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1、,B,TUTORIALA GENERAL DESIGNPROCEDURE FOR BANDPASSFILTERS DERIVEDFROM LOW PASS PROTOTYPEELEMENTS:PART I,andpass filters serve a variety of func-tions in communication,radar and in-strumentation subsystems.Of the avail-able techniques for the design of bandpass fil-ters,those techniques based upon the
2、 lowpass elements of a prototype filter have yield-ed successful results in a wide range of appli-cations.The low pass prototype elements arethe normalized values of the circuit compo-nents of a filter that have been synthesized fora unique passband response,and in some cas-es,a unique out-of-band r
3、esponse.The lowpass prototype elements are available to thedesigner in a number of tabulated sources 1,2,3and are generally given in a normalized for-mat,that is,mathematically related to aparameter of the filter prototype.,thesized for a unique filter parameter.A num-ber of illustrated examples are
4、 offered to vali-date the design procedure.LOW PASS PROTOTYPE FILTERSLow pass prototype filters are lumped ele-ment networks that have been synthesized toprovide a desired filter transfer function.Theelement values have been normalized with re-spect to one or more filter design parameters(cutoff fre
5、quency,for example)to offer thegreatest flexibility,ease of use and tabulation.The elements of the low pass prototype filterare the capacitors and inductors of the laddernetworks of the synthesized filter networks asshown in Figure 1.This diagram also depicts,Fig.1 Circuit topologiesof low pass prot
6、otypefilters.w,This article presents a general design pro-cedure for bandpass filters derived from lowpass prototype filters,which have been syn-,Bandpass filters serve a,g0=1.00,g2=L1,gn1,gn+1=1.00,variety of functions in,+e1(s),g1=C1,g3=C2,gn2,gn,+e2(s),communication,radar andinstrumentation subsy
7、stems.,+e1(s),g0=1.00,g1=L1,g3=L2g2=C1,gn2,gngn1,n+1+e2,=1.00,K.V.PUGLIAM/A-COM Inc.,Lowell,MAReprinted with permission of MICROWAVE JOURNAL from the December 2000 issue.2000 Horizon House Publications,Inc.,ATTENUATION(dB),SPARAMETERS(dB),C1=,R 1,C2=,R 1,and,r,10,1,(),e2(),e1(s),f=,TUTORIAL,L(f),(f)
8、,L(f)10,L1=9.78 nH,L2=9.78 nH,be adjusted(de-normalized)in accor-dance with,01020304050,C1,C2=,C3=,R0 1R0 2109L1=0R0 2109,g1g2,60708090100,0,1,2,3,4,5,01020,S11,S21,R0 1R0 2109L 2=0R0 2109,g3g4,v Fig.2,NORMALIZED FREQUENCY f(Hz)Low pass Chebyshev filter,3040,C3=,R0 1R0 2109,g5,prototype response.,50
9、0,0.5,1.0,1.5,2.0,In addition to the tabulated data of,Fig.3 Low pass prototype filterschematic.wL=g2 L2=g4,FREQUENCY(GHz)v Fig.4 Chebyshev low pass filterwith a 1.0 GHz cutoff.,low pass prototype filter elements,the values may be computed via exe-cution of the equations found inMatthaei,et al.1,and
10、 repeated in Ap-pendix A for convenience.The schematic of the filter,which,C1=,C2=g3,C3,g5,=log1 dB 1,was derived from Chebyshev,the lowpass prototype elements and the asso-ciated frequency response are shown,the two possible implementations ofthe low pass prototype filter topolo-gies.In both cases,
11、the network trans-fer function isTs=wheres=+j,the Laplace complexfrequency variableClearly,the transfer function,T(s),isa polynomial of order n,where n isthe number of elements of the lowpass filter prototype.The illustrated circuit topologiesrepresent a filter prototype contain-ing an odd number of
12、 circuit ele-ments.To represent an even numberof elements of the prototype filter,simply remove the last capacitor orinductor of the ladder network.For purposes of illustration,an ex-ample representing a Chebyshev fil-ter is offered.The power transferfunction of the Chebyshev filter maybe represente
