A numerical method for the optimization of symmetrical lumped element low-pass and band-pass filters with a generalized Chebyshev response is considered. By exploiting the fact that a network based on generalized Chebyshev prototype has a prescribed number of turning points in the insertion loss and an identical number of independent parameters which can be assigned as variables to adjust their levels, the method gives fast convergence

Budimir, D.

Lalehparvar, L.

Nikolic, N.

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Design of generalized Chebyshev lumped element filters by computer optimisation

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    WestminsterResearch http://www.wmin.ac.uk/westminsterresearch   Design of generalized Chebyshev lumped element filters by computer optimisation.  Djuradj Budimir Laleh Lalehparvar Natasa Nikolic  School of Informatics, University of Westminster   Copyright © [2002] IEEE. Reprinted from the proceedings of the 11th Mediterranean Electrotechnical Conference, 2002. MELECON 2002, pp. 475-478.  This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Westminster's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.   The WestminsterResearch online digital archive at the University of Westminster aims to make the research output of the University available to a wider audience.  Copyright and Moral Rights remain with the authors and/or copyright owners. Users are permitted to download and/or print one copy for non-commercial private study or research.  Further distribution and any use of material from within this archive for profit-making enterprises or for commercial gain is strictly forbidden.      Whilst further distribution of specific materials from within this archive is forbidden, you may freely distribute the URL of the University of Westminster Eprints (http://www.wmin.ac.uk/westminsterresearch).   In case of abuse or copyright appearing without permission e-mail wattsn@wmin.ac.uk. IEEE MELECON 2002, May 7-9,2002, Cairo, EGYPT. Design of Generalized Chebyshev Lumped Element Filters by Computer Optimisation D. Budimir, L. Lalahparvar and N. Nilcolic Wireless Communications Research Group, Department of Electronic Systems, University of Westminster, 115 New Cavendish Street, London W1 W 6UW, United Kingdom. Tek +44 207 9115139; Fax: +44 207 91 15089 budimid@,wmin.ac.uk Abstract A numerical method for the optimiZation of a Symmetrical lumped element lowpass end baodpass flters with Generalized Cheyshev response is considered. By exploiting the fad that a network based on generalized Chebyshev protolype has a prescribed number of turning points m the insidion loss and an identical number of iradependent parameters which can be assigned os variables to adjust their levels the method gives fast convergence. Key-Words: - Low-pass Filters, Bandpass Filters, LC Filters, Lumped Element, Generalized Chebyshev Filters, Optimization 1. INTRQDUCTIQN When a commofl approach to the design of filters d t s  in a design passband which differs cousidembly from that which is specified, optimizatioSis required to tune the filter elements to achieve a design that meets certain requirements. Most RF and microwave filters have not yielded exact optimum synthesis. Taking into account parasitic effects, high fresuency operation, fresuency dependent elements, a narrow range of element values, and so on, a common approach to design provides, at besf only approximate answers. Not infrequenty, a common approach may be used to great advantage in providing the initial points for optimizatioa In this paper, we introduce an optirnizaton procedure based on equal ripple optimisation to Optimise filters based upon Generalized Chebyshev fimction prototype. Thismethodsearchesfortuningpointsinthefiltertransfer fundion and forces the ripple levels at these points to have quxifkd values. The method requires knowledge of the ater insettion orreann loss at these points. The method will generate a set of equations which are solved to give a new set of parameter values. The cycle is then npated, until the filter characteristic is within an arbitrarily close value to the daired specification. This technique quires less calculation of the eledrical paramden3 of filter discontinuity than generalised optimization routines so far applied [ 13. 