# Faster technique for the modal analysis of a coaxial cable with misaligned inner conductor

## Abstract

[1] In this article we present a new technique for the modal analysis of a coaxial conductor with misaligned inner rod. The value of this method lies in the fact that, being based on linear techniques, it is possible to achieve the cutoff frequencies of the propagating modes without losing precision when the order of the modes increases. The differences in the results, obtained using 225 and 1600 basis functions, is quite negligible. Moreover, the procedure is completely analytical up to the evaluation of projection integrals, and the numerical part is restricted only to the matrix inversion. Therefore the run time needed is very low. This method was implemented in Fortran software written on this purpose by the authors. Numerical results are presented, keeping into account the computational time and complexity, and finally showing good agreement with other results already present in literature.

## 1. Introduction

[2] Coaxial waveguides are used very frequently for both ground and satellite microwave equipment. They have been studied in the past with a variety of approaches, including modern numerical techniques such as finite element and finite difference (FD). An important application of the coaxial technology is in the implementation of low-pass and bandpass filters. The correct design of this type of hardware requires the knowledge of the modal structure of the coaxial waveguide.

[3] The calculation of the modal fields of an uniform hollow conducting waveguide and the corresponding cutoff wave numbers is equivalent to the determination of the resonant modes and frequencies of a two-dimensional resonator [Marcuvitz, 1951]. Several numerical techniques have been developed for this purpose in the last decades. Most of these techniques may be grouped into two classes. The first consists of techniques based on the finite element method (or finite difference method) [Itoh, 1989], which leads to either large-size standard eigenvalue matrix problem or to multistep iterative schemes. The second class consists of techniques based upon the solution of integral equations by algorithms such as the method of moments [Itoh, 1989; Conciauro et al., 1984], the null-field method, the point matching method, or the auxiliary source method. All these techniques lead to the solution of small-size nonalgebraic eigenvalue problems [Spielman and Harrington, 1972]. The techniques belonging to the former class require an “ad hoc” choice of the location, the shape, and the number of elements inside the cross section of the waveguide, particularly when irregular and pointed boundaries are to be dealt with. In addition, their application requires time consuming procedures and/or large computer memory availability.

[4] Storage requirements are strongly reduced for the techniques belonging to the latter class. Nevertheless, computational time is still fairly long (especially when many modes must be computed), due to the need of finding zeros of determinants of matrices whose elements are transcendental functions of the frequency (nonlinear eigenvalue problems).

[5] The modal analysis of a perfectly symmetrical coaxial cable can be found in literature [e.g., Gimeno and Guglielmi, 1997, but the modal analysis of an off-center coaxial cable has not yet been analyzed using techniques belonging to the latter class. By building junctions between two asymmetric coaxial cables it is possible to realize interesting devices such as filters whose behavior is different from the behavior of a filter realized with two perfectly symmetric coaxial cables. Moreover this work is interesting because using this theory it is possible to foresee the variations of the behavior of a symmetric coaxial cable when the offset of the inner conductor is due to manufacturing tolerance.

## 2. Theory

[6] The geometry we have to study is characterized by two parallel cylindrical conductors misaligned by an amount Δ with a given difference in potential (see Figure 1). In order to find the eigenvalues of the TM and TE modes, we start from a static problem. We can reproduce the potential in the section between the two conductors by considering two wires with opposite charges and certain linear density set at a distance of 2d from each other.

[7] We put these two wires in the points F and F′ (see Figure 1) where F′ is the image of F, with respect to the outer conductor. The following relation is then satisfied:

where a is the radius of the outer conductor. In this way we are sure that on the outer conductor we satisfy the boundary conditions.

[8] Following the same way we enforce the boundary conditions on the inner conductor by verifying that:

where b is the radius of the inner conductor. The shift on the inner conductor is equal to Δ; we have

Combining equations (1) and (2) we obtain a system of two equations in two unknowns:

which solved gives the point F, and consequently F′, in which to place the two electrical wires.

[9] Now we can start from the system of the two wires. The equipotential surfaces are cylinders not concentrical to the wires and the lines of strength are orthogonal to the cylinders. At this point, as the equipotential surfaces are cylinders, we can transform the circular domain (in the plane Z = x + jy) into the rectangular domain in the plane W = ζ + jΘ (see Figure 2).

[10] In order to do that, we use the conformal transformation:

so we have the following transformation of coordinates [see Morse and Feshbach, 1953]:

In this way if we choose for the outer circle the line ζ = ζa,a > 0) we have:

In the same way if we choose for the inner conductor the coordinate ζb we have:

From equations (6) and (7) we obtain ζa, ζb,

and the relation CF·CF′ = a2 holds.

[11] It is important to note that when the shift of the inner conductor tends to zero, the mapping expressed by equation (2) is no longer valid because the position of the two wires tends to infinity. Now let us see how the Helmholtz equation changes when we are moving in the plane mapped by the transformation equation (2). We can start from the Helmholtz equation in the Z plane:

After some algebra, the Helmholtz equation becomes:

where ϕ is the scalar potential.

