Introduction
This page provides an interactive tool for design-optimization/diagnosis of microwave multiple-coupled resonator filters. This tool allows user to design a filter with general response and finite transmission zeros. This is a powerful method to provide very sharp response near passband eadges, required in many wireless and satellite communication systems.

Design Process
Diagnosis Process
Special Notes
Disclaimer
External links
Contact

Design Process
The process of design, consists of two main stages of synthesis and approximate modeling. The synthesis stage divides into two steps of response synthesis and circuit synthesis.

1- Response Synthesis
In the response synthesis step, the user inputs the initial data such as degree N, center frequency, bandwidth, RL, etc. Next, the program trasnforms the response characteristics to the low-pass and then calculates the coefficients of polynomials in the reflection and transmission functions based on characteristics of general Chebyshev approximation function in an iterative process. These polynomials are defined below:

A, B, and D are polynomials of degree N. For the lossless filters, these two functions can be combined in one equation for transmission function as

For general Chebyshev response, K should be

where complex frequency is the location of nth complex transmission zero. Using this equation for K, all coefficients of polynomials A, B, and D can be calculated in an iterative procedure. The resultant response will be equiripple in passband with ripple value of and at least two transmission zeros at . The locations of finite transmission zeros are arbitrary, though the maximum number of transmission zeros in the current version of program should not be greater than N-2. The output of program for the first step is the plot of synthesized response. If the response does not satisfy the required filtering mask, the location of transmission zeros can be changed and the new response can be generated. The program has also the capability of predicting the effect of loss on the response at this stage. By providing the Q-factor of resonators, an appropriate amount of shift in the real values of all complex poles and zeros of transmission and reflection functions are calculated and applied. The response calculated from these new sets of poles and zeros simulates the response of lossy filter. Using this method we can predict the effect of loss before circuit synthesis step.

2- Circuit Synthesis
The next step would be circuit synthesis, in which the parameters of the prototype low-pass equivalent circuit, shown below, are computed.

The coupling matrix M is symmetrical and its diagonal elements represent the deviation from center frequency. The synthesis is performed using an optimization procedure. The values of C and L will be set to 1F and 1H to normalize the center frequency to one. The unknowns will be the elements of coupling matrix M and normalized input/output resistance R₁ and R₂. The objective function used for optimization is defined as

where are zeros of polynomials A and B found in previous step. In comparison to using frequency mask, this objective function decreases the number of frequency calculations at each iteration considerably. In the demo version, computation is based on the canonical topology for the resonators. The program automatically finds the minimum number of coupling elements required for realization of synthesized response. In the full version, user can set the topology and optimize the coupling matrix independently to achieve the same response. The computed coupling matrix elements and input/output resistance can be denormalized later to achieve the related bandpass values. To provide additional information about the filter characteristics, program also plots the pole-zero distribution of transmission and reflection functions at low-pass.

3- Approximate modeling
The approximate modeling section is based on coaxial resonators structure as shown below. Adjacent and cross couplings are realized by apertures. This module has two capabilities. User can either find the electrical parameters of each resonator by entering the physical dimensions (analysis), or find the dimensions from the given circuit parameters (synthesis) (This part is under construction.)

Top view of a 6-pole filter with cross-coupling between resonator 2 and 5.

Diagnosis Process
In this section, user can upload the S-parameters of a measured faulty filter in standard formats such as Touchstone or Citi and calculate the coupling matrix and input/output resistors. This will be compared with the original synthesized data and the source of fault will be determined. This is performed based on model based parameter estimation and multilevel optimization (This part is under construction.)

Special Notes:

• The web interface works based on CGI concept. In order to limit the workload of server, the degree of filter is limited to 5. If you need to design a higher degree filter, send me an E-mail.
• The interface is in its experimental stage and does not have all the complexities to handle extreme combinations of input data. If you get a very strange result or nothing at all, there is a good chance that your input data is not within the reasonable range. Double check your data and run the program again. However, if you consistently get the busy server error, or the plots are not shown or updated, send me an E-mail and I will reset the error files.
• The main engines of this program are developed at RF/microwave group, dept. of Electrical and Computer Engineering, University of Waterloo.

Disclaimer
Data and information is provided for informational and educational purposes only, and is not intended for commercial purposes. RF/microwave group shall not be liable for any errors in the content and information provided here.

External links
Other sites with relevant information about design of multiple coupled resonator filters are listed below:

• Guided Wave Technology
• Plonix RF & Microwave Design Center
• Coupled Resonator Filter Design