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Analytical solution of higher order modes of a dielectric-lined eccentric coaxial cable

Published online by Cambridge University Press:  15 July 2020

Mehdi Gholizadeh Open the ORCID record for Mehdi Gholizadeh [Opens in a new window]  and
Farrokh Hojjat Kashani
Mehdi Gholizadeh*
Affiliation:
Department of Electrical Engineering, Iran University of Science & Technology, Narmak, Tehran, Iran
Farrokh Hojjat Kashani
Affiliation:
Department of Electrical Engineering, Iran University of Science & Technology, Narmak, Tehran, Iran
*
Author for correspondence: Mehdi Gholizadeh, E-mail: mehdi.gholizadeh1991127@gmail.com
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Abstract

This study provides an analytic method for the calculation of the cutoff frequencies and waveguide modes of a partially filled eccentric coaxial cable. The method is based on the expressions of the involved electromagnetic fields in bipolar coordinate systems and the validity range of the solution is discussed. It is shown how the waveguide geometry and dielectric parameters may be selected to engineer the lined waveguide's spectral response. Numerical results are included which show good agreement with the corresponding results from full-wave simulations by commercial software.

Keywords

Analytical SolutionHybrid ModesCutoff FrequenciesDielectric-Lined Eccentric Coaxial Cable

Information

Type
EM Field Theory
Information
International Journal of Microwave and Wireless Technologies , Volume 13 , Issue 3 , April 2021 , pp. 247 - 254
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Copyright
Copyright © Cambridge University Press and the European Microwave Association 2020

Introduction

Without modifying the dimension of conductors, the characteristic impedance of a coaxial cable is adjustable by laterally changing the offset of the inner conductor. This technique can be used to realize a quarter-wave matching element that forms one of the sections in a multisection quarter-wave transformer for broadband-matching applications [Reference Chakrabarty, Sharma and Das1]. Besides, the analysis of cavities excited by thin probes can be simplified using eccentric circular metallic waveguide structures with a small ratio of inner-to-outer conductor dimensions [Reference Davidovitz and Lo2]. Despite of these interesting applications, the shape of boundaries severely limits the possibility for analytical solutions of eccentric circular metallic waveguide configurations [Reference Kotsis and Roumeliotis3, Reference Kotsis and Roumeliotis4]. The investigations of this type of waveguide have initiated interest of researchers for a long time [Reference Chakrabarty, Sharma and Das1–Reference Gholizadeh, Baharian and Kashani15]. Various techniques have been used to obtain numerical results: point-matching [Reference Yee and Audeh5], conformal transformation [Reference Abaka6], related addition theorem [Reference Roumeliotis, Hossain and Fikioris7], a combination of the conformal mapping of the cross-section with the intermediate problems method to obtain the lower bounds for the cutoff frequencies and the Rayleigh–Ritz method for the upper bounds [Reference Kuttler8], perturbation techniques [Reference Davidovitz and Lo2], transforming eccentric coaxial into coaxial configuration using bilinear transformation [Reference Das and Vargheese9], a combination of the polynomial approximation and quadratic functions with the Rayleigh–Ritz [Reference Lin, Li, Yeo and Leong10], a combination of conformal mapping with the finite-element [Reference Yang and Lee11], a combination of conformal mapping with the finite-difference [Reference Chakrabarty, Sharma and Das1, Reference Das and Chakrabarty12, Reference Das, Chakrabarty and Mallick13], a combination of the fundamental solutions and particular solutions methods [Reference Fan, Young and Chiu14], a combination of the perturbation method with the separation of variables' technique followed by the well-known cosine and sine laws [Reference Kotsis and Roumeliotis3], and the separation of variables' technique in bipolar coordinate systems (BCSs) [Reference Gholizadeh, Baharian and Kashani15]. All these investigations have been concentrated on the evaluation of the higher-order modes and their cutoff frequencies without any dielectric support between the inner and outer conductors. In [Reference Dey, Agnihotri, Chakrabarty and Sharma16], eccentric circular metallic waveguide supported by dielectric slab between the inner and outer conductors has been investigated using the finite element approach. Propagation in composite cylindrical structures, composed of a bianisotropic cylinder embedded in an unbounded bianisotropic space and enclosing an array of parallel bianisotropic rods, has been studied in [Reference Vardiambasis, Tsalamengas and Kostogiannis17]. In [Reference Gholizadeh and Hojjat-Kashani18], higher order modes of two wire waveguides have been investigated using BCS and separation of variable technique.

This paper presents an analytical solution of higher order modes in a dielectric-lined eccentric circular metallic waveguide. In [Reference Gholizadeh, Baharian and Kashani15], the solution of the Helmholtz equation in BCS has been obtained using the technique of separation of variables and the validity range of the solution has been discussed. In this study, this solution has been applied to analyze the higher order modes of a dielectric-lined eccentric circular metallic waveguide. Rather than TE and TM modes, this structure also supports hybrid electric (HE) and hybrid magnetic (EH) modes which are similar to the transversal TE and TM modes of a homogeneously filled eccentric coaxial cable, except that the longitudinal electric and magnetic fields do not generally vanish. The cutoff frequencies of the higher order modes (TE, TM, HE, and EH modes in this case) have been determined by enforcing the boundary conditions and the continuity of the tangential electric- and magnetic-field components at the boundaries of the structure and an analytical expression is proposed for the electromagnetic fields. Moreover, it will be shown that this approach significantly works better than the method presented in [Reference Gholizadeh, Baharian and Kashani15] in calculating the cutoff frequencies of a homogeneously filled eccentric circular metallic waveguide. The paper is organized as follows. Section “Problem formulation and solution” presents the formulation of the problem. The obtained results are discussed in section “Results and discussion”. Finally, section “Conclusion” concludes this research.

Problem formulation and solution

Figure 1 depicts the geometry of the partially filled eccentric coaxial line under consideration. A hollow infinite perfect electric conductor (PEC) cylinder of radius R 1 (the outer conductor) in the z-direction eccentrically surrounds a dielectric cylinder (of radius R 2 with permittivity ɛ 2 = ɛ r2ɛ 0 and permeability μ 2 = μ r2μ 0) which itself eccentrically encloses another PEC cylinder of radius R 3 (the inner conductor). The remaining space inside the waveguide is filled by another dielectric material with permittivity ɛ 1 = ɛ r1ɛ 0 and permeability μ 1 = μ r1μ 0. The offset of the inner dielectric cylinder (region 2) and the PEC core are denoted D 1 and D 2, respectively. This type of waveguide can be easily described using BCS. In this paper, it has been supposed the circles in Fig. 1 lie to the right of the y-axis (0 ≤ ζ < +∞). The relations between the parameters of BCS and the dimensions of the waveguide can be written as follows:

(1)$$R_1 = \displaystyle{{a\;} \over {\sinh \lpar \zeta _1\rpar }}\comma \;\quad R_2 = \displaystyle{{a\;} \over {\sinh \lpar \zeta _2\rpar }}\comma \;\quad R_3 = \displaystyle{{a\;} \over {\sinh \lpar \zeta _3\rpar }}$$
(2)$$\;D_1 = a\lpar \coth \lpar \zeta _1\rpar -\coth \lpar \zeta _2\rpar \rpar \comma \;\quad D_2 = a\lpar \coth \lpar \zeta _1\rpar -\coth \lpar \zeta _3\rpar \rpar $$

where a is an arbitrary positive real number and 2a shows how far apart the poles (P1(a, 0) and P2(−a, 0)) of the BCS lie. It is obvious that ζ = ζ 1, ζ = ζ 2, and ζ = ζ 3 determine the boundaries of the waveguide in BCS, where ζ 1 < ζ 2 < ζ 3. The Helmholtz equation in BCS can be written as:

(3)$$\displaystyle{{\partial ^2\varphi \lpar {\zeta \;\comma \eta } \rpar } \over {\partial \zeta ^2}} + \displaystyle{{\partial ^2\varphi \lpar {\zeta \;\comma \eta } \rpar } \over {\partial \eta ^2}} + \displaystyle{{a^2k_c^2 \varphi \lpar {\zeta \;\comma \eta } \rpar } \over {{\lpar {\cosh \lpar \zeta \rpar -\cos \lpar \eta \rpar } \rpar }^2}} = 0.$$

and its solution is as follows:

(4)$$\eqalign{& \varphi \lpar \zeta \comma \;\eta \rpar = \varphi _\zeta \lpar \zeta \rpar \varphi _\eta \lpar \eta \rpar = \lpar A_1\sin \lpar n\eta \rpar + A_2\cos \lpar n\eta \rpar \rpar \cr & \quad \lpar B_1J_n\lpar 2ak_ce^{-\zeta }\rpar + B_2Y_n\lpar 2ak_ce^{-\zeta }\rpar {\rm \rpar \comma \;\ }\vert \zeta \vert \ge 3} $$

where φ represents the scalar function that illustrates the longitudinal component of the field, n is the azimuthal mode index, A 1, A 2, B 1, and B 2 are the amplitude coefficients, Jn(x) and Yn(x) are the Bessel functions of order n of first and second kinds, respectively, $k^2_c = k_{}^2 -\gamma ^2$, $k_{} = \omega \sqrt {\mu _{}\varepsilon _{}}$, and γ is the axial propagation constant [Reference Gholizadeh, Baharian and Kashani15]. For the structure shown in Fig. 1, |ζ| ≥ 3 means D 1/R 1 = 1 or D 2/R 1 = 1. In other words, (4) is valid for either D 1/R 1 = 1 and every arbitrary value of D 2/R 1, or D 2/R 1 = 1 and every arbitrary value of D 1/R 1. By suppressing the propagation term (e −γz) and the time–harmonic nature of the fields (e jωt), since the tangential electric-field components must vanish at the outer PEC boundary (ζ = ζ 1), the electric- and magnetic-field components in the z-direction in region 1 (ζ 1 ≤ ζ ≤ ζ 2) can be written as:

(5)$$E_{1z} = C_1F_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cos \lpar n\eta \rpar $$
(6)$$H_{1z} = C_2G_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \sin \lpar n\eta \rpar $$

where

(7)$$\eqalign{F_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar = \,& Y_n\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar J_n\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad -J_n\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar Y_n\lpar 2ak_{1\zeta }e^{-\zeta }\rpar } $$
(8)$$\eqalign{G_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar = \,& {{Y}{^\prime}_{\!\!\!n}}\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar J_n\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad -{{J}^{\prime}}_{\!\!n}\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar Y_n\lpar 2ak_{1\zeta }e^{-\zeta }\rpar } $$

Here, C 1 and C 2 are the amplitude coefficients, $k_{1\zeta } = k_1^2 -\gamma ^2$ and $k_1 = \omega \sqrt {\mu _1\varepsilon _1}$. Similarly, the solution in region 2 (ζ 2 ≤ ζ ≤ ζ 3) has the following form:

(9)$$E_{2z} = C_3F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cos \lpar n\eta \rpar $$
(10)$$H_{2z} = C_4G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \sin \lpar n\eta \rpar $$

where

(11)$$\eqalign{F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar = & Y_n\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar J_n\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad - J_n\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar Y_n\lpar 2ak_{2\zeta }e^{-\zeta }\rpar } $$
(12)$$\eqalign{G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar = & {{Y}^{\prime}}_{\!\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar J_n\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad -{{J}^{\prime}}_{\!\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar Y_n\lpar 2ak_{2\zeta }e^{-\zeta }\rpar } $$

Here, C 3 and C 4 are the amplitude coefficients, $k_{2\zeta } = k_2^2 -\gamma ^2$ and $k_2 = \omega \sqrt {\mu _2\varepsilon _2}$. The other field components in each region can be extracted from the following equations:

(13)$$E_\zeta = \displaystyle{{-\gamma } \over {hk_\zeta ^2 }}\left({\displaystyle{{\partial E_z} \over {\partial \zeta }} + \displaystyle{{\,j\omega \mu } \over \gamma }\displaystyle{{\partial H_z} \over {\partial \eta }}} \right)$$
(14)$$E_\eta = \displaystyle{{-\gamma } \over {hk_\zeta ^2 }}\left({\displaystyle{{\partial E_z} \over {\partial \eta }}-\displaystyle{{\,j\omega \mu } \over \gamma }\displaystyle{{\partial H_z} \over {\partial \zeta }}} \right)$$
(15)$$H_\zeta = \displaystyle{{-\gamma } \over {hk_\zeta ^2 }}\left({\displaystyle{{\partial H_z} \over {\partial \zeta }}-\displaystyle{{\,j\omega \varepsilon } \over \gamma }\displaystyle{{\partial E_z} \over {\partial \eta }}} \right)$$
(16)$$H_\eta = \displaystyle{{-\gamma } \over {hk_\zeta ^2 }}\left({\displaystyle{{\partial H_z} \over {\partial \eta }} + \displaystyle{{\,j\omega \varepsilon } \over \gamma }\displaystyle{{\partial E_z} \over {\partial \zeta }}} \right)$$

where h = a/(coshζ − cosη) is the transversal scale factor.

