Design of RF and Microwave Filters
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1 Desn of RF and Mcrowave Flters
2 차례. Introducton ; types of Flters Characterzaton of Flters Approxmate Desn Methods Lowpass Prototype Network Impedance Scaln frequency Expanson Lowpass to hhpass transformaton Lowpass to bandpass transformaton Lowpass to bandstop transformaton Immtance Inverters Bandpass flters usn J-, K- nverters LC flters Dstrbute flters Mcrowave & Mllmeter-wave Lab.
3 . 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
4 . Flter Characterzaton Two-port Network ; Input H( ) Output H( ) H( ) e ( ) F. Two-port Network Mcrowave & Mllmeter-wave Lab. 4
5 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
6 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
7 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
8 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
9 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
10 Approxmaton methods : ) Maxmally Flat (Butterworth) response ) Chebyshev response 3) Ellptc Functon response Mcrowave & Mllmeter-wave Lab.
11 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.
12 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
13 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
14 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
15 4. Lowpass Prototype Flter R L, rad / s c ; normalzed to R a N =... a' R a N = a'... F. 5 Lowpass prototype Mcrowave & Mllmeter-wave Lab. 5
16 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
17 Table. Element values for Butterworth and chebyshev flters (n=5) Element No Type Butterworth. db rpple Chebyshev.5 db rpple Chebyshev Mcrowave & Mllmeter-wave Lab. 7
18 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
19 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
20 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.
21 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.
22 Mcrowave & Mllmeter-wave Lab. 5.4 Lowpass to bandpass transformaton (low cutoff freq., hh cutoff freq. ) mappn functon ; ) ( f and, ' '
23 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
24 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, , 4. step : apply mpedance scaln L Z H, C / Z. F, L 3Z 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
25 Mcrowave & Mllmeter-wave Lab Lowpass to bandstop transformaton (low cutoff freq., hh cutoff freq. ) mappn functon ; nverse of bandpass mappn functon ) ( f and, ' '
26 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
27 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
28 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
29 R a 4... n Z' n (') 3 5 n+... n- a' F. Lowpass network and bandpass network Mcrowave & Mllmeter-wave Lab. 9
30 Mcrowave & Mllmeter-wave Lab. 3 3 n n n Y n n n Y Input admttance of LPF Prototype
31 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
32 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
33 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
34 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
35 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
36 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
37 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
38 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
39 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 3.3 GHz : 5 db Mcrowave & Mllmeter-wave Lab. 39
40 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
41 7 db Mcrowave & Mllmeter-wave Lab. 4
42 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
43 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
44 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
45 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
46 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
47 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
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