Reflection coefficient of a waveguide with slightly uneven walls

First-order results are derived for the reflection coefficient of a waveguide with slightly uneven walls. Specific analytical and numerical results are given for rectangular waveguides and coaxial transmission lines. Simple upper bounds are given for reflection coefficients in terms of the maximum deviation of the waveguide. For typical tolerances the reflection coefficients are very small ( >


I. INTRODUCTION
ONUNIFORM waveguides have been studied for N some time with application to antennas [I] and tapers [2]. The generalized telegrapher's equations [3], [4] provide a useful starting point, and Solymar [2] has worked with coupled traveling waves for studying spurious mode generation.
In t h s paper we use Solymar's formulation to derive first-order results for waveguides with small nonuniformity or wall roughness. Section I1 treats the reflection and transmission of the dominant mode and the generation of higher order modes for a waveguide of arbitrary cross section. Section I11 contains specific results for the reflection coefficient of the TE,, mode in a rectangular waveguide. Section IV contains similar results for the reflection coefficient of the TEM mode in a coaxial line. For typical tolerances the reflection coefficients are very small, but the results are important in precise six-port measurements [5].

FIRST-ORDER SOLUTION
We consider a perfectly conducting waveguide with a nonuniform section of length L as shown in Fig. 1. Starting with the generalized telegrapher's equations of Reiter [4], Solymar has derived the following differential equations for coupled traveling waves [2]: 1 d(lnK,) traveling waves, subscript i refers to the ith mode. p, is the wavenumber of the i th mode, K , is the wave impedance of the ith mode, SI; and SI,, are forward and b ac .k ward coupling coefficients, and the p summations are over all waveguide modes. The time dependence is exp(jot). In (1) it is assumed that K , # 0 and K , # m . and this means that p, f 0. Thus (1) is not valid for modes at the cutoff frequency (p, = 0). However, (1) remains valid for modes below cutoff where 6, and K , are pure imaginary.
If the waveguide is fed by a single mode m , the boundary conditions at the ends of the nonuniform section are for i + t n . (2) Because we are interested in the effect of small waveguide roughness or imperfections. we assume that the waveguide cross section is nearly constant. Consequently we are able to use a perturbation solution. This is in contrast to the work of Solymar where intentional waveguide tapers were considered. The zero-order perturbation solution to ( I ) and (2) is the solution to the uniform guide, and only the forward-traveling m th mode is nonzero: U.S. Government work not protected by U.S. copyright A first-order analysis of (1) and ( 2 ) shows that the first-order amplitudes satisfy the following differential equations: Here the superscript (1) refers to first-order quantities. The solutions to the differential equations in (4) are We are actually most interested in the wave amplitudes at the ends of the nonuniform section; these are given by The results in (6) are slightly different from those Solymar, but they are equivalent to first order.

A . Integral Form
In this section we derive the reflection coefficient for the dominant TE,, mode in a rectangular waveguide. The geometry for a rectangular waveguide of width a and height b is shown in Fig. 2. In our model a and b are allowed to vary with z , but the guide cross section is always rectangular. Using the notation of Solymar, we write the scalar mode function + where superscript A indicates a unit vector and V, is the transverse gradient operator. The wavenumber Prlo, is where k = 27r/X and A is the wavelength in the medium filling the guide. The wave impedance Kllol is where p and c are the permeability and permittivity of the medium inside the guide.
The expression for the backward coupling coefficient where the integration is over the waveguide perimeter C(z), ds is an element of C(z), and tan6 is the slope of the waveguide wall in the z direction. The sidewalls (1 and 3) do not contribute to the integral in (12) because the s derivative is zero. If we substitute (8) into (12) and perform some algebra, we can simply SiolilOl to -1 db To evaluate (7), we also require the following derivative: Equation ( I 3) is consistent with Solymar's rectangular waveguide example, but he did not require (14) because he did not consider any variation in a.
In keeping with our first-order analysis, we write the guide width and height in the following form: and where A c l / a o < < 1. Ab/hO < < 1, and a. and bo are independent of z . In addition we require that A,(O) = A,(O) = A(,( L ) = Ah( L ) = 0 so that the waveguide surface is continuous. Using (13)-(15) in (7), we obtain the following first-order expression for Sll: The expression in (16) provides a formal solution for S I , , but its form is inconvenient because it requires the derivatives of A, and A,,. Using integration by parts, we can rewrite (16) in the following form:

