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: [email protected]

Nadia Benabdallah

Physics Engineer

(e-mail: [email protected])

Nasreddine Benahmed

Professor of Communications Systems

(e-mail: [email protected])

Yamina Bekri

Telecommunications Engineer

(e-mail: [email protected])

University of Tlemcen, Universit 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* _{t}V(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 ??e

_{0}e

*(3)*

_{r}E_{t}.E_{t}^{*}dxdywhere 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:

C = | 4W_{em} | (F/m) (4) |

(V_{1} - V_{2})^{2} |

where V_{1} and V_{2} = the fixed potential on the conductors.

The characteristic impedance, Z_{0}, can be calculated by applying Eq. 5:

Z_{0} = | 1 | (Ω) (5) |

v_{φ}C |

Where:

v_{φ} = | 3.10^{8} | (m/s) (A) |

√e_{r} |

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When the even- and odd-mode characteristics impedances (Z_{oe} and Z_{oo}, respectively) for the STC are known, the coupling coefficient (k) can be found from Eq. 6:

k = | Z_{oe} - Z_{oo} | (6) |

Z_{oe} + Z_{oo} |

On the basis of the previous theory as applied to the shielded bandline analysis, the authors created a computer-aided-design (CAD) program based on the FEM to show the effects of thickness on the even- and odd-mode impedances (Z_{oe} and Z_{oo}, respectively) and on the coupling coefficient (k) of the STC.

Fig. 2 shows the even-mode characteristic impedance (Z_{oe}) as a function of the window angle, , with outer-toinner- conductor ratio (r_{b}/r_{o}) as a variable parameter and for various values of thickness-to-inner-conductor ratio (t/r_{o}). Fig. 3 shows the influence of thickness on the odd-mode characteristic impedance for different values of window angles and outer-to-innerconductor ratios.

The FEM analysis approach was also used to determine the effects of the thickness on the coupling coefficient for the STC, with the results shown in Fig. 4. It is apparent from these plots that the thickness has considerable effect on the even- and odd-mode characteristic impedances and consequently on the coupling coefficient (k) of the STC, which exhibits coupling that varies between 1.2 and 26 dB.

The results of the FEM analysis were used to design and build a 10-dB slotted-tube directional coupler. All four ports of the coupler (Fig. 5) were matched to 50 Ω. Fig. 6 shows plots of dielectric constant ( εr) as a function of the window angle (θ) with outer-to-inner-conductor ratio (r_{b}/r_{o}) as a parameter and for different values of thickness-to-inner-conductor ratio (t/r_{o}). The fixed parameters of the 10- dB slotted-tube directional coupler include a characteristic impedance of Z_{c} = 50 Ω and an operating frequency of 2 GHz. The features of the coupler obtained from the analysis results include a dielectric constant of 4, an outer-to-inner-conductor ratio (r_{b}/r_{o}) of 3, a thickness-to-inner-conductor ratio (t/r_{o}) of 0.1, a window angle of 80 deg., a coupler length of 18.75 mm, even- and odd- mode characteristic impedances, Z_{oe} and Z_{oo}, respectively, of 67.36 and 35.54 Ω.

Using an adapted numerical model,^{4} the resulting scattering parameters (with respect to 50 Ω) were plotted from 0.2 to 3.9 GHz in Fig. 7. The results indicate the desired 10-dB coupling occurring from 1.5 to 2.5 GHz, with minimum directivity of 32 dB. The results shown for the directional coupler take into account the effects of conductor thickness on the coupler's EM behavior, and show the effectiveness of the authors' FEM-based CAD program.

REFERENCES

- N. Ben Ahmed and M. Feham, "Analyzing EM parameters for shielded bandline," Microwaves & RF, vol. 45, No. 3, March 2006, pp.86-92.
- N. Ben Ahmed and M. Feham, "Finite element analysis of RF couplers with sliced coaxial cable," Microwave Journal, vol. 43, No. 11, November 2000, pp 106-120.
- N. Ben Ahmed, M. Feham and M. Kameche, "Finite element analysis of planar couplers," Applied Microwave & Wireless, vol. 12, No. 10, October 2000, pp 28-38.
- A.R. Djordjevic, M. Bazdar, G. Vitosevic, T. Sarkar and RF. Harrington, Scattering parameters of microwave networks with multiconductor transmission lines, Artech House, Norwood, MA, 1990.