Introduction to Surface Acoustic Wave (SAW) Devices

Transcription

1 Introduction to Surface Acoustic Wave (SAW) Devices Part 4: Resonator-Based Filters Ken-ya Hashimoto Chiba University

2 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design

3 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design 3

4 LC Filter Topology R L C L C E S L C R BPF Composed of 3 LC Resonators Limited Q Factor for L and C Large Insertion Loss and Wide TB Mutual Inductive Coupling 4

5 What is Q? Steepness of Resonance Peak Stored Energy Q = π Energy Dissipated in Half Cycle G max G Y Y max Q=ω r /δω G max / δω Y max / δω ω r ω ω r ω 5

6 Impulse Response exp(-t/t) t Q=.5ω r T Determine Impulse Response Length 6

7 RF Surface Acoustic Wave (SAW) Resonator Interdigital Transducer (IDT) λ Reflector (Al) Piezo-Substrate (4 o YX-LiTaO 3 ) Mass Production by Photolithography High Frequency, Low Loss Good Temperature Stability Cheap(?), Small(?) 7

8 RF Bulk Acoustic Wave (BAW) Resonator resonator cavity resonator cavity (a) Substrate Etching (Fujitsu Labs.) (b) Surface Micromachining (Avago (Agilent), STMicro) resonator (c) Multi-Layer Reflector (Infineon, NXP(Philips), EPCOS), SMR (Solidly- Mounted Resonator) 8

9 Application of Electric Field to Piezoelectric Material, - M i dv dt ηv k vdt = F V v k - M Piezoelectric Material C L C V F η C R (a) Electric Mechanical Circuit (b) Electrical Equiv. Circuit Analogies between V F and I v reduce to M L, η R, k /C 9

10 Resonance Characteristics Admittance [ms] 5-5 B ω r G ω a Frequency [GHz] Resonance Frequency ω r =/ C L Anti-Resonance Frequency ω a =/ L(C - C - ) -

11 C L k η M C R k (a) Electrical Equiv. Circuit F (b) Equiv. Mechanical Circuit Resonance Frequency ω r =/ C L Anti-Resonance Frequency ω a =/ L(C - C - ) - Resonance Q (Steepness of Resonance) Q=ω r L/R Determine Insertion Loss and Skirt Characteristics Capacitance Ratio (Weakness of Piezoelectricity) γ=c /C =[(ω a /ω r ) -] - Determine Insertion Loss and Bandwidth

12 Influence of Parasitics C e (a) L m C G C e m R m (a) Capacitance ω a BW Reduction (b) L m C C m R m (b) Conductivity Q a (c) L e L m C C m R m (c) Wire Inductance ω r BW Expansion (d) R e L m C C m R m (d) Wire Resistance Q r Loss Increase

13 Figure of Merit, M Impedance [Ω] 3 M a =Q a /γ M r =Q r /γ Frequency [GHz] Determining Filter Insertion Loss 3

14 Thickness Resonance st nd 3rd h Resonance at nλ/ V=8,m/s, f=ghz, n= h=μm For even n, zero excitation efficiency Reduction of Efficiency by/n 4

15 Role of Rx Filter RF-BPF Jammer Elimination (Saturation and Inter-modulation) LNA Spectrum frequency Spectrum ω frequency ω Spectrum frequency ω Front End Should be Low Loss, Low Noise, and Linear Resonator-Based Passive Filters 5

16 Trade off Between Temperature Stability and Transition Bandwidth (TB) T drift (a) Wide TB Necessary ω T drift (b) Narrow TB Acceptable ω Narrow TB Efficient Use of Frequency Resource 6

17 Role of Tx Filter Spurious Signal Suppression in PA Output Spectrum PA Tx frequency RF-BPF Rx Spurious ω Insertion Loss of db = % Reduction in Output Power Current Consumption Heat Generation Ultimate Loss Reduction Required! 7

18 How about Software Defined Radio? Jammer Elimination (Saturation and Inter-modulation) Anti-Aliasing RF-BPF BPF ADC Baseband Signal Processing Bit Error Rate Deterioration Spectrum frequency ω Spectrum frequency ω Front End BPF Should be Low Loss, Low Noise & Linear 8

