditions
E_-y = E-y,x(x, y, - z).i + E-y,y(x, y, - z)fj
(45)-(47) are used, the material tensors will have the
- E-y,z(x, y, - z)z same restrictions.
H_'Y = -H-y,x(x, y, - z).i - H-y,y(x, y, -z)fj 4) The variational expression is in the form of standard
+ H-y,z(x, y, - z)z (45) generalized eigenvalue equation where the propagation
constant appears explicitly as the desired eigenvalue.
with E-y, x representing the x component of the electric field Each of the variational formulations proposed previously
and likewise for the other field components. Second, consid has more restrictions than (48). These restrictions can be
ering the waveguide with rotation symmetry about the x axis, classified into the following three types:
1770 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 8, AUGUST 1995
1) variational formulations which are restricted to lossless Thus, after some manipulation of (53), we have for
media [3], [9], [10];
2) variational formulations which only apply to media with 2 = (e-rlA · h-y)
1
(e-rlB · h-y)
the longitudinal part of the medium tensor decoupled
from the transverse part [7]-[9]; (-w2 · z x Ji,· h - 'Vt~ Vt· (z x h)IA · h-r)
3) variational formulations which do not enforce the fzz
divergence-free condition [3], [6]. (-w2 · z x µ, · h - Vt~ Vt· (z x h)lz x h-y).
Hence, we have derived in (48) for the first time a varia fzz
(54)
tional formulation for the propagation constant satisfying the
divergence-free condition in lossy inhomogeneous anisotropic
Equation (54) is in the form of standard generalized eigenvalue
yet reciprocal waveguides whose media tensors have full
equation where the propagation constant appears explicitly
nine components as long as the waveguide is bi-directional.
as the desired eigenvalue. The longitudinal components of
With X~ replacing X_.x in (41), the formulation in (41)
electric and magnetic field will be given by the divergence
is variational even in lossy nonreciprocal problems as long
free condition. The divergence-free condition is enforced in
as the waveguide and its adjoint waveguide is mutually bi
(54). Davies has given a formulation similar to (54) which
directional, which is equivalent to the condition that the matrix
does not apply to media with inhomogeneous permeability
D in (44) is nonsingular.
[7]. On the other hand, (54) works for the case of media with
For some applications, it is not preferable or necessary to
inhomogeneous permeability and permittivity as A contains
avoid the previously mentioned restrictions. We will derive the
spatial derivatives of permeability.
corresponding variational formulations for such applications.
The formulation given in (54) contains second order deriva
Even with the same restrictions with a decoupled longitudinal
tives in space. In (54) the second order derivatives can
part of the medium tensor, the following derived formulations,
be reduced to first order by integration by parts if (54) is
which can be used in more general problems or yield a
represented in terms of both e-y and h-y. In reciprocal systems,
better approximation for the propagation constant, will have
the material tensors must be symmetric. There are no other
some advantages over the corresponding formulations in the
restrictions on the elements of the material tensors. They may
literature.
be complex, functions of frequency or functions of position.
When the transverse part of the media tensors are decoupled
We can also give the variational expression in terms of full
from the longitudinal part, i.e.,
magnetic field. It will be also possible to reduce the second
order derivative in space to a first order derivative. In the
following derivation, we again assume that the transverse part
of the media tensors are decoupled from the longitudinal part,
(49) as in (49). Using (3), (5), (50) and the adjoint field solution [5]
(55)
we can derive a variational form for 12 in terms of only h.
By using the relationship \7 · B = 0, we can represent Hz we have
in terms of h
'Y 2 -_ ( - 2( ZX h -y1f= - 1 ·v" tX Z-1 v" t·µ=· h-y )
ZA ·µ= - 1 ·ZA 'Vt·J:-l·h A A µzz
Hz = . (50)
- (Vt x h-rl€ -1 ·Vt x Hzz)
J'Y
+(Vt x Hzzl€- 1 ·Vt x Hzz) + (h-ylw2Ji, · h-y)
By using (3) with (50), to represent Hz in terms of h, and
premultiplying (3) with € · zx, we get [8] -(Hzzlw2Ji,·Hzz))j(z x h-rlE-1 ·z x h-y)· (56)
122 x h - z x Vtz . µ,-1 . z'Vt. µ,. h - w2€. 2 x µ, Davies [6] has derived a variational formulation for the propa
gation constant in terms of full magnetic field vector. However,
·h-E·zX 'Y't XE-1 ·'Y'txh=O. (51)
(56) will yield a better approximation for the propagation
constant since the Euler equation of Davies' formulation (37)
Note that (51) is a form of (6), which can be expressed as
A · h-y - 12 B · h-y = 0. Hence, from (15) we have in [6] is not the vector magnetic field equation [12]. (56) is in
the form of standard generalized eigenvalue equation where
2 _ (h~IA · h-y) the propagation constant appears explicitly as the desired
' - (h~IB·h-r). (52) eigenvalue. Spurious modes can occur in using the functional
given in (56), since the vector that satisfies (56) does not au
Using the result in [8], we have h~ e-y. The transverse tomatically satisfy the divergence-free condition. The penalty
electric field, e-y, must satisfy parameter method [11] can be used to remove the spurious
modes. The second order derivative in (56) can be reduced
to first order using integration by parts. The elements of the
·z µ.
