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_ BIBLIOTHEEK TU Delft P 1250 5237 347092BY A CLASS OF WAVEGUIDE DISCONTINUITIES
BY A CLASS OF WAVEGUIDE DISCONTINUITIES
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C.J.D.M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUUR-KUNDE, VOOR EEN COMMISSIE UIT DE SENAAT
TE VERDEDIGEN OP WOENSDAG 6 DECEMBER 1967 TE 14 UUR DOOR
HANS]ANFRANKENA ^^^^ S^JS2
ELEKTROTECHNISCH INGENIEUR GEBOREN TE HAARLEM 1967 "BRONDER-OFFSET" ROTTERDAMPROF. DR. IR. A. T. DE HOOP
i
I
i
CHAPTER I INTRODUCTION 1
1. Scope of the present investigation and survey of
the literature 1
CHAPTER II THE JUNCTION OF TWO CYLINDRICAL WAVEGUIDES 7
2. Description of the configuration 7
3. The field within a uniform section of a filled,
cylindrical waveguide 8
4. Some separable configurations 24
5. The coupling of two uniform waveguides through a
junction of arbitrary shape 39
6. The coupling of two aligned uniform waveguides with
walls of equal geometrical cross-section 53
CHAPTER III SCATTERING BY SHEETS OF VANISHING THICKNESS IN
RECTANGULAR WAVEGUIDES 72
7. LSE- and LSM-fields 72
8. Scattering by a single strip in a rectangular
waveguide 81
9. Scattering by two symmetrically located and
iden-tical strips inside a rectangular waveguide 89
10. Computation of the transmission and reflection
factors describing the scattering by strips in a
rectangular waveguide (LSE-fields) 95
CHAPTER IV SCATTERING BY ANGLED BENDS IN RECTANGULAR WAVEGUIDES 112
11. Scattering by an arbitrary angled junction 112
12. Scattering by an unsymmetrical H-plane angled
junction 117
13. Scattering by symmetrical H-plane angled junctions 120
14. Scattering by an H-plane right-angle bend 128 15. Scattering by a symmetrical H-plane Y-junction of
factors describing the scattering by H-plane
angled bends and by an H-plane Y-junction 140
APPENDIX
A. The Lorentz reciprocity theorem for fields of harmonic time dependence
B. On the non-existence of a class of fields in some transversely inhomogeneous waveguides
C. The scattering of incident LSM-fields by a single strip in a rectangular waveguide
D. Details of the computation of eigenvalues for a special strip-loaded waveguide
147 147 151 155 160 REFERENCES 170 SUMMARY 174 SAMENVATTING 176 LEVENSBERICHT 179
INTRODUCTION
7. Saope of the present investigation and survey of the literature
The main object of the present thesis is the calculation of the
scattering of electromagnetic fields in a structure consisting of two uniform cylindrical waveguides which are coupled through a junction.
Details concerning the structure under consideration are given in Sec-tions 2 and 3.
If in the waveguides a known electromagnetic field is incident
upon the junction, in general outgoing fields are generated. The part of the outgoing field which is due to the incident field in the same
waveguide is called the reflected field; the part of the outgoing field which is due to the incident field in the other waveguide is called the transmitted field. The technical features of the junction are usually expressed in terms of the waves in the waveguides, while the detailed knowledge of the field in the junction is of less interest. Therefore,
we shall emphasize the description in which the influence of the junc-tion is expressed in terms of the field amplitudes in the uniform
guides.
In the guides commonly used in practice (with rectangular, circu-lar, elliptical cross-section or the coaxial guide, and filled with a
linear, isotropic, homogeneous medium) we can expand the fields in
terms of the well-known transverse electromagnetic (TEM), transverse
electric (TE) and transverse magnetic (TM) modes; these are given in
the standard textbooks of VAN B L A D E L I , COLLIN^, JOHNSON^, JONES'* and MARCUVITZ^. As, however, we also want to consider electromagnetic
phe-nomena in guides with isotropic, but transversely inhomogeneous or stratified media (although uniform in the axial direction), a
differ-ent field description must be used. In a rectangular guide filled with
a medium whose electromagnetic properties vary only in a direction per-pendicular to one pair of its walls (the x-direction, say), it is
the longitudinal section magnetic (LSM) modes. The former type has its
electric field vector perpendicular to the ar-direction, while the
lat-ter type has its magnetic field vector perpendicular to the x-direc-tion. These fields, originally introduced by BUCHHOLZ°, are described in, e.g., the textbooks by COLLIN^ and FLUGGE^ and in the reports by FELSEN and MARCUVITZ^, Applications of these mode expansions can be found, e.g., in PRACHE^ for waveguides containing a dielectric slab
and in COCHRAN and PECINA^° with regard to field calculations in a curved rectangular waveguide.
Through a separation in the various field functions of the axial coordinates from the transverse ones, the dependence of the field vec-tors on the axial coordinate is expressed in terms of two scalar func-tions only. These two funcfunc-tions turn out to satisfy equafunc-tions of the
same type as those which are found for the voltage and the current along a two-wire transmission line (compare SCHELKUNOFF^-^). This simi-larity leads to a description in which the propagation of each mode through the guide is symbolized by the propagation of the voltage and
current along a transmission line, while the junction of two guides is described in terms of a linear electrical network coupled to the lines. For detailed information concerning this transmission line analogy the reader is referred to the textbooks by MARCUVITZ^, FLUGGE^ and
MONT-GOMERY, DICKE and PURCELL^^, where the case of cylindrical guides fill-ed with homogeneous mfill-edia is coverfill-ed. FELSEN and MARCUVITZ^, too, con-sider the transmission line analogy for inhomogeneous media in cylin-drical waveguides, whereas for curved waveguides MORGAN^^ develops a
description in terms of coupled transmission lines.
Those parts of the field functions that, after the separation of variables has been performed, contain the transverse coordinates have to satisfy differential equations and boundary conditions having the character of a (complex and vectorial) Sturm-Liouville problem (compare
CODDINGTON and LEVINSONI**, COURANT and H I L B E R T I ^ , JOHNSON^). This type of problem has a non-trivial solution only for a restricted set of val-ues of a certain parameter; the admissible valval-ues of this parameter are called eigenvalues. In our case the relevant parameter turns out to be the propagation constant of the corresponding mode. Such a mode is the electromagnetic field corresponding to a particular eigenvalue.
The properties of the modes as regards the orthogonality and the
com-pleteness will be freely used here, in particular, they enable us to expand an arbitrary field in terms of the modes and to determine the
amplitudes of the modes in this expansion. A detailed account of
Sturm-Liouville problems involving real scalar functions has been given by TITCHMARSH^^ and by ZAANEN^^; an investigation of the eigenvalues of
(homogeneously filled) waveguides of arbitrary shape can be found in a paper by ZELBY'^^. In the following, only those properties of the
eigen-values and modal functions will be derived which are of direct interest
in subsequent calculations.
Let the fields in the uniform cylindrical waveguides be expressed
in terms of incident and outgoing modes. Then, it is possible to relate the technically important properties of the junction to the linear
re-lation which exists between the outgoing modes at one hand and the
in-cident modes at the other hand. In this way, the scattering matrix and the impedance matrix formulations are introduced (see, e.g., VAN
B L A D E L I , JOHNSON^, JONES**, MONTGOMERY, DICKE and PURCELL^^) . The influ-ence of symmetry in the geometrical configuration upon the properties
of either the scattering or the impedance matrix has been analyzed in
detail by MONTGOMERY, DICKE and PURCELL^^^ KERNS^^ and PANNENBORG^". As we shall see, linear algebraic equations between the field
ampli-tudes in the two uniform waveguides can be procured by a judicious ap-plication of Lorentz's reciprocity theorem for electromagnetic fields
that vary harmonically with time (published by a.o. VAN BLADEL^ for an
inhomogeneous anisotropic medium, HARRINGTON and VILLENEUVE^^ for gyro-tropic media while a related reciprocity theorem for lossless
aniso-tropic media can be found in VILLENEUVE^^). In those cases where the vector problem reduces to a scalar one, the relevant equations can be
obtained with the aid of the (scalar) Green's theorem, see e.g. the
pa-per by HURD and GRUENBERG^^. In this way we arrive at an infinite sys-tem of linear algebraic equations in which both the amplitudes of the
incident and the outgoing modes in both guides occur. For known inci-dent fields, the unknown amplitudes of the outgoing modes have to be
determined from this system of equations.
