Desn of RF and Mcrowave Flters
차례. Introducton ; types of Flters --------------------------------------. Characterzaton of Flters ------------------------------------------ 3. Approxmate Desn Methods -------------------------------------- 4. Lowpass Prototype Network --------------------------------------- 5.. Impedance Scaln ------------------------------------------------ 5.. frequency Expanson ---------------------------------------------- 5.3. Lowpass to hhpass transformaton ----------------------------- 5.4. Lowpass to bandpass transformaton ---------------------------- 5.5. Lowpass to bandstop transformaton ----------------------------- 5.6. Immtance Inverters ---------------------------------------------- 5.7. Bandpass flters usn J-, K- nverters --------------------------- 6.. LC flters ----------------------------------------------------------- 6.. Dstrbute flters --------------------------------------------------- 3 4 8 5 8 9 5 7 8 33 4 Mcrowave & Mllmeter-wave Lab.
. Introducton Types of Flters A. Lowpass Flters B. Hhpass Flters attenuaton attenuaton passband transton band stopband stopband transton band passband freq c ; cutoff cutoff C. Bandpass Flters D. Bandstop Flters freq atten transton band transton band atten transton band transton band stopband stopband passband passband passband stopband freq freq f f f f Mcrowave & Mllmeter-wave Lab. 3
. Flter Characterzaton Two-port Network ; Input H( ) Output H( ) H( ) e ( ) F. Two-port Network Mcrowave & Mllmeter-wave Lab. 4
Characterstcs of deal bandpass flters ; H( ) for for f f f f, f f f and ( ) d lh()l () Freq. F. Characterstcs of deal bandpass flter not realzable approxmaton requred Mcrowave & Mllmeter-wave Lab. 5
Practcal specfcatons ; ) Passband ; lower cutoff frequency - upper cutoff frequency ) Inserton loss lo H( ) ( db) ; must be as small as possble 3) Return Loss lo ( db) ; deree of mpedance matchn 4) Rpple f f ; varaton of nserton loss wthn the passband Mcrowave & Mllmeter-wave Lab. 6
d ( ) 5) Group delay d d ; tme to requred to pass the flter 6) Skrt frequency characterstcs ; depends on the system specfcatons 7) Power handln capablty Mcrowave & Mllmeter-wave Lab. 7
3. Approxmate Desn Methods ) based on Ampltude characterstcs A. Imae parameter method B. Inserton loss method a) J-K nverters b) Unt element - Kuroda dentty ) based on Lnear Phase characterstcs Mcrowave & Mllmeter-wave Lab. 8
3. Flter Desn based on the nserton loss Defnton of Power Loss Rato (P LR ) ; mpedance matchn as well as frequency selectvty P n [S ] P ref l P trans F. 3 General flter network P refl P n S P n P trans T P n S P n P LR P P n tran N( ) D( ) network synthess procedures are requred Mcrowave & Mllmeter-wave Lab. 9
Approxmaton methods : ) Maxmally Flat (Butterworth) response ) Chebyshev response 3) Ellptc Functon response Mcrowave & Mllmeter-wave Lab.
3. Approxmaton Methods A. Maxmally flat response P LR Chebyshev Maxmally flat.5.5 c P LR k c N Where, k ;passband tolerance N ; order of flter Usually k deree of freedom= (order N) F. 4 Comparson Between Maxmally Flat and Chebyshev response Mcrowave & Mllmeter-wave Lab.
Mcrowave & Mllmeter-wave Lab. B. Chebyshev response : equal rpple response n the passband : Chebyshev Polynomal of order LR T N k P N T N ) ( ) ( ) ( 3 4 ) (,, ) ( 3 3 x T x xt x T x x x T x T x x T n n n ; rpple (. db,. db, etc.) ; order of flter deree of freedom= (rpple and order) k N
attenuaton P LR s +k - c Chebyshev Response, N=4 p s Ellptc functon response N=5 F. 5 Chebyshev and Ellptc Functon response Mcrowave & Mllmeter-wave Lab. 3
C. Ellptc Functon response equal rpple passband n both passband and stopband s s : stopband mnmum attenuaton : transmsson zero at stopband deree of freedom=3 (order N, rpple, transmsson zero at stopband s ) Mcrowave & Mllmeter-wave Lab. 4
4. Lowpass Prototype Flter R L, rad / s c ; normalzed to R a N... 6 4 5 3 =... a' R a... 7 5 3 6 4 N = a'... F. 5 Lowpass prototype Mcrowave & Mllmeter-wave Lab. 5
Mcrowave & Mllmeter-wave Lab. 6 Maxmally Flat response ; Equal Rpple response ; R P L N LR ), (,...,,, sn F H N N even odd, ) ( N k k k N R T k P L N LR ln, snh, sn, 4 k k N b N a b a a
Table. Element values for Butterworth and chebyshev flters (n=5) Element No Type Butterworth. db rpple Chebyshev.5 db rpple Chebyshev.68.468.758.68.37.96 3..975.548 4.68.37.96 5.68.468.758 Mcrowave & Mllmeter-wave Lab. 7
5. Impedance Scaln and Frequency Mappn 5. Impedance Scaln R L RL 5 Impedance level 5 ; same reflecton coeffcent mantaned seres branch (mpedance) elements ; 5 5 shunt branch (admttance) elements ; r r / 5 r r /5 Mcrowave & Mllmeter-wave Lab. 8
5. Frequency Expanson cutoff frequency lowpass cutoff frequency c mappn functon ; f ( seres and shunt branch elements ; ) c c c Mcrowave & Mllmeter-wave Lab. 9
P LR P LR ' - (a) Lowpass Prototype response - c c (b) Frequency expanson P LR P LR - c c (c) Lowpass to Hhpass transformaton - (d) Lowpass to Bandpass Transformaton F. 6 Varous mappn relatons derved from lowpass prototype network Mcrowave & Mllmeter-wave Lab.