13、d byT(f)=10log1+cos2ncos1(f)for f 1.0T(f)=10log1+cosh2ncosh1(f)for f 1.0,whererdB=inband ripple factor in decibelsThese equations represent thepower transfer function of theChebyshev low pass prototype filterwith normalized filter cutoff frequen-cy f of 1.0 Hz.A graphical represen-tation of the powe
14、r transfer functionof the Chebyshev low pass prototypefilter is shown in Figure 2.The low pass prototype filter para-meters for the low pass Chebyshevfilter example aren=5R0=1.0 andrdB=0.5A schematic representation of theprototype Chebyshev filter is shownin Figure 3.The prototype elementsare from M
15、atthaei,Young and Jones1where the normalized cutoff frequen-cy is given in the radian format 1=1.0=2f1.If this five-section prototype filterwere constructed from available ta-bles of elements and a circuit simula-tion performed,the transfer functionwould be exactly as represented inthe schematic.To
16、construct the filterat another frequency(1.0 GHz,forexample)and circuit impedance level(R0=50),the element values must,in Figure 4.Note that the 0.5 dB in-band ripple results in a return loss of10 dB as expected.A low pass filter may be convertedto a bandpass filter by employing asuitable mapping fu
17、nction.A map-ping function is simply a mathemati-cal change of variables such that atransfer function may be shifted infrequency.The mapping functionmay be intuitively or mathematicallyderived.A known low pass to band-pass mapping function may be illus-trated mathematically asf0 f f0 f f0 f wheref0=
18、f1f2f=f2 f1and f0,f1 and f2 represent the center,lower cutoff and higher cutoff fre-quencies of the corresponding band-pass filter,respectively.If the substi-tution of variables is made within theChebyshev power transfer function,the power transfer function of thecorresponding bandpass filter may be
19、determined as shown in Figure 5.The schematic diagram of thebandpass filter,which was derivedfrom the low pass prototype filter viathe introduction of complementary el-ements and producing shunt and se-,ATTENUATION(dB),Z,E,10,0,Zin C,Yin,Yin,C,L,Yin,C,C,0 dX(),=,=,0C,0L,(cot 0+0 csc 0,f0,Lf=10 log,1
20、 l,Y0,2,),TUTORIALtributed resonators are shown in Fig-,L(fa),(fa),L(fa)10,ure 7.Resonators may be characterized,by their unloaded quality factor Qu,which is the ratio of the energy stored,20304050607080901005 4 3 2 1 0 1 2 3 4 5NORMALIZED FREQUENCY fa(Hz)v Fig.5 Chebyshev bandpass filtertransfer fu
21、nction.ries resonators,is shown in Figure 6.This is a basic low pass to bandpasstransformation,and unfortunatelysometimes leads to component values,which are not readily available or haveexcessive loss.As described later,themapping function need not be consid-ered as part of the bandpass filter de-s
22、ign procedure.It is presented here asa supplement to the filter theory.It bears repeating that the lowpass prototype filter elements,that is,the g-values,are the result of networksynthesis techniques to produce a de-sired characteristic of the prototype,v Fig.6 Chebyshev bandpassfilter schematic.LZ,
23、EE/4 E=/2Zin OPEN Zin(a)LZ,EE=(b)v Fig.7 Typical resonant circuit;(a)seriesand(b)shunt resonators.filter transfer function.These desiredcharacteristics might include a flatamplitude response,maximum out-of-band rejection,linear phase re-sponse,Gaussian or other amplituderesponse,minimum time sidelob
24、esand matched signal filters.RF AND MICROWAVERESONATORSRF and microwave resonators arelumped element networks or distrib-uted circuit structures that exhibitminimum or maxi-,to the energy dissipated per cycle ofthe resonant frequency.Resonatorsare also characterized with respect totheir reactance or
25、 susceptance slope parameters,which are defined,respectively,as2 d=0and0 dB()2 d=0These are important resonator para-meters because they influence Quand the coupling factor between res-onators in multiple resonator filters.Table 1 provides the reactance andsusceptance slope parameters ofsome common