19 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design 9

20 Response of Linear Circuits e e i i Z = Z Y = Y Z Z i i e e Impedance (Z) Matrix e i i Y Admittance Y (Y) Matrix e Z ji =Z ij & Y ji =Y ij for Passives Attention to Current Direction! Z=Y -, Y=Z - i i i i Z a Y a e e e e Z b Y b Series-Connection Z=Z a Z b Parallel-Connection Y=Y a Y b

21 Cascade Connection F=F a F b i e e i = = i e D C B A i e F F F F i e F a F b i e i e Attention to Current Direction! [ ] [ ] Λ Λ Λ = = = X X X X Λ XΛ X F N m N N N N N O N-th Order Cascade of Identical Circuits (F=XΛX - ) Four Terminal (F, ABCD) Matrix

22 Low Frequency Design CD player Amp. DAT High Z in and Low Z out Design High Frequency Design Source Load Source Source R S E S R L R S E S? R L R L =R S for Maximum Power Transfer Impedance Matching

23 Scattering Matrix (VNA Output) a a b b b S S a? = b S S a a m : Amplitude of Input Signal b m : Amplitude of Reflected Signal S ji =S ij for Passives (Reciprocity) Normalized Amplitude : a m Corresponds to Unit Power Unit: dbm (mw) dbμ (μw) a b n n = = V V n n R R R R I I n n Y = R S [ I S][ I ] I: Unit Matrix 3

24 S : Transmission Coef. S : Reflection Coef. Insertion Loss: -log S Return Loss: -log S S S S S Dissipated Power Relative to Input Power 4

25 Unitary Condition = m b m a m m a a * * [ ]* = [ ]* b b a a b b (Input Power=Output Power) Fundamental Assumption for Simulation & Design. S S * S S * t S S S S S S * * S S * * S S S S = S S S S * S S S S * = = = Unitary Matrix: S - =S t* ( Λ = for All Eigen Values Λ ) 5

26 From Symmetry and Unitary Properties, S =S, S =S, S S =, S S * S * S = Y Z S = ± j S S S S S = G Y = G ( S) S ( S ) S S S S = R Z = R ( S) S ( S) S Y = G S S S * S * ± j S Determination of Y or Z from Given S Z = R S S S * S * m j S Determination of Circuit 6

27 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design 7

28 H (db) N-th Order Butterworth Filter Maximally Flat Response: H (n) ()= for n=,,---n H ( ω) = N ( ω / ω) N=3 N=5-8 N=7 - N= Relative frequency H (db) - N=3 - N=5 N=7-3 N= Relative frequency Gradual Skirt & Group Delay Characteristics 8

29 S (db) N-th Order Chebyshev Filter Equal-Ripple Characteristics (ε: parameter) H ( ω) = ε T N ( ω / ω) cos( N cos Chebyshev polynomial T N x) cosh( N cosh -.db -4-6 N=3 N=5-8 N=7 N= Relative frequency S (db) ( = - x) x).db N=3 - N=5 N=7 N= Relative frequency Good Skirt but Bad Group Delay Characteristics x x 9

30 7 Element T-type Butterworth LPF Various values R =5Ω, f =MHz 3

31 L C' L' C' L' L' C' L C C' L' L' C' C' L' C' C' L' Original LPF (Cut-off ω c ) LPF to HPF ω C' c ω L' c = / ω L LPF to BPF LPF to BEF c = / ω C ω L' C' =, ω L' C' = ( ω ω ) L' /( ω ω ) C' c ( ω ω ) C' c ( ω ω ) L' c /( ω ω ) L' ω L' C' =, ω L' C' = ( ω ω ) C' /( ω ω ) L' c c /( ω ω ) C' c c c c = ω L c = ω C c = / ω L c = / ω C c3

32 R =5Ω, Δ=%, f c =MHz 3

33 Infinite Q Q= for Parallel-L Q= for Series-L Round Left Shoulder Increased Insertion Loss Round Right Shoulder 33

34 Influence of Mutual L M L L L -M L -M M V V di = L dt di = M dt M L di dt di dt = ( L = ( L di M ) M dt di M ) M dt d dt ( I d ( I dt I I ) ) 34

