8 th Order Dielectric Resonator Filter with Three Asymmetric

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1 Application Article CST AG th Order Dielectric Resonator Filter with Three Asymmetric Transmission Zeroes The dielectric resonator filter (Figure 1) is a high-performance filter design which is well-suited for applications where compactness and power are important. These sorts of filters are widely used in communication systems for example, in mobile phone base-stations. The number of independent design parameters that need to be optimized makes higher-order dielectric resonator filters challenging to tune. Simulation software can therefore be used to make it easier to design and tune these filters. Specification Value Center frequency 2 MHz Fractional bandwidth at -26 db % (6 MHz) Elements 8 Transmission zeroes 1867 MHz, 1921 MHz, 1945 MHz Table 1: The design specifications for the filter. This article explains the design and tuning of a dielectric resonator bandpass filter to fulfil the specifications shown in Table 1, with three transmission zeros (TZs) placed at critical frequencies in the lower stop band region in order to obtain a steeper transition from the passband to stopband. Since the first TZ is close to the passband the cross-coupling has an essential influence on the main signal path, which can lead to potential degradation in the passband performance if not carefully handled. The second tricky aspect is the bandwidth requirement. In this case, the center frequency is 2 MHz with a % fractional bandwidth at -26 db this is a particularly narrow bandwidth and quite a design challenge because of sensitivity to the manufacturing tolerances. Figure 1: An 8 th order dielectric resonator, including two cross-coupling taps.

2 Application Article CST AG 8 th Order Dielectric Resonator Filter with Three Asymmetric Transmission Zeroes Initial Design (without TZs) Filter Tuning (without TZs) Filter Tuning (with TZs) Single Cavity Design (Eigenmode f res optimization) Central Frequency Correction (Exploiting cavity f res sweep) Introduction of the X-couplings (Parameter sweep on X-coupl.) Inter-resonator Coupling (Eigenmode CBW except 1-2) Tuning of symmetric filter (1/2 parameters, coarse mesh) Optimization of S 11 In/Out Coupling (Eigenmode - Ext Q, Loaded f res ) CBW 1-2 Cavity (Eigenmode - Corrected DR1) Filter Assembly (coarse mesh) Figure 2: Filter design workflow. Lastly, we include the transmission zeros. Splitting the TZs from the filter design reduces the number of variables that need to be optimized simultaneously, and therefore the complexity of the optimization. Manufacturing tolerances and variations in material properties mean that the filter still has to be tuned after manufacturing our goal is to produce a design that is as close to the specification as possible and therefore reduce this post-manufacturing tuning effort. To simplify the problem, it is possible to split the workflow into three easy-to-manage sections (Figure 2). The first step is the initial design without TZs. We want to tune the resonant frequency of the single cavity (eigenmode solver), and then go on to look at the inter-resonator coupling and the input/output couplings, as well as correcting for the effect of introducing the feed. The first stage finishes with the assembly of the filter as a single model. Secondly, we move onto the tuning of the filter as a whole. For this, we need to optimize multiple design parameters simultaneously. The powerful Trust Region Framework (TRF) can be used to finetune the whole filter at once. Initial Design The first step in the design is to look at the individual components which come together to make this filter and take into account how it will be excited. This means building up the coupling matrix initially, without taking the TZs into account. CST STUDIO SUITE includes a built-in macro for finding the relevant Chebychev coefficients which relate to our filter specification. This macro gives the external Q-factor of the filter, which is related to the coupling to the first cavity, and the coupling bandwidth (CBW) between each pair of resonators for the inter-resonator coupling. Only four CBWs are required since the filter topology (without TZ structures) is symmetrical. These numbers serve as the basis of the design. Figure : Model of a single resonator cavity. The dielectric resonator is the blue cylinder (highlighted on the right). 2

