Modal Analysis of Directional Coupler and Its Equivalent Circuit

Modal analysis for the directional coupler is presented in this paper. The analysis is based on the mode-matching technique. The theory is verified by Ansoft HFSS software at a directional coupler with standard X-band (WR 90) rectangular waveguide ports. Element values of equivalent circuit model are computed by using the S-parameters obtained from the presented method in this paper. In addition, this paper has corrected two formulations used in two previous works which have been published.


Introduction
Duo to its equal power splitting, high isolation, low VSWR and accurate 90 phasing, the directional coupler [1] is a very attractive microwave circuit element for many applications such as in the duplexer, balance mixers, broad band-band switches, and H-plane magic tee [2,3]. The structure of a narrow wall directional coupler is shown in Fig. 1, the two waveguides are coupled through the slot in the narrow wall. The key to designing the coupler is to obtain the width of the slot and the width of the coupling region [4,5], so a large number of directional coupler simulated to find out the best one which realizes a -3dB coupler. This process is a time-consuming course. Therefore fast computing method is needed to analyze the transmission characteristic of the directional coupler. This paper uses the mode-matching technique directly to design the directional coupler. The design method is based on the field expansion into the normalized incident and scattered waves [6]- [9]. The theory yields the modal S-matrix of the directional coupler directly. There are a few works which have obtained equivalent circuit of the waveguide directional coupler with a large aperture. Hirako et al. proposed a new approach which uses a transformer to the equivalent circuit for this structure [10], but in this paper, capacity and inductor are used instead of the transformer. In this paper is presented an accurate circuit model for the structure. The topology of the model is the same as that introduced by Marcuvitz [11]. This method calculates the admittance matrix of the directional coupler from its scattering matrix that is obtained by the modematching technique.

Modal analysis
The geometry of the narrow wall directional coupler is shown in Fig. 1, the two waveguides are coupled through the slot in the narrow wall. The directional coupler is decomposed into five regions ( Fig. 1(b)), and discontinuity change only in waveguide width. The total scattering matrix of the directional coupler is formulated by suitable direct combination of the related individual modal scattering matrices. Because there is no y variation introduced by this discontinuity, a #$ % wave incident in the main waveguide 1excites only &$ % waves. Therefore, for each subregion v= I, II, III, IV, and V, the fields [12]: Are derived from the z component of the Hertzian vector potential () * , which is assumed to be a sum of suitably normalized eigenmodes satisfying the corresponding boundary conditions: has been changed from negative to positive. The field expansion equations obtained from (1) and (2) in subregions 1, 2 and 4 are: Matching of the tangential field components at z=z1 are: When the same procedure is done at z=z2, equations (10)-(12) will be obtain: In all of the above relations, n=1, 2, …N. By solving equations (7)-(12) for N=K=L using Matlab software, the amplitudes of modes are obtained. Scattering matrix S c can be obtained by using the method proposed in [8] according to the amplitude of modes. Then the overall scattering matrix can be obtained by the direct combination of all the local scattering matrixes S c and S w . Where S w is the scattering matrix of waveguide I, III, IV and V, and can be written as: (13) where L is the distance of discontinuity to the port of waveguide.

Numerical results
For the verification of the theory, the scattering parameters of the directional coupler with the parameters of a=22.9mm, t=1mm, and z2-z1=36mm are calculated and compared with Ansoft HFSS software (Fig. 2). The results agree well with that obtained by MMM (mode-matching method) in this paper. According to the obtained results, there is a good convergence for N=4 with 0.3 percent error. Phase characteristic of the directional coupler is shown in Fig. 3 that confirm the precise of 90° phasing.

Equivalent circuit
The circuit model of the directional coupler with admittance parameters is shown in Fig. 4. For deriving an equivalentcircuit representation, the admittance matrix (Y) of the structure under consideration is computed by: with the normalized admittance matrix:

Conclusion
A mode-matching method is presented to the modal analysis of the directional coupler. The explained theory is confirmed at scattering parameters of the directional coupler by HFSS software. Furthermore, in this paper, a novel equivalent circuit was presented that its element values are computed by using the S-parameters obtained from the presented method in this paper.