Progress In Electromagnetcs Research B, Vol. 17, 213 231, 29 DIMENSIONAL SYNTHESIS FOR WIDE-BAND BAND- PASS FILTERS WITH QUARTER-WAVELENGTH RES- ONATORS Q. Zhang and Y. Lu School of Electrcal and Electroncs Engneerng Nanyang Technologcal Unversty Sngapore 639798, Sngapore Abstract Ths paper presents a dmensonal synthess method for desgnng wde-band quarter-wavelength resonator bandpass flters. In ths synthess method, an alternatve lowpass prototype flter and the edge frequency mappng method are proposed and appled. The mproved - and J-nverter model wth the exponent-weghted turns rato s also proposed n order to ncorporate the frequency dependence of nverters. Based on the edge frequency mappng method and the mproved nverter model, an teratve dmensonal synthess procedure s then presented. As desgn examples, a four-pole rectangular coaxal bandpass flter wth 63% fractonal bandwdth s desgned and fabrcated. The smulaton and measurement results show good equal rpple performance n the passband. 1. INTRODUCTION Desgn technques for wde-band bandpass flters wthout excessve global optmzaton have become more and more mportant. Wth the mprovement of these technques, the accuracy, tme, and complexty of the desgn procedure can be mproved sgnfcantly. And the excessve use of optmzaton can be avoded especally n complcated structures where there are numerous dmensons to be optmzed. In the classc bandpass flter theores [1], the network wth mmttance nverters s employed. By usng mpedance or admttance nverters, mcrowave bandpass flters can be desgned convenently. However, the nverters have to be frequency-ndependent theortcally, whch s not true for practcal cases. One of the most serous degradatons n frequency response of a bandpass flter from the deal Correspondng author: Q. Zhang (e722@ntu.edu.sg).
214 Zhang and Lu one s due to the frequency senstvty of the nverters. The general desgn procedure for drect-coupled flters was frst devsed by Cohn [4]. And then Matthae, Young and Jones [1] ntroduced a bandwdth contracton factor and a devaton of center frequency for drectcoupled cavty flters to predct the change caused by the frequencydependent nverters usng desgn graphs. Levy, [3], suggested deal transformers wth frequency-dependent turns rato on both sdes of the nverters. These technques dscussed above are employed to desgn bandpass flters wth half-wavelength resonators whch, however, have the dsadvantage of spurous response below 2f (twce of the passband frequency). Comparatvely, bandpass flters wth quarter-wavelength resonators have advantages that the length of the flter s shorter and the second passband center s at 3f nstead of 2f [5]. So far, however, quarter-wavelength resonator bandpass flters are desgned accordng to the classc method n [5] and no mproved synthess method was reported. In ths paper, we propose a dmensonal synthess method for desgnng wde-band bandpass flters wth quarter-wavelength resonators wthout global optmzaton. It s realzed n rectangular coaxal structures, whch have the advantage of low delectrc loss, low radaton loss and weak cross couplng wth other crcuts n a system [6]. As desgn examples, rectangular coaxal bandpass flters are desgned usng the proposed synthess method, and the results show good equal rpple performance n the passband. 2. THEORY 2.1. Alternatve Lowpass Prototype Flter Immttance nverters have the ablty to shft mpedance or admttance levels dependng on the choce of or J parameters. Makng use of these propertes enables us to convert a flter crcut to an equvalent form that would be more convenent for mplementaton wth mcrowave structures. Fg. 1 shows the classc lowpass prototype La1 L a2 Lan Z,1 1,2 2,3 n, n+1 Z n+1 Fgure 1. nverters. The classc lowpass prototype flter wth mpedance
Progress In Electromagnetcs Research B, Vol. 17, 29 215 flters wth mpedance nverters and the calculated as n [1] and [2]: Z L a1, 1 g g 1 L a L a(+1), +1 g g +1 L an Z n+1 n, n+1 g n g n+1 1,2,...,n 1 parameters can be, (1) where the g are the normalzed elements of the lumped-element lowpass prototype flter [1]. The element values Z, Z n+1 and L a n Fg. 1 may be chosen arbtrarly and the flter response wll be dentcal to that of the orgnal prototype, provded that the parameters are specfed as ndcated n (1). However, the lowpass prototype flters n Fg. 1 have only nverters and thus t can be only transformed nto bandpass flters wth half-wavelength resonators. So we should propose an alternatve lowpass prototype flter wth alternatng mpedance nverters and admttance nverters, whch can be transformed nto bandpass flters wth quarter-wavelength resonators. The mpedance nverter wth seres nductance can be equvalent to an admttance nverter wth parallel capactance and two deal nverter added on both sdes as shown n Fg. 2. The transfer matrx of the deal nverter s gven by [ ] Z, (2) Z where Z s the nput termnatng mpedance n Fg. 1. The transfer matrx of the left sde of Fg. 2 s [ ] [ ] [ ] 1 ωl, (3) 1 Z ωl and transfer matrx of the rght sde s [ ] [ ] [ ] [ ] [ ] Z J 1 Z Z 2 J. (4) J ωc 1 Z Z 2 J To make (3) equvalent to (4), the followng condton should be satsfed: J Z Y L Z C, (5) Y where Y 1/Z. ωc J
216 Zhang and Lu L J C Fgure 2. The equvalence of two nverter networks. L a1 Z,1 J1,2 C 2,3 n, n+1 C an Z n+1 a2 (a) L a1 L an Z,1 J1,2 C a2 2,3 J Z n, n+1 n+1 (b) Fgure 3. Alternatve lowpass prototype flters: (a) n s even, (b) n s odd. Accordng the equvalence ntroduced above, we can replace the even number mpedance nverters of the lowpass prototype flter n Fg. 1 by the admttance nverters and obtan the alternatve lowpass prototype flters as shown n Fg. 3. By substtutng (1) wth (5), the parameters and J parameters n Fg. 3 can be expressed as,1 Z La1 Z g g 1,+1 Ca L Z a(+1) n,n+1 Z CanZn+1 g ng n+1 g g +1 2,4,6,..., J,+1 La C Y a(+1) g g +1 1,3,5,... n s even, J n,n+1 Y 1 LanZn+1 Z g ng n+1 n s odd. (6) Here we only dscuss the cases that the frst nverter s the mpedance nverter and the other two cases that the frst nverter s the admttance nverter can be derved n a smlar way. 2.2. Equvalent Network for the Quarter-wavelength Bandpass Flter Fgure 4 shows the classc quarter-wavelength resonator bandpass flter, whch have alternatng hgh and low mpedance levels on two ends of the quarter-wavelength transmsson lnes [5]. Here we only dscussed the case that the frst nverter s mpedance nverter and the resonator number s even. Other cases can be derved n a smlar way.
Progress In Electromagnetcs Research B, Vol. 17, 29 217 In order to derve the equvalent network for the quarterwavelength resonator bandpass flter, we should frst derve the equvalent network for the quarter-wavelength transmsson lne, whch has been dscussed n [5]. As shown n Fg. 5, the nput mpedance can be expressed as Z n Z ZL + Z tan πω Z + Z L tan πω. (7) If Z L Z, Equaton (7) can be approxmated by Z n Z ZL + Z tan πω Z L tan πω X(Ω) + Z2 Z L. (8) Accordng to (8), the equvalent network can be shown n Fg. 5(a) and the seres reactance s X(Ω) Z cot πω. (9) If Z L Z, Equaton (7) can be approxmated by Z n Z Z tan πω Z + Z L tan πω 1 B(Ω) + Z L Z 2. (1) λ /4 λ /4 λ /4 Z,1 1,2 n-1, n n, n+1 n+1 J J Z L H H L H H L Fgure 4. The bandpass flter wth quarter-wavelength resonators (n s even). X (Ω) λ /4 Zn L Z Z L (a) B (Ω) Z Z n λ L /4 Z L (b) Fgure 5. Equvalent network for the quarter-wavelength transmsson lne: (a) Z L Z, (b) Z L Z.
218 Zhang and Lu X 1(Ω) Z,1 1,2 J B 2(Ω) J n-1, n B (Ω) n n, n+1 Z n+1 Fgure 6. Equvalent network for the quarter-wavelength resonator bandpass flter. Accordng to (1), the equvalent network can be shown n Fg. 5(b) and the shunt susceptance s B(Ω) Y cot πω. (11) By applyng the equvalent network n Fg. 5, we can easly derve the equvalent network for the quarter-wavelength resonator bandpass flter as shown n Fg. 6. 2.3. Edge Frequency Mappng Method Snce the lowpass prototype flter n Fg. 3(a) and the equvalent network for the quarter-wavelength resonator bandpass flter n Fg. 6 have been obtaned, the transformaton from the lowpass flter to the bandpass flter can be derved. Comparng the two networks, we can get the mappng functon ωla X f : (Ω), 1, 3,..., n 1. (12) ωc a B (Ω), 2, 4,..., n The followng condton should be mposed: X (Ω ), B (Ω ) X (Ω 1 ) ω 1 L a, B (Ω 1 ) ω 1 C a. (13) X (Ω 2 ) ω 1 L a, B (Ω 2 ) ω 1 C a where ω 1 s the cutoff angular frequency of the lowpass flter, and Ω, Ω 1, Ω 2 denote the center angular frequency, lower and upper edge angular frequency of the bandpass flter, respectvely. By solvng (12) and (13), we can get La X (Ω 1 )/ω 1 C a B (Ω 1 )/ω 1 odd even, (14) X (Ω 1 ) + X (Ω 2 ) B (Ω 1 ) + B (Ω 2 ) odd even. (15)
Progress In Electromagnetcs Research B, Vol. 17, 29 219 Equaton (15) denotes the condton mposed on the center frequency, lower and upper frequency of the bandpass flter. By substtutng (6) wth (14), the parameters and J parameters can be calculated. If the -nverters and J-nverters n Fg. 6 are consdered as deal nverters, we can get a smple case. By substtutng (14) and (15) wth (9) and (11), we can get La Z C a Y 1 ω 1 cot πω 1 cot πω 1 + cot πω 2. (16) It s noted from (16) that the edge frequency mappng method employs the resonator nformaton both at the center frequency and edge frequency, whch s dfferent from the classc mappng method [1] n whch the resonator reactance value and the slope parameter at the center frequency are consdered. 2.4. Frequency-dependent Inverter Model The result n (16) s calculated on the condton that all the nverters are deal; however, the practcal nverters are all frequency-dependent. So we present a frequency-dependent nverter model here, whch s based on the turns rato concept n [3]. However, the turns rato we propose here has a more general defnton and we also brng n the weght exponent n the decomposton of the nverter. The turns rato here s defned as,+1 (Ω)/ m,+1 (Ω),+1 (Ω ),, 2,..., n J,+1 (Ω)/J,+1 (Ω ), 1, 3,..., n 1. (17) It s noted from Fg. 6 that there s an deal nverter on the left sde of every frequency-dependent nverter. So we consder them together. The transfer matrx of the frequency-dependent -nverter together wth the deal nverter on the left can be presented as [ ] [ Z Z [ m /n,+1 (Ω),+1 (Ω),+1 (Ω) m /n,+1 (Ω) [ m /n 1,+1 (Ω) m 1 /n,+1 (Ω) ] ] [ Z,+1 (Ω ) ] [ Z,+1 (Ω),+1(Ω ) Z,+1(Ω) Z ]. (18) Accordng to (18), the decomposton for the -nverter s shown n Fg. 7. It can be seen that the transformer s added only on one ]
22 Zhang and Lu 1: m,1(ω),1 (Ω) (Ω ),1 (a) /n 1: (Ω) m, +1 1-/n 1: m (Ω), +1, +1 (Ω), +1 (Ω ) (b) 1: m (Ω) n, n+1 (Ω) (Ω ) n, n+1 n, n+1 (c) Fgure 7. Decomposton of the frequency-dependent nverters. (a) The frst nverter, (b) The ( + 1)th nverter, (c) The last nverter. sde for the frst and last nverter by usng the weght exponent. So the frequency dependence of the two end nverters can be dstrbuted equally to all the others. Smlarly, the transfer matrx of the frequency-dependent J-nverter together wth the deal nverter on the left can be presented as [ ] [ Z ] [ ] Z J J,+1 (Ω),+1 (Ω) 1 Z J,+1 (Ω) Z J,+1 (Ω) [ ] m /n,+1 (Ω) [ Z ] J,+1 (Ω) m /n,+1 (Ω) 1 Z J,+1 (Ω) [ ] m 1 /n,+1 (Ω) m /n 1,+1 (Ω). (19) The decomposton of the frequency-dependent J nverter s shown n Fg. 8. The turns rato can be absorbed by the adacent dstrbuted resonator elements. Fg. 9 shows the turns rato absorbed by the seres
Progress In Electromagnetcs Research B, Vol. 17, 29 221 reactance. The transfer matrx of the new resonator elements s gven by [ ] m ( 1)/n 1 [1 ] [ ] 1, (Ω) X (Ω) m /n m 1 ( 1)/n,+1 (Ω) 1, (Ω) 1 m /n,+1 (Ω) [ m ( 1)/n 1 1, (Ω) m /n,+1 (Ω) m( 1)/n 1 1, (Ω) m /n,+1 (Ω) X (Ω) m 1 ( 1)/n 1, (Ω) m /n,+1 (Ω) ].(2) Snce the turns rato m s very close to 1, the followng approxmaton can be made: m ( 1)/n 1 1, (Ω) m /n,+1 (Ω) 1. (21) So the new dstrbuted resonator elements can be also regarded as a seres reactance. By applyng (9), t s expressed as X (Ω) m ( 1)/n 1 1, (Ω) m /n,+1 (Ω) X (Ω) ( Z m ( 1)/n 1 1, (Ω)m /n πω,+1 (Ω) cot ). (22) Smlarly, we can derve the turns rato absorbed by the shunt susceptance as shown n Fg. 1 and the new shunt susceptance can be expressed as B (Ω) m ( 1)/n 1 1, (Ω) m /n,+1 (Ω) B (Ω) ( Y m ( 1)/n 1 1, (Ω)m /n πω,+1 (Ω) cot ). (23) -/n, +1 1: m (Ω) /n-1 1: m (Ω), +1 J (Ω) J, +1(Ω ), +1 Fgure 8. Decomposton of the frequency-dependent J nverters. 1-(-1)/n -/n 1: m -1, (Ω) 1: m, +1(Ω) X (Ω) X *(Ω) Fgure 9. Turns rato absorbed by the seres reactance.
222 Zhang and Lu By applyng the edge frequency mappng method to the quarterwavelength resonator bandpass flters wth the frequency-dependent nverter model, the mappng functon can be expressed as f : ( ) ωl a Z m ( 1)/n 1 1, (Ω)m /n,+1 (Ω) cot πω ( ) ωc a Y m ( 1)/n 1 1, (Ω)m /n,+1 (Ω) cot πω By applyng (13), we can get L a Z C a Y 1 ω 1 m ( 1)/n 1 1, (Ω 1 ) m /n 1 ω 1 m ( 1)/n 1 1, (Ω 1 ) m /n m ( 1)/n 1 1, (Ω 1 )m /n,+1 (Ω 1) cot +m ( 1)/n 1 1, (Ω 2 )m /n,+1 (Ω 2) cot,+1 (Ω 1) cot,+1 (Ω 1) cot ( ) πω1 ( ) πω1. ( ) πω1 ( ) πω1 odd even. (24) odd even, (25) (26) By substtutng (6) wth (25), the parameters and J parameters can be calculated. It can be seen from (26) that the edge angular frequences Ω 1 and Ω 2 satsfy dfferent equatons for dfferent dstrbuted resonators. Ths s why the bandpass flters wth frequency-dependent nverters can never acheve the same deal wdeband performance as the lumped-element bandpass prototype flter. However, f Ω 1 s fxed, the Ω 2 values calculated usng (26) for every dstrbuted resonator are very close to each other and the performance of the flters desgned usng ths method s stll very close to the deal performance. The desgn example n Secton 4 wll show that, by usng the fxed Ω 1, the bandpass flters can stll acheve good equal rpple performance n a very wde bandwdth. n / -1, m, +1 (-1)/n-1 1: m (Ω) 1: (Ω) c d b B (Ω) B * (Ω) a Fgure 1. Turns rato absorbed by the shunt susceptance. Fgure 11. Cross secton of the rectangular coaxal cable.
Progress In Electromagnetcs Research B, Vol. 17, 29 223 3. RECTANGULAR COAXIAL FILTERS SYNTHESIS Rectangular coaxal cable has the advantage of low delectrc loss, low radaton loss and weak cross couplng wth other crcuts n a system [6]. They can be fabrcated usng the mcromachnng technques and many applcatons are reported n [6 8]. Fg. 11 shows the cross secton of the rectangular coaxal lne. The dmensons are chosen accordng to [9] and [1] to make the characterstcs mpedance s close to 5 ohms and the cutoff frequency of the hgher modes s above the frequency band of the desgned flter. 3.1. Realzaton of -nverter and J -nverter In order to apply the theores n the last secton to the rectangular coaxal flters synthess, t s necessary to derve two rectangular coaxal structures as the -nverter and J-nverter. We employed an nductve rs structure wth two compensated transmsson lnes added on both sdes as the -nverter as shown n Fg. 12. Its equvalent model conssts of a frequency-dependent -nverter and two extra transmsson lnes on both sdes. The extra transmsson lnes are added on both sdes dueto the compensated transmsson lnes, whose phases wll change wth the frequency as the wavelength changes wth the frequency. Fg. 13 shows the capactve gap structure as the J- nverter and ts equvalent model. The extra transmsson lne phase can be expressed as ϕ (Ω) ϕ (Ω) ϕ (Ω ). (27) ϕ (Ω ) ϕ (Ω ) ϕ (Ω), + 1(Ω) ϕ (Ω) Fgure 12. Realzaton of the -nverter and ts frequency-dependent model. ϕ (Ω ) ϕ (Ω ) ϕ (Ω) J, +1 (Ω) ϕ (Ω) Fgure 13. Realzaton of the J-nverter and ts frequency-dependent model.
224 Zhang and Lu The frequency-dependent nverter can be decomposed nto frequencyndependent nverter wth exponent-weghted turns rato on both sdes as s ntroduced n the last secton and the extra transmsson lnes can be absorbed by the adacent quarter-wavelength resonators. 3.2. Flter Synthess By applyng the equvalent model for the nductve rs and capactve gap structure ntroduced above together wth the decomposton of frequency-dependent nverter n the last secton, we can derve the expresson for the seres reactance and shunt susceptance, whch absorb the turns rato and the extra transmsson lnes from the adacent nverters. They can be expressed as X (Ω)/Z m ( 1)/n 1 1, (Ω) m /n,+1 [ (Ω) ] πω cot + (φ 1 (Ω) φ 1 (Ω )) + (φ (Ω) φ (Ω )) odd. (28) B (Ω)/Y m ( 1)/n 1 1, (Ω) m /n, +1 [ (Ω) ] πω cot + (φ 1 (Ω) φ 1 (Ω )) + (φ (Ω) φ (Ω )) even. (29) For dervaton convenence, we take the followng notatons: X (Ω)/Z E (Ω) odd B (Ω)/Y even, (3) La /Z D odd C a /Y even, (31), A, +1 (Ω) +1 (Ω)/Z, 2,..., n J, +1 (Ω)/Y 1, 3,..., n 1. (32) Substtute (17) wth (32), we can get m, +1 (Ω) A, +1 (Ω)/A, +1 (Ω ). (33) The mappng functon for the transformaton from the lowpass prototype flter n Fg. 3(a) to the bandpass flter n Fg. 6 can be wrtten as f : ωd E (Ω). (34) By applyng (14), (28), (29) and (3), we can get D 1 [ ] A 1, (Ω 1 ) ( 1)/n 1 [ ] A,+1 (Ω 1 ) /n ω 1 A 1, (Ω ) A,+1 (Ω ) [ ] πω1 cot +(φ 1 (Ω 1 ) φ 1 (Ω ))+(φ (Ω 1 ) φ (Ω )). (35)
Progress In Electromagnetcs Research B, Vol. 17, 29 225 By substtutng (6) wth (31) and (32), we can get A,1 (Ω ) D1 g g 1 A,+1 (Ω ) D D +1 g g +1 A n,n+1 (Ω ) Dn g n g n+1 1,2,...,,n 1. (36) It s noted from (35) and (36) that the calculaton of parameters D nvolves parameters A,+1 (Ω ) and the calculaton of parameters A,+1 (Ω ) nvolves parameters D. In order to solve ths problem and calculate A,+1 (Ω ), we employ an teraton procedure. A (),+1 (Ω ), A (),+1 (Ω 1), φ () (Ω ) and D () are used to represent all the parameters after teratons. The parameters after teratons can be calculated usng the parameters after ( 1) teratons as D () 1 ω 1 ( + [ ( 1) A 1, (Ω ] ( 1)/n 1 [ ( 1) 1) A,+1 A ( 1) 1, (Ω (Ω 1) ) A ( 1),+1 (Ω ) φ ( 1) 1 ) ( (Ω 1) φ ( 1) 1 (Ω ) + A (),1 (Ω ) A (),+1 (Ω ) A () n,n+1 (Ω ) D () 1 g g 1 D () D () +1 g g +1 D () n g n g n+1 φ ( 1) 1,2,...,,n 1 ] /n [ πω1 cot )] (Ω 1 ) φ ( 1) (Ω ), (37). (38) It can be seen that (37) and (38) gve the calculaton formulae for D () and A (),+1 (Ω ) but the formulae for A (),+1 (Ω 1) and ϕ () (Ω ) are not provded. Actually, these two parameters can be calculated from A (),+1 (Ω ) usng the cubc splne data nterpolaton functons, whch wll be ntroduced n the next part. Snce the synthess s an teraton procedure, we should set ntal parameters for the teraton. The ntal parameters A (),+1 (Ω ), A (),+1 (Ω 1), φ () (Ω ) and D () can be calculated from the deal nverter model. For the deal nverter case, the exponent-weghted turns rato and the extra transmsson lnes are not consdered. So the calculaton of D () s a compact form of (35) and t s wrtten as D () 1 ( ) πω1 cot D. (39) ω 1
226 Zhang and Lu Substtutng (36) wth (39), we can get A (),1 (Ω ) A (),+1 (Ω ) A () n,n+1 (Ω ) D g g 1 D 1,2,...,,n 1 g g +1 D g ng n+1. (4) The other two ntal parameters A (),+1 (Ω 1) and ϕ () (Ω ) can be also calculated from A (),+1 (Ω ) usng the cubc splne data nterpolaton method ntroduced n the next part. 3.3. Element Parameters Extracton Fgure 14 depcts the quarter-wavelength resonator bandpass flter realzed n rectangular coaxal structures. In order to acheve the requred Chebyshev response, the rs wdths, gap wdths and the resonator lengths must be derved. For a gven nductve rs or capactve gap dmenson, we can calculate the scatterng parameters usng the full-wave smulaton or mode-matchng program. The parameters and the compensated transmsson lne phase can be derved as,+1 (Ω) Z 1 S11 (Ω) 1+ S 11 (Ω) φ (Ω) 1 2 ( S 11(Ω) π), (41) where S 11 (Ω) s the reflecton coeffcent of the nductve rs element. The J parameters and the compensated transmsson lne phase can be also calculated as J,+1 (Ω) Y 1 S11 (Ω) 1+ S 11 (Ω) φ (Ω) 1 2 S, (42) 11(Ω) where S 11 (Ω) s the reflecton coeffcent of the capactve gap element. Irs wdth( w,+1) Irs thckness(g) gapwdth( v,+1) Resonator length( l ) