Electromagnetics Generalized Telegraphist's ...

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Electromagnetics

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Generalized Telegraphist's Equations for Deformed Waveguides

Laurens Weissa; Wolfgang Mathisa a EE Dept. Institute for Measurement Technology & Electronics IPE University of Magdeburg, Magdeburg, Germany

To cite this Article Weiss, Laurens and Mathis, Wolfgang(1998) 'Generalized Telegraphist's Equations for Deformed

Waveguides', Electromagnetics, 18: 4, 353 — 365 To link to this Article: DOI: 10.1080/02726349808908594 URL: http://dx.doi.org/10.1080/02726349808908594

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GENERALIZED TELEGRAPHIST'S EQUATIONS FOR DEFORMED WAVEGUIDES

Downloaded By: [German National Licence 2007] At: 14:58 9 February 2011

Laurens Weiss Wolfgang Mathis EE Dept. Institute for Measurement Technology & Electronics IPE Universify of Magdeburg 39016 Magdeburg, Germany

ABSTRACT Only in a few cases of special symmetries, an analytic solution of Maxwell's Equations together with appropriate boundary conditions can be found. Among other things, difficulties occur because Maxwell's Equations r e p resent a set of coupled partial differential equations (PDEs). Therefore, methods have been developed to convert the PDEs with boundary conditions into a set of ordinary DEs. This simplification usually goes along with a loss of accuracy. In this paper, the prerequisites for the derivation of (stochastic) Generalized Telegraphist's Equations (GTEs) for (irregular) deformed waveguides are discussed. It turns out that a systematic approximative derivation of GTEs is only possible in rare cases. In any case, further knowledge about the fabricational cause of the surface imperfections is needed to fix the stochastic properties of the GTEs. The method's application to an irregular deformed helix is used as an illustrating example.

1. INTRODUCTION In many cases it is possible to describe the propagation of electromagnetic waves in waveguides by equations analogous with those for coupled transmission lines. For "regular" geometries, Schelkunoff has developed a method to convert Maxwell's Equations with appropriate boundary conditions into these GTEs [I], [Z]. Instead of solving a system of coupled PDEs with boundary conditions one can solve a system of coupled ordinary DEs without boundary conditions. For complex voltages V ,and currents I, associated

Electromagnetics. 18:353 - 365. 1998 Copyright Q 1998 Taylor B Francis 0272.6343 I 0 8 $1 2.W + .W

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L. WElSS AND W. MATHIS


with the m-th transmission line the GTEs describing the steady state are

Besides distributed series impedances Z,, and shunt admittances Ymn,(1) contains "voltage and current transfer coefficients" T:,, TL,. The latter couple the voltages Vm and currents I, of the m-th line to all other voltages V, and currents I,, respectively. This additional coupling makes the difference between classical Telegraphist's Equations and GTEs. In [Z] seve r a l a s e s of waveguides with variable cross-sections were treated. In (31, [4], [5], the method was adopted to investigate irregular deformed waveguides, i.e., to derived stochastic GTEs with explicit expressions for the stochastic coupling coefficients between different modes. At first glance, the results, although approximative, seem to be unique and well adapted to the physies of the problem. A second view reveals difficulties which question the applicability of the method to deformed waveguides. In the sequel we therefore discuss these prerequisites needed to derive GTEs for irregular deformed waveguides and use a helix (see [3]) to demonstrate our ideas.

2. GENERALIZED TELEGRAPHIST'S EQUATIONS In this section we work out the prerequisites of the derivation of GTEs for waveguides, where we are especially interested in irregular variations of the guide's surface. Irregular here means that only statistical properties of the surface's deviations from ideal geometry are known. First of all, a general mathematical model for a waveguide that is filled with a homogeneous medium, e.g., vacuum, is needed. We assume that the relevant physical properties of the guide are completely characterized by geometry and conductivity of its surface, and that the prescribed conductivities result in two "ideal" boundary conditions

The ideal boundary conditions (2) refer to an ideal geometry, i.e., to the g e ometrically undisturbed guide. Usually, the equations (2) 6 x the transverse and the longitudinal electrical field a t the boundary, respectively. S means the guide's nominal cross-section and OS its nominal cross-sectional surface, see FIGURE 1. Next, we introduce a set of appropriate locally orthogonal generalized cylindrical coordinates (u, u , r ) which are defined by the element of length


TELEGRAPHIST'S EQUATIONS FOR WAVEGUIDES

FIGURE 1. Cylindrical waveguide For convenience we let the z-axis coincide with the axis of the guide. Thus, u and v are transverse coordinates, whereas z is the longitudinal one. Now source free Maxwell's Equations

formulated in (u, v, r ) , have to be solved together with the boundary conditions (2). The electromagnetic field inside the guide can be derived from the general solution of a scalar Helmholtz equation A(u.v.r)$

+ k2$

= 0.