[12] At this point we apply the method of moment and we arrive to the definition of two matrices Lγ and Qγ by the following internal product:

where Qs is the linear differential operator defined by:

and the ψi is the basis function. The details of these matrixes are given in Appendix A.

## 3. TE Modes

[13] After these considerations, we are able to apply MoM for evaluating the cutoff frequencies of TE modes for the off-center structure. The basis functions used to obtain the TE modes are:

in which the cosine function refers to the TE-even modes (which corresponds to γ = e) and the sine function refers to the TE-odd modes (which corresponds to γ = o). Equation (13) automatically satisfies the Neumann boundary conditions:

Now we performed a mapping on the parameters s and n in a single parameter ordered with m. Using this basis function for the expansion of the scalar potential we obtain:

where

and

Finally, the Helmholtz equation, in the transformed domain, becomes

Expanding the scalar potential ϕ as previously outlined (equation (15)) and applying Galerkin's method we obtain a matrix linear eigenvalue equation.

The elements of the matrices and can be evaluated numerically and are contained in Appendix A.

## 4. TM Modes

[14] The theory for TM modes follows exactly the theory already outlined for TE modes. The basis functions used to obtain the TM modes are:

where the cosine function refers to the TM even modes (which corresponds to γ = e) and the sine function refers to the TM odd modes (which corresponds to γ = o). Equation (19) automatically satisfies the Dirichlet boundary conditions:

Using this basis function for the expansion of the scalar potential we obtain:

where

and

Finally the Helmholtz equation, in the transformed domain, becomes:

Expanding the scalar potential ϕ as previously outlined (equation (21)) and applying Galerkin's method we obtain a matrix linear eigenvalue equation.

The elements of the matrices and can be evaluated numerically and are contained in Appendix A.

## 5. TEM Mode

[15] The evaluation of TEM modes in the transformed domain is straightforward. The conditions that the scalar potential in the Helmholtz equation has to satisfy are:

In order to find the potential ϕ(ρ, θ) we consider the Helmholtz equation for TEM modes: ∇t2ϕ = 0 and apply the conformal mapping to the circular domain, and we obtain the Helmholtz equation in the rectangular domain:

with its associated boundary conditions (equation (25)),

The above equation is separable and can be solved by standard techniques. The solution is:

Equation (4) allows going back to the Z plane.

## 6. Numerical Results

[16] First of all, we started evaluating, in order to validate the technique, the cutoff frequencies of a perfectly symmetric coaxial cable. To do this we must give an infinitesimal shift to the inner conductor because of the theoretical considerations in section 2. The geometry considered in this first simulation has the following parameters:

We used 225 basis functions and the computation time was only 15 s on a RISC 4000 Sun Machine. The results are shown in Table 1 compared with Marcuvitz [1951].

Table 1. Zeros of Jm(cχ)Nm(χ)-Nm(cχ)Jm(χ) = 0
ModeOur ProgramX-Maple VRelative DifferenceReferencea
TM013.13711963.13711962.23134e-83.137
TM113.15497313.15497336.79143e-83.155
TM213.20792593.20792585.68691e-83.208
TM313.29425163.29425168.76617e-93.294
TM026.28092186.28092413.75685e-76.281
TM126.28997286.28997247.21677e-86.290
TM226.31704966.31704936.61134e-86.317
TM326.36192716.36192669.12148e-86.362
TM039.42326729.42326551.91016e-79.423
TM139.42931739.42931561.89746e-79.429
TM239.44744499.44744312.05488e-79.447
TM339.47758289.47758101.90782e-79.478
TM0412.56523912.5652353.18180e-712.565
TM1412.56978112.5697773.57395e-712.570
TM2412.58339712.5831934.09988e-712.583
TM3412.60605812.6060552.76756e-712.606

[17] As is clear, the results are much closer with the results of Maple and practically the same as in the work of Marcuvitz [1951]. We repeated the simulation to see the precision for the evaluations of the cutoff frequencies of the TE modes. We used 225 basis function (BF) and the computation time was 15 s. The results are shown in Table 2.

Table 2. Zeros of Jm(cχ)Nm(χ)-Nm(cχ)Jm(χ) = 0
ModeOur ProgramX-Maple VRelative DifferenceReferencea
TE023.15539563.15539971.29935e-63.155
TE123.17495333.17495474.40951e-73.174
TE223.22897553.22896981.76526e-63.229
TE323.32015163.32015201.20476e-73.319
TE036.29307486.29307825.40276e-76.290
TE136.30599286.30599361.26863e-76.299
TE236.34706066.34706781.13438e-66.326
TE336.41300716.41300831.87119e-76.412
TE049.42920709.42920126.15110e-79.429
TE149.43593639.43593902.86140e-79.435
TE249.45424099.45424112.11545e-89.454
TE349.48578689.48578776.48787e-89.484
TE0512.57028912.5702907.95526e-812.570
TE1512.57397112.5739731.59058e-712.574
TE2512.58799712.5880023.97203e-712.588
TE3512.61069812.6107023.17190e-712.611

[18] It is important to note that the differences between X-Maple and our results are smaller than the error introduced by the given shift. In the previous two tables, the differences between even and odd modes cannot be appreciated because smaller than the ninth decimal digit (which is the highest precision achievable in our software). Now we simulate a different geometry [see Das and Chakrabarty, 1995]:

We used again 225 basis functions and the computational time was again 15 s. The results are shown in Table 3.

