An aluminum coaxial re-entrant resonator is designed for high Q to meet specifications on insertion loss. High rejection is achieved with a symmetrical pair of transmission zeros implemented using an electric probe. The filter is implemented with the well known folded topology which is fairly insensitive to manufacturing tolerances. Port tuning is introduced which cuts simulation time and makes the connection to the synthesized coupling matrix transparent.

**Specification**

- Passband from 9850 MHz to 10150 MHz (10000 MHz at 3% bandwidth).
- Less than 1.5 dB insertion loss.
- 20 dB return loss.
- 40 dB rejection at +/- 250 MHz from center.
- Group-delay not critical.
- To be machined in aluminum.

**Matrix Synthesis**

Using Couplings Designer to synthesize a chebyshev response, a filter of order 8 is found to achieve the required rejection level. Meeting the demand for low insertion loss would require 8 large resonators of Q 2400. A more appropriate alternative is to add a pair of symmetric transmission zeros at 9750 Mhz and 10250 MHz and reduce the order to 6. Thanks to a lower order we can meet the specification with a Q of 1800! Transmission zeros deteriorates the phase linarity but so does a higher order and the latter solution actually offers better group delay flatness. Thanks to the pair of transmission zeros we can enjoy good rejection, larger spur-free region and a flatter group delay with reduced complexity and size of the filter.

The synthesized coupling matrix arranges our 6 resonator filter in a folded topology which makes it easy to realize the cross-coupling between resonators 2 and 5. This topology is also quite insensitive to tolerances but it requires a negative non-adjacent coupling to support the pair of transmission zeros. All main-line couplings will be realized with irises allowing magnetic coupling, the cross-coupling will be capacitive by connecting the resonators with a probe.

The response shows high rejection on both sides of the passband thanks to the symmetrical transmission zeros. Click on the image to zoom.

**Resonator Design**

A coaxial re-entrant resonator with one end shorted is resonant when the coaxial line is approximately a quarter wavelength long. However the resonator is contained inside a cavity and the open end of the coax center conductor will experience strong electric fields between it and the housing which loads it capacitively, reducing the resonance frequency. A tuning screw can be added to adjust this capacitance. It is tedious to analytically compute the exact resonator frequency and Q, it is best done with an eigen-mode solver.

An optimum characteristic impedance of 75 Ohm has been found to give the highest Q and translated to dimensions it means that an outer and inner condutor radius ratio of 3.6 is optimum. The resonator itself can be of any size, typically the larger you make it the higher the Q and the shape may be cylindrical, square, rounded-square, helical, pretty much arbitrary as long as it supports the fundamental TEM mode. One should however watch out for higher order modes, typically the quasi-TE111 mode, but a long tuning screw will also form a coaxial line and may resonate too close to the passband.

To design an air-filled re-entrant coaxial resonator at 10 GHz, start by estimating the center conductor length. A quarterwave in air is 7.5 mm and to have a large spur-free region without higher order modes the outer radius should be made small enough compared to this length. Keeping the optimum ratio of 3.6, make the outer radius 5 mm and inner radius 1.4 mm. Allowing the housing to be twice as high as the outer radius, 10 mm, one can estimate the Q with a simple expresion as Q=5.34*r*sqrt(f), where r is the outer radius and f the resonant frequency with all metal parts of aluminum. The expression can be scaled to fit other length/radius ratios and materials. In this case Q is estimated to be 2670.

Entering all dimensions in an eigen-mode solver tells us that the actual center conductor should be 5.7 mm to resonate at 10 GHz with a Q of 2500. The screw allows tuning the resonator 2% in each direction. The screw itself resonates at 18-25 GHz, far from our passband and a quasi-TE111 mode is located at 21 GHz. Entering a Q of 2500 into Couplings Designer brings the loss down to 1.1-0.6 dB however in reality one can expect the Q to be roughly 25% smaller due to tuning screw and roughness effects.

**Extraction of Couplings**

External Q has been extracted following the same principle described in the Combline Cavity Filter tutorial. However in this filter a probe is coupling to the resonator’s electrical field (TEM) by proximity effects. The closer the gap the harder it couples and the lower the external Q. Keep in mind that this coupling is loading the resonator, reducing its frequency and one will need to adjust the center conductor length to bring it back up to 10 GHz. A QE versus gap spacing table is constructed and regression methods are applied to connect the dots. A spacing of 0.75 mm is found to be a good start.

Magnetic couplings (positive) are extracted as explained in the Combline Cavity Filter tutorial from the split even/odd mode resonance using an eigen mode solver. A table is constructed by adjusting the iris aperture. An iris of 6 mm, 5 mm and 5.25 mm between resonators 1-2/5-6, 2-3/4-5 and 3-4 respectively is a good start.

Electric couplings (negative) are extracted in a similar way by adjusting the length of a probe suspended by a teflon brick. The probe couples to the electric fields of the resonators. Once again a table is constructed and a length of 6 mm seems to do the job.

**Port Tuning**

All physical dimensions has now been extracted to realize the coupling matrix synthesized by Couplings Designer. An initial simulation will most likely not produce what we expected but rather a severely distorted return loss. The resonant frequency of each resonator is super sensitive to any changes to its field composition and our iris/probe modification has made them asynchronous, not resonating at the same frequency and the external and internal couplings we extracted may not be as exact as we had hoped for. Therefore tuning is almost always necessary!

It is possible to directy optimize the filter in a fullwave simulator but it may require plenty of iterations before it converges and can literally take days to compute. In addition one does not have full control of this iterative process, it may converge into something ugly and suboptimal. Instead one can tune the filter, in real time, with a s-parameter simulation. This only requires connecting extra lumped ports from each resonator to the housing in the fullwave simulation and apply a high port impedance, say 100 kOhm. The N-port S-parameter file is then hooked up with loading capacitors to each resonator-port and tuned/optimized until the filter response is optimal. A resulting negative capacitance indicates that the resonator frequency was too high, a positive capacitance indicates that it was too low. A table translating capacitor value to mm can be constructed and the offsets are applied to each resonator in the fullwave simulation. It may take a couple of iterations to get it perfect.

In a similar way the external/internal couplings and their offsets can be found by connecting inverters between resonator ports in an S-parameter simulation. Each inverter is tuned/optimized and changes required are reflected back to the fullwave simulation. One can also extract the exact external quality and couplings directly from the fullwave simulation. External quality is computed from the group delay at port 1 and 2. Internal couplings are computed from the resonator succeptance slope parameter (derivative of resonator admittances, Y11 and Y22, at center frequency) and admittance, Y21, connecting two resonators. The resulting couplings are compared to the synthesized matrix and offsets found from the previously extracted tables are applied to the fullwave simulation.

**Final Design**

The tuned 10 GHz filter is shown below with its 6 resonators connected in a folded topology with two symmetrical transmission zeros generated by a cross-coupling between resonators 2 and 5. All metal parts are aluminum and the passband loss is 0.6 – 1.2 dB.

Do you sell this software to operate in windows based environment? If so, please quote price and delivery.

Thank you,

Tony Hayek