(5)

Separation of the transverse variables (u, v) leads to a transverse Helmholtz equation

TA of (6) are independent of z. They are usually specified The solutions Tn, via boundary conditions for perfectly conducting walls, i.e., von Neumann (TE waves) and Dirichlet problems (TM waves), respectively. Expanded in terms of (derivatives of) T,,TA,the transverse fields are

k is the intrinsic propagation constant of the waveguide interior, hn are axial propagation constants. The twodimenional first order differential operators grad and flux are defined by

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The z-dependence in (7) is put into the expansion coefficients. The ones belonging to the transverse electric fields we call voltages, the ones for the transverse magnetic fields we call currents. This is due to the normalization of the T,,TA,which can be chosen so that

holds. Note that the integral (9) extends over the nominal cross section S of the guide, i.e., 0 < u 5 a0 and 0 < v 5 vo. In case of finite conductance or deviations from ideal geometry these equations will be coupled. The constants dn and thus the series for the transverse fields are adapted to the case of geometrically ideal boundary conditions (2). This can be done by substitution of the series (7) for E, into one of the ideal boundary conditions, e.g. bideO'. As a consequence we can make use of the orthonormality condition TA,later on, which of course refers to an ideal geometry. The (9) for the Tn, remaining first ideal boundary condition b:&O1 determines the characteristic equation for the guide, which fixes the discrete separation constants

Substitution of the transverse expansions (7) into Maxwell's (rot),-equations leads to expansions for the longitudinal fields

Inserting all field expansions into Maxwell's divergence equations we get a set of uncoupled classical Telegraphist's Equations

that determine Vnand In. Solved for the Vs and I s the undeformed boundary value problem is completely determined. Let there be small deviations from ideal geometry. That will change boundary conditions. We will refer to the new situation as the "disturbed case" with "disturbed boundary conditions". Instead of (2) we have

where means the cross-section of the deformed guide. An analytic solution of Maxwell's Equations together with (13) will mostly be impossible, especially, when the disturbances are irregular and the equations (13) become stochastic equations. Instead of solving Maxwell's Equations , we use the transverse expansions (7) to derive GTEs , i.e., a system of coupled first


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order DEs for the expansion coefficients V n ( z ) ,I n ( z ) . Since we adapted the series ( 7 ) for E, to the ideal boundary condition bipea', this series cannot represent the new, disturbed field E, at the guide's surface but only inside the guide. As a consequence of their derivation the same holds for both longitudinal expansions ( 7 ) . To derive GTEs for the deformed guide we explicitly write down Maxwell's rot-equations

and substitute the transverse expansions into the four transverse rot-components. The first pair of equations ( 1 4 ) , ( 1 5 ) then contains terms dV,,/dz, the second pair ( 1 7 ) , ( 1 8 ) terms d I n / d z . Each pair is combined to one set of GTEs. To derive the first one, add -f luz,,Tm d,,,$grad,,T; times (14) and -fluz,Tm +d,,,$grad,~; times ( 1 5 ) , insert ( 7 ) and integrate over the nominal cross section S. With the help of ( 9 ) and (10) one gets

+

The intermediate result (20) has been obtained without the help of the disturbed boundary conditions (13). Therefore, it clearly represents the ideal, non-deformed waveguide. As ( 2 0 ) has a non-vanishing right-hand side we call it "inhomogeneous GTEs". There are two tasks left now: Since we are interested in the physics of the deformed guide we have to bring in boundary conditions ( 1 3 ) . Therefore, we are not allowed to insert the series ( 1 1 ) for E,. The use of (11) would make the right hand side of (20) equal to zero and thus prevent us from describing the deformed guide. We have seen

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L. WEBS AND W. MATHIS

above that the resulting set of uncoupled classical Telegraphist's Equations (12) describes the geometrically ideal waveguide. Note that we do not use convergence but physical arguments here (in contrast, see e.g. [3], p. 250 or [2]). Of course we have to and will take care that our calculations are mathematically correct. As a second task we have to eliminate the inhomogeneity E,. Obviously, both tasks must be combined: At least one of the disturbed boundary conditions (13) has to be used to express E,. This can be done by solving for example the first one for E,


EzIaS

=

(b'Y)-l(~t,~)(aj.