5.3 Lowpass to Hhpass transformaton (lowpass cutoff freq. hhpass cutoff freq. ) mappn functon ; f ( ) c / seres branch (mpedance) elements ; c ( c / ) /( c shunt branch (admttance) elements ; r ( c / ) r r /( cr ) ) R... 5 ' 3 ' ' N ' 4 ' ' R L =... F. 7 Hhpass flter derved from lowpass prototype Mcrowave & Mllmeter-wave Lab.
Mcrowave & Mllmeter-wave Lab. 5.4 Lowpass to bandpass transformaton (low cutoff freq., hh cutoff freq. ) mappn functon ; ) ( f and, ' '
Mcrowave & Mllmeter-wave Lab. 3 seres branch element : mpedance shunt branch element : admttance s s C L ; p p r r r r r L C ;...... R C N R L = C L L 3 L 5 C 4 L 4 C 5 C C 3 L L N F. 8 Bandpass flter derved from the lowpass prototype
Example : Desn a bandpass flter havn a.5db equal-rpple response, wth N=3. The f s GHz, bandwdth s %, and the nput and output mpedance 5Ω. step : from the element values of lowpass prototype (.5dB rpple Chebyshev).5963,.967, 3.5963, 4. step : apply mpedance scaln L Z.59635 79.85 H, C / Z. F, L 3Z 79.85 H step 3 : apply bandpass transformaton L ' L C L C ' C / 7 nh ' / L ' / C.99 pf C.76 nh / 34.9 pf ( ) / L 3 ' 3 ' R=5 L 3 '=7nH C 3 '=.99pF L '=.76nH C '=.99pF L '=7nH R L =5 C '=34.9pF Mcrowave & Mllmeter-wave Lab. 4
Mcrowave & Mllmeter-wave Lab. 5 5.5 Lowpass to bandstop transformaton (low cutoff freq., hh cutoff freq. ) mappn functon ; nverse of bandpass mappn functon ) ( f and, ' '
Mcrowave & Mllmeter-wave Lab. 6 seres branch element : admttance shunt branch element : mpedance s s L C ; - p p r r r r r C L ; - F. 9 Bandstop network derved from the lowpass prototype...... R C N R L = C L L 3 L 5 C4 L 4 C 5 C C 3 L L N
5.6 Immtance Inverters Z n (or Y n ) K(or J) Z L (or Y L ) mmttance nverter F. Immtance nverter K ; mpedance nverter J ; admttance nverter Z K / n Z L Y J / n Y L ex. smplest form of nverter : λ /4 transformer seres LC J-nverter + shunt LC shunt LC K-nverter + seres LC Mcrowave & Mllmeter-wave Lab. 7
5.7 Bandpass flters usn J-, K-nverters Lossless Lossless Z n () or Lowpass n+ R Bandpass R n+ low Network Network Z n '() or band F. Equvalent Network for lowpass prototype and bandpass network Reflecton coeffcent ; lowpass : Low Z Z n n ( ') / ( ') / bandpass : Band Z Z n n ( ) / R ( ) / R If Z n ( ') / Zn ( ) / R Low( ') Band ( ) (mappn relaton) Mcrowave & Mllmeter-wave Lab. 8
R a 4... n Z' n (') 3 5 n+... n- a' F. Lowpass network and bandpass network Mcrowave & Mllmeter-wave Lab. 9
Mcrowave & Mllmeter-wave Lab. 3 3 n n n Y 4 3 3 n n n Y Input admttance of LPF Prototype
Input admttance of BPF usn J-Inverter Y n B Y n r B r J B r 3 J J J J B / B J 3 rn / B J 34 J G n, n G B n, n B r / BrBr J 3 / Br Br 3 J 34 r 3 Br 4 / B rn J L C B r L Y G n A Y G n G B G bw A r A B B b b J W J r r, nn, b =ω C r G B G b W B rn B n n n Mcrowave & Mllmeter-wave Lab. 3
Mcrowave & Mllmeter-wave Lab. 3 From the partal fracton expanson ncludn bandpass mappn relaton : fractonal bandwdth, : center frequency In the same manner, J-nverter values are derved as,,,, n n n n n n L W R K L L W K L R W K,,,, n n n n n n C W G J C C W J C G W J W
Typcal mmttance nverters ; -C -C -L -L C L K / C K L Z X=neatve Z X=postve =postve =neatve F. 3 Impedance(K-) nverters Mcrowave & Mllmeter-wave Lab. 33
C L -C -C -L -L J C J / L / / / / Y B=postve Y B=neatve =neatve =postve F. 4 Admttance(J-) nverters Mcrowave & Mllmeter-wave Lab. 34
6.. LC flters A. C-coupled bandpass flters Y... L L C C J J J n,n+ L4... L n C n Y b F. 4 Bandpass flter network usn deal J-nverters Y C... J L L C C -C L n C n J n,n+ Y b J-nverter F. 5 Bandpass flter network contann practcal nverters... Mcrowave & Mllmeter-wave Lab. 35
Y n Y n C -C a ' Y a J Y a C L L C a ' C F. 6 Inverter of frst and last staes Y n J Y a C a ' Y n / Y a / C C C / Y C / Y / Y a a C a By equatn the real and manary part of C a ' C, J C f C Y a Yn and Y n Mcrowave & Mllmeter-wave Lab. 36
C C C 3... C n+ Z a C p C p L p L p L pn C pn Z b F.7 C-coupled Bandpass flter B. L-coupled bandpass flter L L L 3... L n+ Z a C p C p L p L p L pn C pn Z b F.8 L-coupled Bandpass flter Mcrowave & Mllmeter-wave Lab. 37
Example : Desn a LC bandpass flter. The f s.8 GHz, bandwdth s 5 MHz, and the nput and output mpedance 5Ω. step : from the element values of lowpass prototype step : apply mpedance scaln step 3 : apply bandpass transformaton usn J-nverters Step 4 : smulaton Mcrowave & Mllmeter-wave Lab. 38
6.8 nh ar-col.7 nh chp 5 pf 6.8 nh ar-col.7 nh chp.5 pf pf 6.8 nh ar-col.7 nh chp.5 pf pf 6.8 nh ar-col.7 nh chp 5 pf.5 pf.5 pf Step 5 : Realzaton pf.5 pf pf.5 pf.5 pf pf.5 pf pf Inserton loss < 3. db Return loss > 5.5 db Attenuaton @ 3.3 GHz : 5 db Mcrowave & Mllmeter-wave Lab. 39
6.8 nh ar-col pf 6.8 nh ar-col pf 6.8 nh ar-col pf 6.8 nh ar-col.7 nh chp 6.8 nh ar-col.7 nh chp.5 pf.5 pf.5 pf pf 6.8 nh ar-col.7 nh chp Step 6. mprovement pf.5 pf.5 pf.5 pf.5 pf.5 pf pf 9.5 nh ar-col 9.5 nh ar-col 6.8 nh ar-col C-coupln LC flter + L-coupln = LC flter Mcrowave & Mllmeter-wave Lab. 4
7 db Mcrowave & Mllmeter-wave Lab. 4
6. Dstrbuted flters At mcrowave frequences : Resonators made of Lumped elements are lossy(low Q) or bulky Dstrbuted Resonators Dstrbuted resonators ; quarter-wavelenth or halfwavelenth transmsson lnes such as mcrostrp lnes, coaxal lnes and waveudes Mcrowave & Mllmeter-wave Lab. 4
A. Comblne flters : cellular base statons as well as handy phone conductor ar or ceramc a F. 9 (a) Top vew of Comblne Flter conductor F. 7(a) Top Vew of Comblne Flter tunn screw L F. 7(b) Sde Vew of Comblne Flter F. 9 (b) Sde vew of Comblne Flter Mcrowave & Mllmeter-wave Lab. 43
Instead of lumped element nductors dstrbuted nductors (L < λ/4) are used. In Y oe (Y oe -Y oo )/ out Y oe Y oo In Y oe Y oe Out F. 8 Coupled lne F. Coupled lne Overall equvalent crcut : F. Equvalent crcut of F. L c C C C 3 C 4 L c L c3 L c4 L c5 C c C c C c3 L L L 3 L 4 F. Equvalent crcut of F. 9 Mcrowave & Mllmeter-wave Lab. 44
B. Mcrostrp flters : Compact, lht weht and low cost F. Sde-couple mcrostrp flter F. 3 Sde-coupled Mcrostrp flter Mcrowave & Mllmeter-wave Lab. 45
Practcal specfcatons ; ) Passband ; lower cutoff frequency - upper cutoff frequency ) Inserton loss lo H( ) ( db) ; must be as small as possble 3) Return Loss lo ( db) ; deree of mpedance matchn 4) Rpple f f ; varaton of nserton loss wthn the passband Mcrowave & Mllmeter-wave Lab. 46
d ( ) 5) Group delay d d ; tme to requred to pass the flter 6) Skrt frequency characterstcs ; depends on the system specfcatons 7) Power handln capablty Mcrowave & Mllmeter-wave Lab. 47