-w x h
e-y = ------'-- (53) material tensors may be complex, functions of frequency or
/ /W functions of position.
LIU AND WEBB: VARIATIONAL PROPAGATION FOR ANISOTROPIC WAVEGUIDES 1771
If the system is lossy and nonreciprocal, the derivation of a L22 terms. With the adjoint solution <I> _-y, the second order
adjoint solution must be investigated for each specific problem. derivative term in Ln can be represented as
We give an example for such waveguide.
With the choice of a real inner product, the adjoint field J 1
e_-y · .!_Vt xi - - ix Vt· e-y ds. (62)
must satisfy Maxwell's equations with transposed permittivity
W µzz
and permeability tensor and with propagation constant -"(
(15]. This requires that the original and adjoint waveguide be Using (61), (62) can be transformed to
mutually bi-directional (17]. Assuming that the media tensors
are of the form
= [ 'Eu
f. =
- f~. tt z
(63)
µtt
=µ - [ (57)
-
-µ~ttz
Similarly, the second derivative term from L can be reduced
22
with Ett and /J,tt symmetric, the adjoint field will be mutually to first order derivative. The term in (A3) involving line
bi-directional to the original waveguide and can be written as integrals vanish if there is a pee or pmc at the boundary or if
[5] the boundary is at infinity [8]. For the variational expression in
(54) with the substitution of (51), the second order derivative
E~ = -e-y + Ezi terms can be represented as
H~ = -h-y + Hzi. (58)
(64)
Hence
(59) and
J
with
e · 'E · i x Vt x 'E -l Vt x h ds. (65)
'nI.' a'Y -_ [j-he-y-y ] . (60)
Using (61), (64) can be transformed to
The material tensor of transversely DC magnetized, low loss,
-j
magnetically-saturated ferrites is of the form (57) (16]. The Vt x e ·ii· µ,-1 · iV t · /J, · h ds
formulation given in (59) works for a lossy nonreciprocal -f
media as long as the adjoint solution exists. For this specific
n·exii·/J,-1 -iVt·/J,·hdl. (66)
waveguide problem, Chen has derived a variational formula
tion ((43) in [5]], which does not satisfy the divergence-free
condition, to demonstrate the application of his variational Using (Al), (A7) can be transformed to
technique. On the other hand, the divergence-free condition
will be satisfied in using (59) since the longitudinal component -j
'E-1vt x h ·Vt xix 'E ·eds
Ez and Hz are given by divergence condition using transverse
-f
field and from solution of (59).
fi .'f-1Vtxhx ix'E·edl. (67)
V. REDUCTION OF THE SPATIAL DERNATIVE
OF THE VARIATIONAL FORMULATION Again, the terms in (66) and (67) involving line integrals
vanish with a pee or pmc boundary or if the boundary is at
In the last part of this paper, the second order derivative
infinity. For the variational expression in (56), the second order
term in the variational formulation will be reduced to first
derivative term can be represented as
order derivative. By the utilization of the vector identity
J f '
A · Vt x B ds = B x A · n dl - 2 JZ X h -y ·f=. -l ·v't"'7t X Z' -1 v't"'7 t·µ=· h -y d S. (68)
J µzz
+ B · Vt x A ds (61)
Using (61), (68) can be transformed to
n '
where is the unit normal vector, the second order derivative
term in the functional can be reduced to first order derivative. - 2 Jz -1v t't"'7· µ=· h ·v't"'7t Xf=. -l ·Z' X h d s
In addition, the boundary line integral in (61) vanishes with the µ z'z 'Y 'Y
application of suitable boundary conditions. For example, the - 2 f z -1v t't"'7· µ=· h -yXf=. - l ·Z' X h -y ·n' dl . (69)
functional in (48) has second order derivatives in the Ln and µzz
1772 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 8, AUGUST 1995
VI. CONCLUSION [ 11) B. M. A. Rahman and J. B. Davies, "Penalty function improvement of
waveguide solution by finite elements," IEEE Trans. Microwave Theory
We have derived a variational formulation for the prop
Tech., vol. 32, pp. 922-928, Aug. 1984.
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Theory Tech., vol. 34, no. 11, pp. 1227-1228, Nov. 1986.
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for waveguides where the longitudinal part of the medium
[14) K. Hayata, K. Miura, and M. Koshiba, "Finite-element formulation for
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[18] P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers,
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electric media based on variational principles," in Dig. IEEE Antennas 1956, in Stawell, Victoria, Australia. He received
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