3 8 th Order Dielectric Resonator Filter with Three Asymmetric Transmission Zeroes Application Article CST AG The first step is to design a single resonator (Figure ). The constraints of the application limit each cavity to 7 mm high and 5 mm wide, and each contains a dielectric tuning disk and a dielectric resonator ring, with a dielectric constant of 4. A ring is used instead of a solid puck because the ring will suppress higher unwanted modes in the resonator enlarging the spurious free out-ofband response. The important design parameter here is the radius of the dielectric ring, which dictates the resonant frequency of the filter. To tune this resonator to 2 MHz, we use the eigenmode solver combined with the Nelder-Mead Simplex Algorithm (a local optimizer which offers good performance for single-parameter optimization). We meshed the structure with a tetrahedral mesh with curved elements which conform to the cylindrical surfaces of the dielectrics. We thus were able to use a relatively coarse mesh to get accurate results, allowing an optimization of the structure dimensions to be completed in only a few minutes. The next element to be designed is the probe (Figure 4), which determines Q ext (i.e. the energy coupled into the first resonator cavity). In this filter, it takes the form of a sickle-shaped metal rod. Other feed topologies did not offer a strong enough coupling. By parameterizing the gap between the probe and the dielectric cylinder, we can optimize it in order to match the cavity Q ext -factor found using the Chebychev macro. Introducing the probe changes the geometry of the cavity, which means that the first cavity needs to be retuned. A single parameter sweep over the range of different aperture heights produces a design curve relating aperture height to CBW (Figure 6). This curve can be used for reading off the aperture height for the central three coupling apertures, but the presence of the feed probe in cavity 1 means that a separate parameter sweep is required in order to extract the height for the coupling aperture between resonators 1 and 2. Aperture width Aperture height Figure 5: Model with two resonators and with an aperture; Inter-resonator coupling. 6 CBW between resonators vs. aperture height 5 CBW / MHz Resonator height (1.6 r) Outer radius (r) Inner radius (. r) Aperture height / mm CBW / MHz CBW 2- CBW -4 CBW 4-5 Resonators Required CBW Aperture height Figure 6: Design curve for the resonators, relating aperture height to CBW. Probe radius Figure 4: Model of the first resonator cavity with the mesh shown, including the probe for Q ext evaluation. The next step is to design the aperture (Figure 5). This time, the design parameter of interest is the size of this aperture. Thanks to the Chebychev macro, we already know what the coupling bandwidth (CBW) should be between each pair of resonant cavities.

4 Application Article CST AG 8 th Order Dielectric Resonator Filter with Three Asymmetric Transmission Zeroes Filter Tuning different values for the dielectric resonator radius in order to find the value for the dielectric resonator radius which compensates for the frequency shift. This is done using the single resonator model. Because the center frequency has dropped by 4 MHz, we increase the goal frequency by the same amount to counteract the cavity interactions. In other words, we want to find the value that gives a resonant frequency of 24 MHz, so that when the filter is assembled, the center frequency is shifted back down to 2 MHz. The coupling is kept the same only the resonant frequency is altered Resonant frequency as a function of resonator radius Figure 7: The assembled filter model, with the cavities numbered. Once each individual element has been designed we can combine them to produce the entire filter layout (Figure 7). The FEM based frequency domain solver, and in particular the fast reduced order model (ROM) implementation, is the optimal way of performing a full D EM analysis of resonant structures such as this filter. In combination with a curved tetrahedral mesh, which resolves the curved geometry of the filter very accurately, we can obtain broadband results of the filter, with an accuracy of better than.1 % (1.2 MHz) in passband center frequency, in under a minute on a laptop. 1 S-Parameter [Magnitude in db] Resonant frequency / MHz , ,5 1 1,5 Resonator radius / mm Figure 9: Results of a parameter sweep over dielectric resonator radius, plotting the radius against the resonant frequency. The highlighted value at 24 MHz corresponds to a 4 MHz increase in resonant frequency. The first and last cavity have a different resonator radius to the other cavities, so we can either adjust then with a separate simulation, or use the following formula: , = 2, 1, 2, Figure 8: S-parameters for the filter before center frequency adjustment. Simulation reveals that this initial design does not behave exactly as expected (Figure 8) the center frequency has decreased by about 4 MHz, the bandwidth is too narrow, and the filter doesn t achieve the -26 db passband performance, due to mutual cavity interactions that are omitted in the partial single or two cavity models in the initial design steps. We can tune the filter to specification in two stages. First, we can shift the center frequency by adjusting the individual resonator elements again. To do this, we perform a parameter sweep over S1,1 S1,2 This gives us a good starting point for performing an optimization. For the filter tuning (without TZs) we consider a minimum set of 8 parameters (4 for the couplings and 4 for the resonant frequency of cavities). The trust region framework (TRF) optimizer is especially good for these sorts of complex problems, since it is efficient and particularly robust against a convergence falling into a local minimum. We set each parameter range to be inversely proportional to how sensitive the S-parameter response of the filter is to variations in that parameter. In this case, the heights of the apertures are allowed to vary by 2%, since they have a relatively small effect on the filter s performance, but the radius of the dielectric resonator is only allowed to vary by 1.6%. The goal of the optimization is simple to reduce S 11 to below -26 db across the range from 197 MHz to 2 MHz. The fast simulation time achieved with the ROM frequency domain solver allows us to perform an optimization of 16 iterations under three hours to get the improved results shown in Figure 1. Note that we do not need a perfect match at this stage, because introducing the TZs will distort the S 11 again. 4

