Filter Design. Filter Design by INSERTION LOSS METHOD controls Γ(ω) to control passband and stopband of filter. sabarina/ppkkp 1

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1 Filter Desin Filter Desin by INSERTION LOSS METHOD cntrls Γ(ω) t cntrl passband and stpband f filter. sabarina/ppkkp 1

2 Filter parameters Passband -- frequencies that are passed by filter. Stpband -- frequencies that are rejected. Insertin lss -- hw much pwer is transferred t lad in passband. Attenuatin -- hw much pwer is rejected (nt transferred t the lad) in the stpband. Cutff rate r attenuatin rate -- hw quickly the filter transitins frm pass-t-stp r stp-t-passbands. Phase respnse -- Linear phase respnse in the passband means that sinal will nt be distrted. sabarina/ppkkp 2

3 Binmial / Butterwrth / Maximally Flat P LR 1+ k 2 ω ω c 2 N N Filter Order ω frequency f interest ωc cutff frequency At ωc, P LR 1+k 2 If the -3dB pint is defined t be the cutff pint (cmmn), k1 Fr ω>>ωc then PLR k 2 (ω/ωc) 2N which means Insertin Lss increases at a rate f 20N db / decade sabarina/ppkkp 3

4 Chebyshev / Equal Ripple Filters P LR ω N ωc k T Where T N are Chebyshev plynmials Ripples are equal-in-size 1+k 2 Cutff Rate is 20N db/decade, same as binmial. Insertin lss in the stpband is (2 2N )/4 reater than binmial. sabarina/ppkkp 4

5 Filter Desin Methd Desin a LP filter fr nrmalized Z,ω Scale Z. Cnvert frm LP t HP r BP as desired. Cnvert frm lumped t distributed elements as desired. sabarina/ppkkp 5

6 Binmial Desin f LP Filter fr Nrmalized Z,ω Step 1: Determine hw many elements are needed (N) Find (ω/ω c ) and lk at the fiure fr attenuatin in the stpband. sabarina/ppkkp 6

7 Example Hw many elements are required t desin a maximally-flat filter with a cutff frequency f 2 GHz if the filter must prvide 20 db f attenuatin at 4 GHz? Fr this case, at bttm axis ω ω c 2 Find N line n filter that is ABOVE the desired attenuatin. Therefre, N4. sabarina/ppkkp 7

8 Step 2: Find resistance r cnductance values frm Table Lk at N ; 2 2.0; 3 1.0; sabarina/ppkkp 8

9 Step 3: Chse LP Filter Prttype sabarina/ppkkp 9

10 sabarina/ppkkp 10 Impedance and Frequency Scalin (nrmalizatin) T build the same filter fr Z R R L and a iven cutff frequency ω c Use the same filter prttypes but scale the values: ; ; Z R Z C Z L Z C Z R L c c c ω ω ω

11 sabarina/ppkkp 11 Impedance and Frequency Scalin (nrmalizatin) ; ; 1 Z R Z L Z C Z L Z R L c c c ω ω ω

12 sabarina/ppkkp 12 Cnvert frm LP t HP ; 1 ; ; n L n L k c k k c k Z R Z R Z R Z R Z L Z C ω ω

13 Example Desin a maximally flat LPF with cut-ff frequency f 20GHz, impedance f 50 hm, and at least 15dB insertin lss at 3 GHz. Cmpute and plt the amplitude respnse and rup delay fr 0 t 4GHz, and cmpare with an 3dB equal-ripple and linear phase filter with the same rder. Find ω ω c N5 (frm table) sabarina/ppkkp 13

14 Slutin sabarina/ppkkp 14

15 Cnvert frm LP t Bandpass r Bandstp Nrmalized bandwidth Δ ω 2 ω ω 0 1 ω 0 ω ω 1 2 where ω 2 upper limit and ω1 lwer limit sabarina/ppkkp 15

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17 Denrmalizatin f standard LPF T arrive at relizable filters, have t denrmalize the afre mentined cefficient t meet realistic frequency and impedance requirements. The standard LPF prttype shuld be cnvertible int HPF r bandpass r bandstp types dependin n applicatin. sabarina/ppkkp 17

18 The cnvertible can be achieved by Frequency transfrmatin T cnvert frm nrmalized frequency t actual frequency. This implies the scalin f the standard inductances and capacitances. Impedance transfrmatin T cnvert standard eneratr and resistances and (n+1) t actual resistances R L and R G. sabarina/ppkkp 18

19 Example An N3 Chebyshev bandpass filter is t be desined with a 3dB passband ripple fr a cmmunicatin link. The center frequency is at 2.4GHz and the filter has t meet a BW requirement f 20%. The filter has t be inserted int a 50 hm characteristic line impedance. Find the inductive and capacitive element and plt the attenuatin respnse in frequency in rane 1 t 4 GHz. sabarina/ppkkp 19

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21 Example Desin a bandpass filter havin a 0.5dB equal-ripple respnse with N3, the bandwidth is 10% and the impedance is 50 hm. sabarina/ppkkp 21

22 Slutin sabarina/ppkkp 22

23 Filter Implementatin Filter desins >>50MHz are difficult t realize with discrete cmpnents because the wavelenth becme cmparable with physical filter element dimensins Resultin in varius lsses severely deradin the circuit perfrmance. Fr practical filters, the lumped cmpnent filters must be cnverted int distributed element realizatin. sabarina/ppkkp 23

24 Tls Richards transfrmatin Cncept f the unit element Kurda s identities sabarina/ppkkp 24

25 Richard s Transfrmatin T accmplish the cnversin between lumped and distributed circuit desin t transmissin line sectins. Richards prpsed a special transfrmatin that allws pen and shrt circuit transmissin line sements t emulate the inductive and capacitive behavir f the discrete cmpnents. sabarina/ppkkp 25

26 The transfrmatin, Ω tan βl l tan ω v p map the ω plane t The reactance f the The susceptance f Ω plane, an inductr; a which capacitr; jx L jb repeat wit C jωl jωc h a perid jl tan βl f jc tan βl ωl v p 2π Ω 1 tan βl ive a stub lenth f l λ/8, where λ at cut - ff frequency, ω c sabarina/ppkkp 26

27 Richard s transfrmatin Fr an inductr t a shrt-circuited stub.. Fr a capacitr t an pen-circuited stub. sabarina/ppkkp 27

28 Cncept Of The Unit Element When cnvertin lumped elements int transmissin line sectins, there is a need t separate the transmissin line elements spatially t achieve practically realizable cnfiuratins. This is accmplished by insertin s-called unit elements (UE). The UE has an electric lenth f and a characteristic impedance, π θ 4 Z UE f f sabarina/ppkkp 28

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30 Kurda s Identities In additin t the UE, it is imprtant t be able t cnvert practically difficult t implement desin t a mre suitable filter realizatin. Can be used t separate filter elements by usin transmissin line sectin. sabarina/ppkkp 30

31 Fur Kurda Identities sabarina/ppkkp 31

32 Example Desin a LPF whse input and utput are matches t a 50 Ohm impedance and that meets the fllwin specificatins; cut-ff frequency f 3GHz; equi-ripple f 0.5 db; rejectin at least 40dB at apprximately twice the cut-ff frequency. Assume a dielectric material that results in a phase velcity f 60% f the speed f liht. sabarina/ppkkp 32