Sllr, = j2/?{~~lC,,/or,A,l( Z )~-J~~I : A I ' dz
and This form is more convenient than (16) because it involves the width and height deviations, rather than their derivatives. We have broken SI, into two parts to illustrate the separate dependencies on A, and A,,.
The form of (17) is similar to that in sea scatter [6] where the backscattered signal has the Bragg diffraction form. The integration in (17) essentially picks out the Fourier components of the surface variations with wavenumber 2/3$',,,. If we are given the width and height variations, A,( z ) and A,( z ) , we can calculate S,, numerically from (1 7).

B. Upper Boiuid
Frequently the actual z profiles of A , and A,, are not known, but an estimate of the upper bound is available. Let us assume that where A max is a known di rnensional toleraiice. Then from (17), lS,ll satisfies IS111 2q;;\,/,(lCl t l~~/ , l )~l l l~l y~ (19) Thus the upper bound on IS, 1 is directly proportional to A,,,, and I,. For realistic profilcs of A,, and Ab, the actual value of ISll\ will norinally be niiich smaller than the upper bound in (19) because uf the oscillatory nature of the exponential factor in (1'7).
To illustrate the order of tnagnitude of the quantities in (19) we consider a six-port application at X band. Typical The JSIlNJ term can be viewed ;is an iriipedancc effect and could be predicted by classical nonuniform transmission theory [8]. However, tlie IS, term is caused by backward coupling into the same mode and i s not predicted by a classical transmission line analysis. I n gencral, the two terms are of the same order of tnagnitude. 'I'heir frequency dependence is different because I( : , I is frequency depcndent and lC/,l is not. I n contrast the forward coupling coefficient is zcro [ 2 ] . and clianges i n /> (10 not affect the transmission coefficient (sec Appetidix H). The treatment here h;is assumed that tlic nonuniform section of wavegriidc i s continuous at thc ends. h,,(U) = A,@) = A,(L) = Ab(L). If there is a discontinuity at either end (as with a junction), then that effect must be addressed separately [9].

C. Sinusoidal Profile
In this section we consider an idealized sinusoidal profile for both A a and Ab. This is a convenient profile to consider because it is zero at both ends ( z = a and z = L ) and the integrations in (17)  If we replace the sine factor by complex exponentials and perform some algebra, we can write I , as The first term in (23) has a peak at Pf&L = n~/ 2 , which is the condition for Bragg scatter.
Using (17) and ( Figs. 4 and 5 indicate that a special profile is required for the magnitudes to even approach the upper bounds, and generally the magnitudes are much lower.

A. Integral Form
In this section we derive the reflection coefficient for the dominant TEM mode in a coaxial line. The geometry for a coaxial line with inner radius p, and outer radius po is  shown in Fig. 6. In our model p, and p,? are allowed to vary with z, but the guide cross section is always coaxial. The electric-field mode function e, for the TEM mode is The wave impedance El, of the TEM mode is the intrinsic impedance of the medium: Since KO does not depend on p, and p,, the derivative term in (7) is zero. Thus the reflection coefficient S,, of the TEM mode obtained from (7) is s,,= -lLs--J 2 k r d z .
(28) 0 The backward coupling coefficient S& of the TEM mode is most directly determined from the general formula of Reiter [4] which is an integral over the cross section of the guide. For the coaxial line t h s integral takes the following form: If we substitute (25) into (29) and carry out the integrations, we obtain The result in (30) is equivalent to that of classical theory for nonuniform transmission lines [7] if we take into account the difference in the definition of voltage and current. In keeping with our first-order analysis, we write the inner and outer radii as where A,/pj0 < < 1, A,/po0 < < 1, and pl0 and po0 are independent of z . Using (30) and (31) in (28), we obtain the following first-order solution for S,,:  where and For realistic profiles of A, and A,, the actual value of lSlll will normally be much smaller than the upper bound in (35) because of the oscillatory nature of the exponential factor in (33).
To illustrate the order of magnitude of the quantities in (35), we consider a six-port application. Typical parame- ters are [7]: p, = 1.52 mm, p, = 3.5 mm, L = 3 cm, A,,, = 0.635 pm, AomU = 1.27 pm, and frequency G 18 GHz. The upper bounds on lSllrI, ISllol, and JSllI are shown in Fig. 7. All three quantities are directly proportional to frequency because of the k factor in (35). The results for a realistic profile would normally have a more complicated frequency dependence because of the exp( -j 2 k z ) factor in the integrals in (33).

C. Actual Profile
The outer diameter of a precision air line was measured with an air gauge. The parameters are the same as in the previous case except that the line is longer ( L =16 cm). We assume that the inner conductor has no variation ( A j = 0). In Figs. 8-10 There is a small increase in lSlll with frequency, but it is much less than the linear increase in the upper bound as shown in Fig. 7. Its also interesting that lSlll does not increase as rapidly with line length as indicated by the upper bound expression in (35). The actual ISlJ stays well below the upper bound expression in (35) for all lengths and frequencies.

APPENDIX A HIGHER ORDER MODES
When only the m th mode is propagating and all higher order modes are below cutoff, the propagation constant mately equal to l/r,(')) to contribute to the integral.
Consequently the amplitudes of the reflected higher order modes are much smaller than that of the reflected dominant mode which is proportional to L as shown by (19).
For a more precise comparison, we would need to evaluate the coupling coefficient S,;, but it is of the same order as To examine the transmitted higher order modes, we substitute (Al) into (6) and obtain the following expression: From (A5) we can obtain the following upper bound: where r,(') is the attenuation constant and is pure real. If we substitute (Al) into (6), we get the following expression for the amplitudes of the reflected higher order modes: Thus the upper bound for the transmitted higher order modes has the same form as (A4) except that the forward coupling coefficient SA appears in place of the backward coupling coefficient. Both coupling coefficients are small The zero-order propagation constant B{pdl is given by (lo), and the first-order propagation constant can be obtained from a Taylor expansion: where Thus 62, is proportioiial to the average value of AI, and is normally much less thaii one. Also, a, , is independent of For the usual case of /i32,1 <r 7 , we can write S,, in the Thus to first order the magnitude of S,, is one, and S2, undergoes an additional phase shift of --a, , .
For the coaxial line, the propagation constant Po of the TEM mode is equal to k and is independent of z. Consequently, the transmission coefficient S,, is the same as that of the uniform line to first order: For iniperfect wall conductivity, the analysis is in general much more complicated, and even the modes for the uniform waveguide are difficult to analyze [lo]. The analysis for the coupling coefficients for nonuniform waveguides is very complicated, hut the simpler two-dimensional case of a parallel-plate waveguide has been analyzed using the surface-impedance boundary condition [ll].
For metal waveguides of high conductivity, the surface impedance is very small, and the mode fields do not differ much from those of the perfectly conducting waveguide. Consequently the coupling coefficients for nonuniform waveguides which depend on the mode fields do not differ much from those of perfectly conducting waveguides. The main effect of imperfect conductivity is to cause attenuation and a small change in the phase constant [9]. For the where vo is the free-space impedance, A , is the free-space wavenumber, and Z,,, is the surface impedance of the waveguide walls. For high wall conductivity, the surface impedance is where p,, is the wall permeability and U,,, is the wall conductivity.
To account approximately for the effects o f finite wall conductivity, we can replace jb$, in Section 111 by y as given by (Cl). In most cases this effect will be negligible. For example [9], a copper waveguide at X band has an attenuation rate a of approximately 1 0~ ' Np/m and a relative change in the real part of P,C$, of less than 10 '. If we make the y substitution for ,jb$)& in (23) and (24)  The correction for finite wall conductivity can be inade in a similar manner for the coaxial line. Here the complex propagation constant, y = (Y + jb, of the TEM mode is approximately [12] where Z,. is again given by ((12). 'IO account approximately for the effects of finite wall conductivity, we can replace j k in Section 1V by y as given by (C3). In most short line applications, the effect will be negligible.

ACKNOWI EDGMbN 1
The author would like to thank D. K. Holt for assistance in generating the numerical results and B. C. Yates for useful discussions and comments on the manuscript. [3] [4] [5] scatter from the sea." rectangular waveguide, the coniplix propagation constant, pp. 2-10. 1972 (C1)