35 Null Generation! Equivalent to L 4 & L 6 Coupling Equivalent to L & L 3 Coupling 35

36 Constant K Filter E S R L C L/C=R R Uniform Impedance Values V L L L L V I C C C I I L L L I C C C C V Cascade-Connection with Inversion (Equivalent Circuit for Transmission Line) 36

37 V L L L L V H = When Equivalent Transmission Line I C C C I Length is nλ/ H (db) N=3 N=3 N=5 - N=5-8 N=7 N=7 N=9 N= Relative frequency Relative frequency H (db) Similar Response to Butterworth Filter 37

38 Fabry-Perot Resonator A ΓA T ΓT L Interference Between Two Reflected Waves % Transmit When L=nλ/ 38

39 Image Parameter Z i, i Z o & ϕ Z i Z i ϕ Z o Z o Z i Z i ϕ Z Z o Example Z s Cascade Connection with Inversion Z p Z i ϕ Z o Z o ϕ Z i Z i ϕ Z o Z i = Z s ( Z s Z p ) F ξ Zi = Z i coshϕ ξ sinhϕ 3ϕ Z o Z iξ sinhϕ ξ coshϕ where ξ=z i /Z o Y o = ϕ = Z o sinh = Y p ( Z / Z ) s ( Y p Y p 39 s )

40 Analysis i i i 3 i M e e e3 [Fe] e M R [Fo] [Fe] [Fo] or [Fo] R e out e in # # #3 #M. Cascade Connection with Inversion ξ cosh Mϕ Z iξ sinh Mϕ = Zi ξ sinh Mϕ ξ cosh Mϕ F cosh Mϕ Zi sinh Mϕ Zi sinh Mϕ cosh Mϕ ( M ( M :odd) :even) 4

41 i i R e F F e E in R F F S = F F R F G F ( ξ ξ )cosh Mϕ ( Ziξ / R = cosh Mϕ ( Zi / R R Rξ / Zi )sinh / Z )sinh Mϕ i Mϕ ( M :odd) ( M :even) When ϕ = jn π / M, S = 4

42 R L Z i = K ( ω / ω ) c E S C R Z o ϕ = = K / j sin ( ω / ω ) ( ω / ω ) c c K = L / C, ω c = / LC When ω<ω c, ϕ Imag., Z i & Z o Real Passband When ω>ω c, ϕ Real, Z i & Z o Imag. Rejection band Zero Insertion Loss ϕ=jnπ/μ ω/ω c =sin(nπ/μ) 4

43 Derived m Filter E S R L C R =L/C R Constant K Filter ml (m - -m)c mc ml R =L/C (m - -m)l mc m Null Due to Series Resonance Null Due to Parallel Resonance 43

44 ml L L ml mc (m - -m)l C C C mc (m - -m)l Combination of Derived m and Constant K Filters H (db) Relative frequency H (db) Relative frequency Steep Skirt and Good Out-Band Rejection - - m=.6 m=.7 m=.8 m=.9 m= 44

45 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design 45

46 Ladder-Type Filter Configuration (Resonator-Based Constant K Filter) Fabrication of Multiple Resonators on a Chip Low Loss High Power Durability Moderate Out-of-Band Rejection 46

47 Performance of Ladder-Type SAW Filter Scattering Parameter S (db) Tx Rx Frequency (MHz) Scattering Parameter S (db) W-CDMA-Rx Fujitsu FAR-F6CP-G4-LM 47

48 Ladder-Type FBAR Filter By Agilent (Avago) R. Ruby, et al., IEEE Microwave Symp. (4) pp

49 - ω r s - Y s ω a s Frequency [GHz] 49 Scattering Parameter, S [db]

50 - ω a p - ω r p Y p Frequency [GHz] 5 Scattering Parameter, S [db]

51 Scattering Parameter, S [db] - - ω r p ω rs =ω a p ω a s Frequency [GHz] Y p Y s Y p 5

52 5 r r r r a a Q j j Q j j C j Y ) / / ( ) / ( ) / / ( ) / ( ω ω ω ω ω ω ω ω ω = Resonator Model for Y p & Y s Resonator Model for Y p & Y s. γ & Q r Identical for Z s & Z p. ω rs =ω a p Assumption R C L C Z s Z p ( ) s p s p p o o p s s i Y Y Y Y Y Z Y Z Z Z Z / sinh ) ( ) ( = = = = ϕ Passband When -<Y p /Y s < (Either Y s or Y p Inductive)