Fig. 1.

Transverse cross-section of the partially filled eccentric coaxial cable.

Therefore, we can write the following relations for each region:

(17)$$\eqalign{E_{1\zeta } = & \displaystyle{{-\gamma } \over {hk_{1\zeta }^2 }}\lpar {-}C_12ak_{1\zeta }e^{-\zeta }{{F}^{\prime}}_{\!\!1n} \lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad + C_2\displaystyle{{nj\omega \mu _1} \over \gamma }G_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \rpar \cos \lpar n\eta \rpar } $$
(18)$$\eqalign{E_{1\eta } = & \displaystyle{{-\gamma } \over {hk_{1\zeta }^2 }}\lpar {-}C_1nF_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad + C_2\displaystyle{{\,j\omega \mu _1} \over \gamma }2ak_{1\zeta }e^{-\zeta }{{G}^{\prime}}^{}_{\!\!1n} \lpar 2ak_{1\zeta }e^{-\zeta }\rpar \rpar \sin \lpar n\eta \rpar } $$
(19)$$\eqalign{E_{2\zeta } = & \displaystyle{{-\gamma } \over {hk_{2\zeta }^2 }}\lpar {-}C_32ak_{2\zeta }e^{-\zeta }{{F}^{\prime}}^{}_{\!\!2n} \lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad + C_4\displaystyle{{nj\omega \mu _2} \over \gamma }G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \rpar \cos \lpar n\eta \rpar } $$
(20)$$\eqalign{E_{2\eta } = & \displaystyle{{-\gamma } \over {hk_{2\zeta }^2 }}\lpar {-}C_3nF_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad + C_4\displaystyle{{\,j\omega \mu _2} \over \gamma }2ak_{2\zeta }e^{-\zeta }{{G}^{\prime}}^{}_{\!\!\!2n} \lpar 2ak_{2\zeta }e^{-\zeta }\rpar \rpar \sin \lpar n\eta \rpar } $$
(21)$$\eqalign{H_{1\zeta } = & \displaystyle{{-\gamma } \over {hk_{1\zeta }^2 }}\lpar {-}C_22ak_{1\zeta }e^{-\zeta }{{G}^{\prime}}_{\!\!1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad + C_1\displaystyle{{nj\omega \varepsilon _1} \over \gamma }F_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \rpar \sin \lpar n\eta \rpar } $$
(22)$$\eqalign{H_{1\eta } = & \displaystyle{{-\gamma } \over {hk_{1\zeta }^2 }}\lpar C_2nG_{1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad -C_1\displaystyle{{\,j\omega \varepsilon _1} \over \gamma }2ak_{1\zeta }e^{-\zeta }{{F}^{\prime}}^{}_{\!\!\!1n} \lpar 2ak_{1\zeta }e^{-\zeta }\rpar \rpar \cos \lpar n\eta \rpar } $$
(23)$$\eqalign{H_{2\zeta } = & \displaystyle{{-\gamma } \over {hk_{2\zeta }^2 }}\lpar {-}C_42ak_{2\zeta }e^{-\zeta }{{G}^{\prime}}_{\!\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad + C_3\displaystyle{{nj\omega \varepsilon _2} \over \gamma }F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \rpar \sin \lpar n\eta \rpar } $$
(24)$$\eqalign{H_{2\eta } = & \displaystyle{{-\gamma } \over {hk_{2\zeta }^2 }}\lpar C_4nG_{2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad -C_3\displaystyle{{\,j\omega \varepsilon _2} \over \gamma }2ak_{2\zeta }e^{-\zeta }{{F}^{\prime}}^{}_{\!\!\!2n} \lpar 2ak_{2\zeta }e^{-\zeta }\rpar \rpar \cos \lpar n\eta \rpar } $$

where

(25)$$\eqalign{{{F}^{\prime}}_{\!\!1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar = & Y_n\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar {{J}^{\prime}}_{\!\!n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad -J_n\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar {{Y}^{\prime}}_{\!\!n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar } $$
(26)$$\eqalign{{{G}^{\prime}}_{\!\!\!1n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar = & {{Y}^{\prime}}_{\!\!\!n}\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar {{J}^{\prime}}_{\!\!n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar \cr & \quad -{{J}^{\prime}}_{\!\!n}\lpar 2ak_{1\zeta }e^{-\zeta _1}\rpar {{Y}^{\prime}}_{\!\!n}\lpar 2ak_{1\zeta }e^{-\zeta }\rpar } $$
(27)$$\eqalign{{{F}^{\prime}}_{\!\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar = & Y_n\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar {{J}^{\prime}}_{\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad -J_n\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar {{Y}^{\prime}}_{\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar } $$
(28)$$\eqalign{{{G}^{\prime}}_{\!\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar = & {{Y}^{\prime}}_{\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar {{J}^{\prime}}_{\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar \cr & \quad -{{J}^{\prime}}_{\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta _3}\rpar {{Y}^{\prime}}_{\!\!n}\lpar 2ak_{2\zeta }e^{-\zeta }\rpar } $$

The continuity of the tangential electric- and magnetic-field components at ζ = ζ 2 relates the coefficients to one another as follows:

(29)$$C_1 = C_3\displaystyle{{F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar } \over {F_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar }}$$
(30)$$C_2 = C_4\displaystyle{{G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar } \over {G_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar }}$$
(31)$$\displaystyle{{C_4} \over {C_3}} = \displaystyle{{\lpar\! \gamma n/2j\omega ae^{-\zeta _2}\rpar \lpar 1/k_{2\zeta }^2 \rpar -\lpar 1/k_{1\zeta }^2 \rpar } \over \matrix{{\lpar \mu _2/k_{2\zeta }\rpar \lpar {{G}^{\prime}}_{\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar /F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar -\lpar \mu _1/k_{1\zeta }\rpar \lpar \lpar {{G}^{\prime}}_{\!\!1n}}\cr{\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar /\lpar G_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar \rpar }}}$$
(32)$$\displaystyle{C_3 \over C_4} = \displaystyle{\lpar \!\gamma n/2j\omega ae^{-\zeta _2}\rpar \,\lpar 1/k_{2\zeta }^2 \rpar -\lpar 1/k_{1\zeta }^2 \rpar \over \matrix{\lpar \varepsilon _2/k_{2\zeta }\rpar \lpar {F}^{\prime}_{\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar /G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar -\lpar \varepsilon _1/k_{1\zeta }\rpar \lpar \lpar {F}^{\prime}_{\!\!1n} \cr \lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar /\lpar F_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar \rpar }}$$

where (31) is used for HE modes and (32) is used for EH modes. Using (31) and (32), the coefficients C 3 and C 4 can be eliminated to obtain the following dispersion relation:

(33)$$QV = \left({\displaystyle{{\gamma n} \over {2\omega ae^{-\zeta_2}}}} \right)^2\left({\displaystyle{1 \over {k_{2\zeta }^2 }}-\displaystyle{1 \over {k_{1\zeta }^2 }}} \right)^2$$

where

(34)$$Q = \displaystyle{{\mu _2} \over {k_{2\zeta }}}\displaystyle{{{{G}^{\prime}}_{\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar } \over {F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar }}-\displaystyle{{\mu _1} \over {k_{1\zeta }}}\displaystyle{{{{G}^{\prime}}_{\!\!1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar } \over {G_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar }}$$
(35)$$V = \displaystyle{{\varepsilon _2} \over {k_{2\zeta }}}\displaystyle{{{{F}^{\prime}}_{\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar } \over {G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar }}-\displaystyle{{\varepsilon _1} \over {k_{1\zeta }}}\displaystyle{{{{F}^{\prime}}_{\!\!1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar } \over {F_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar }}$$

The characteristic equations of TE0m and TM0m modes can be obtained by setting n = 0 in (33) as follows:

(36)$$\eqalign{& F_{20}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \lpar \mu _2 k_{1\zeta } {{G}^{\prime}}_{\!\!20}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar G_{10}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar \cr & \quad -\mu _1 k_{2\zeta } {{G}^{\prime}}_{\!\!10}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar G_{20}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar = 0} $$
(37)$$\eqalign{& G_{20}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \lpar \varepsilon _2 k_{1\zeta } {{F}^{\prime}}_{\!\!20}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar F_{10}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar \cr & \quad -\varepsilon _1 k_{2\zeta } {{F}^{\prime}}_{\!\!10}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar F_{20}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar = 0} $$

where the roots of (36) and (37) represent the cutoff frequencies of the TE0m and TM0m modes, respectively.