5 8 th Order Dielectric Resonator Filter with Three Asymmetric Transmission Zeroes Application Article CST AG S-Parameter [Magnitude in db] Initial Corrected Optimized The model is optimized once more to fine-tune the strength of the cross-couplings in order to place the transmission zeroes properly, and to repair the response of the filter in the passband which had been degraded by the introduction of the TZs in such close frequency proximity (Figure 14). Again, the frequency domain ROM solver and the TRF algorithm are used. The parameters that were optimized were the positions of aperture windows and cross-coupling dumbbells. In this case the tuning screws were included in the design, but not selected for optimization. They were simulated at the mean height, so that after manufacture there is the possibility of tuning the filter manually Figure 1: Comparison of the S-parameters for the initial design, the corrected design and the optimized filter. Transmission Zeroes 2 1 Figure 12: An illustration of the coupling triplets in the filter. Frequency corresponding to S2,1 minimum as a function of dumbbell length ,5 24,1 24,15 24,2 24,25 24, 24,5 24,4 24,45 24,5 Dumbbell length / mm Figure 1: Results of a parameter sweep for the length of the central dumbbell (triplet 1). Figure 11: Cross-section of a cross-coupling. Finally, we add the transmission zeroes to the model. These can be implemented in different ways, for example by introducing a form of dumbbell element between non-adjacent resonators (Figure 11), thus forming a coupling triplet so that each TZ is formed by capacitive cross-coupling. In this example, we ve chosen a topology with three triplets, where two are realized using dumbbell and the third one is formed by offset coupling windows as shown in Figure 12, and we use a parameter sweep over the length of the first dumbbell coupling to tune the frequency of the first transmission zero (Figure 1). The third triplet can t be investigated/designed using a circuit simulator only, but the D EM simulation gives us full information about the structure behavior including the higher order effects S-Parameter [Magnitude in db] Figure 14: Results for the final tuned filter. Each TZ is numbered with the corresponding triplet, as shown in Figure 12. S2,1 S1,1 5

6 Application Article CST AG 8 th Order Dielectric Resonator Filter with Three Asymmetric Transmission Zeroes Conclusion The basic techniques described in this article can be applied to the design of a wide range of cavity filter types. A general workflow for any such filter will follow the same basic steps: first, the Chebychev coefficients are calculated in order to produce a series of optimization goals. The individual element is designed and optimized to produce the correct resonant frequency, and the feed elements are optimized for external Q-factor. Next, the inter-resonator coupling is optimized according to the calculated coupling bandwidth (CBW), taking into account the effect of the feed. Once each individual element has been designed, the entire structure is then assembled in D and optimized. Global optimization methods allow multiple independent design parameters to be optimized effectively over a complex parameter space. By first designing the elements individually, and then optimizing the final models, filters can be designed and tuned quickly and efficiently. The step-by-step process allows design parameters to be optimized individually rather than immediately beginning with a daunting multi-variable optimization. This retains the performance benefits of a full-system optimization to fine tune the filter at the end of the design process. This divide and conquer approach to design makes simulation a powerful tool for filter design. Author Dr. Vratislav Sokol Senior Application Engineer, CST AG (Branch Office Prague) CST AG Bad Nauheimer Str Darmstadt Germany info@cst.com Trademarks CST, CST STUDIO SUITE, CST MICROWAVE STUDIO, CST EM STUDIO, CST PARTICLE STUDIO, CST CABLE STUDIO, CST PCB STUDIO, CST MPHYSICS STUDIO, CST MICROSTRIPES, CST DESIGN STUDIO, CST BOARDCHECK, PERFECT BOUNDARY APPROXIMATION (PBA), and the CST logo are trademarks or registered trademarks of CST in North America, the European Union, and other countries. Other brands and their products are trademarks or registered trademarks of their respective holders and should be noted as such. 6