One way of solving the infinite set of equations is to reduce it
finite square system and solving this system exactly. This method has been used by MITTRA^"* and by KARJALA and MITTRA^S. Other authors have
applied perturbation techniques to obtain solutions of the equations describing the field in coupled waveguide systems (FOX and MAGNUS^^, REITZIG^^, RICE28). CLARRICOATS and SLINN29 have solved such systems of
equations directly by numerical methods, however, without giving de-tails as to their procedure.
In special cases, the coupling of two uniform waveguides can be
formulated in terms of integral equations for the unknown field quanti-ties; when these equations are of the Wiener-Hopf type they can be
solved by the Wiener-Hopf technique (see, e.g., NOBLE^O). Further, the configuration can be such that the linear algebraic equations mentioned
before are of a type that can be solved by making use of the theory of complex functions. This method, used a.o. by BERZ^l, COLLIN^, WHITE-HEAD^2 and WU and GALINDO^^, is discussed in Chapters II and III; it is
applied to the calculation of the scattering by resistive strips of vanishing thickness parallel to the narrow side of a rectangular
wave-guide. A waveguide section with such strips acts as a microwave
attenu-ator. Normally, only one strip is used, but the structure in which two (equal) strips are placed symmetrically with respect to the waveguide
axis is considered here, too. As will be described in Sections 8 and 9, the analytical solutions for the reflection and transmission factors
for these structures can be written as infinite products; in these
products the various propagation constants of the empty as well as the strip-loaded guide occur. In Section 10, we shall give a detailed
ac-count of the numerical techniques by which these propagation constants can be obtained.
The values of the reflection and transmission factors, describing
the scattering of the field in a rectangular waveguide by a perfectly oonduoting strip of vanishing thickness, placed parallel to either the narrow or the broad side of the guide, have been calculated by
MARCUVITZ^ with the aid of the Wiener-Hopf technique. In the case that the strip is placed at the center of the waveguide and is situated
par-allel to the narrow side, KADEN solved the (truncated) set of linear
algebraic equations for these factors numerically. His results led ATIYA^^ to the calculation of the properties of a bisected waveguide
section of finite length. The structure containing a single perfectly
conducting axial strip at an arbitrary location in the guide has also
been investigated by HURD and GRUENBERG^^. They have obtained analyti-cal solutions of the linear equations mentioned before by the
applica-tion of funcapplica-tion theoretical methods; their results are in agreement with Marcuvitz's. MITTRA^^ constructed approximate solutions from the
(truncated) set of equations given by Hurd and Gruenberg, discussing
the approximation method thoroughly. The scattering by n equally spaced and perfectly conducting strips inside a rectangular waveguide
leads to a set of simultaneous Wiener-Hopf equations which have been solved exactly by IGARASHI^^, The theory of a trifurcated waveguide
(with the strips at arbitrary locations parallel to the narrow sides of
the guide) has been reduced to that of a bifurcated waveguide by PACE and MITTRA^^.
The scattering of electromagnetic fields by a single absorbing strip placed at the center of a rectangular waveguide has been consid-ered by NOBLE " who, however, used a boundary condition at the strip
different from the condition we shall impose in Chapters II and III. The solution of that problem (involving two independent Wiener-Hopf
equations) was obtained exactly. This structure has also been studied
by PAPADOPOULOS^^, whose statement of the problem is in better agree-ment with that of our Chapter III. In the analytical solutions, the
values of the propagation constants in both the empty and the strip-loaded guide occur. In the latter guide, these values have to be
de-termined as the solutions of a transcendental equation. For one
cen-tral resistive strip, analytical solutions related to the dominant mode have been given by MARCUVITZ^. Numerical work in this field,
re-stricting the calculations to the two lowest modes in a guide contain-ing one strip at an arbitrary location has been performed by MUSHIAKE
and ISHIDA'*", who published papers on the same subject at an earlier
date'*^»'*^. Similar computations for a waveguide loaded with two equal resistive strips located symmetrically with respect to the centre of
the guide have been published by the same authors'*^; there, however, only the dominant mode is taken into account. For the calculation of
the propagation constants of higher modes, the reader is referred to
Chapter IV presents the calculation of the properties of corners
in rectangular waveguides. The reflection of such angle bends can be reduced, at least within a restricted frequency range, by the
inser-tion of certain plane and perfectly conducting reflectors (see RAGAN'*'*
and DE RONDE**^). Therefore, it is of interest to analyze such matching structures in corners; simple angle bends and mitred angle bends are considered in Sections 11-14, where infinite sets of linear algebraic equations are obtained for the reflection and transmission factors,
The numerical solution of these systems of equations is described in
Section 16. For the theory of the simple bends we further mention the formulas given by MARCUVITZ^, RICE'*^ and UDAGAWA and MIYAZUKl'*^, who
all obtained their solutions in the quasi-static approximation employ-ing conformal mapping techniques. A consideration of the mitred bend can be found in RAGAN'*'* and JOHNSON^. Finally, several authors discuss
the case where the waveguide is only slightly tilted at the "corner"; then, approximate solutions can be obtained (see ROWE and WARTERS**^) .
The related problems of slight tilts in circular waveguides have been considered by IIGUSHl'*^ and by NODA^O.
THE JUNCTION OF TWO CYLINDRICAL WAVEGUIDES
2. Description of the configuration
In this chapter we present the general properties of the
electro-magnetic fields that occur in the structure which arises when two
cy-lindrical waveguides are coupled together. We consider field quantities which vary harmonically with time. The complex representation of field
quantities is used; the complex time factor exp(jojt) (j = imaginary unit, Ü) = angular frequency, t = time) is omitted throughout. In each
waveguide a (local) Cartesian coordinate system is introduced with its
2-axis parallel to the axis of the guide (see Fig.l).
Fig.l. A uniform section of a filled, cylindrical waveguide.
The wall of each waveguide may consist of various parts, one of
which (the outer wall) surrounds the remaining ones. The
electromagnet-ic properties of the wall, assumed to be independent of the s-coordi-nate, will be given in terms of linear relations between the tangential
components of the local electric and magnetic field vectors; these re-lations have the form of an impedance boundary condition. The domain in between the outer wall and the other parts of the wall is the interior
of the guide; its cross-section perpendicular to the s-axis may be mul-tiply connected. The boundary of this domain is a set W of closed, bounded and non-intersecting curves of which one (the cross-sectioi of
the outer wall) surrounds the others completely.
The electromagnetic behaviour of the medium in each of the
wave-guides is assumed to be linear and isotropic, the properties of the me-dium are independent of the 2-coordinate. Across a finite number of
cy-lindrical surfaces parallel to the waveguide axis, the electromagnetic properties of the medium may be discontinuous. The cross-section of the
surfaces of discontinuity form a (possibly disjoint) set of curves D. The condition that across these surfaces the tangential components of the electric and magnetic field vectors be continuous, is the continu-ity condition. The behaviour of the medium is described in terms of the complex scalar permittivity e=e(x,j/;ii)) and the complex scalar
perme-ability \i=\i(x,y;iXi); these functions are piecewise continuous and show finite jumps across D.
Inside the medium filling the waveguide a finite number of
cylin-drical sheets of vanishing thickness and parallel to the 2-axis may be present. The cross-sections of these sheets form a (possibly disjoint)
set of curves S; their electromagnetic properties will be described in terms of two linear relations between the jumps across the sheet in the tangential component of the field vectors. These conditions will be
re-ferred to as the jump conditions.
In Section 3 we shall describe the general properties of the
elec-,tromagnetic field in a uniform section of a cylindrical waveguide of
the type under consideration. In Section 4, those uniform cylindrical waveguide configurations are studied where in the differential
equa-tions for the electromagnetic field vectors a separation of the trans-verse coordinates from each other can be carried out. Section 5 deals
with the configuration that arises when two cylindrical waveguides are
coupled to a junction. In Section 6, we consider the special case in which two different waveguides with the same axial direction and with
outer walls of equal cross-sections (but with possibly different elec-tromagnetic properties), are directly connected to each other.
3. The field within a uniform section of afilled, cylindrical waveguide
We consider a uniform, filled waveguide of the type described inSection 2. At its wall, we impose the following impedance boundary con-dition:
Here, E=E_(x,y,z;m) and Hj=B(x,y,z;i^ are the complex electric and mag-netic field vectors and n is the unit vector in the direction of the normal to U, pointing into the interior of the waveguide. The complex quantity Z =2 (x,i/;ijd) is the surface impedance of the wall. If the wall
w w
is made of absorbing material, it absorbs power and hence the component
of the time-averaged power flow density, pointing into the surface of the wall, has to be positive. Consequently,
Re{K-(E X ff*)} < 0 (x,y on W), (3.2)
where an asterisk denotes the complex conjugate value. With the aid of (3.1), we can conclude from this, that
Re(Z ) > 0 at absorbing walls. (3.3) w
If the waveguide wall is non-dissipative, the component of the time-averaged power flow density pointing into the wall vanishes; this leads to
Re(Z ) = 0 at non-dissipative walls. (3.4) w
At the sheets which are embedded in the medium filling the waveguide, we impose jump conditions which, on the supposition that electrical surface currents occur in the sheets, are of the type
(x,y on S). (3.5) (K
X ff]^ =
Z^' (nX
E)X
n,Here, n is the unit vector normal to the sheet, pointing from side 1 to side 2 (see Fig.2). The symbol [aj de£ a„-a, stands for the difference
between the value a- of a at side 2 and its value a at side 1. The complex quantity Z =Z (a;,y;D) is called the surface impedance of the sheet. If the sheet is made of absorbing material and hence absorbs power, the component of the time-averaged power flow density pointing
waveguide w«ll
Fig.2. The position of the normal n at the sheets.
into the sheet has to be positive. This implies that
Re{w'(S X [2*]^)} < 0 (x,y on S); (3.6)
with the aid of (3.5) we then obtain the condition
Re(Z ) > 0 at absorbing sheets. (3.7) s
At sheets of lossless material, where the component of the time-aver-aged power flow density pointing into the sheet must vanish, we have
Re(Z ) = 0 at lossless sheets. (3.8) s
On a sheet consisting of perfectly conducting material, we have the condition nxE=0; in that case we must replace (3.5) by nx£'=0 at both sides of S.
The continuity conditions at the surfaces of discontinuity of the medium have the form
{n-E]] = 0,
(x,y on D); (3.9)
the medium is absorbing, the permittivity and permeability have a
nega-tive imaginary part, for non-absorbing passive media these quantities are real. For physical reasons, the electric and the magnetic field
en-ergy density in this medium are positive for non-vanishing field
vec-tors; this condition requires the real parts of e and y to be positive. Consequently,
Re {e (a, J/; ID) } > 0, Im{e(a:,!/;(jj) } é 0,
(3.10) Re[\i(x,y;üi)} > O, Im{\j(x,y;bi)) è 0.
At an interior point of the medium, the complex field vectors ff
and H have to satisfy the complex Maxwell equations
r o t fi = jiJieE,
( 3 . 1 1 ) r o t E = - j'oiuff,
which hold in the absence of external electric and magnetic currents.
Now, the waveguide is used as a duct through which electromagnetic
power is transmitted in the axial direction. Therefore, the axial com-ponent of the power flow density, i.e. l_ • (ExH_ ) , is of interest. Since the axial components of E and ff do not contribute to this quantity, these components can be expected to play a role which differs from the
one that the tangential components do. We therefore make the separation
E = E^ + i E , — —T —3 3' H = H^ + i H , — —T —zz'
(3.12)
where the transverse parts of the field vectors are
£•„ def i E + i E , —T = —XX -^ y
(3.13)
H^ def i H + i H . -T = — -XX -y y
The geometrical configuration is cylindrical with the axis
paral-lel to the 2-axis; consequently we may expect the field vectors to have a functional dependence on z that essentially differs from the
func-tional dependence on the transverse coordinates. Therefore, we also make in the nabla-operator the separation
V = Vj, + i^O/dz), (3.14)
with
V de^ijd/ix) •>• iO/dy). (3.15)
I X y
Substituting (3.12) and (3,14) in (3.11) we are led to
±^ X OH^/dz) + grady(fl^) X i^ = ji^^E^,
(3.16)
Vy xE^ = - j^^lH^,
i X (dE^/dz) + grad^(E^) x i^ = - juuff^.
From the first and the third equation of (3.16) we see, that E
and H are completely determined by fi_ and 5„, respectively. They can, therefore, provide us with no further information than is known when the transverse components are determined; we shall, wherever possible, eliminate the axial components from our calculations. If we eliminate
E from the first and the last equation of (3.16), and H from the sec-ond and third equation of (3.16), we obtain a pair of coupled vector
differential equations in the transverse components only:
3Sy/32 = jiA€(i^ X E^) - grady{(jü)y) divy(i^ x E^)],
dE^/dz = jm(H^ X 4 ) - g r a d y l O O " ' diVy(^y x i^)}.
(3.17)
vari-ables from the longitudinal one. Accordingly, we write
E^(.^jy,Si'^) = V(z;m) e(x,y;oi),
(3.18)
:ffy(^.2/.2;a)) = I(z;ii>) h(x,y;a).
We remark, that in these formulas V, e_, I and h are not completely de-termined; this freedom can be used to simplify relations at a later
stage of the calculations,
The separation of the transverse and axial coordinates according
to (3,18) leads to solutions of (3.17) in the form of a set of charac-teristic functions, the waveguide modes. These modes will be used later on when we expand any field in the waveguide in terms of these
func-tions.
If we introduce the quantities Y="y('»j)> 2=Z(a)) and Y='ï(bs) = \IZ by letting Y-5' and yZ to be the separation constants, we obtain
- dlldz = ylV,
- dV/dz = yZI,
(3.19)
ylh = juic(i^ X e) - gradj,{ (Juu)" diw^(i^ x e)},
yZe = jia\i(h x i ) - grady{(joje) div^(h x i^)}.
Insertion of (3.18) in the impedance boundary condition (3.1), the jump
conditions (3.5) and the continuity conditions (3.9) leads to a set of linear homogeneous relations between e^,h and their derivatives at the sets of curves W, D and S. The homogeneous differential equations for e and ^ in (3.19) have only non-trivial solutions satisfying these bound-ary conditions for an infinite, enumerable set of values for y, called
the eigenvalues. We denote the eigenvalues by y(u)=y (u) (n=l,2,...). The other separation constant Z=l/J does not follow from this
eigenval-ue problem; it remains to be chosen properly. This fact can be used to
obtain certain symmetries in the matrices that represent the properties of a waveguide junction (see Section 5 ) .
Corresponding to each eigenvalue we have a solution of the bounda-ry value problem, a so-called eigenfunction; we denote them by V= =V (2;Ü)), J = J ( 2 ; U ) , e=e (x,y;iii) and h=h (x,y;bi) while, for the eigen-function to which the number n is assigned, we denote the separation constants Z and Y by the symbols Z (u) and Y (oj), respectively. The electromagnetic field formed from y , Z , V , I , e and ^ will be called the n-th mode; these modes form a complete system, i.e. any electromagnetic field inside the cylindrical waveguide under consider-ation can be written as a linear combinconsider-ation of these modes (compare COLLIN^ and HEYN^l). The differential equations and the boundary condi-tions for e and h are homogeneous; therefore, these functions can be
-n -n
determined to within a constant factor. This constant factor will be chosen later on by imposing a proper normalization condition (see
(3.39) and (3.48)).