For hybrid modes (n ≠ 0 and at cutoff (γ = 0)), the right-hand side of the dispersion relation (33) becomes zero and the HE and EH modes are decoupled. This results in the following characteristic equations for hybrid modes:

(38)$$\eqalign{& F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \lpar \mu _2 k_{1\zeta } {{G}^{\prime}}_{\!\!2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar G_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar \cr & \quad -\mu _1 k_{2\zeta } {{G}^{\prime}}_{\!\!1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar = 0\comma \;\quad n\ne 0} $$
(39)$$\eqalign{& G_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \lpar \varepsilon _2 k_{1\zeta } {{F}^{\prime}}_{\!\! 2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar F_{1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar \cr & \quad -\varepsilon _1 k_{2\zeta }{{F}^{\prime}}_{\!\! 1n}\lpar 2ak_{1\zeta }e^{-\zeta _2}\rpar F_{2n}\lpar 2ak_{2\zeta }e^{-\zeta _2}\rpar \rpar = 0\comma \;\quad n\ne 0} $$

where the roots of (38) and (39) represent the cutoff frequencies of the HE and EH modes, respectively.

Results and discussion

To check the accuracy of the calculations, two special cases have been investigated. The first corresponds to partially filled coaxial cable. To achieve this, D 1, D 2 → 0 must be considered. The cutoff frequencies can be calculated for different values of R 1, R 2, and R 3. For these calculations, we can put D 1/R 1 = 10−10 and D 2/R 1 = 10−15. Comparison with the exact results reported in [Reference Lewis and Kharadly19] gives agreement to seven significant digits. The second case is related to a homogeneously filled eccentric coaxial cable, where the fields inside the region are either TE or TM. Such a case can be achieved by considering ɛ 1 = ɛ 2 = ɛ. In [Reference Gholizadeh, Baharian and Kashani15], it was mentioned that the solution is valid for D 2/R 1 ≤ 0.05. However, in this study, we can obtain the results for any arbitrary values of D 2/R 1 in homogeneously filled eccentric coaxial cable (by considering ɛ 1 = ɛ 2 = ɛ). Because, there are more parameters in the analysis of a partially filled eccentric coaxial cable than a homogeneously filled eccentric coaxial cable which can satisfy the main limiting condition in BCS (cosh (ζ) ≫ 1 or |ζ| ≥ 3), without limiting the values for D 2/R 1. In other words, we can keep D 1/R 1 negligible instead of D 2/R 1 so that the values of ζ remain larger than 3. To show this, in Table 1, the cutoff wavenumbers (k nm) of the homogeneously filled eccentric coaxial cable are calculated for several different values of D 2/R 1 and R 3/R 1 using our method, and the results are compared with those given in [Reference Das and Vargheese9] and [Reference Gholizadeh, Baharian and Kashani15], and also their simulation values. For these calculation we put D 1/R 1 = 10−24. Full-wave frequency-domain simulations (using CST Microwave Studio) have been used for the calculations, where 6 743 881 elements in the mesh were used. As it can be found from Table 1, one of the substantial features of our method is that it remains valid even for larger values of D 2/R 1, where the methods used in [Reference Das and Vargheese9] and [Reference Gholizadeh, Baharian and Kashani15] have shown a significant weakness. It is noteworthy to mention again that the limiting condition for the method used in [Reference Gholizadeh, Baharian and Kashani15] is D 2/R 1 = 1. Moreover, the method used in [Reference Das and Vargheese9] is based on transforming eccentric coaxial into coaxial configuration using bilinear transformation expressed in terms of mutually inverse points and gives acceptable results as long as the values of ${\rho }^{\prime}_{\!\!1}/R_3$ and ${\rho }^{\prime}_{\!\! 2}/R_3$ are negligible, where ${\rho }^{\prime}_{\!\! 1}$ and ${\rho }^{\prime}_{\!\! 2}$ are the radius of the inner and outer cylinders of the transformed coaxial configuration, respectively. However, as D 2/R 1 increases the values of ${\rho }^{\prime}_{\! \! 1}$ and ${\rho }^{\prime}_{\! \! 2}$ become comparable with R 3, which brings about large errors in the final results for large values of D 2/R 1.

Table 1.