53 Relative admittance Y p ω r p Null Y s ω a s ω ap =ω r s Null ω Passband When -<Y p /Y s < (Either Y s or Y p Inductive) Larger Y s Offers Wide Bandwidth 53

54 54 At the Resonance(ϕ=) ) ( ) / / ( η η ω ω = NrM Z R R Z N S p s s r where s C p C R ω η = Loss Minimum Condition η= /γ Q r M = (FOM) s p C C r / = = = NrM S s r η ω ω Low Loss When NrM - «= :even) ( )sinh / / ( cosh :odd) ( )sinh / / ( )cosh ( M M Z R R Z M M M Z R R Z M S i i i i ϕ ϕ ϕ ξ ξ ϕ ξ ξ

55 Bandwidth δω s ω r γ ( γ )( r γ ( γ )( r ) γ ) r γ ( γ )( r where ) p s r = C / C Out-of-Band Rejection When η=, S cosh( Nϕ) = T N ( ) r where T N (x)=cosh(ncosh - x): Chebyshev Polynomial N Influences IL and OoB Rejection r Influences IL, Bandwidth and OoB Rejection 55

56 Scattering Parameter, S [db] -5 - N= -5 N= - N=3 N=4-5 r=.4 N= Frequency [GHz] 56

57 Scattering Parameter, S [db] - - r=.4 N= N= N=3 N=4 N= Frequency [GHz] 57

58 Scattering Parameter, S [db] -5 - N=5-5 - r=. -5 r=.4-3 r=.6 r=.8-35 r= Frequency [GHz] 58

59 Scattering Parameter, S [db] - - N=5 r=. r=.4 r=.6 r=.8 r= Frequency [GHz] 59

60 Scattering Parameter, S [db] N=5 r=.4 δ=mhz δ=mhz δ=4mhz δ=6mhz δ=8mhz Frequency [GHz] 6

61 Scattering Parameter, S [db] N=5 r= d=mhz Q p =, Q s = - Q p =, Q s = Q -.5 p =, Q s = Q p =Q s = Frequency [GHz] 6

62 Influence of Common Impedance L i SAW Device Chip S S S 3 L o wire SAW chip wire C i C e P C e P C e3 P 3 C e4 C o L e L e L e feedthrough L c Origin of L c Equivalent Circuit 6

63 Scattering Parameter, S [db] r=.4 δ=mhz N=5 L c = L c =.nh L c =.nh L c =.3nH Frequency [GHz] 63

64 Scattering Parameter, S [db] r=.4 δ=mhz N=5 L c = L c =.nh L c =.nh L c =.3nH Frequency [GHz] 64

65 Design of Ladder-type Filters Resonator γ Limits Achievable Band Width For IL Minimization, ω r C p C s R S For Wide bandwidth, ω rs should be a little larger than ω p a With smaller r= C p /C s, IL and Band Width Improved but OoB Rejection Deteriorated. With N, OoB Rejection Improved but IL Increases IL is Very Sensitive to Motional Resistance of Y s Very Sensitive to Parasitics Q Influences IL and Shoulder Characteristics 65

66 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design 66

67 Lattice Filter Resonator Y o R Y e R e out Balanced I/O e in Y o Y o / Y e / :- Equivalent Circuit Unbalanced I/O 67

68 RF Front End FBAR Integration M.A. Dubois, et al., IEEE J. Solid State Circuits, Vol (6) pp. 7-6

69 Performances of Integrated FBAR Balanced Topology Offers Improved OB Rejection 69

70 Y o -Y e Y o -Y e Y e -Y o Y e -Y o :- Y e Y e Y o Y o Modified Equivalent Circuit Equivalent to -Stage Cascaded L-type (or π-type) Filter Res. Freq. for Series Arm Anti-Res. Freq. for Parallel Arm Y e =, Y o = or Y o =, Y e = 7

71 Resonance Condition Res. Freq. for Y o Anti-Res. Freq. for Y e Or Res. Freq. for Y e Anti-Res. Freq. for Y o S = R ( Yo Ye ) ( Y R )( Y e o R ) At Resonance η( M S o ω ω r η η( M M ) ) = M IL Min. Condition η= Where M=Q r /γ, η=ωc R S o ω = ω r M M 7

72 Frequency Response S (db) Y o 5 - Y e S Relative frequency Good OoB Rejection Relative admittance (db) 7

73 Wideband When ω ro >ω a e -5 - S (db) S (db) Relative frequency -3-4 Δ= Δ=.% Δ=.4% Δ=.6% 73

74 Better OoB Rejection When C o >C e - S (db) Relative frequency Null Generation Near Passband r= r=.95 r=.9 r=.85 r=.8 74

75 Relative admittance ω Y e ω r o Y o ω a e ω ao =ω r e ω ω S = R ( Yo Ye ) ( Y R )( Y e o R ) Null Generation When Y e =Y o 75

76 Influence of Mutual Coupling L I SAW Device L L I Cc C C V L 4 L 3 L 3 L L 4 33 L 44 I 3 I L 4 34 V Very Small Values but May Not Negligible 76

77 Influence of Inductive Coupling EM Feedthrough (db) C =C =3.5pF (b) C c =.5fF, L 34 =.5pH (a) C c =.5fF Frequency (GHz) Tiny Coupling Gives Significant Influence 77

78 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design 78

79 Stacked FBAR Filter Electrodes Si Good OoB Rejection Moderate IL L m C m R m R C Z C R Equivalent Circuit for Fundamental Resonance E Static Capacitance When Parallel-Connected 79 -Port Resonator When Parallel-Connected with Inversion

80 Frequency Response of Stacked Resonator Filter From TFR Technologies 8

81 i L m C m R m e e C C Transfer Function S Z i ){ ZR ( = ( jωc R i i Where jωc R Y = Y Y Y = Z = ) } Y Y jωc = Z jωl e e m Z R m / jωc m Ideal Case (R m =), ZR ( jωc R ) = /( jωc R ) At ω Satisfying S jω C R jωc R Z = ( ω C R = S jωcr = ) (Loss Less) 8

82 8 }] ) ( /{ / ) ( [ ) ( C G G R L j C j G G S m ω ω ω ω ) ( ) ( R C R C j R L j Z m ω ω ω ω Gives }/ ) ( { = R R C R S m ω ω ω = = M R C S m ω ω ω = R C ω IL Min. Condition γ ω ω γ / / Q M R L R C Q C C m m m m m = = = = Determined by M

83 When ω C = G S j[ jγ ( ω ω ) / ω M ] When ω = ω± = ω[ ± γ ( M -3dB Bandwidth ω ω = γ ( M ω S ω = ω ω ) ω ω γ Determined by M and γ ) / ] S [ M γ = C / C Q = ω C R m M = Q / γ m m ] ωl = R m m 83

84 Response of Stacked Resonator Filter S [db] R =.5 Ω R =5 Ω Frequency [GHz] ZnO (L=.6 μm, S=6 6 μm ) S is Set to Z-Match at GHz 84

85 Cascaded Stacked Resonator Filters L m C m R m L m C m R m L m C m R m R C C C C C C R E in R R E in F N = = jr jr cosθ e cos Nθ e sinθ sin Nθ jre sinθ cosθ N jre sin Nθ cos Nθ where cosθ = jr jr e e Z = sinθ = Z sinθ = jωl m jωc jωc R m Z / ( jωc jωc 85 m Z)

86 S [db] N= N= N=3 N= Frequency [GHz] ZnO (L=.6 μm, S=6 6 μm ) Widened Passband Steeper Skirt Characteristics 86

87 Contents Basic Resonator Theory Linear Two-Port Circuit Basic Filter Design Ladder Type Filter Design Lattice Filter Design Stacked Resonator Filter Design Coupled Resonator Filter Design 87

88 Uncoupled State k M M k Resonance Frequency: ω r =(k/m).5 Coupled State k M k c M k Resonance Frequency Symmetric Resonance: ω rs ={(kk c )/M}.5 Antisymmetric Resonance: ω ra =(k/m).5 88

89 Coupled Resonator M M V - dv dt dv dt ηv i v ηv k k v v η M k - dt dt k - k c k c M ( v ( v η v V - v ) dt ) dt v i V V C V F k c - F V C (a) Electric Mechanical Circuit 89