From (3.19) we can see that a simultaneous reversion of the signs of y, y and Z gives the same differential equations. As none of the boundary conditions contains y, Y or Z explicitly, we may conclude that the solution of our boundary value problem is independent of such a re-version, This result holds in isotropic media (compare CHORNEY^^ for considerations about reciprocal anisotropic media and VILLENEUVE22 £ Q ^ a treatment of the field in non-reciprocal media). We now can, without loss of generality, impose the condition Im(Y)=0.
The four equations (3.19) are valid for the functions describing one distinct mode; by elimination of the function J we obtain from the first and second equation
dh Idz^
- y V = 0 . (3.20)
The solutions of (3.20) that have the character of travelling waves in the 2-direction are exp(-y s) and exp(Y s ) , hence,
V =A exp(-y z) + B exp(y z); (3.21)
n n ^ n n ^ n '
here the constants A =A (m) and ^„-^y,(-^) ^^^ determined by the boundary conditions at suitably chosen values of 2. From the second equation of (3.19) we obtain, with (3.21),
With the foregoing choice lm(Y )=0 and our time factor exp(j(i)t), the first terms at the right-hand sides of (3.21) and (3.22) describe trav-elling waves propagating in the positive 2-direction and the second terms describe travelling waves propagating in the negative a-direc-tion.
The first and second equations of (3.19) have the same form as the relations between the complex voltage and current for a uniform two-wire transmission line. By analogy with the terminology used there we
call V the mode voltage, I the mode current, y the propagation
con-stant, Z the mode impedanoe and Y the mode admittance of the n-th
' n '^ n
mode. The possibility to use a transmission line analogy in relation to the propagation of waves in a uniform cylindrical waveguide has found many applications. For a general outline of the introduction of trans-mission line analogies in waveguide theory we refer to the literature
(COLLIN^, MARCUVITZ^, FELSEN and MARCUVITZ^).
Form the last two equations of (3.19) we obtain through elimina-tion of h and e , respectively, the following vector differential equations in e and h only:
- n - n . - '
^iz " 8'^^'^r^y ^^^T^iz "" S-n^^ ~ gradyie divy(ee^) } +
(3.23) - ci^ X grady{e~ div^(h^ " is^-^ ~ S ^ ^ ^ T ^ ^ ~ '^i'^y^H^^^^ "^
- (fe'
-
\ ^Jin
=
0'where k is given by (see (3.10))
k^ = I/EV (Re(fe) è 0, Im(?c) £ 0 ) . (3.24)
The solution of these equations for given boundary conditions in some special cases will be discussed later on.
determination of the constants A and B exist. In general, their
val-n val-n & >
ues can be calculated if we prescribe at two different cross-sections of the uniform waveguide a linear relation between V and I . A special
^ n n '^
case of this occurs if we give the values of V (2;ijj) at two different
s n '
cross-sections, e.g. V (2.;u))=7. and V (z-,tii)=V„, with z <z„. Then, if
° n A' A n B B A B
z„~z =1, we obtain from (3.21)
D A
^„ = [VA ^ X P ^ V B ^ " ^B ^""P^VA^^/^ sinh(Y„Z),
(3.25)
B^ = {V^ exp(-T„2^) - V^ exp(-y^2^)l/2 sinh(y^^);
substitution in (3.21) and (3.22) yields
l'^(2;to) = {V^ sinh{y^(2g-2)} + V^ sinh{y^(2-2^) }}/sinh(y^Z) , (3.26)
I^(z;i,) = YjV^ cosh{y^(2^-2)} - V^ cosh{y^(2-2^) }}/sinh(y^Z) .
Analogously, if the values of J at two cross-sections are known, the values of A (u) and B (u) can be determined from (3.22). Insertion of these results in (3.21) and (3.22) leads, if J (2 ;OÜ)=J and I (z ;a)) = =Ig, with 2^<2g, to
F (2;u) = Z {l. cosh{Y (2-3)} - I„ cosh{y (z-2,)}}/sinh(Y J ) .
n 71 H Yl D u n /± n,
(3.27) I^(2;(i)) = {jj sinh{y^(2^-2)} + I^ sinh{y^(2-2^) }}/sinh(y^i) .
Besides this, we can determine A and B if both V and I have known
n n n n
values at a single cross-section, e.g. at z=z.. Then, if ^„(2 • ;ii))=F. and I (2. ;oo)=J., we obtain
^n- '^K ^^n'A^
-P^V^)'
(3.28)
B = \[V, - Z I,\ exp(-Y 2.),
n ^^ A n A' ^ ^ 'n A"
V^(z;^) = V^ cosh{Y„(2-2p} - Z^I^ sinh{Y„(2-2^) },
(3.29) ^„(3;^) = - V ^ sinh{Y^(2-2^) } + J^ C0Sh{Y^(2-2^)}.
From (3.16), (3.18), (3.21) and (3.22) we can obtain for each mode the 2-components of both the electric and magnetic field vectors as a sum of two different kinds of terms, one kind describing waves travel-ling in the positive 2-direction, the other describing waves propaga-ting in the negative 2-direction. With this, we finally arrive at the following expansion of the electric and magnetic field vectors:
^ = I , ^ .
— '•«=1 —n
(3.30)
^ = I , ^ .
— ''n=l —n
in which the electric field vector E and the magnetic field vector H
—n * —n
of the n-th mode can be written as
E = E * exp(-Y z) + E exp(yz), —n —n '^^ 'n —n '^ 'n
(3.31)
S = H exp(-y z) + H exp(y 2 ) . —n —n '^^ 'n —n '^ 'n
Here, the amplitudes of the travelling waves propagating in either direction can be calculated to be
E * = \e + (ey ) ~ ' i div„(ee )} A , —n ^—n n —2 T —n ' n E ~ = [e - (Ey ) ~ ' i div„(ee )} B , —n ^ ^ n —2 T -n ' n (3.32) H '^ = Y {h + (yy ) ~ ' •£ div„(y;z )] A , —n n'—n 'n —2 T —n ' n' ü~ = Y [- h + (py ) ~ ' i div^(ii;! )} B , —n n'- -^ 'n —2 T —n ' n
where A ,B (n=\,2,...) are constants to be determined from boundary
conditions at suitably chosen cross-sections as has been shown above. In order to express the coefficients A and B in terms of the
transverse components of the actual field at a cross-section, we need the orthogonality properties of the different modes. Applying the Lo-rentz reciprocity theorem (see Appendix A) to a section of the uniform cylindrical waveguide between 2=2. and z=z^ (z.<z„), we then obtain, (see (A.9)),
ƒ/, (^''^Z, - 4x4)-12
^' =
//E (V^Z> -
ib^'Sa^-^z ^^'
(3.33)
A B
where Z, is the cross-section of the interior region of the waveguide at 2=2. and Z_ is the corresponding cross-section at z=z^. We now sup-pose E ,H to be the electromagnetic field of the m-th mode in this configuration travelling in the positive 2-direction and E,,IL to be the electromagnetic field of the n-th mode travelling in the negative
2-direction. Explicitly, we have from (3.31)
la
= 4 ^ exp(-Y^2).
4 =4^
^^p(-v>-(3.34) E-, = E exp(y 2 ) ,—b
—n
'^ 'n '
H, = H exp(Y 2) ,—b
—n
^ 'n '
where m and n are arbitrary but fixed. We need, however, in the inte-grands of (3.33) only the transverse components of the field vectors. Hence, with the aid of (3.32), we obtain
Sk/ia'^h - h^'^^-iz ^'
= - Vn^^P^-(V^n>^^>
"
X
K
//z ( W - i 2 ^ *
'm lk,^^n-^?'iz
^^1'
A Aikjia^'Hb - h^^^-i-z '^
= - V n «-P^-(W^B> ^
(3.35)X fy ff^
^ n
^'t„^—m—n^
(e xh )'i dS + Y
—2
m ^'l„—n—m' —z
ff,
(e xh )-i ds}.