26、lumped element anddistributed resonators.Qu may also be defined in terms ofthe reactance or susceptance slopeparameter asQu=RshRsewhere,TABLE IRESONATOR SLOPE PARAMETERS,mum real imped-ance at a single fre-quency or at multi-ple frequencies.,Rse=resonator series resistanceRsh=resonator shunt resista
27、nceTogether these resistances represent,Resonator Type Reactance Slope,1Series LC 0L or,Susceptance Slope,The resonant fre-quency f0 is the fre-quency at which theinput impedance or,the resonator loss.The bandpass fil-ter design examples will illustrate theutility of the slope parameters.In many ban
28、dpass filter applica-,Shunt LCShunt,0C or2 0C,1,admittance is real.The resonant fre-quency may be fur-ther defined interms of a series or,tions,particularly those applicationswhere the filter is deployed at thefront end of a receiver,it is importantto know the Qu for the resonators inorder to accura
29、tely estimate the in-,/2 line(short)/2 line(open)/8 line(short+C)/4 line(short)/2 WG line(short),Z 02 Z0 g0 2 0,Y02Y04,2,shunt mode of reso-nance;the seriesmode is associatedwith small values ofinput resistance atthe resonant fre-quency,while theshunt mode is asso-ciated with largevalues of resistan
30、ceat the resonant fre-quency.Some typi-cal lumped and dis-,sertion loss of the filter.The insertionloss of a single transmission resonatorcan be mathematically represented as 2 1+2Ql f f0 Q 2 Qu At f=f0,this equation may furtherreduce to,La(f)(dB),0,()(),Y0,2,1,4,70,1,1,Qu,f0,f,J,f,TUTORIAL,1942,195
31、8,COUPLING,RESONATOR,COUPLING,1,0.470,234,3.470,Rsh,0/4,C,51925 1935 1945 1955 1965 1975,FREQUENCY(MHz),v Fig.10 The single resonatorsequivalent circuit.,v Fig.8,/8A single resonator at 1960 MHz.,v Fig.9 The single resonators measuredperformance.imum insertion loss and minimumsize as critical design
32、 parameters.Rectangular,coaxial/8 resonatorswere determined to offer the mini-mum volume in an eight-resonatorfilter consistent with the maximum,where=cot 0+0 csc 2 0=0.01836for 0=and Y0=Mhosor,L()=10log 2 Ql Qu Solving for Qu,yields Ql L(f0)1 10 20 Therefore,a measurement of thesingle resonator ins
33、ertion loss at theresonant frequency L(f0)and the 3dB bandwidth f is sufficient to accu-rately determine Qu of any resonantstructure.The loaded quality factorQl may be determined from a mea-surement of the resonant frequencyand the 3 dB bandwidth from theequationQl=wheref=3 dB bandwidthThis measurem
34、ent technique mayalso be employed to compare thequality factors(Qu)of different typesof resonant structures or as a methodof comparing various plating or man-ufacturing techniques for the filter ifinsertion loss is a critical parameter.Consider an example.A PCS1900transmit filter was required with m
35、in-,insertion loss requirement of lessthan 1.0 dB at the center frequencyof 1960 MHz.A single resonator wasconstructed of the type anticipated tobe used within the filter.The singleresonator is shown in Figure 8.A single resonator structure wasfabricated and plated with silver inorder to obtain the
36、maximum Qu.The resonant structure was tuned tothe desired center frequency and theinsertion loss L(f0)and 3 dB band-width were measured.The measure-ment data is shown in Figure 9whereL(f0=1.950 GHz)=0.533 dBf=14.50 MHzQu 2250.In executing the measurement,threenotes of caution are required in theinte
37、rest of accuracy:The couplingprobes to the cavity should be equaland minimized to avoid load andsource resistance across the res-onator;the source and load SWRsshould be kept low;and the inputSWR at f0 should be minimized toavoid mismatch loss.The equivalent circuit of the singleresonator is shown i
