79INTL JOURNAL OF ELECTRONICS AND TELECOMMUNICATIONS, 2017, VOL. 63, NO. 4, PP.375-380 Manuscript received January 4, 2017; revised September, 2017. DOI: 10.1515/eletel-2017-0051 Design and Optimization of Rectangular Waveguide Filter based on Direct Coupled Resonators Mehdi Damou, Keltouma Nouri, Mohammed Feham and Mohammed Chetioui Abstract—The waveguide filter structure is treated by two The coupling matrix is used to represent the coupled resonator softwares (HFSS (High Frequency Structure Simulator) and CST circuit. Each matrix entry value refers to a physical dimension (Computer Simulation Technology)). Numerical example is given of the circuit. The response of the circuit can also be calculated in this article to demonstrate, step by step, the application of the of the circuit. The response of the circuit can also be calculated approach to the design of resonator, direct coupled waveguide and microstrip filters based on electromagnetic (EM) simulations. For by using the coupling matrix. Different methods are developed this design procedure, the filter structure is simulated by to generate the coupling matrix. This work presents design of successively adding one resonator at a time. To continue the work the coupled resonator based filter by using the coupling matrix illustrates how to design a fourth order coupled resonator based local optimization technique. The initial values of the matrix, rectangular waveguide circuit in the traditional way. With a large which is used as the input of the optimizer, have a great effect number of variables, such tuning work consumes a lot of time and on the convergence of the final result of the optimization. A the convergence of the final result is not guaranteed. A fourth order X-band bandpass filter with a center frequency of 11 GHz proposed tuning technique, called Step Tune method, is also and a fractional bandwidth FBW = 0,0273 is designed using this presented in this work. Instead of conventionally tuning the procedure and presented here as an example. The simulated results whole structure, we simulate and tune just part of the circuit by by CST are presented and compared withthe results simulated by using the new method. As only limited number of physical a high-frequency structure simulator. Good agreement between dimensions is tuned each time, the final result is more reliable. the simulated HFSSand simulated results by CST is observed. In this article, we present an EM-based design approach for Keywords—Coupling Matrix, Filter, Waveguide, technology, determining the physical dimensions of a coupled waveguide Electromagnetic, Bandpass Filter filter with any type of topology. A design case of direct coupled waveguide filter will be discussed. Extractions for physical I. INTRODUCTION dimension by external quality factors and coupling coefficients are presented during the design procedures. Finally a proposed microwave filter is a two-port network used to transmit optimization technique, called Step Tune method has been A and attenuate signals in specified frequency bands. Traditionally, the design methods for direct coupled filters have developed, a design case of a direct-coupled waveguide filter been applied to extract the dimensions for direct coupled filters. will be presented in the end of this article. This design process usually involves the following four main II. DESCRIPTION OF THE DESIGN steps: (i) identify the filter order and filter functions according to specification requirements; (ii) synthesis or optimise the The method, which is based on electromagnetic (EM) coupling coefficients (Mi,j) and external quality factors (Qe) simulation, will be called the step tune method. Instead of that can realize the desired filter function; (iii) choose the filter traditionally altering all the parameters of the circuit in each type (waveguide, microstrip, etc), and obtain dimensions which optimizing iteration, the step-tune method simulates only one can achieve desired specified Qe and Mi,j from electromagnetic resonator of the device in the first step. When finishing tuning (EM) simulations on one resonator and two weakly coupled the first resonator, one more resonator is added and then the resonators; (iv) construct the filter in the simulator to get its circuit is tuned or optimized again. More resonators are added initial responses [1, 2]. Coupled resonator circuits are of successively to tuning at each step. For each step, a new importance for design of RF/microwave filters with any type of coupling matrix is required for the tuning. As limited number of resonator regardless its physical structure. physical dimensions needs to optimize in each step, the optimizing process works more efficiently and generates more reliable solutions. The key point of this method is to