'
sides of both equations (3.35) are equal because both domains E and E
H D
are identical and the functions e , e , h and h are independent of 2. —m' — n ' —m —n "^
The right-hand sides of (3.35) have to be equal for any choice of 2 and 2 and consequently the relevant factors have to be zero, which amounts to
\
n ,
ie^^hn^'iz dS . Y^
//^ (£„xi2„)-i3
dS
= 0, (3.36)
a c
provided that y ^'y . In this equation, the symbol E stands for the cross-section of the interior region of the uniform cylindrical wave-guide.
From the same calculations in the case that the field E.,]J_-u repre-sents the n-th mode travelling in the positive 2-direction, we find analogously to (3.36),
Y fL (e xh )'i dS - Y ƒƒ„ (e xh )-i^ dS = 0, (3.37)
n •' -^ E W7! — n — 2 m "Ï. —n ~m —z \ • < /
c a
provided that y +y ^ 0 . Combination of (3.36) and (3.37) leads to
" m n
ƒƒ
(exh)'i as = 0
if
yj- + yj-.
(3.38)
'' L —m —n —2 m n
a
If we assume that y ^y if m^n (the non-degenerate case), (3.38)
m n
holds for any m and n if m^. Then, the fields e ,h and e ,h are
—m—m —n—n
called bi-orthogonal. In the degenerate case (i.e. when y =y for
cer-tain m^), again a procedure can be followed by which the
eigenfunc-tions become bi-orthogonal (compare COLLIN^, COURANT and H I L B E R T I ^ ) . in the case m=n Lorentz's reciprocity theorem yields an identity; the val-ue of the left-hand side of (3.38) is then not determined by the dif-ferential equations and boundary conditions. This ambiguity is used to supplement the bi-orthogonality relations by the normalization condi-tion
[\^ (e xh )-i dS = N . (3.39)
c
The bi-orthogonality property and the normalization condition can be combined to
jf (exh)-i ds - N& (m,n=l,2,...), (3.40)
L —m —n —2 m m,n
c
where N (m=],2,...) are normalization constants that will be chosen
m
properly in the sequel and where
0
(mM),
6 = (3.41)
m,n
1 (m=n),
is the Kronecker symbol. Now, consider an electromagnetic field E_,H_
that is represented by the expansions (3.30). The orthonormality rela-tion (3.40) now will provide us with the tool to determine the coeffi-cients occurring in the expansions of this field in modes.
The case in which the uniform waveguide is lossless (no losses present in either the walls, the medium or the sheets) deserves special attention. Then, we can make use of the reciprocity theorem for non-dissipative configurations (A.13) which leads to
[L (E xH* + E*xH )'i dS = / L (E xB* + E*xH )-i dS, (3.42)
•'•'E. -a-o
-b -a
-2
^'Z -n-b —b —a —z
where E. and E_ have the same meaning as in (3.33). We assign to theA D
fields E ,H and E,,H, the values given in (3.34). After substitution
—a —a •=*'-% *
of (3.32), we then obtain from (3.42)
exp{-(y -y *)Z,]{Y ff^ (e *xh )-i dS - Y *ï{^ (exh *)'i dS] =
^ m n A '• m"Z.-^n -m —z n "Z.—m —n—z *
= exp{-(y -y *)Z„){Y ƒƒ, (e *xh )-i dS - Y */ƒ, (e xh *)-i dS].
"^ 'm 'n ' B ^ m'' Z„-^ ^m —z n •'•'E_ -m —n —z '
^ ^ (3.43)
If, however, the field E,,IL is taken to be the field of the n-th mode propagating in the positive 2-direction, we obtain
exp{-(y -^y *)Z,}{Y ff^ (e *xh )'i dS + Y *jL (e xh *)-i dS] =
*^ m n A ^ m" Z. —n -m —2 n •' •' E. -m -n—z '
A A
= exp{-(Y n * ) 2 R } { ^ \ L (e *xh )-i dS + Y *jL (e xh *)-i dS].
m 'n B ' m"Z^-^ -m —2 n ''Z„-m-n —z ^
In both (3.43) and (3.44), the left-hand and the right-hand side have to be equal independent of the choice of z. and z . Then, if both y 5^
^y and y j^-y , in both (3.43) and (3.44) the factors multiplying the exponential function at the left-hand side and at the right-hand side have to be zero; for non-vanishing mode admittances this then leads to
jj^ (e xh*)'i dS = 0 if Y 2 ?« y *^. (3.45)
•^ •' E ^—m —n — 2 m n
c
Here, the integrals over E. and E_ , occurring in (3.43) and (3.44),
A D
have been replaced by integrals over the cross-sectional domain E , which is permitted because the relevant integrands are independent of the axial coordinate. Through (3.45), the bi-orthogonality of the se-quences of functions e ,h and e ,h is mathematically expressed.
^ -m'-m -m '—m ^ ^
From (3.43) and (3.44) we conclude in the case that n=m,
exp{-(Y^-Y/)2^} l m ( y = exp{-(y^-y/)2g} Im(f2^),
exp{-(y^n/)2^} Re(«^) = e^{-(y^^y*)z^} R e ( y
(3.46)
for any value of 2. and 2 , where
A D
n def y •ƒƒ (e x^*)-i d5. (3.47)
m W E —m —m —z c
Rejecting the trivial solution e =0, h =0 of the eigenvalue problem, we conclude from (3.46), since for non-trivial solutions no eigenvalues Y =0 are admissable, that
either: y = y and Re(fi ) = 0
m m m
or: y = -y and Im(fi ) = 0.
m m m
Thus, the value of the left-hand side of (3.45) in the case n=m is not completely determined by the differential equations and boundary condi-tions. We use this indefiniteness to normalize the mode functions now through the condition
|f^ (e xh *)'i dS = P , (3.48)
^' E -m ~m ' —z m'
a
where P (m=l,2,...) is the so-called power normalization constant. From the conclusions drawn from (3.46) we then have
either: Im(y ) = 0 and Re(Y *P ) = 0
'm m m
or: Re(Y ) = 0 and Im(y *P ) = 0.
'm m m
Combination of (3.45) and (3.48) leads in the non-degenerate case (i.e.
\L(exh*)'i dS = P & (m,n=l,2,...), (3.49)
•' •' E —m —w — 2 m m,n ' ' ' '
a
where 6 is given by (3.41). With (3.49), we can derive for the time-m,n * J ^ ' V / . averaged power flow in the positive 2-direction through a cross-section
i ƒƒ, (ExH* + E xH)-i dS = k I , Re(V I P ) . (3.50)
* •' •' E — — —2 ' ^m= \ m m m
c
The physical interpretation of this equation is that in a non-dissipa-tive waveguide the time-averaged power flow through a cross-section caused by the total field is equal to the sum of the contributions from the modes separately. If it is possible to put all constants P equal to unity, the right-hand side of (3.50) assumes exactly the form valid for the time-averaged power flow through an (infinite) system of trans-mission lines with mode voltages V and currents J . If the field E,H
^ m m — —
in (3.50) consists of only the m-th mode, propagating in the positive 2-direction with Re(Y ) = 0, we can prove with the aid of (3.32) and
(3.19) that the left-hand side of (3.50) is positive in the case that the guide is filled with a homogeneous medium and the waveguide wall is perfectly conducting; then we have R.e(Y P ) > 0. For more general non-dissipative structures, we can conclude that (since y is real) there exist real solutions for both equations given in (3.23). With the aid of (3.16), we can for each mode reduce the boundary conditions (3.1),
(3.5) and (3.9) to relations between the mode functions e and h with
—m —m
be chosen to be real for any non-dissipative configuration.