(21)

To create boundary terms including E,, we evaluate the cross-sectional integral in (20) using integration by parts

Substitution of (11) and (21) leads to the first set of GTEs

+

To get the second set of GTEs multiply (17) by -(flux,Tm dmgra&TA) and (18) by -(flux,Tm+dmgrad,T~), add both equations, insert the series (7) and integrate over the cross section of the ideal waveguide:

Again we have to eliminate the inhomogeneity on the right hand side of (24), this time using the second disturbed boundary condition to express Hz. As the boundary condition (13) has not yet been included in (24) the series (11) for Hz must not be inserted because this would make the right-hand side vanish. The resulting set of uncoupled GTEs (12) belongs to the

bp


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geometrically ideal case. After integration by parts grad H z is changed into


Hz

Then we solve

bp for H z ,

and insert (26) in (25). The result is a second set of GTEs

Striking feature of the results (23), (27) is the coupling between different modes. For irregular deformations of the waveguide's surface, only a statistical knowledge of the behavior of the disturbances can be given. Then, the GTEs (27) represent a system of stochastic DEs.

3. ESSENTIAL REQUIREMENTS In the last section we made several assumptions concerning the structure of the disturbed boundary conditions (13) in order to derive GTEs : (al) One disturbed boundary condition (13) can be solved for E,, the other one for Hz (at least approximately). When not directly solvable, in certain cases both equations (13) can be combined to new, solvable equations. In other cases Maxwell's Equations have to be used. Equations (13) cannot be solved for E, and H z , respectively, if - the equations do not include E, and H z , respectively - E, and Hz are coupled via (13) and cannot be decoupled When a solution is impossible even with the help of Maxwell's Equations no GTEs can be derived.

360



(a2) After E, and Hz from the disturbed boundary conditions have been inserted into (22), (25), the right-hand sides of (22), (25) include only field components that have a series representation valid at the boundary as. In other words the right-hand sides must not include "inhomogeneities" like E, or partial derivatives of E, or Hz. In some of those cases "inhomogeneities" vanish when interest is limited to a certain type of modes, e.g., to TEmodes. Otherwise, no GTEs can be derived. (a3) There is a unique way to include the boundary conditions (13) into (22) and (25). Otherwise the resulting equations are more or less arbitrary. Not only that the coupling coefficients T;,, TA, can have numerically different values in different cases but they can vanish in one case and be non-zero in another. (a4) The mistake made by using the nominal cross section S instead of the deformed one aS in the integrals in (20), (24) must be "small". The error depends of course on the specific type and on the largeness of the surface deformations. At this state of our investigations we can state the following: Whether a derivation (always approximative) of GTEs is possible or not essentially depends on the structure of the disturbed boundary conditions (13). The more complex they are, i.e., the more field components they relate to each other, the smaller is the possibility to derive GTEs.

4. APPLICATION TO A DEFORMED HELIX Let us consider an irregular deformed helix waveguide ([3], [4]) in order to illustrate our thoughts. As a mathematical model for a helix an anisotropically conducting sheath is used, which has a surface impedance Z in longitudinal direction and conducts perfectly in circumferential direction. Since an ideal helix has cylindrical geometry we choose cylindrical coordinates (u, v , z) = (r, p, z), i.e. el = 1 and ez = r . (Geometrically) ideal boundary conditions are

Inserting the series (7) for E, into (29), d , is determined


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In case of (irregular) geometric disturbances the nominal inner radius a0 will vary with rp and z , i.e., a = ao(l+6(rp,z ) ) ,where 6 expresses deviations from roundness, see FIGURE 2.

FIGURE 2. Deformations of the cross section The deformations impose new, disturbed boundary conditions on the fields at r = a . To simplify the following calculations we suppress the zdependence. We assume 6 = 6(rp), i.e., the deformations do not change along the z-axis. When 6 is differentiable we have

For irregular surface deformations, 6 is a stochastic process, and the equations ( 3 1 ) , (32) are stochastic equations. As adapted to (29) the series ( 7 ) for E, does not represent the disturbed field at the boundary r = a0 itself but is valid only for 0 < r < ao. The same holds for the two longitudinal series. The one for Hz has been obtained by differentiating the series for E,. As the functions T,,, determined in the limit to a perfectly conducting metallic waveguide, vanish at r = ao, the series the one for E, vanishes at r = ao, whereas E,, obeying (31), does not vanish there. In order to include the disturbed boundary conditions into (22) and ( 2 5 ) , respectively, equations (31) and (32) are developed in a Taylor series at r = ao. Assuming 6 , d6/dv