calculate the S parameters in each step and apply the responses as the This work was co-financed by LTC. Damou Mehdi, Nouri Keltouma and Chetioui Mohammed are with objective ones for the physical optimizing. For this design Laboratory Technology of Communication, Faculty of Technology University method, the middle stage S-parameter responses are calculated Tahar Moulay of Saida, Algeria ([email protected]; keltoum_nouri@ from their corresponding coupling coefficients, and act as the yahoo.fr). Feham Mohammed Head of STIC Laboratory, Laboratory Systems and objective responses for the tuning. To plot the desired responses Technologies of Information and Communication University Tlemcen, Algeria at each stage, the inner coupling coefficient needs to be ([email protected]). converted into external quality factor. For instance, at Step 1, 376 M. DAMOU, K. NOURI, M. FEHAM, M.CHETIOUI 풬푒2 should be calculated from 푀12. After expressing both the ➢ Lumped LC elements of resonators [5]: external quality factor (풬푒) and internal coupling coefficients (푀푖,푖+1) using inverter value K [2], the relationship between 푀 and 풬 can be found as: 푍01=푍45 =Z(ohm) (9) 푖,푖+1 푒 푍 푍 푍12=푍34 = , 푍23= 푀12푄푒 푀23푄푒 2 1 푀푖,푖+1. 풬푒푖 = 푛휋 휆 2 (1) ( 푔) 퐹퐵푊 2 휆 IV. APPLICATION OF THE DESIGN METHOD Where 휆 is the guided wavelength of the resonant frequency To demonstrate the technique for using the coupling matrix 푔 filter circuit model and an EM simulator to synthesize the and λ is the free-space wavelength, n is the number of half- physical dimensions of the filter, we consider the design of the wavelengths of the waveguide resonator cavity. proposed four-poles filter which. The Figure 1 illustrates the The coupling coefficients between external ports and inner topology and the structure of this filter. resonators can be calculated by 1 푀푖,푝2=√ 풬푒푖,푝2 (2) i refer to the resonator number connecting to the output ports (푝2). Substituting (1) into (2), we have 푛휋 휆 푀 =√ 퐹퐵푊 ( 푔)푀 푖,푝2 2 휆 푖,푖+1 (3) Where 푀푖,푝2is the equivalent external coupling coefficient of the internal coupling iris. HYSICAL OF OUPLING ATRIX III. P C M After determining the normalized coupling matrix [m] for a coupled resonator topology, the actual coupling matrix [M] of a Fig. 1. Illustration of a four order X-band. These four resonators are operating at TE101 mode All the irises have the same thickness t of 2 mm, a = 22.86 mm, coupled resonator device with given specification can be b = 10.16 mm calculated by prototype de-normalization of the matrix [m] at a desired bandwidth, as follows [3]: The filter is a BPF at the center frequency 11GHz, its pass band is 300MHz (2.7272%) and reflection loss of 20 dB at the (4) Mi,i+1 mi,i+1 .FBW For i=1 to n-1 passband, the out-of-band rejection is 45dB. These specifications can be achieved by Chebyshev response, since The actual external quality factor Qe is related to the there is no transmission zeros in the response. With Chebyshev response, [6] for the desired bandwidth, the ripple in the pass normalized quality factor qe by [3]: band and the rejection, we found that the filter must be of the (5) qe forth degree at least. The n× 푛 coupling matrix cannot be used Qe FBW to derive the responses [8]. An n+2 coupling matrix [푀]푛+2 is The coupling elements M and external quality factors ( applied as i, i+1 Q , Q ) are related to the lumped element lowpass prototype 0 −0.0282 0 0 0 0 e1 en −0.0282 0 0.0249 0 0 0 elements푔 ,푔 ,푔 ….푔 as follows [4]: 0 0.0249 0 −0.0191 0 0 0 1 2 푛+1 푀 = 0 0 −0.0191 0 0.0249 0 0 0 0 0.0249 0 −0.0282 푔0푔1 푔푛푔푛+1 푄 = , 푄 = [ 0 0 0 0 −0.0282 0 ] 푒1 퐹퐵푊 푒푛 퐹퐵푊 (6) Qe1 Qe2 Qe 34.21 퐹퐵푊 푀푖,푗+1 = for i=1 to n-1 √푔푖푔푖+1 The circuit parameters can be related to the bandpass filter V. STRATEGIC OF THE STEPS design parameters by the following equations: The process of tuning the 4th order filter can be divided into four steps. For convenience the selected steps can be ➢ Lumped LC elements of resonators [5]: represented by means of the pertinent diagram, where each 푄푒 12 (7) circle represents a resonator and 푀 describe the coupling 퐶0 = × 10 (pF) 푖,푖+1 휔0푍 between resonators. The calculation of the physical dimensions 푍 9 퐿0 = × 10 (nH) (8) 휔0푄푒 for the fourth order pass-band filter shown in Figure 1comprises DESIGN AND OPTIMIZATION OF RECTANGULAR WAVEGUIDE FILTER BASED ON DIRECT COUPLED RESONATORS 377 the following steps. In this way, the four stages selected in this obtained in step 3, since resonator 4 has a negligible impact on design are represented schematically in figure 2. resonator 1. 1. Calculate the approximate initial dimensions for all the resonators and irises using the equivalent circuit models based V.1 First Step: on the coupling matrix as described in [1, 6, 7] The first step in the design process takes exclusively into 2.