With the aid of the normalization conditions, we can give expres-sions for the constants A and B (m=l,2,...) in the expansions (3.32).
m m "
We have discussed earlier (see (3.25) and (3.28)), how these constants
can be calculated if we have some knowledge about the functions F(2;(JJ) and J(2;II)) at two different cross-sections (in special cases: a single
cross-section). Now, we finally shall have to show how the values of 7(2;u)) and I(z;m) follow from a knowledge of the field components in the cross-sections under consideration. In the general case, when our
modal functions satisfy (3.40), we can proceed as follows. The trans-verse components of the total electric and magnetic field can be
writ-ten as (compare (3.18), but see (3.21). (3.22), (3.30), (3.31) and (3.32), too)
IT
= Cl
^m
^•
^T " ^m=l ^m ^ •
With the aid of (3.40) we derive from this
-1
(3.51)
^m'^^A'"'^ " ^m J'^E i^rC^.i/.^^;'^) ^ h^(x,y;i^)\-i^ dx dy,
c
•'•m^^A'^^ " ^m ^U tfrn^^-y;'^) " Eq,(^,y,Zj^;'^)]'i^ dxdy,
c
(m=l,2,...). (3.52)
These formulas give us the possibility of calculating V (2. ;ii)) and J^(2^;'^) when £y(a;,2/,2^;ü)) and H^(x,y,Z^;Ü^) are given.
In the case that all materials in the configuration are lossless we can employ the orthonormality condition (3.49) instead of (3.40).
Then, from (3.49) and (3.51), we derive
^w^^A'"^ " ^m J'-^E i?y(^.2/.2^;") X h^(x,y;ii,)]-i^ dxdy,
c
(m=l,2,.,.). (3.53)
As with (3.52), we can in this case calculate V (z.;hi) and J (z ;ÜJ) if lit /i III /i, Em(x,y,z.;iii) and H_„(x,y,z.;lii) are known. The relations (3.25)-(3.28) then lead to expressions for A (in) and B (m) in terms of £_,(a;,j/,2. ;u)),
£y(x,j/,2^;(jj), E^(x,y,z^;üi) and H^(x,y,z^;a), in which the values of z^
and z„ can be chosen arbitrarily.
Let a uniform section of the filled cylindrical waveguide extend along the 2-axis in the interval 3.<z< z„. By definition, we shall
de-A D
note at z-z the part of the field travelling in the positive 2-direc-tion as the incident field; at s=2_ the part of the field propagating in the negative 2-direction will be denoted as the incident field. If at z=z . the incident part of any mode is given, we know all coeffi-cients A in the field expansions (3.30)-(3.32); analogously, we know all coefficients B in these representations if the incident field at
n "^
z=z„ is known. Consequently, the field as given by (3.30)-(3.32) is uniquely determined by the transverse components of the incident part
of the electromagnetic field at both z=z . and z=Zj,. If, e.g., the inci-dent field consists of a single mode, propagating in a certain
direc-tion along the 2-axis, the total field, too, consists of only that mode travelling in the same direction; this can be seen directly from the
field expansions (3.30)-(3,32),
If, in the waveguide structure under consideration, one or more
perfectly conducting sheets extend in a cross-section from one point of the outer wall to another, these sheets essentially separate various
distinct waveguides from each other. Each of these guides fulfills all requirements mentioned in the beginning of Section 2; the problem in
that case has to be treated as the propagation through a number of dis-tinct, parallel waveguides.
4, Some separable configurations
In the previous section the electromagnetic field within a uniform section of a cylindrical waveguide has been expanded in a set of modes
be solutions of the differential equations (3.19) or, equivalently, of
(3.23). In the present section we shall investigate under which re-strictions to be laid upon the field vectors and on the configuration a
separation of the two transverse variables from each other occurs. To this aim we replace the system of Cartesian coordinates x,y,z, used in Section 3, by a right-handed system of orthogonal, cylindrical coordi-nates u,V,z. Such a procedure has been used previously by MORGAN and by SPENCER^^. Let the metrical coefficients (see, for their defini-tions, e.g. STRATTON^"*) of this new coordinate system be a ,c and 1, respectively. Then we can write for any set of modal functions £ ,^ , omitting for the moment the subscript indicating the mode number,
e = 1 e + X e , — —uu —V V
h = i h + i h , — —uu —V V
(4.1)
where i and i are the unit vectors in the directions of the u- and V-—u —V
axes, respectively. Further, if (j) denotes some scalar quantity and if a=i a *i .a . denotes some vectorial quantity, we have
u u —V V ^ •" g r a d _ ( é ) = i c B d i + i . e 3 * , B yVT/ _ j ^ ^ yT —vv V d i v „ ( a ) = (a o ) " ' { 3 (c a ) + 3 ( c a , ) } , ( 4 . 2 ) T— u V ^ u V u V u V ' , - 1 d i v „ ( i xa) = (a o ) {-3 ( e a ) + 3 ( e a ) } , T^—z— uv ' u V V V u u '
where the operators 3 and 3 denote partial differentiations with
re-U V
spect to u and V, respectively. Substitution of (4.2) in the two first-order differential equations for ^ and h in (3.19), leads to the dif-ferential equations (valid at an interior point of the cross-sectional domain E introduced in Section 3)
c
yYh = -J UES - c "'3 l(juye,e,)"'{3 (e e ) - 3 (c e ,) }}, u V u M ' U V V u u u V V '
yYh = jinee - a 3 { ( i u y c a ) { 3 ( o e ) - 3 ( < ? e , ) } } , ' V u V u ' u V V u u u V V '
- 1 , v-1 yZe = ji^vh - a 3 { ( J u e c c ) ~ {i (a h ) - 'è (c h )] , ' u " *^ V u u^^" u V u^ V v' V u u ' ' yZe = -ji^]ih - c ' s { ( j u e c a )~\d (a h ) - d (a h )}]. ' V " ^ u V v^" u V u V V V u u ' Analogously, -ue 3 *^ V V c 3 u u u u^ - o , 3 V V -ea 3 V V - C - ' , u u EC "'s { - a 3 V V 3.23) yields (yc c ) {3 (c e ) - 3 (c e )}} + u V V u u u^ V V '
<^V^>"'t3«(^Vu) -^ \^^%%'^^^ - (^'"^') ^.. = °'
yc c ) " {3 (c e ) - 3 (c e )}} +^'%%^~'^\^'%%^ " ^y(^v«)^} - (^'"^'> % = °'
(ec e )"'{3 (c ;? ) - 3 (c ;z )}} + ^ u V V u u u^ V V '(VC o ) '{3 (yc ;z ) + 3 (^o h ))] - (fe^+y^) ^ = 0, M V U V U V^ U V ' ' ' u ' zc c ) h'è Ac h ) - ^ (a h )}] •*•
u V V u u u V V '
(yc c ) '{3 (ye ;2 ) + 3 (yc h ) }] - (k^+y^) h = 0. ^'^ u V u V u V *^ u V ' V
(4.4)
We now obtain (see SPENCER^^) a system in which a separation of the
variable u from the variable V is possible, if we put equal to zero in each left-hand side of (4.4) the sum of the terms that do not contain
the unknown function occurring in the last term of that left-hand side. If this is done for the first and third equations of (4.4) we arrive at
the conditions
ye "'3 {(\ic a) '3 (a e )} - e ' s {(ec c ,) '3,(Ee e ,)} = 0, V V u V u V V u u u V V u V
ec "'3 {(Ee e,)~'3 (e ^z )} - o "'3 {(ye e )~'3 (yo ;? ) } = 0. V V u V u V V u u u V v^ u V
(4.5)
Here, a considerable simplification is obtained if we suppose E and y
to be functions of u and ID only (similarly, such simplified equations are obtained from the second and the fourth equation of (4.4) if we had
assumed e and y to be functions of V and ti), only). Then we observe that (4.5) is satisfied identically if either c and e are independent of u and V (as is the case for Cartesian coordinates) or e »e , as well as e and h are independent of V and e is independent of M (as is the
V V U
case for rotationally symmetric fields in circular cylindrical
coordi-nates). In the remainder of this section we confine our attention to these two important cases. Then, from (4.4) and (4.5), we obtain the
following scalar differential equations for e and h only:
(4.6)
We can take the best advantage of the separation of the variables in cases where the boundary conditions can be formulated on parts of the coordinate planes exclusively. Therefore, we suppose the wall of the waveguide to be located such that in the impedance boundary condi-tions (3.1) either n=*i or n=±-i at the different elements of W. The
— —U — —V
condition that the medium is homogeneous in the u-direction requires
the elements of the set of curves D, introduced in Section 2, to be parallel to the u-axis. The technically most interesting problem is
that in which the resistive sheets mentioned in Section 2 are parallel
to the y-axis (see COLLIN^^ and HARVEY^^), hence we can choose in the jump conditions (3,5) n to be H at the different elements of 5.