Cutoff wave numbers (k nm) of the homogeneously filled eccentric coaxial cable (by considering ɛ r1 = ɛ r2 = 1)

Comparison with [Reference Das and Vargheese9], [Reference Gholizadeh, Baharian and Kashani15] and simulation results (R 2/R 1 = 0.6, D 1/R 1 = 10−24).

In Table 2, the cutoff frequencies of a dielectric-lined eccentric coaxial cable are calculated for a number of the higher order modes (TE, TM, HE, and EH) using our method, and the results are compared with their simulation values. The good agreement between the data justifies the validity of the analysis. Figure 2 illustrates the variations of the cutoff frequencies of HE11-mode versus the different physical and geometrical parameters of the problem, where in Figs 2(a)–(e), the liner is considered to be on the outer conductor (ɛ r2 = 1), and in Figs 2(f)–2(j) it is put on the inner conductor (ɛ r1 = 1). As one can see, for the both cases, even though the dielectric liner only occupies a small portion of the total cross-section of the waveguide, it is to be expected that a thin epsilon-positive liner with permittivity larger than unity slightly lowers the natural cutoff frequency, and reversely, that an epsilon-positive liner with permittivity smaller than unity will increase it. In other words, the cutoff frequency is to a low extent dependent on large positive permittivity values whereas it increases significantly for small ones, suggesting that the waveguide is thrust more deeply into cutoff as the liner permittivity is positive and tending to zero (Figs 2(a) and 2(f)). Besides, the cutoff frequency is to a high extent dependent on small eccentricities (D 1/R 1, D 2/R 1 < 0.05) while it almost remains stable for larger ones (Figs 2(b), 2(c), 2(g) and 2(h)). It is noteworthy to mention that the variation of eccentricity has its maximum effect on the cutoff frequency when the liner is on the outer conductor and D 2/R 1 is changing (Fig. 2(b)). Moreover, the cutoff frequency can be decreased by considering either thicker liner (Figs 2(d) and 2(i)) or the inner conductor with larger radius (Figs 2(e) and 2(j)). It should be mentioned that in Fig. 2(j), we see gradual decline in cutoff frequency, because as R 3/R 1 increases, the liner thickness decreases accordingly. Figure 2 may be employed in choosing the value of permittivity and dimensions required to achieve a desired cutoff frequency of HE11 mode in a partially filled eccentric coaxial cable.

Fig. 2.

HE11-mode cutoff frequency versus the different physical and geometrical parameters of the problem. (a) ɛ r1 (D 2/R 1 = 0.2, D 1/R 1 = 0.0001, R 2/R 1 = 0.9, R 3/R 1 = 0.3, ɛ r2 = 1), (b) D 2 (D 1/R 1 = 0.0001, R 2/R 1 = 0.9, R 3/R 1 = 0.3, ɛ r1 = 3.6, ɛ r2 = 1), (c) D 1 (D 2/R 1 = 0.0001, R 2/R 1 = 0.55, R 3/R 1 = 0.1, ɛ r1 = 3.6, ɛ r2 = 1), (d) R 2 (D 1/R 1 = 0.0001, D 2/R 1 = 0.1, R 3/R 1 = 0.1, ɛ r1 = 3.6, ɛ r2 = 1), (e) R 3 (D 1/R 1 = 0.0001, D 2/R 1 = 0.1, R 2/R 1 = 0.9, ɛ r1 = 3.6, ɛ r2 = 1), (f) ɛ r2(D 2/R 1 = 0.01, D 1/R 1 = 0.0001, R 2/R 1 = 0.45, R 3/R 1 = 0.3, ɛ r1 = 1), (g) D 2 (D 1/R 1 = 0.0001, R 2/R 1 = 0.8, R 3/R 1 = 0.3, ɛ r1 = 1, ɛ r2 = 3.6), (h) D 1 (D 2/R 1 = 0.0001, R 2/R 1 = 0.55, R 3/R 1 = 0.1, ɛ r1 = 1, ɛ r2 = 3.6), (i) R 2 (D 1/R 1 = 0.0001, D 2/R 1 = 0.1, R 3/R 1 = 0.1, ɛ r1 = 1, ɛ r2 = 3.6), (j) R 3 (D 1/R 1 = 0.0001, D 2/R 1 = 0.1, R 2/R 1 = 0.9, ɛ r1 = 1, ɛ r2 = 3.6).

