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1 MWE 5 WEB- Introduction to Theory, Analysis, and Design of Microwave Filters in New Era of Wireless Masataka OHIRA Graduate School of Science and Engineering, Saitama University () () (3) () (5) Q Attenuation (db) 3 Return loss Insertion loss Specifications f f Δf f L Rmin L Amax LAtt M S M S M SN M SL M S M M M N M L M S M M M N M L [M]= M SN M N M N M NN M NL M SL M L M L M NL 5 Dielectric substrate s 3 lx 6 s w 3 ly Input l s ly Via hole lx Abstract In this tutorial lecture, the basic theory, analysis, and design method of microwave bandpass filters are described for easy learning from the knowledge acquired through study of electric circuits For the future step to the design of advanced microwave filters, the lecture is based on the coupling matrix used for the designs of recent coupled resonator filters More specifically, it contains five parts: () typical transfer functions employed for approximation of filter responses, () examples of coupling topologies to realize the transfer functions, (3) circuit analysis using the coupling matrix, () circuit synthesis, and (5) physical dimension design of microwave filters based on coupling coefficients and external Q factors Design examples are provided to demonstrate microstrip filter designs with the effective use of EM simulators Keywords Microwave filters, bandpass filters, transfer functions, coupling topology, coupling matrix Background knowledge Electric circuits and distributed circuits

2 Source Input (port) (port) 5 [] [7] classical filter [5], [7] [7] [9] advanced filter S S insertion loss return loss L A(ω) = log S (ω) (db) () L R(ω) = log S (ω) (db) () passband db stopband db () () (3) Insertion loss (db) Insertion loss (db) Passband Stopband f c Stopband Passband Stopband f S Filter S Load f Insertion loss (db) Insertion loss (db) Stopband Passband f c Passband Stopband Passband f (d),,, (d) microwave filter () lowpass filter LPF f c f c () highpass filter HPF f c f c (3) bandpass filter BPF f f () bandstop filter BSF bandrejection filter BRF f f 3 () narrowband filter % f

3 Attenuation (db) 3 Return loss Insertion loss Specifications f f f Δf L Rmin L Amax LAtt 3 attenuation 3 db L Att () wideband filter or broadband filter % % 3 3 design specification 3 () center frequency f f f f = f f f f f = () frequency bandwidth Δf =f f f f fractional bandwidth FBW = Δf f f Δf % (3) 3 db L Amax () 3 db L Rmin group delay = Q = = 3 () N N N () Q Q u unloaded quality(q) factor Q Q Q 3 Q Q (3) transfer function db db () reflection zero RZ attenuation zero () transmission zero TZ attenuation pole

4 S, S S, S 8 6 Δf f S S Reflection zero S, S 8 6 Δf f S S Reflection zeros S S 8 6 S S Tranmission zeros Reflection zeros f S, S 8 6 S S Transmission zeros Reflection zeros f 5 S S (A) (B) (A-) (A-) Butterworth response maximally flat response Δf Chebyshev response equiripple response Δf N (B-) elliptic function response 5 (B-) general Chebyshev function response 5 3 N [] 6 Q Q u = N N Q Q u = N 6 Q 6 Q Q u N 6(d) N N =5 N =8 pseudo-elliptic function filter

5 Insertion loss (db) Insertion loss (db) N= 3 FBW= f f Q u = Q u = N=5 Insertion loss (db) Insertion loss (db) N= f (d) Q u = FBW=5 w/o TZs Q u = N=5 w/ TZs N=8 N=5 6 N Q Q u =, N Q u =, Q u =, (d) Q u = 5 coupling topology = = 7 direct coupled resonator filter source load,,, N Q 3 () coupling coefficient m i,i+ i (i +) i=,,,n () Q external Q factor Q es Q el 3 k [M] m m f m3 3 m3 m5 m m6 S Q Source es m Q es S Source 7 m3 m Q es m Q el m3 mn-,n S N L Source Load 3 m3 m5 5 m56 6 QeL m5 (d) m56 5 L Load m m58 m Q es S Source m78 Q el 8 L Load S Source m3 m3 m m3 m56 Folded CT (d) CQ, (e) Q msn ms ms N (e) m SL mln ml ml Q el L Load 7 (e) 7 CT cascaded trisection (d) CQ cascaded quadruplet (3) nonadjacent coupling m ij j = i + 7(e) transversal array filter [] m SL N n max = N n min [7] N n min n min = n max = N L Load

6 C L C L S C L C L L Q es m Q el S L i L L i R L e Source Load S 8, () direct source/load coupling R S input/output coupling m SL circuit analysis S i i =, L i C i i i i L R S R L e S R [ ] [ ] S+jωL + jωl jωc jωl R L+jωL + i es = (3) i jωc 9 L S L L L S L L L S = L L = [Z][i] =[e] () [i] [e] [i] =[i S,i,i,i L] T, [e] =[e S,,, ] T (5) Q es m Q el S L Source ms ml Load msl R S e S i S L S L S R S e S C i L L S i S L L SL L S C i L jb jb L L i L L S L L L SL 9,, T [Z] R S jωl S jωl S jωl SL [Z]= jωl S jωl + jωc jωl jωl L jωl S jωl jωl + jωc jωl L (6) jωl SL jωl L jωl L R L L ij = L ji 6 z z 33 L=L =L C =C =C jb i 5 [7] ω jωl i+ jωl+ jωc i jωc + =ω FBW L jb i i L L L Ω c i L L L R L i L L L (jω jx i)(7) FBW ω =/ LC ( ) Ω= Ωc ω ω FBW ω ω (8) ω Ω [] [7] Ω c = lowpass prototype filter LPF Q [Z] 5 Frequency Invariant Reactance (FIR) [7] R L

7 [ Z] Δ = Ωc FBW RS Δ ω L Δ ω L RL RS Δ ω L Δ ω L RL R S jωl S jωl S jωl SL jωl S jωl + jωc jωl jωl L jωl S jωl jωl + jωc jωl L jωl SL jωl L jωl L R L =[ Z] (9) 6 3 LPF [ Z] ω ω ω ω [5] [ Z] [ Z] =[R]+jΩ[U] +j[m] () L C R G [ Z] [Ȳ ] C M () M ii (i=,,,n) i self coupling Ω+M ii = LPF Ω i = M ii M ii = () M ij (i, j =,,,N, i = j) [R]=diag[,,, ] [U]=diag[,,, ] mutual coupling (3) M S,i (i =,,,N) i M S M S M SL M S M M M L [M] = () () M i,l (i =,,,N) i M S M M M L M SL M L M L (5) M SL [M] coupling matrix LPF [M] N S S [R] [U] (N +) (N +) S (Ω) = [ ] Z(Ω) (9) [R]=diag[,,,,, ], [U]=diag[,,,,, ] () S (Ω) = [ ] Z(Ω) () N+, [M] (N +) (N +) [5], [7] [ Z] ij M S M S M SN M SL [ Z] (i, j) M S M M M N M L 6 Q M S M M M N M L [M]= (3) 6 LPF M SN M N M N M NN M NL M SL M L M L M NL Q 9 m ij = L ij/ L il j M ii = X i (i =,,,N) () m ij (5) LPF M M ij = Ωc L ij ij (i, j =,,,N, i = j) (5) FBW LiL j m ij = FBW M ij () Ω Ωc ω c L S,i M S,i = (i =,,,N) (6) FBW ωl ir S Ωc ω L L,i M i,l = (i =,,,N) (7) FBW ωl ir L L ij M SL = ωlsl T (8) RSR L T F ABCD

8 C i L ij C j C i L i L ij L ij L j C j L S,i C i L i C i L i C i L i L j C i L i K ij i j L j L ij, T, K [ ] A B ±jωl ij [F ]= = () C D ±j ωl ij K ij = ω L ij K ij K [] [7] K m ij m ij = ω L ij K ij = (3) ωl iω L j ω LiL j 6 Q R S i m S,i m S,i = C j ωls,i K S,i = () ωl ir S ωl ir S R S K K S,i R in () F / R S / K S,i π/ R in R in = K S,i R S + jk S,i tan π K S,i + jr S tan π = K S,i R S (5) K R S () m S,i = ωlirs K S,i = ωli R in = Q es,i (6) Q Q es,i m S,i Q Q es,i Q el,i LPF R S e S L S (=) L i R S e S K S,i i, K, Rin R in e S M S,i M i,l (6) (7) Q es,i = Ω c = m S,i FBW MS,i Q el,i = Ω c = m i,l FBW Mi,L (7) (8) J K [] 6 3 ω [M BPF] m S m S m SN m SL m S m m m N m L m S m m m N m L (9) m SN m N m N m NN m NL m SL m L m L m NL [M BPF]= LPF Ω [M] m ij = FBW M ij Ω c (3) Q Q es,i = Ω c = m S,i FBW MS,i (3) Q Q el,i = Ω c m = i,l FBW Mi,L (3) m SL = M SL (33) Ω+M ii = ( ) m ii = M ii = Ω = Ωc ω ω (3) FBW ω ω ω i ω i LPF Ω [M] (3) (33) Q

9 7 circuit synthesis 7 L C v (t) v (t) V (s) V (s) s s=jω H(s)= V(s) V (s) [] P max = E S /(R S) P = V /R L T (s) T (s) T (s) =T (s)t (s)=t (s)t ( s)= P = RS P max V(s) (35) R L Γ(s) E S Γ(s) = T (s) (36) (R S)/R L T (s) T (s) = + Ψ N (s) (37) Ψ N N LPF N N Ψ N(s) = Γ(s) / T (s) T (s) Γ(s) S S L A L R L A(s)= log T (s) = log ( + ΨN (s) ) (db) (38) L R(s)=log (db) (39) Γ(s) 7 Γ(s) Z in(s) E S V LC circuit V Source R S Z in Γ(s) = Zin(s) RS Z in(s)+r S = zin(s) z in(s)+ R L Load () z in(s) z in(s) =Z in(s)/r S z in(s) positive real function [] Γ(s) Γ(s) =± F (s) E(s) () s F (s) E(s) z in(s) z in(s) = E(s) ± F (s) E(s) F (s) () [6] T (s) T (s) = P (s) E(s) (3) E(s) F (s) P (s) E(s)E( s) F (s)f ( s) =P (s)p ( s) () () E(s) 6 () F (s) F (s) (3) P (s) Ψ N(s) Ψ N (s) LPF () Ψ N (s) 6 () s () (3) E(s) s

10 g = g g g N T N(Ω)= { cos(n cos Ω) for < = Ω < = cosh(n cosh Ω) for Ω > (9) g = Z in g g g g N- g N+ g N g N+ g g N- N: odd N: even g N LPF g g [] [7] (A-) 3 Z in g g 3 g N N: odd g N+ g N- g N+ N: even LC LPF g, g () E(s)E( s)=+ Ψ N (s) = P (s)= (3) E(s) () z in(s) (5) z in(s) () z () in (s)=gs + g s + g 3s + g s + z () in (s)= z () in (s) = g s+ g s+ g 3s+ (5) (6) g i i =,,,N + g (6) 3 LC 3 g g N+ (5) (6) g i (A-) Ψ N(s) =Ω N (7) (A-) Ψ N(s) =ɛt N (Ω) (8) ɛ ɛ= L Ar L Ar db T N N g = (5) ( ) (i )π g i =sin for i=,,,n N (5) g N+ = (5) (A-) g = (53) g = ( π ) γ sin (5) N ( ) ( ) (i )π (i 3)π g i = sin sin N N ( ) g i (i )π γ +sin N for i =, 3,,N (55) forn odd ( ) g N+ = β (56) coth for N even ( ) β =ln coth LAr (57) 737 ( ) β γ =sinh (58) N L Ar db 7 3 K (5) (6) J K LPF z in K z in = Zin = K R S R S K jωl a + K3 jωl a + jωl a3 + (59) ω =ΩΩ c K ΩcR SL a K = (6) g g

11 R S R S Z in Z in C a C a C an J J J 3 J N,N+ L a L a L an K K K 3 K N,N+ J LPF, K LPF L a,il a,i+ K i,i+ =Ω c (i =,,,N ) (6) g ig i+ Ω cl an R L K N,N+ = (6) g N g N+ J [] [7] 7 LPF [M] M S M S M M M 3 [M] = M 3 M NL M NL R L R L (63) 7 LPF positive function [7] C N [6] [9] C N T (Ω)T (Ω) = [ j ɛ ɛ R C N(Ω) Ω TZi i=,,,n TZ < = N f x i(ω) = Ω /ΩTZi Ω/Ω TZi (7) (6) (6) M S = (6) gg M i,i+ = (i =,,,N ) (65) gig i+ F (s) P (s) M NL = (66) gng N+ E(s) E(s) F (s) P (s) ABCD g i LPF g Q LPF [M] Q es = Ωc gg (67) M S M S M S3 M SN M SL FBW m i,i+ = FBW M S M M L (i =,,,N ) (68) M Ω c gig i+ S M M L [M]= M S3 M 33 M 3L (73) Q el = Ωc gn gn+ (69) FBW M SN M NN M NL M SL M L M L M 3L M NL ][ +j ɛ ɛ R C N(Ω) ] (7) ɛ ɛ R C N [ N ] C N(Ω) = cosh cosh (x i(ω)) (7) i= x i(ω) N TZ

12 Input Q es m Q el m3 S 3 L f f f l s s s 3 Dielectric substrate l w l3 s 5 3, 7(d) (N +) [7] 7 5 [3] K K = [M] Q [5], [] / l i i =,, 3 s i i =,, 3 Q s i i =, 8 Current Voltage 6 Input Resonant frequency mm l Length l (mm) 7, () f = (GHz) () Δf = (MHz) FBW= 5 (3) N =3 L Ar = (db) g (53) (58) g = g =3 g =7 g 3 =3 g = LPF [M] (63) (66) [M]= (7) Q (3) (33) Q es = =6 (75) Q el = Q es =6 (76) m =5 99 = 6 (77) m 3 = m =6 (78) Q Sonnet em ε r =6 mm 8 3 f = (GHz) ε r t

13 ε eff [5] ε eff =5 l Input 5 mm s 5 mm l = λ ε eff =5 (mm) (79) λ f l l 7 S (db) -7 Weaker coupling -8 5 w = (mm) Gap s (mm) l 7 f = (GHz) l =5 (mm) 8 m (= m 3 ),, 8 3 s Input 8 f s S - 5 mm 5 synchronize -5 3dB 8 Δf 3 f p f p -3 6 f p f p f 7 m S (db) m = f p fp fp + f (8) p 9 Q Q es (= Q el ), Q, f p >f p 8 Q Q 9 asynchronize 9 m = ( ) f Δf Q loaded Q ( f + f f p fp ) ( ) f f f fp +f f factor Q L Q p f +f (8) Q Q es Q Q u Q Q el [5] f f (8) = Δf = + + Q L f Q es Q u Q el f =f (8) (8) Q u Q Q el Q es s (8) m Q el s m Q s 8 Q es = f Δf (83) m (= m 3) =6 s (= s 3)=5 (mm) Stronger coupling f p f p Coupling coefficient m External Q factor Q es Gap s (mm)

14 Input 7 Attenuation (db) EM simulated Return loss Insertion loss Synthesized Return loss Insertion loss unit: mm , 9 s Q Q es s s (= s )=7 (mm) Q Q es(=q el)= () f = (GHz) () Δf =5 (MHz) FBW= 5 (3) L R > = (db) () N = (5) (6) f TZ=95 (GHz) f TZ =5 (GHz) 7 [M] [7] [9] [M]= (8) Folded [7] S MS MS3 MS 3 ML M3L ML MS ML L S M3 3 M M M3 M S M M L L S Q es m m3 3 m m3 Q el, M Folded M Folded [M]= (85) M = M [3] [M]= (86) M Folded Q (3) (33) Q es = =885 (87) 5 3 Q el = Q es =885 (88) m =5 875 = 38 (89) m 3 =5 767 = 38 (9) m 3 = m =38 (9) m =5 ( 73) = 86 3 (9) GHz m 8 / [5] / / 3 L

15 Dielectric substrate s 3 lx Dielectric substrate 5 w 3 ly 85 s 5 Input l s ly Input Via hole lx Via hole 5 unit: mm Coupling coefficients m and m 8 6 / Voltage Current 3 / 38 5 m m 86x Gap s / s (mm) Coupling coefficient m Gap s 3 (mm) m m,m 3 External Q factor Q es Tap position l(mm) 5 Q Q es / [] S S m s m 3 m m Q / f Q Attenuation (db) 6 Attenuation (db) 3 5 Synthesized Return loss Insertion loss EM simulated Return loss Insertion loss Synthesized Return loss Insertion loss EM simulated Return loss Insertion loss , ε r = 635 mm ANSYS HFSS m m 3 m s =5 (mm) s 3 = (mm) s = (mm) Q Q es l 5 Q Q l = (mm) 8 l

16 Q Q Q N ω ΔL A [], [], [5] ΔL A(ω )=33 ω Δω N i= g i Q u,i (db) (93) g i Δω/ω Q Q u,i Q N ΔL A [5] 3 9 [5] J-S Hong and M Lancaster, Microstrip Filters for RF/Microwave Applications (Second Ed), Wiley, [6] P Jarry and J Benneat, Advanced Design Techniques and Realizations of Microwave and RF Filters, New York: Wiley, [7] RJ Cameron, CM Kudsia, and RR Mansour, Microwave Filters for Communication Systems: Fundamentals, Design, and Applications, New York: Wiley, 7 [8] RJ Cameron, General coupling matrix synthesis methods for Chebyshev filtering functions, IEEE Trans Microwave Theory Tech,vol7,no,pp33,Apr 999 [9] RJ Cameron, Advanced coupling matrix synthesis techniques for microwave filters, IEEE Trans Microwave Theory Tech, vol5, no, pp, Jan 3 [] H Kayano and M Ohira, Fundamentals of microwave filters: synthesis theory and design techniques, Tutorial Material in 3 Thailand-Japan Microwave (TJMW 3), Dec 3 [],,,, (C), volj96-c, no, pp7 79, Dec 3 [],,, 997 [3] S Amari, U Rosenberg, and J Bornemann, Adaptive synthesis and design of resonator filters with source/loadmultiresonator coupling, IEEE Trans Microwave Theory Tech, vol5, no8, pp , Aug [],, Open-loop,, C--79, p, Mar [5] M Ohira and Z Ma, A parameter-extraction method for microwave transversal resonator array bandpass filters withdirectsource/loadcoupling, IEEE TransMicrowave Theory and Tech, vol6, no5, pp8 8, May 3 mohira@mailsaitama-uacjp [] GL Matthaei, L Young, and EMT Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures New York: McGraw-Hill, 96 [],,,,, 7 [3],,, 9 [],, Microwave Workshops & Exhibition (MWE ), Dec

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