Table 3. TE Frequency kj for Eccentric Coaxial Line 225
ModeOur ProgramDas and Chakrabarty [1995]
TE01e00–0
TE02e1.40792151.401–1.407
TE03e2.68607002.659–2.684
TE04e3.92953883.918–3.925
TE10e5.01747574.982–5.103
TE01o1.35086051.344–1.351
TE02o2.68793772.657–2.682
TE03o3.96174433.892–3.925
TE04o5.01489484.872–4.994
TE10o5.80748415.512–6.103

[19] As clear from the table most of the results are inside the range of Das and Chakrabarty [1995]. We repeated the last simulation using a very large number of basis functions: 1600. The total computing time was about 5 min and the results are shown in Table 4. Even if not all the results are in the range of Das and Chakrabarty [1995], we can believe that the precision of our cutoff frequencies is higher than Das and Chakrabarty [1995]. We repeated the same simulations for the TM case using both 225 and 1600 basis functions. The results of the simulations are in Tables 5 and 6.

Table 4. TE Frequency kj for Eccentric Coaxial Line1600
ModeOur ProgramDas and Chakrabarty [1995]
TE01e00–0
TE02e1.40793181.401–1.407
TE03e2.68609412.659–2.684
TE04e3.92958273.918–3.925
TE10e5.01756544.982–5.103
TE01o1.34087161.344–1.351
TE02o2.64797042.657–2.682
TE03o3.86180913.892–3.925
TE04o5.01500764.872–4.994
TE10o5.70757765.512–6.103
Table 5. TE Frequency kj for Eccentric Coaxial Line 225
ModeOur ProgramDas and Chakrabarty [1995]
TM01e4.7904.792–4.809
TM02e6.0736.127–6.170
TM03e7.2977.386–7.391
TM04e8.3838.445–8.489
TM10e9.3049.158–9.269
TM01o5.4805.485–5.509
TM02o6.6536.729–6.796
TM03o7.9497.931–7.956
TM04o9.0208.897–8.999
TM10o9.4459.391–9.453
Table 6. TE Frequency kj for Eccentric Coaxial Line 1600
ModeOur ProgramDas and Chakrabarty [1995]
TM01e4.7904.792–4.809
TM02e6.146.127–6.170
TM03e7.3727.386–7.391
TM04e8.5098.445–8.489
TM10e9.3159.158–9.269
TM01o5.4815.485–5.509
TM02o6.7696.729–6.796
TM03o7.9537.931–7.956
TM04o9.0248.897–8.999
TM10o10.0459.391–9.453
TM01e4.7904.792–4.809
TM02e6.146.127–6.170
TM03e7.3727.386–7.391
TM04e8.5098.445–8.489
TM10e9.3159.158–9.269
TM01o5.4815.485–5.509
TM02o6.7696.729–6.796
TM03o7.9537.931–7.956
TM04o9.0248.897–8.999
TM10o10.0459.391–9.453

[20] Only for the higher-order cutoff frequencies it is possible to note a slight difference in the results. This should prove that 225 BF are already enough to reach the convergence for the lowest-order modes.

[21] In Figures 3–10 it is shown a convergence analysis for two eigenvalues (of low and high order) both for TE and TM case. It is clear that more than 40 basis functions can guarantee that the convergence is reached for low-order modes (both TE and TM) and around 200 basis function are sufficient for the convergence of the TM high-order modes (around 100 are sufficient for TM modes).

[22] In the following figures there are the relative variation, expressed in percent, of the cutoff frequencies of TE and TM modes, plotted for three different values of the mutual shift between the inner and outer conductor of the cable. The bigger oscillations for the lower-order modes could be due to machine roundoff errors in the evaluation of the cutoff frequencies.

[23] From the previous figures it can be notices that the cutoff frequencies will increase faster depending on the inner conductor shift. In the last two figures it is shown the separation of the even and odd modes, for a shift of 0.3 mm, both for TE and TM modes.

## 7. Conclusions

[24] In this paper we presented a new and faster technique for the modal analysis of a coaxial cable with off-center inner rod. The value of this work lies in the fact that the method used is a linear method and gives good results, compared with the ones already presents in literature, with very low requirement of time. Moreover, the technique, being completely analytical, can provide us with accurate results in very limited time because of the absence of numerical evaluation of the integrals. The simulations run show that the convergence of the method is already achieved using a low number of basis functions for the expansion of the scalar potential. The method is very simple concerning the theoretical structure and is very easy to implement in a calculator.

## Appendix A

[25] All the integrals here given, are solvable analytically using commercial mathematical software, such as Mathematica or X-Maple. That is why the analytical expression is not given here.

[26] 1. Matrices and for the TE-even modes.

[27] 2. Matrices and for the TE-odd modes.

[28] 3. Matrices and for the TM even modes.

[29] 4. Matrices and for the TM odd modes.