model is used. Another very efficient approach is to employthe coupling matrix (CM) as a surrogate for an optimizationroutine [1].

In the commercial simulation software CST Studio Suite®a technique exists for extracting the CM directly from S-parameter results [2]. Moreover, in the module Filter Designer3D a filter topology need to be specified and, based on this ref-erence, a polynomial fitting is applied to the transmission andreflection coefficients from the full-wave simulation model.The tool generates a new CM which is compared to the idealCM and from the coefficient errors the problem areas can beidentified in the 3D model – essentially providing a view ofthe filter operation without having to dissect the model itself.

This extracted CM is utilized by the built-in optimizerto minimize all the coupling errors, leading to a responsewith an equal ripple passband and transmission zeros in thespecified locations. If parasitic couplings exist in the 3D modelthey are also included in the reference CM. Since they areunintended and generally difficult to control, the errors of thesecoefficients are simply ignored for the purpose of retaining anequal ripple in the passband return loss.

An example of a 6th order combline cavity filter is shown inFig. 1. The topology consists of two cascaded triplet sections,each producing a transmission zero on the high-side of thepassband. This model is optimized from a detuned state wherethe tuning screws are not properly inserted and the model iscompletely mismatched as seen from the results in Fig. 1(c).In this model there are 14 parameters to optimize. By usingthe CM as a goal, an optimal state is achieved within only 23full-wave simulation steps.

This particular model was also manufactured as shown inFig. 1(b). It is usual that these cavity type filters also requirepost-production tuning, hence the inclusion of tuning screws.To this end Filter Designer 3D was used to also tune up thisparticular fabricated device. The software reads the real-timeS-parameter data from the network analyzer and indicates howfar each coupling coefficient is detuned from the ideal state. Bymanually tuning the corresponding screws of the device verygood agreement is achieved between simulated and measuredresults as seen in Fig. 1(c).

III. DIPLEXER DESIGN

It has been shown that the coupling matrix can also beused to synthesize a diplexer filter, where the common port isconnected to the channels either through a star or a resonatorjunction [3]. In Fig. 2 the latter is used in a combline cavitydiplexer for a typical GSM uplink and downlink. One impor-tant goal with diplexers is to achieve high isolation betweenthe channels. Therefore in this case two transmission zerosare placed between the passbands by using a box topology foreach channel filter.

To tune this model the same approach can be followed asoutlined in the previous section for the bandpass filters. TheTrust Region Framework algorithm, used for the optimization,relies on perturbations of the parameter space before buildingits "trusted region" [4]. Since this diplexer model has 22

(a)

(b)

Fig. 2. (a) A combline cavity diplexer model with (b) the optimization results.

parameters which will result in a costly perturbation process,a more attractive approach is to instead use the sensitivityanalysis in CST Studio Suite. This technique calculates thefirst-order derivative of each parameter in terms of the S-parameters, which means that the optimizer can effectively usethis without the need of full-wave simulations for building itstrusted region.

With this approach the diplexer is successfully optimizedwithin only 14 full-wave simulation steps and the results areshown in Fig. 2(b). Note that this number of steps actuallyalso includes the perturbation of 3 of the model parametersthat did not utilize the sensitivity analysis.

IV. MANIFOLD MULTIPLEXER DESIGN

Manifold-coupled multiplexers (MUX), as shown in Fig.3, are widely used for output multiplexing applications dueto its compact size and low insertion loss. With contiguouschannels it is beneficial to utilize singly-terminated filters ina manifold layout, since each filter has an input conductanceapproaching zero outside its passband while the input suscep-tance is conjugately-matched to the adjacent passbands. Thiscontiguous nature makes it impossible to optimize the channelfilters independently.

In [5] a piecewise optimization approach for manifold-coupled multiplexers is outlined. The problem is that withsuch large models the computational effort is considerableand the routines become time-consuming. Space-mapping isone approach to try and speed up the process, but unfortu-

(a)

(b)

Fig. 3. (a) The output MUX with a rectangular E-plane coupled manifoldand 6 input channels, each realized with a TE113 circular cavity filter. (b)The solid curves represent the return loss of each channel in the optimizedMUX. The dashed curves represent the model that includes the optimizedmanifold and CMs, but where the CM of channel 6 alone is replaced with itssynthesized 3D filter model.

nately mapping can quickly become a cumbersome task toautomate. An alternative solution is to replace each channelfilter with a corresponding CM. Then, after optimizing themanifold and the coupling coefficients of each channel, thefilters are synthesized based on the optimized matrices. Thecommercial software tool CST Studio Suite Fest3D [6] offerssuch a synthesis approach for circular cavity filters – includinghigher-order TE11z mode designs – which are typically usedin satellite output MUXs.

