This numerical method accounts for thickness effects in determining the coupling coefficient as well as the even-and odd-mode impedances for slotted tube couplers.

Abdelhafid Lallam
Assistant Professor of Physics
University of Mascara, BP 305 Route de Mamounia
29000 Mascara, Algeria; (045) 80-41-68/69, FAX: (045) 80-41-64, e-mail: ab_lallam@yahoo.fr

Nadia Benabdallah
Physics Engineer
(e-mail: n_benabdallah@yahoo.fr)

Nasreddine Benahmed
Professor of Communications Systems
(e-mail: n_benahmed@yahoo.fr)

Yamina Bekri
Telecommunications Engineer
(e-mail: mina_souf1@yahoo.fr)
University of Tlemcen, Université Abou Bekr Belkaid-Tlemcen, B.P. 119, (13000) Tlemcen, Algeria

Analysis of high-frequency structures by means of the finite-element method (FEM) can reveal a great deal about the expected electromagnetic (EM) behavior of those structures. As the authors have already shown just two years ago for shielded bandline structures, ¹ FEM studies can show how the EM behavior of a high-frequency structure can be affected by physical parameters, such as conductor thickness. To continue those studies, the authors have performed an analysis on the effects of conductor thickness on the EM parameters of slotted-tube couplers (STCs) using the FEM.

Fig. 1 shows a schematic representation of an STC. For the purposes of this analysis, the coupler is assumed to be lossless. It is filled with dielectric material having a relative dielectric constant of e_r. The cross-sectional view of the coupler shows that it has an inner conductor radius r_o, thickness t, outer shield radius r_b, and window angle θ.

Electrically, the STC can be described in terms of its even-mode (Z_oe) and odd-mode (Z_oo) impedances and its coupling coefficient, k. A study of the structure shown in Fig. 1 is based on resolution of Laplace’s equation in two dimensions for the even-mode and odd-mode behavior as shown in Eq. 1:

div [?_tV(x,y)] = 0 (1)

where the subscript t indicates the cross section of the structure.

For the even mode, V = 1 V on the two conductors and V = 0 on the shield. For the odd mode, V = 1 V on one conductor and V = -1 V on the other conductor, while V = 0 on the shield. Solution of the equation can be found by applying the FEM, a general but powerful method of analysis.^{2, 3}

The solution of this equation represents the distribution of potential V in the structure. When potential V is known, it is possible to calculate the even- and odd-mode characteristic impedances and the coupling coefficient for the STC. Applying the lossless line theory makes it possible to determine a number of unknowns from potential V: the electric field, the magnetic field, and the electrical energy accumulated in the coupler structure, Wem. All the characteristic impedances for the two transverse-electromagnetic (TEM) modes can then be found from the energy W_em. Consequently it is important that potential V be found with high precision.^2,3 The procedure for studying a given TEM mode is fairly straighforward, based on the next few equations. For example, the electrical field can be found by simple derivation from the potential V by using Eq. 2:

E^t = -?_t V(x, y) (2)

The electrical energy accumulated in the structure can be deduced from the electrical field by applying Eq. 3:

W_em = 1/4 ??e₀e_rE_t.E_t^*dxdy (3)

where the asterisk indicates the conjugate vector.

Equation 4 describes how to deduce the capacitance per unit length (in F/m) directly from the electrical energy:

where V₁ and V₂ = the fixed potential on the conductors.

The characteristic impedance, Z₀, can be calculated by applying Eq. 5:

Where: