Design of all-pole microwave filters. Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department

Transcription

1 Design of all-pole microwave filters Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department

2 In-line filters with all-equal resonators R L eq, f L eq, f L eq, f L eq, f R K 1 K 1 K 3 K n,n+1 From the general design equations: K k L qq, 1 qq, 1 eq L K K R eq 1 nn, 1 1 QE with: k qq, 1 B 1 f g g q q1 Q E g1 B f

3 / waveguide resonators coupled with inductive reactances L eq, f / Zc=1 L eq g ZC K q,q+1 q,q+1 Zc=1 jx q,q+1 q,q+1 Kqq, 1 Zc Zc=1 x K 1 Z qq, 1 qq, 1 1 qq, 1 qq, 1 C tan x L 1 L L n. x 1 x 1 x 3 x n-1,n x n,n+1 L q 1 g q1, q q, q1

4 Inductive reactances in rectangular waveguide Iris with finite thickness b Aperture W Zc=1 jx s Zc=1 a Equivalent circuit (reference sections: symmetry axis) je e S11 S1 e, o S11 S1 e o 1 xs tan o e S 11, S 1 : scattering parameters (computed numerically with EM simulators) j o

5 Evaluation of iris equivalent parameters.4 x.3. X Electrical Length delta (deg) Reactance x W (mm) W.5 With the previous formulas, the curves of x and vs. W are drawn (see the graph above). For the required value X the aperture W is first obtained through the blue curve. The corresponding electrical length is derived from the red curve

6 Filter dimensioning L 1 L L n. W 1 W 1 W 3 W n-1,n W n,n+1 a Top view The apertures W i,i+1 are evaluated as explained in the previous slide from the K i,i+1. The separation of the iris are obtained from the lengths L q corrected with the irises parameters q,q+1 : 1 L L q q q1, q q, q1 g

7 Example of design Specifications: f =4 GHz, B=4 MHz Return Loss in passband 6 db A s1 45 db for f>4.5 GHz A s 6 db for f>3.9 GHz Chebycheff characteristic Being f s1 and f s not geometrically symmetric, the requested order n of the filter is obtained by selecting the larger between the following ones: As 1 RL6 As RL 6 n n 4.58 log 6 log 6 s1 n=6 s

8 Step 1: Evaluation of equivalent circuit param. Selected waveguide: a=5mm, b=5mm Waveguide parameters: Cutoff frequency: f c =v/a=3 GHz g g mm,.86 1 f f c Prototype normalized coefficients: g q ={.7919, , 1.7, , 1.589,.7163} Coupling reactances and resonator lengths: x q,q+1 ={.3,.346,.36,.,.36,.346,.3} L q ={5.5, 55.61, 55.83, 55.83, 55.61, 5.5}

9 Filter response (ideal inductive reactance) Ideal waveguide Ideal Chebycheff Frequency (GHz) The reactances are assumed to vary with frequency as g / g

10 Step : Irises dimensioning W q,q+1 ={1.7, 1.86, 9.46, 9.19, 9.46, 1.86, 1.7} mm q,q+1 ={1.69,.87,.683,.665,.683,.87, 1.69} Corrected lengths of resonators: L q ={51.49, 55.14, 55.41, 55.41, 55.14, 51.49} mm Example of fabricated device (4 resonators sample)

11 Computed filter response (Mode Matching) -1 - Ideal reactances Iris (MM) Frequency (GHz)

12 Broad frequency range response -5-1 Waveguide filter Ideal Chebycheff Frequency (GHz)

13 Effect of filter bandwidth on accuracy Normalized Bandwidth: 1% Normalized Bandwidth:.5% Frequency (GHz) Frequency (GHz) Normalized Bandwidth: 5% Normalized Bandwidth: 1% Frequency (GHz) Frequency (GHz)

14 In-line filters with assigned inverters G C eq,1, f C eq,, f C eq,3, f C eq,n, f G J 1 J J 1 J J G J 1 Each resonator is loaded with two conductances with value J 1.The loaded Q of q-th resonator is obtained from the general design equations: Q q Ceq, q 1 J g q B f 1

15 Inverters with /4 line sections C eq,1, f C eq,, f C eq,3, f C eq,n, f G Y C Y C1 Y C1 Y G C. /4 /4 /4 /4 1) Assign Y ) Compute J c1 1 Y G Y c c1 3) Compute C Y Q Y eq, q c1 q c1 g q B f

16 Resonators with shorted transmission lines C eq,q, f Y cs,q /4 C Y 4 eq, q cs, q Y Y cs, q c1 4g q B f NOTE: The choice of Y c1 affects the computed values of Y cs,q. These latter must be physically realizable with the chosen fabrication technology of the filter

17 Example: filter in microstrip technology Let assume the following specifications: f =1 GHz, B/f =.1, RL=15 db, n=5 Assuming the Chebycheff characteristic: g={ } Now suppose that the range of realizable characteristic impedance is -15 Ohm. Then we assume Y c1 =1/15=.67. Using the previous formula: Y cs,q ={ } 1/Y cs,q ={ } The computed impedances are not realizable with microstrip technology!

18 Possible solution to allow the feasibility Increase the imposed return loss (not too much effective) Use two stubs in parallel with twice Y cs,q Choice a different implementation of resonators (for obtaining smaller Y cs,q for the same equivalent capacitance). For instance, using open-circuited, / stubs the characteristic impedances are doubled. Another choice is the use of capacity-loaded resonators: short L< /4 Ycs CS Ceq 4 C s Y cs, q c1 Y 4gq Y B f

19 Implementation with double / resonators Substrate: Duroid ( r =.16, H=1. mm, t=5) 1/Y cs,q ={38.3, 34.67,.88, 34.67, 38.3} Yc GYc (1 Yc 86.6) Y c1 =1/15=.67 Layout: Discontinuities!

20 Computed response Microstrip Ideal Lines (Discontinuities Transmission Chebycheff Compensated) Lines Frequency 1 (GHz) Frequency (GHz)

21 Attenuation produced by losses An estimate of the overall attenuation at center frequency f can be obtained with the following formula: A db 1log 1 Q Q q1, n q 1 Where Q q is expressed as function of the prototype parameters g q : 1 gq Qq B f Note: Narrow band filters require high Q resonators to reduce passband attenuation

22 Physical structure: Coupled-line filters Z /4 Z e,1 Z o,1 d i,i+1 /4 Z e,1 Z o,1 /4 Z Z m,(i,i+1) Z m,(i,i+1) Basic Block: /4 Z e,3 Z o,3 /4 /4 w i,i+1 Open K i,i+1 Open w i,i+1 d i,i+1 Z e,i,i+1 Z o,i,i+1 1 Zm,( i, i1) Ze,( i, i1) Zo,( i, i1) Z,( ii, 1) Ze,( ii, 1) Zo,( ii, 1) K(, ii1), Z,(, ii1) sin L

23 Typical dependance of Z m and Z on d/w Zm Zdelta d/w For d/w >1 Zm is practically independent on d for assigned w and practically coincides with the characteristic impedance of the isolated line with the same w.

24 Equivalent circuit (inner blocks) Z m,(i-1,i) Z m,(i-1,i) Z m,(i,i+1) Z m,(i,i+1) K i-1,i K i,i+1 i-th resonator L Z Z Z 4,,( 1, ),(, 1) L eq, f eq i m i i m i i c Design Equations: k K Z L L qq, 1,( ii, 1) qq, 1 mii, 1 Zm,( i, i1) eq, q eq, q1 m k ii, 1 ii, 1

25 First (last) block Z m,(,1) Z m,(,1) Z m,(1,) X 1 Z m,(,1) Z m,(1,) /4 /4 jx 1 R R K >>X 1,1 K,1 R Q E L K eq,1 R,1 X K f Z cot 1 1 m,(,1) R f f K 1 f X1 X1 cot Zm,(,1) Zm,(1,) Zc cot f R f Z eq E L R Q k c,1 1, Leq,1, m,1, m1, Zc Zc Zc R Q 4 R Q E E

26 Dimensioning of the filter 1. The dimension w of the lines is first assigned. The values of m i,i+1 are computed with the previous formulas 3. Using a graph of m vs. d (like the one shown below) the distances d i,i+1 are evaluated m d (mm)

27 Example of microstrip filter design Specifications: f=1 GHz, B= MHz, RL=6 db, n=7 g={.87, 1.397, 1.758, 1.634, 1.758, 1.397,.87} Assigned substrate parameters: r =1, H=1.mm, t=1 Assigned width of lines (for Zc 85 Ohm): w=.5 mm Computed m i,i+1 : m={.415,.58,.1,.186,.186,.1,.58,.415} Computed distances di,i+1 (with the previous graph): d={.7,.649,.861,.93,.93,.861,.649,.7} Evaluation of length of blocks (about /4 for a not coupled line): L=3.44mm

28 Computed filter response (circuit model) Microstrip (including discontinuities) Microstrip optimized Chebycheff Ideal Frequency (GHz) (GHz)

29 Filters with an array of coupled-lines Interdigital: Comb:

30 Basic block: array of coupled lines n+1 n+ n+3 n n+1 n+ n+3 n. L 1 3. n 1 3 n Admittance Matrix (nxn). 1 3 n The admittance matrix Y (nxn) can be evaluated from the static capacitance matrix p.u.l C (dimension n x n): Y Y vc vc Y, Y, Y Y Y jtan( L) jsin( L) is the phase velocity

31 Matrix Y of Interdigital Array n+ n Admittance Matrix (nxn) Admittance Matrix Y I (nxn). n 1 3 Let assume that C i,j = for j>i+1 (no coupling between non-adjacent lines). Elements of matrix Y I are then given by: vc ii, jtan( L) i j y i j vc jsin( L) j i j i 1 ii, 1 I (, ) 1 Matrix Y I is tri-diagonal

32 Condition on the Y matrix for representing a filter n Y Y,3 Y 3,4 Y n-1,n Y1,1 1, Y, Y 3,3 Y 4,4 Y n,n Equivalent parameters (Y i,j =jb i,j ): vc B L L ii, ii, tan( L) 4 vc J B C ii, 1 ii, 1 ii, 1 ii, 1 sin( L) 1 B C C 4 ii, eq, i i, i

33 Coupling with external loads 1 J,1 R B 1,1 Y J,1 1,1 jb1,1 G Q E 1 B1,1 C C eq,1 4 J G Re Y Y Re,1 1,1 1,1 ii,

34 Equations for interdigital filters design k ii, 1 J 4 C C, 4 C C C C Re ii, 1 ii, 1 QE 1,1 Y1,1 eq, i eq, j i, i i1, i1 Typically the coupled lines are assumed of equal width (assigned a priori). The unknowns are then represented by the distances d i,i+1 between the lines and by the dimensional parameter (d e ) of the external coupling structure. The solution is obtained by solving the system of nonlinear equations in the variables d i,i+1 and d e : 4 k, 1 (, 1),, 1, ii F d Q U d d ii E 4 e

35 Matrix Y of Comb Array. Admittance Matrix (nxn) 1 3. n Admittance Matrix Y C (nxn). n 1 3 With no coupling between nonadjacent lines: vc ii, j Csi, tan( L) i j vc ii, 1 yc (, i j) jtan( L) j i1 j i 1 To get y c (i,i+1) : L

36 Equations for comb filters design (l=/4) J B C, C C 4 C 4 C 4 kii, 1 C C Re ii, 1 ii, 1 ii, 1 eqi, ii, 1,1 ii, 1, QE ii, i 1, i 1 Y 1,1 As in the case of interdigital filters, assigning all the lines with the same width, the solution is obtained by solving the system of non-linear equations in the variables d i,i+1 and d e : 4 k, 1 (, 1),, 1, ii F d Q U d d ii E 4 e The tuning capacitances are obtained by the resonance condition: vc ii, Cs, i vci, i tan( 4)

37 Approximated evaluation of Capacitance Matrix Single line: Two coupled lines: C C / C / C / C / C a / C a / C C even odd C a C C C a C C C C a C C even odd C even Y C, Y C even even odd odd

38 Array of coupled lines: evaluation of C C,(1,) C C,(,3),(3,4) C / C / C C a,(1,) C a,(1,) C C C C C C C C a,(3,4) a,(3,4) a,(,3) a,(,3) C C Cii, 1 C,( ii, 1) even,( i, i 1) odd,( i, i 1),(1,),(1,),(1,) 1,1 a C odd even 1, Ca,( i1, i) Ca,( i, i1) Cii, C 1, C, 1 i i i i Ceven,( i1, i) Codd,( i1, i) Ceven,( i, i1) Codd,( i, i1) C If the lines are not too close: C C ii,

39 Design equations Being: C C Y Y even,( i, i1) odd,( i, i1) even,( i, i1) odd,( i, i1) The design equations for comb and interdigital filters become: 1 Yeven,( i, i1) Yodd,( i, i1) Y kii, 1, QE M M Y Re Y1,1 with: M (Comb) M (Interdigital) 4 4

40 Example: Comb Filters in slab-line d. 1 3 n h Array of coupled rods (coupled slab-lines) C=Capacitance matrix p.u.l s i,i+1 Tuning capacitances L Inline comb filter configuration One-side short circuited array of coupled rods