As an example consider the 6 channel multiplexer in Fig. 3,consisting of a rectangular E-plane manifold that is configuredin a herringbone layout. Each channel has a 4th order filterwith 40 MHz bandwidth in the Ku-band while each guardband is 6 MHz wide. The singly-terminated coupling matricesare calculated with Filter Designer 3D and directly importedinto Fest3D, to form the complete MUX circuit as shown inFig. 4. Each component is solved with modal techniques toquickly calculate the response of the entire model.

The initial dimensioning of the manifold consist of normalE-plane T-junctions and λg/2 sections – i.e. between thejunctions, at the short-circuit end and connected to each CM

(a)

(b)

Fig. 4. (a) The multiplexer circuit in Fest3D with coupling matrix blocks (an-notated as "CM"). (b) The return loss curves before and after the optimizationsteps.

(called the ’channel stub’). Here the guided wavelength isdefined at 11 GHz – the frequency of the lowest channel.The initial common port return loss (CPRL) is depicted bythe dashed curve in Fig. 4(b) and it is clear that even withthe aforementioned systematic design the response is far fromperfect.

In the first step the manifold is optimized to get to a CPRLgreater than 10 dB. Then the CM coefficients are includedin the optimization process, where it is approached channelby channel. After this process the multiplexer is completelyoptimized (refer to Fig. 4(b)), in that the 3D model of themanifold is combined with the correct CM coefficients foreach channel. What is important to highlight here is thatthe simulation is extremely fast since all parts in the circuitbasically have analytical or quasi-analytical solutions.

Each of the channels have different CMs after the optimiza-tion and can now be transformed into actual physical filters.This can be done for any type of filter technology, as outlinedin the Section II, but in this case the automatic design tool fordual-mode filters in Fest3D is used. The result for Filter 6in the circuit (refer to Fig. 4) is shown in Fig. 5. Note thatthe CM is no longer symmetrical after the optimization andof course the response is detuned in isolation.

(a)

(b)

(c)

Fig. 5. (a) The optimized coupling matrix for channel 6 and (b) thecircular cavity filter synthesized directly from this CM in Fest3D. (c) Thecorresponding S-parameter results.

Next the CMs are replaced systematically in the circuit bythe synthesized 3D filter models. Starting with Filter 6, thereturn loss is indicated by the dashed lines in Fig. 3(b) and itis clear that this passband as well as the adjacent bands requiresome tuning. The detuning is caused by the loading effect ofthe input iris of the filter. Therefore during optimization thedimension that is predominantly changed is the length of thechannel stub. Once this is done for all the filters the final MUXresponse is obtained, as depicted by the solid curves in Fig.3(b).

V. CONCLUSION

It was shown how the coupling matrix can effectivelybe used for both direct optimization routines of filters and

diplexers, as well as in a piecewise optimization of manifold-coupled multiplexers. All of this was seamlessly achieved bybringing together the right software tools and solvers for theindividual workflows.

The filter and diplexer designs illustrate that the couplingmatrix extraction approach can be applied to a model in ahighly detuned state at the outset of the direct optimization.This effectively means that less accuracy is required duringthe design of each coupling mechanism – provided of coursethat the distributed model allows the range of the electricalproperties.

In the manifold-coupled multiplexer workflow a key partis the direct incorporation of the coupling matrix in the 3Dmodel simulation. An accurate MUX design is finally ensuredwith the direct synthesis of each channel filter based on thecoupling matrix.

REFERENCES

[1] T. S. Beukman, "An efficient design workflow of a compact ridgedwaveguide filter," 2017 IEEE AFRICON, Cape Town, 2017, pp. 569-573.

[2] Dassault Systèmes, 64289 Darmstadt, Germany. CST Studio Suite®2019. [Online]. Available: https://www.3ds.com/products-services/simulia/products/cst-studio-suite/

[3] G. Macchiarella and S. Tamiazzo, "Novel approach to the synthesis ofmicrowave diplexers," IEEE Transactions on Microwave Theory andTechniques, vol. 54, no. 12, pp. 4281-4290, Dec. 2006.

[4] S. Koziel, F. Mosler, S. Reitzinger and P. Thoma, "Robust microwavedesign optimization using adjoint sensitivity and trust regions," Interna-tional Journal of RF and Microwave Computer-Aided Engineering, vol.22, no. 1, pp. 10-19, Dec. 2011.

[5] R. I. Cameron and M. Yu, "Design of manifold-coupled multiplexers,"IEEE Microwave Magazine, vol. 8, no. 5, pp. 46-59, Oct. 2007.

[6] Dassault Systèmes, 46022 Valencia, Spain. Fest3D 2019. [Online].Available: https://www.3ds.com/products-services/simulia/products/fest3d/

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