38、n Figure 10.Note that the circuit element,whichrepresents the resonator loss Rsh,hasbeen included.The value of Rsh maybe determined with the aid of thesusceptance slope parameter fromQu=Rsh,Rsh=122.5 kThis measurement technique forestimating the value of Qu iscompletely general and applies tolumped
39、element and distributedresonators.RESONATOR COUPLINGResonator coupling represents oneof the most significant factors affect-ing filter performance.There are sev-eral methods to couple resonators.For ease of manufacturing and tun-ing,a common resonator type andcoupling method is generally prefer-able
40、.Matthaei1 proposes what havebeen termed J(admittance)and K(impedance)inverters both to permita common type of resonator and toserve as coupling elements for theresonators.The J inverters may be represent-ed as the admittance of the elementor the value of the characteristic ad-mittance of a quarter-
41、wavelength linein the equivalent circuit that couplesthe resonators.Similarly,the K in-verters may be represented as the im-pedance of the element or the valueof the characteristic impedance of aquarter-wavelength line that couplesthe resonators.This permits the gen-eral expression of the coupling b
42、e-tween resonators to be mathematical-ly written ask i,i+1=i,i+1,2,=,2,2,/4,(b),/4,/4,/8,Ki,i+1,C C C,20C 2C,()=,kc=,()(),Yse,2,kc=,=,=,=,TUTORIAL,L,C,L,/4,For the special case where 0=/4,the coupling may be written as,C,CC,CrC,C,kc=,2YshYse 1+,=,(a),kc/8,(a),Yse=vCm,2Cm 2CmCg 1+Cg 1+,Another very p
43、opular type of res-onator,which is frequently used inmicrowave bandpass filters,is thequarter-wavelength resonator.Figure,v Fig.11 Coupled(a)-type LCresonators and(b)/8 distributed resonators.,13 illustrates the coupling of sym-metrical/4 resonators and theequivalent circuit.Note that/4 res-onators
44、must be grounded at opposite,kc,(b),Ysh=vCg,ends to prevent the null couplingcondition caused by cancellation ofthe electric and magnetic field,v Fig.13 The symmetrical/4 coupledresonators(a)proximity coupled/4 lineand(b)equivalent circuit.,modes.The coupling coefficient for thisconfiguration may be
45、 written as,(a)Z se=1/vCm/8/8/8Z sh=1/vCg(b)v Fig.12 The coupled/8 transmissionline resonators(a)proximity coupled lineand(b)equivalent circuit.andk i,i+1=for shunt-type and series-type res-onators,respectively,where the cou-pling between the ith and the ith+1resonators is represented by ki,i+1.A si
46、milar approach to the generaldesign of bandpass filters employingcommon types of resonators proposesspecific coupling elements in the caseof lumped resonator bandpass filters,or specific proximity methods of cou-pling in the case of distributed res-onator bandpass filters.To illustrate,consider the
47、coupled-type of L-Cresonators and the coupled/8 trans-mission line distributed resonatorsshown in Figure 11.In the case of the coupled-typeL-C resonators,a series capacitor isinserted between the resonators toperform the coupling function,inwhich case the coupling coefficient isk=0=0=In the case of
48、the coupled/8 trans-mission line resonators,the equivalentcircuit shown in Figure 12 is useful indetermining the coupling coefficient.The coupling coefficient may be deter-mined from the capacitive matrix para-meters associated with the coupledlines,that is,the capacitance per unitlength to ground C
49、g and the mutual ca-pacitance per unit length between theconductors Cm,where is the velocityin the dielectric medium.The couplingcoefficient may be written asYsh cot 0Ysh cot(0)cot 0+0 csc 2 0,Cm Cm 4Cm 4Cm Y0 Cg Cg4For a given coupled line geometry,the/4 lines offer closer couplingthan the comb-lin
50、e configuration.The input,output and adjacentresonator coupling in a multi-elementbandpass filter are the parametersthat determine the amplitude,phaseand SWR over the passband of thefilter.This statement understates theimportance of resonator coupling tothe bandpass filter parameters.Recall that the