If now e (w,y;ü)) and h (u,v;ia) could be calculated from (4.6) and the various boundary conditions, the first and the third equation of
(4.3) would be linear, inhomogeneous second-order differential equa-tions for e and h , respectively. With the positions of the wall, the discontinuities and the sheets as indicated above, we are led to a set of linear, inhomogeneous boundary conditions for e and h . The rele-vant boundary value problems have a unique solution (compare COURANT and HILBERT^^); therefore e and h are, in the cases under
consider-' V V ^
ation, uniquely determined by e and h . As has been remarked before (see Section 3 ) , the values of the components E and H can be calcu-lated uniquely from the other field components, so the complete
h , V and J for that mode. Thus, in contrast to the case in Section 3 u' '
where the field was determined by an infinite sequence of two scalar and two vectorial functions (viz.. V ,1 ,e and ^ ) , we now have a
prob-' nprob-' n'—n —n
lem in which the field is determined by an infinite sequence of four scalar functions.
As all differential equations and boundary conditions involved are linear, the electromagnetic field in a uniform waveguide that satisfies the restrictions imposed above can be constructed through superposition of the fields occurring in the three cases (a) e ïO, h sO; (b) e iO,
" u u u
h =0; (c) e =0, h fO. It can be proved (see Appendix B) that the fields in case (a) (where e 50, h sO) are identically zero.
Therefore, we consider the cases where either e =0 or h =0
sepa-' u u
rately when we deal with the configurations under consideration; super-position of the two types of fields then leads to the complete electro-magnetic field in that (uniform) structure.
In the case e =0, h tO we are led from the second equation of (4,4) to a differential equation for e only:
yc "^3 {(yc,)"'3 (e e )} + o ~^3 ^e + (fe^+Y^) e = 0 U U V U V V V V V V
if e^EO, (4.7)
in which the eigenfunction e and the propagation constant y correspond to the same mode. With the aid of (4.2) we can derive from (3,1) and (3.16) the following impedance boundary conditions for e at the wall of the guide e, = *Z,(j(jjyc c ) 3 (e ,e„) if n=H , V w^ u V u V V — —u 3 (c e ) = 0 if n=±i and Z 5^0, M y V — —y w (u,v on f/) ; (4.8) here the upper and lower signs in the first condition and the restric-tion menrestric-tioned behind it go together. If Z =0, i.e. on perfectly
con-w
ducting parts of the wall, the second condition has to be replaced by the condition that the tangential component of £ vanishes. Replacing E
have
3,,(e„) = 0 if n=H ^ and Z =0 (u,v on W). (4.9)
The continuity conditions (3.9) are replaced by
''v^\
(u on D). (4.10)
• 1 . / s i 2
fy 3
(c e
)]; = 0,
^^ u^ V v'^ \ '
At the absorbing sheets of vanishing thickness the jump conditions (3.5) lead in the present structure to the jump conditions
(u on S). (4.11)
((jiAuc c ) 3 ( e e ) l , = Z e ,
'-^'' ^ u v^ u V V ' 1 s y'
Here, i points from side 1 to side 2 of the sheets. As all conditions as well as the differential equation are homogeneous, the function e can at best be determined to within a constant factor. If e is known,
V
h andu V h can be calculated, from (4.3) and (4.7), to be V / \ / >
h^ = -Z(j.^y)-hc^-h^\ . y \ } ,
if e,EO, (4.12)
K = 2 0 v Y C ^ c / ) " ' 3 ^ 3 y ( e ^ e y ) , 2.-1. "
where Z is the mode impedance for the mode under consideration. With this, the undetermined constant factor in e can be expressed in terms of the normalization constant for the relevant mode as introduced in
(3.39) or (3.48).
We now perform the separation of the transverse coordinates u and
V from each other by insertion in (4.7) of
e (u,y;n)) = <|>(W;(JÜ) /(y;(jj). (4.13)
2
^u\^^^''v^~^'^uK'^^^
* (fe^Y^-a\"2) * = 0,
dj^f
+
aV
=
0,
(4.14)
where now d and d are operators which perform an ordinary differenti-ation with respect to u and v, respectively. The solution of the second
equation has the form
/(v;u) = /j cos(ay) + f^ sin(ay), (4.15)
where f^. and ƒ_ are constant coefficients and where a is chosen as the 2
square root of a with Re(a)^0 (we remark that a reverse m the sign of o at the right-hand side of (4.15) does not lead to new solutions). From (4.8)-(4.11) we now obtain the impedance boundary conditions
ƒ = 0 if n=H^ and Z jtO, (u,V on W), (4.16)
df =0 if n=H and Z =0,
the continuity conditions
W ? = o.
(y-'d^(e^*)); = 0,
and the jump conditions
W ? = 0.
(u on Ö) (4.17)
(u on 5). (4.18)
((j^ye^e^)-'d^(e^<t.))f = 23"'*,
We have remarked in Section 3, that only for a specific enumerable set of values for y (the propagation constants y (n=l,2,...)) the boundary value problem has non-trivial solutions (the eigenfunctions
e ,h («=1,2,...)). The problem formulated in (4.14)-(4.18), which is a special case of the problems stated in Section 3, has the same proper-ty.
The function of v at the right-hand side of (4.15) can satisfy the impedance boundary conditions only for an enumerable set of values of a, say 0=0 (q=\,2,...); to each value of a there corresponds a so-lution /=f(y;oj) of (4.15). The first differential equation of (4.14), together with the boundary conditions (4.16)-(4.18), form a Sturm-Liou-ville eigenvalue problem (see COURANT and HILBERT^^); this can be solv-ed for any o only for an enumerable set of values of the constant
2 2 2 -2 i
Udef (fe +y -a Q ) . We label the eigenvalues by the integer p (p=l,
4 V
2,...) and denote them by W . A s in the mathematical problem stated before only w plays a role, we can - arbitrarily - restrict the signs of the eigenvalues by the condition Im(u )m0. For any value of the pair of integers p,q one value of y is found; in Section 3 we de-noted these characteristic values of the propagation constant by y . For the structures under consideration, the system of numbering using n is a condensed form of the numeration system using the pair p,<7, we al-ways assume a one-to-one relation to exist between both systems. Thus we obtain, with the remark preceding (3.20) that we can choose Im(Y)=0, the following relation between the n-th eigenvalue '^ and the corre-sponding propagation constant y
'^n = ("n^ ' ^^ * °q\~^^^ (Im(Y„)iO). (4.19)
while the corresponding eigenfunction e _ can be written as
e (w,y;u) = ((> (u;w) f (y;a)). (4.20)
n;v ' n •'n From (4.12), we then obtain
^n;M<"'"5"> = -2«('''^^'V"'(^n^ " " Z ^ " ^ ) ^n^"'") fn^^'''^'
(4.21)
h^,^(u,v;o^) = Z^(ia)yy^e^e^2)-'d^(ey*^(2.;a))] d^„(ü;a)).
been expressed in terms of the functions (|) and ƒ .
In the case h =0 we obtain (compare (4.7)) as the differential
equation for h ,
<\^^'''v'^~'\K''v'>^
"
% \ \
"
(
^
'
^
^
'
)
\ =
0
if h^^O, (4.22) w i t h t h e i m p e d a n c e b o u n d a r y c o n d i t i o n sK = '^^-^°u%^w^ \K\'>
d (c h ) = 0 U V V i f n = ± i , — —u' i f n=H , — —y (u,V on W), ( 4 . 2 3 )the continuity conditions
W = °.
fe '3
(o
h)1? = 0,
^ w^ y y -• 1 ' (u on D) (4,24) and t h e jump c o n d i t i o n s(^
V%V)I
= °'
( ; : ] ? = ( J u e o c Z ) ' 3 (ah), *• y-* 1 " u V s u V V (u on S ) . ( 4 . 2 5 )The notation in these formulas is the same as the one that has been used in (4.8)-(4.11). The mathemauical problem to obtain h from these relations is for the most part analogous to the problem of solving e from (4.7)-(4.11). We can determine h only to within a constant fac-tor; deriving, in addition, from (4.3) and (4.22) the relations that express e and e in terms of h :
'^ U V V
e^ = Y(j..y)-\c^-W.,\},
if h =0u
a^ = -y(i.Eye^e/)-'3^3^(e^;.^),
(where y denotes the mode admittance for the mode under consideration), we can relate this undetermined constant factor to the normalization constants in (3.39) or (3.48).
Performing the separation of the transverse variables from each other by writing
h^(u,v;iii) = ^(u;(n) ö'(y;cü) (4.27)
we obtain from (4.22), through the introduction of the separation con-,- ^ 2
stant T ,
£C^ \{(tc^)~^d^(c^^)] + (fe^y^-T^Cy ^) .p = 0,
2 2
dy g + -^ g = 0.
The solution of the second equation is written as
(4.28)
g(v;i:i) = g. cos(Ty) + g„ sin(Ty), (4.29)
where g and g„ are constants and Re(T)^ 0, Substitution of (4.27) in the boundary conditions (4.23)-(4,25) leads to the impedance boundary conditions ii = ±(,7(i)Ee c Z ) d (c üi) * ^ u V w u u g = 0 if n=H if n=H , - -v' (u,v on W), (4.30)
to the continuity conditions
( E - V ^ ^ ) ) 2 = 0,
(u on D), (4.31)
(E-'dJe^^))2 = 0.
(u on S). (4.32)
W ? = (j^Ee^e^zp-'d^(e^,|;).
Similar to the case e =0, only non-trivial solutions exist for an enu-" 2 2 2 -2 i merable set of values T = T (q=\,2,...) and Wdef (k +y - T C ) ^ ^„
(p,q=\,2,...). Again returning to a system of numbering using only one
subscript n (which is supposed to have a one-to-one relation with the numeration system using the pair of integers p,q), we now have, with
(4.27):
\.y(u,v;bi) = IC„(M;CD) g^(v;in). (4.33)
We denote the propagation constant corresponding to the mode number n by r ; its relation to the eigenvalue W is then (compare (4.19))
^n = '^^n " ^^ * \^%~^^^ ( l m ( y > 0 ) . (4.34)
From (4.26), we can write the transverse components of the electric field of the n-th mode as
-1 2 2 - 2
e^.Ju,v;in) = y^(j^^r^) (r„ - T ^ c^ ) ^^(u;(.) g^(v;,n),
(4.35)
e (u,v;tj^) = -Y (juizT c c ) d fc \li (M;(II)) d q (y;u). n;v ' ' n^"^ n u v u'- v n ' ' v^n '
Now, the transverse field components in the case h =0 have been ex-pressed in terms of the functions i)j and g .
We now have seen, that in each of the two cases the functions e
' —n and h are determined uniquely. As the set of fields e ,h in each of
— n 1 J —n'—n
the two separate cases is complete, the two sets together must lead to a complete set of modes for the general case. Furthermore, as this com-bination satisfies entirely the same conditions, it must be identical to the set of modes which results (see Section 3) from the more general vectorial problem. We can say that, under the restrictions imposed on the configuration and the fields in the present section, the vectorial problem of Section 3 gives rise to eigenfunctions for which either
e so or /z sO.
n;u n;u
We conclude this section with the consideration of two waveguide configurations of great technical importance, which satisfy such
condi-tions that the theory of this section is applicable.
(i) Equations for the field in a rectangular waveguide. For a waveguide
with rectangular cross-section, we employ the Cartesian coordinate
sys-tem used before; we take the transverse axes parallel to the various parts of the wall of the guide. Then, we have c =c =1. u=x and V=y. Let
u V '
the cross-section occupy the domain 0 < x < a , 0<y<b (see Fig.3). We suppose the discontinuities of the medium to be situated at the sur-faces x=d , 0<y<b (m=l,2 M) and the infinitely thin sheets occupy the surfaces x=Sy, 0<y<b (Z=l,2 L). Now, in the case e =0, (4.16) can be satisfied by (4.15) only if /,=0 in the case of non-perfectly
conducting walls and if / T = 0 in the case of perfectly conducting walls; for the separation constant a we obtain in both cases the values
a =qTt/b (q=\ ,2,...) . The function (fi has, with (4.14), to satisfy p(d/&){y"'d*^/dx} + (^^+Y„^-a^^) <(>„ = 0, (4.36)
with the impedance boundary conditions
Vdiscontinuitier
(|> = Z (iüjy) di /dx ^n w n *„ = -Zy(jü)y) d<^^/dx at a:=0, at x=a, (4.37)
the continuity conditions
(y"'d<f^/c&)5 = 0,
and the jump conditions
2 1
*^ = Z^[(ja)y)"'d<t^/da;)^, . N - 1
at a;=cf (m=l,2 M) (4.38)
at a:=S7 (U\,2,...,L). (4.39)
As far as the fields in the case h =0 are concerned, we see from (4.29) and (4.30), that ^",=0 and T =q-n/b (q=\,2,...); further, (4.28) becomes
£(d/dx){e ^d^\)Jdx] + ('^^+r„^-'^(j^) '('„ = 0- (4.40)
For this configuration we have in the case h =0 the impedance boundary conditions
i|j^ = (j'oeZy)" di>^/dx
-1,
^^ = -(ja)£2y) difjdx
at a;=0,
at x=a,
(4.41)
the continuity conditions
[e-^d^^ldx]]
= 0,
at x=d^ (m=l,2 M) (h.lil)
(E
^d^Jdx)] =
O,
at x=s^ (l=\,2 L). (4.43)
(^jj = (j,n^Z^)~^d^^^/dx,
With this, we have obtained the equations for the functions <() and l^
' ^ n n
pertinent to a rectangular waveguide.
(ii) Equations for rotationally symmetrical fields in a circular
wave-guide. For rotationally symmetrical fields within a waveguide with a
circular wall (or coaxial circular walls) we have, if circularly cylin-drical coordinates are introduced with the axis of the guide as polar axis: u=r,v=Q,o =1 and c =r (see Fig.4, where the cross-section of the
' ' U V " '
guide occupies the domain r. <r < r „ ) . Then, ƒ (e;ijj) in (4.20) is a con-stant and a=0 in (4,14). We assume that no resistive sheets are pres-ent, while the discontinuities of the medium occur at the coaxial
cir-Fig.4, A filled circular cylindrical waveguide structure,
cularly cylindrical surfaces r=d (m~\,2,...M) . The wall of the guide
m
consists of an outer and an inner wall which are circular cylinders coaxial to the 2-axis with radii r^ and r., respectively. Now, we ob-tain in the case e =0 from (4.14) the differential equation for
i>n^v;<ji):