90 i η v M k - k - M η v i C V F k c - F V C (a) Electric Mechanical Circuit η M k - k - M η C k c - C (b) Mechanical Circuit 9

91 η M k - k - M η C k c - C η M k - k - M η C (k c ) - (k c ) - C (a) Symmetric Resonance η M k - k - M η C C (b) Antisymmetric Resonance 9

92 Transversally Coupled Double-Mode Filter Electrodes Si Symmetrical & Antisymmetrical Resonances R in C E in L m L m a C m a s C m s a R m s R m :- Equivalent Circuit C R out -Port Resonator When Parallel- Connected Another -Port Resonator When Parallel-Connected 9 with Inversion

93 Equivalent Circuit of Double-Mode Filter Y e =(Y me jωc ) Y o =(Y mo jωc ) Y me, Y mo :Motional Y, jωc :Static Capacitances Equivalent to Lattice Filter Y mo / C Y me / :- C Y o / Y e / :- 93

94 Fundamental-Mode Quartz Resonator Etching of Quartz Plate Superior Temperature Stability Use of Single Crystal High Q (TOYOCOM TF-DAD6) 94

95 Design Principle of Wideband DMS Filters Coincidence of Multiple Resonance ω for Y e (or Y o ) with Antiresonance ω for Y o (or Y e ) ω r o ω a o ω r e ω ω e a ω r o ω ω a o ω Employing More Resonances Results in More Bandwidths 95

96 Double Mode SAW (DMS) Filter Scattering parameter. S [db] Frequency [MHz] Symmetrical & Anti-symmetrical Resonances Scattering parameter. S [db] Fujitsu FAR-F5EB-94M5-B8E 96

97 Balanced/Unbalanced Transmission - Unbalanced Input and Output Balanced Input and Output - 97

98 DMS Filter (Ideally No Common Signal) Acoustically Coupled but Electrically Isolated V in V out V out- Common Signal Generation by Parasitics 98

99 Z-conversion by DMS Filter V out V in V out- V in V out V out- 99

100 Cascaded Coupled FBAR Filter Electrically Isolated By Infineon G. G. Fattinger, et al., IEEE Microwave Symp. (4) pp.97-93

101 FBAR with Balun & Trans. Function 5Ω:5Ω Input Parallel, Output Series 5Ω:Ω By Infineon

102 Coupled Resonator Filter (CRF) Input Output Si S [db] Overcoupling -5 - Criticalcoupling -5 - Undercoupling Frequency [GHz] ZnO (L=.6 μm, S= μm ) Coupler R=GRayls (Very hard),.8λ How to Realize?

103 Typical Acoustic Materials Material Wave Velocity Acoustic Impedance Coupling Factor Dielectric Constant V L [km/s] R L [MRayls] k t [%] ε 33S /ε ZnO AlN SiO Al Au W Water MRayl= 6 kg/m s k t =e 33 /c 33D ε 33 S 3

104 S [db] S [db] N= -8 N= N= Frequency [GHz] N= - N= N= Frequency [GHz] Cascaded CRF ZnO (L=.6 μm, S= μm ) Coupler GRayls,.8 λ Extremely hard ZnO (L=.6 μm, S=6 6 μm ) Coupler MRayls,. λ Extremely soft 4

105 Realization of Extremely Large or Small Z Z e A r β R A i L Z Γ=A r /A i Γ = Γ exp( jβ ) L Γ Γ = A A = Z Z R R = Γ Γ = jr tan( βl) R Z e R R jz tan( βl) Z Z e =R /Z When βl=π/ (L=λ/4) 5

106 Reflectivity (SiO /ZnO) n /SiO Reflector N= N= N=3 N=4 N=5 N= Frequency [GHz] Rˆ s Zeq Rˆ Layer Thickness=λ/4 = s Rˆ z n 6

107 Design of Coupled Resonator Filters Band Width Governed by γ For Loss Minimization, ω r C R S Coupling Layer Should be Designed to Adjust Multiple Poles to Coincide with Nulls (But Very Critical!) Slight Difference Between Poles & Nulls Effective Cascade-Connection is Effective IL is Sensitive to Motional Resistances & Parasitics 7