Flexible Approximation Functions for Broadband Matching

Introduction. Intensive use of broadband signals in RF devices for various purposes is associated with the need to develop broadband elements of RF systems. Iterative methods for designing such elements are frequently uninformative and ineffective, while analytical methods give solutions only for simple models. The problem is the small set of classical approximations, which impedes dealing with complex models of elements.
Aim. Development of a wide-band matching technique based on generalized Darlington synthesis using flexible approximating functions (AF) for load models with zeros of transmission at infinity.
Materials and methods. The paper is based on the generalized Darlington synthesis method. To extend the capabilities of the method, approximating functions with increased variation properties are used. In order to use the results in engineering practice, a synthesis algorithm was developed, which includes three stages: formation of the frequency response, control of analyticity of the used functions and limits of matching. The method is analytical and does not use iterative procedures. The mathematical apparatus of the method is based on the analysis of residues in the zeros of transfer function of load resistance.
Results. Flexible approximating functions proved to be an effective tool for designing matching circuits with multiple transfer zeros in infinity. Variative properties of the function facilitate the realization of both smooth and wave frequency characteristics. A combination of both is also possible, ensuring the best properties of both. The proposed approximating functions allow a smooth change in the frequency response, while preserving the normalization characteristic of classical approximations. Application of such functions allowed us to virtually remove the limitations inherent in the classical AF on the minimum values of the load capacitance and more than 30 % of the limiting values of inductance in the above examples.
Conclusion. The developed methodology makes the process of wideband matching physically transparent and can be applied to other classes of loads.

1. Yarman S. B. Design of Ultra Wideband Power Transfer Networks. NY, Wiley, 2010, 774 p. doi: 10.1002/9780470688922

2. Devyatkov G. N. Automated Synthesis of Broadband Matching Devices Coupling Arbitrary Immittances of Signal Source and Load. Scientific Bulletin of NSTU. 2004, vol. 1 (16), pp. 155–165. (In Russ.)

3. Samuilov A. A., Cherkashin M. V., Babak L. I. The Technique of "Visual" Design of Circuits on Concentrated Elements for Broadband Matching of Two Complex Loads. Proceedings of TUSUR University. 2013, vol. 2 (28), pp. 30–39. (In Russ.)

4. Voropaev Yu.P., Vasil'ev A. D. Application of Target Transmission Matrices and Complicated Elementary Cascades in the Synthesis of Broadband Matching-Filtering and Modeling Schemes. Doklady BGUIR. 2010, vol. 6 (52), pp. 35–42. (In Russ.)

5. Pegasin D. V. Synthesis of Matching Chains with Prescribed Level of Transducer Power Characteristics on Basis of the Levenberg–Marquardt Algorithm. Doklady BGUIR, 2010, vol. 3 (49), pp. 17–23. (In Russ.)

6. Yantsevich M. A., Filippovich G. A. Technique for the Synthesis of Broadband Matching Devices Using Bounded-Flat Approximating Functions. Proc. Of Francisk Scorina Gomel State University. 2021, vol. 6 (129), pp. 154–158. (In Russ.)

7. Mahata S., Herencsar N., Kubanek D. Optimal Approx-Imation of Fractional-Order Butterworth Filter Based on Weighted Sum of Classical Butterworth Filters. IEEE Access. 2021, vol. 9, pp. 81097–81114. doi: 10.1109/ACCESS.2021.3085515

8. Ali A. S., Radwan A. G., Soliman A. M. Fractional Order Butterworth Filter: Active and Passive Realizations. IEEE J. on Emerging and Selected Topics in Circuits and Systems. 2013, vol. 3, no. 3, pp. 346–354. doi: 10.1109/JETCAS.2013.2266753

9. Mahata S., Kar R., Mandal D. Optimal Fractional-Order Highpass Butterworth Magnitude Characteristics Realization Using Current-Mode Filter. AEU – Intern. J. of Electronics and Communications. 2019, vol. 102, pp. 78–89. doi: 10.1016/j.aeue.2019.02.014

10. Filippovich G. A. Shirokopolosnoe soglasovanie soprotivlenii [Broadband Impedance Matching]. Minsk, Voennaya akademiya RB, 2004, 176 p. (In Russ.)

11. Abrie P. Design of RF and Microwave Amplifiers and Oscillators. Boston, Artech House, 1999, 480 p.

12. Filippovich G. A., Belevich V. F. Necessity and Sufficiency of a System of Restrictions on Broadband

13. Matching. Physics of Wave Processes and Radio Systems. 2006, no. 2, pp. 31–36. (In Russ.)

14. Babak L. I., Cherkashin M. V., Zaitsev D. A. "Visual" Design of Corrective and Matching Circuits of Semiconductor Microwave Devices. Proceedings of TUSUR University. 2007, vol. 1 (15), pp. 10–19. (In Russ.)

15. Burenko E. A. An Approximation of the Amplitude and Frequency Characteristics and Synthesis of High-Order Radio Filters Based on Legendre, Gegenbauer, and Jacobi Polynomials. International Research Journal. 2021, vol. 6 (108), pp. 49–63. (In Russ.)

16. Shashok V. N. Synthesis of Circuits with Rising Wave Transfer Function. Doklady BGUIR. 2011, vol. 8 (62), pp. 52–58. (In Russ.)