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Microwave Filter

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Filter Design
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  RF Engineering – Passive Circuit Microstrip Filter Design 1.0 Introduction to Filter A filter is a network that provides perfect transmission for signal with frequencies incertain passand region and infinite attenuation in the stopand regions! uch idealcharacteristics cannot e attained# and the goal of filter design is to appro$imate the idealrequirements to within an acceptale tolerance! Filters are used in all frequenc% rangesand are categori&ed into three main groups' ã (ow)pass filter *(PF+ that transmit all signals etween DC and some upper limit ω c and attenuate all signals with frequencies aove ω c ! ã ,igh)pass filter *,PF+ that pass all signal with frequencies aove the cutoff value ω c and re-ect signal with frequencies elow ω c ! ã .and)pass filter *.PF+ that passes signal with frequencies in the range of ω /  to ω 0  andre-ect frequencies outside this range! 1he complement to and)pass filter is the and)re-ect or and)stop filter!2n each of these categories the filter can e further divided into active and passive t%pe!1he output power of passive filter will alwa%s e less than the input power while activefilter allows power gain! 2n this la we will onl% discuss passive filter! 1he characteristicof a passive filter can e descried using the transfer function approach or the attenuationfunction approach! 2n low frequenc% circuit the transfer function *,* ω ++ description isused while at microwave frequenc% the attenuation function description is preferred!Figure /!/a to Figure /!/c show the characteristics of the three filter categories! 3ote thatthe characteristics shown are for passive filter! Figure 1.1A  – A low)pass filter frequenc% response! F! 4ungMa% 0556/ A Filter  ,* ω + 7 / * ω +7 0 * ω + ( ) ( )( ) ω ω ω  /0 V V  H   = ( )( )        −= ω ω  /0/5 05 V V  Log n Attenuatio ω c 8,* ω +8 ω /1ransfer functionAttenuation9d. ω 5 ω c :/505:5;5  RF Engineering – Passive Circuit Microstrip Filter Design Figure 1.1B  – A high)pass filter frequenc% response! Figure 1.1C  – A and)pass filter frequenc% response! 2.0 Realization of Filters At frequenc% elow /!5<,&# filters are usuall% implemented using lumped elements suchas resistors# inductors and capacitors! For active filters# operational amplifier issometimes used! 1here are essentiall% two low)frequenc% filter s%ntheses techniques incommon use! 1hese are referred to as the image)parameter method *2PM+ and theinsertion)loss method *2(M+! 1he image)parameter method provides a relativel% simplefilter design approach ut has the disadvantage that an aritrar% frequenc% responsecannot e incorporated into the design! 1he 2PM approach divides a filter into a cascadeof two)port networks# and attempt to come up with the schematic of each two)port# suchthat when comined# give the required frequenc% response! 1he insertion)loss method egins with a complete specification of a ph%sicall% reali&ale frequenc% characteristic#and from this a suitale filter schematic is s%nthesi&ed! Again we will ignore the image parameter method and onl% concentrate on the insertion loss method# whose design procedure is ased on the attenuation response or insertion loss of a filter! 1he insertionloss of a two)port network is given %' ( ) 0 //loadtodeliveredPower sourcethefromavailalePower ω  Γ −=== load inc IL  P  P  P  *0!/+=here Γ   is the reflection coefficient looking into the filter *we assume no loss in thefilter+! F! 4ungMa% 05560 ω Attenuation9d. 5 ω c :/505:5;5 ω c 8,* ω +8 ω /1ransfer function ω / 8,* ω +8 ω   /1ransfer function ω 0 ω Attenuation9d. 5 ω / :/505:5;5 ω 0  RF Engineering – Passive Circuit Microstrip Filter Design Design of a filter using the insertion)loss approach usuall% egins % designing anormali&ed low)pass protot%pe *(PP+! 1he (PP is a low)pass filter with source and loadresistance of / Ω  and cutoff frequenc% of / Radian9s! Figure 0!/ shows the characteristics!2mpedance transformation and frequenc% scaling are then applied to denormali&e the (PPand s%nthesi&e different t%pe of filters with different cutoff frequencies! Figure 2.1  – A normali&ed (PP filter network with unit% cutoff frequenc% */Radian9s+!(ow)pass protot%pe *(PP+ filters have the form shown in Figure 0!0 *Analternative network where the position of inductor and capacitor is interchanged is alsoapplicale+! 1he network consists of reactive elements forming a ladder# usuall% knownas a ladder network! 1he order of the network corresponds to the numer of reactiveelements! 2mpedance transformation and frequenc% scaling are then applied to transformthe network to non)unit% cutoff frequenc%# non)unit% source9load resistance and to other t%pes of filters such as high)pass# and)pass or and)stop! E$amples of high)pass and and)pass filter networks are shown in Figure 0!: and Figure 0!; respectivel%!R  Figure 2.2  – (ow)pass protot%pe using (C elements! F! 4ungMa% 0556: A Filter  ,* ω + 7 / * ω +7 0 * ω + R   >/R  (  >/Attenuation9d. ω 5 ω c  > / :/505:5;5(   / >g 0 ( 0 >g ; C / >g / C 0 >g : R  ( > g  3?/ /(   / >g / ( 0 >g : C / >g 0 C 0 >g ; R  ( > g  3?/ g 5 > /  RF Engineering – Passive Circuit Microstrip Filter Design Figure 2.3  – E$ample of high)pass filter# note the position of inductor and capacitor isinterchanged as compared with low pass filter! Figure 2.4  – E$ample of and pass)filter# the capacitor is replaced with parallel (Cnetwork while the inductor is replaced with series (C network! 3.0 Brief Overview of ow! ass rotot#$e Filter %esign &sing u'$ed (le'ents 1here are a numer of standard approaches to design a normali&ed (PP of Figure 0!: thatappro$imate an ideal low)pass filter response with cutoff frequenc% of unit%! Among thewell known methods are' ã Ma$imall% flat or .utterworth function! ã Equal ripple or Che%shev approach! ã Elliptic function!=e will not go into the details of each approach as man% ooks have covered them!2nterested reader can refer to reference @:# which is a classic te$t on network anal%sis or @;# a more advance version! 1he asic idea is to appro$imate the ideal amplituderesponse 8,* ω +8 0  of an amplifier using pol%nomials such as .utterworth# Che%shev#.essel and other orthogonal pol%nomial functions! 1his is usuall% given as' +*/+* +*+* ω ω ω ω   N ooio  P C  K V V  H  +== *:!/+,ere 4  o  and C o  are constants and P  3 * ω + is a pol%nomial in ω  of order 3! 4  o  and C o  areusuall% dependent on the t%pe of pol%nomial used! A comparison of appro$imating the(PP amplitude response with .utterworth# .essel and Che%shev pol%nomials isillustrated in Figure :!/! F! 4ungMa% 0556; ( 0 ( / C / C  3 C 0 ( 0 ( / C / ( : C : C  3 (  3
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