Application Note
Fast Design of Crosscoupled Filter Substructures
Crosscoupled narrow bandpass filters with asymmetric transmission zeros can be realized in cascaded trisection topology.1,2,3 Each trisection creates a single transmission zero below or above the passband. The realization of resonator trisections is relatively difficult due to the interaction between the couplings. This article shows how threedimensional electromagnetic (3DEM) analysis, together with extraction of an equivalent circuit model, can speed up the realization process of cascaded trisection bandpass filters.
Dieter Pelz
RFS Australia
Kilsyth, Victoria, Australia
In the realization stage of a narrow bandpass filter design, a given set of coupling values (coupling matrix) must be translated into the physical domain in the form of coupling apertures, loop or probecoupling dimensions. A special difficulty exists in resonator trisections where coupling exists between all three resonators (Figure 1 ). At the same time, trisections offer unique features as they can realize a single finite frequency transmission zero, located either below or above the filter's passband. The normalized impedance matrix of a narrowband trisection can be written as
(1)
where
and the X_{v} are constant reactances representing the offset of each resonator from the filter's nominal center frequency. Typical response curves of a trisection are given in Figure 2 . Cascaded trisection filters are attractive because they allow a quasiindependent fineadjustment of transmission zeros. However, given the usually large diameters of the high Q resonator cavities (either TEM or dielectric resonators), the size of the coupling apertures tends to be very large, so that there is only a relatively small residual center "post" which is common to all resonators. Changes in the dimensions of this central post affect all coupling factors and hence the design becomes quite difficult. 3DEM analysis can analyze such filter structures accurately. At the same time, there are very simple and accurate equivalent circuits which can be matched to a given twoport Smatrix obtained from 3DEM analysis, as shown in Figure 3 .


Fig. 1 Threedimensional EM model of a resonator trisection that is part of a multiresonator filter.  Fig. 2 Typical response curves of a narrowband resonator trisection filter substructure. 
The Process
The fundamental principle of the process is that a 3DEM simulationgenerated Smatrix contains all the information about a suitable associated equivalent circuit. 3DEM simulation models of the proposed filter substructures can be constructed relatively easily. Initial dimensions of coupling apertures are found from experience or by simulation experiment. The same is true for the input and output coupling elements. Coarse resonator tuning of the 3D model is required; however, there is no need for producing good return loss responses. The given substructure does not have to have defined twoport properties as long as it realizes the desired coupling factors at the desired filter passband frequencies. The validity of separating a trisection from the remainder of the overall filter is given for narrowband filters with relatively weak coupling between resonators.
Smatrix Identity

Fig. 3 Equivalent circuit of a cascaded trisection filter. 
With correct element values, a suitable equivalent circuit has the same twoport Sparameters as obtainable from a 3DEM analysis. It is hence valid to write that
[S]_{3D EM analysis}  [S]_{Equivalent circuit} = 0 (2)
The Smatrix of the equivalent circuit is a function of the circuit element parameters. For a twoport network, four Sparameters exist which can be utilized to match a given equivalent circuit to EM simulation Sparameter data. Stateoftheart circuit analysis packages provide automatic Sparameter fitting facilities; however, it is also relatively easy to set up such fitting facilities manually. Optimization is used to minimize the Smatrix differences, where the variables are the circuit elements. It is important to refine the circuit model, so as to include major realworld effects. The quality of the extracted equivalent circuit is marked by a very small residual difference between the Sparameter responses, as shown in Figure 4 .
Accuracy Requirements

Fig. 4 Extracted circuit response vs. 3DEM simulation. 
The accuracy requirement for the 3DEM simulation is moderate. It was found that even a relatively rough field solution provides Sparameter data, which contains accurate information on the coupling factors between the resonators. Refinement of the field solution mainly affects the resonator frequencies, which are only of secondary interest, because the coupling factors are not strongly frequency dependent. In the impedance matrix (Equation 1) even the coupling impedances are frequency independent, which is valid in the narrowband case for frequencies in and near the filter passband.
Accuracy of the Method The parameters of different ninepole and sevenpole TEMresonator bandpass filters realized in the 900 and 1800 MHz range were established by extraction from measured Sparameter data. Subsequently the filter's trisection geometries were analyzed by 3DEM simulation (using Agilent HFSS). This involved the inclusion of an additional port probe coupling on one side of the structure, where there would normally be an adjoining resonator. The Sparameter data generated by the EM simulator was then used for extracting the filter parameters of the trisection substructure. Table 1 shows the excellent agreement between the EM simulationextracted filter parameters and the measurement extracted data. The field distortion caused by the additional port coupling element obviously has only a negligible effect.
Table 1  
Coupling Value  Prototype Measurementderived Value  3D EM Trisection Simulation Extracted Value 
k_{12}  0.0200  0.0201 
k_{23}  0.0126  0.0123 
k_{13}  0.0133  0.0134 
Conclusion
It has been established that 3DEM analysis of multiresonator filter substructures with equivalent circuit extraction can yield accurate information on the couplings between resonators. Trisections can therefore be designed by iterative EM analysis with subsequent parameter extraction. This technique replaces lengthy prototype iterations and speeds up the filter realization process. Modifications of the structure geometry can be carried out during the 3DEM modeling and their effect on the filter parameters can be accurately obtained within minutes. The described method also greatly reduces the need for coupling adjustment means.
References
1. Rafi Hershtig, Ralph Levy and Kawthar Zaki "Synthesis of Cascaded Trisection (CT) Dielectric Resonator Filters," EuMC, Jerusalem, 1997.
2. C. Bell, R. Levy, A. Atia and W.C. Tang, Microwave Filter Synthesis and Equivalent Circuit Extractions , 2000 IEEE MTTS International Microwave Symposium, WSC Workshop Notes.
3. N. Yildirim, M. Karaaslan, Y. Sen and O.A. Sen, "Filpro: A Synthesis and Circuit Transformations Software for Filters and Multiplexers," Web Page http://www. eee.metu. edu.tr/~nyil).
Dieter Pelz (Dipl. Ing., Germany, 1975) joined RFS Australia in 1995 after working for both SELAlcatel and Rohde & Schwarz. He has been working in the field of microwave filter synthesis and design for over 20 years. He holds four patents on filters and diplexers. Mr. Pelz can be reached via email at dpelz@compuserve.com.