Table 2.

Cutoff frequencies (GHz) of the dielectric-lined eccentric coaxial cable

Comparison with simulation results.

Conclusion

The problem has been investigated in a fully analytical manner and the analytical expressions have been obtained for the electric and magnetic field functions and the cutoff frequencies. The presented method gives accurate results for either D 1/R 1 = 1 and every arbitrary value of D 2/R 1, or D 2/R 1 = 1 and every arbitrary value of D 1/R 1. An excellent agreement has been observed between the calculated cutoff frequencies with those obtained by full-wave simulations. The combination of accuracy, analyticity and ease of implementation makes this method an appropriate candidate for the analysis of eccentric coaxial line structures. Moreover, it has been shown this method significantly works better than the method presented in [Reference Gholizadeh, Baharian and Kashani15] in calculating the cut off frequencies of a homogeneously filled eccentric circular metallic waveguide.

M. Gholizadeh was born in Ahvaz, Iran, in 1991. He received a degree in electrical engineering from the Faculty of Engineering, Shahid Chamran University, Ahvaz, in 2015. He is currently pursuing his master's degree in electrical engineering-fields and waves at the Iran University of Science and Technology, Tehran, Iran. His current research interests include fundamental electromagnetic theory, electromagnetic waves propagation, electromagnetic scattering, microwave circuits, microwave transmission lines, and cavities.

F. H. Kashani was born in Mashhad, Iran, in 1941. He received his bachelor's degree in electrical engineering from the Faculty of Engineering, University of Tehran, Tehran, Iran, in 1963, and his master's and Ph.D. degrees in electronics from the University of California at Los Angeles (UCLA), Los Angeles, CA, USA, in 1969 and 1971, respectively. He is currently a Professor of electrical engineering with the Iran University of Science and Technology, Tehran. His master's and Ph.D. research area was dispersion of waves and measuring the picosecond waves. He has taught telecommunication courses at UCLA and The University of Sydney, Sydney, NSW, Australia, for some time. After returning to Iran, he has spent many years teaching the undergraduate, postgraduate, and Ph.D. courses at the Sharif University of Technology, Tehran, University of Science and Technology, and Islamic Azad University, Tehran. He has also supervised many students in telecommunications and electronics.

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Figure 0

Fig. 1. Transverse cross-section of the partially filled eccentric coaxial cable.

Figure 1

Table 1. Cutoff wave numbers (knm) of the homogeneously filled eccentric coaxial cable (by considering ɛr1 = ɛr2 = 1)

Figure 2

Fig. 2. HE11-mode cutoff frequency versus the different physical and geometrical parameters of the problem. (a) ɛr1 (D2/R1 = 0.2, D1/R1 = 0.0001, R2/R1 = 0.9, R3/R1 = 0.3, ɛr2 = 1), (b) D2 (D1/R1 = 0.0001, R2/R1 = 0.9, R3/R1 = 0.3, ɛr1 = 3.6, ɛr2 = 1), (c) D1 (D2/R1 = 0.0001, R2/R1 = 0.55, R3/R1 = 0.1, ɛr1 = 3.6, ɛr2 = 1), (d) R2 (D1/R1 = 0.0001, D2/R1 = 0.1, R3/R1 = 0.1, ɛr1 = 3.6, ɛr2 = 1), (e) R3 (D1/R1 = 0.0001, D2/R1 = 0.1, R2/R1 = 0.9, ɛr1 = 3.6, ɛr2 = 1), (f) ɛr2(D2/R1 = 0.01, D1/R1 = 0.0001, R2/R1 = 0.45, R3/R1 = 0.3, ɛr1 = 1), (g) D2 (D1/R1 = 0.0001, R2/R1 = 0.8, R3/R1 = 0.3, ɛr1 = 1, ɛr2 = 3.6), (h) D1 (D2/R1 = 0.0001, R2/R1 = 0.55, R3/R1 = 0.1, ɛr1 = 1, ɛr2 = 3.6), (i) R2 (D1/R1 = 0.0001, D2/R1 = 0.1, R3/R1 = 0.1, ɛr1 = 1, ɛr2 = 3.6), (j) R3 (D1/R1 = 0.0001, D2/R1 = 0.1, R2/R1 = 0.9, ɛr1 = 1, ɛr2 = 3.6).

Figure 3

Table 2. Cutoff frequencies (GHz) of the dielectric-lined eccentric coaxial cable