2 PROBLEM FORMULATIQN Figure 1. Generalised Chebyshev low pass P*totYpe This characteristic m tarns ofinsertion loss,L, is given by where the transmission zeros are of order (A) at 0-7803-7527-0/02/$17.00 02002 IEEE. 475 w=iwo and one at infinity. n is an oddnumber equal L and RL. is the minimum return loss level (a) in the passband A typical insertion loss respaase is illustrated in Figure insertion loss level in IIS stopbana and w, is the badedge fiequeasy oftbe stopband. la, W h  wm is the frequency of the mini" (3) (4) Em E,,, are also furactiaae of the m w - 1  pameter valuesofthesy"etli~1 filter. ~ '0 ,  i,=1,2,3 .... pa . This is a system of m=n-I nonbear equations in m=n-l variables iix the sy"&d case. Solving (6) gives the pamnekr values of a filter rntisfvinp (5). The El (i = I ,  .... m) can be regarded as the canied out by equasiageacb ofthese annponents to zero (a vector process) nuber than " k i n g  the magnib& ofthe vectcr (a scalar process). Thus equal ripple optimizatioa c a n b e r e g a d e d a s a v e c t o r  o f a u m d h " l e r n n v c ~ .  optirmzatlan is procedure whereas g d  purpose aptimipition routines are scaiar p e d r a e s  usuauy the convergence &on applied m galend puspose optimization routines is that the grad- Wih resped to the filter elemas, of the magnitude of the ernn vector is zero. However a zero p d k m t  may colrespoodto a local "utnandthe  e n m m a y  not betnr ly l l lhhkd  Theconv~~critaion8pplied m equal ripple o p h k t i m  isthateachcomponeaSof theerrorvectorhzeau.Tcmconvugeacetheemr is rpmrPn to zem. The problem of local minime does mt anise.. To apply an itehative nodkar equation solver it is neoessary for a given set of filter parameter values to know tbe insertion loss c d y  at tbe bandedge fiequea3cy,f, (minimum) ancl at the ripple "a, However, the frequencies at vvhich the ripple o w b a  OOcLlt are MLaown aod als fimdons of the ater pat.ameter values. The Newton-Raphson m e d d  [4] is a rapidly convergent technique fbr tbe solution of a system of "U equations if a goodl initial y o n  availabk. Tbe number of times the hchon IS e v a l d  m the process of firding its root is tbe usual measure of awlputational effbtt This includes fimction evaluations reqllired to calculate derivatives numerically. The Newton-Rilphsonmetbodl has the geaeml fom [s]  here R is the itmation lllsnl~r *1,2, ..I and J' is the mverse ofthe I x m  Jacobian matrix evaluated at 2'.  he above ~ e s  the regions within the passband which need to be sampled m otder to calculate E&) (and J(i?)). 'h response aod emas a k e a c h i t e "  arecomputedagainwiththenew cmectedpa",  until the e" are judgedto be sufticiently small. 476 3 NUMERICAL RESULTS In ordm to illustrate our approach, a fifth order lumped element low-pass and band-pass filters have been designed. The low-pass filter can be d e s c r i i  by 4 parameters: inductors (Ll=U, L2=U, U) and capacitor (C24X) as marked in Figure 2. We used equal ripple optimization with L1, U, L3 and C2 as variables for filter shown in Figure 2. Figure 3a shows the calculated retum loss (dashed line) and insertion loss (solid line) of filter be& optimization. The return loss (dashed line) and insertion loss (solid line) calculated using the filter elements obtained on convergence are shown in Figure 3b. The band-pass filter can be described by 6 parameters: inductors (Ll=U, L2=L4, L3) and capacitors (C13C5, C2=C4, C3) as marked in Figure 4. We used equal ripple optimization with L1, L2, L3 and C1, C2, C3 as variables for filter shown in Figure 4. Figure 5a shows the calculated return loss (solid line) and insertion loss (dashed line) of filter before optimization. The return loss (solid line) and insertion loss (dashed line) calculated using the filter elements obtaind on convergence are shown in Figure 5b -1 -12400 I 1 Figure 3a. Simulated Insertion aod Return loss of generalized chebyshev lav pass lilter b e f o r e o p t h b i h  -2ooMI OM) 10.00 2oM) 3O.m 40.00 Frequency (GHz) Pigure 3b. Simulated I” aod Return lossofgeneralizedchebysbev low pass filter afteroptimization c1 L1 CO L3 c s  Ls Figure4.Ge”d * Chebyshev band-pass filter 477 References 450.00 - -uw).w - -250.00 i lnsertkn loss (dB0 l ' l ' 4 ' i  0.00 10.00 20.00 Sod0 40.00 Flequency (0%) P i  Sa Simulated Insertion and Retunr loss of generalized Chebyshev band-pass filter before optimization 4 CONCLUSION [l] Bandler, J. W., and S.H. Chen, "Circuit Optimization: The State of the Art," IEEE Trans.Microwav Theory & Tech., Val-MTT- 36, 1988, pp.424-443. 123 Rhodes J. D. and S .  A. Alseyab, "The Generalized Chebyshev Low~-pass Prototype Filten), Mcwwave Theory wrd Applicatwns, icrowave Journal., September 1989, pp.197-201 Computer Optimization", ISBN 0-89006- [3] . B u d i d ,  D., "Generalized Filter Design by 579-9, Artech HOW BO&, 1998. [4]. Gupta K C, R.Garg and RX'hadha, 'CAD of Microwave Circuits", Artecb House, Wham MA, 1981. The method presented hem offers a simple but reliable filters witb G e n a a W  Chebyshev fmction prototype. The method provides fast umvergence. mew for optimization of low-pass and band-pass 

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Design of generalized Chebyshev lumped element filters by computer optimisation

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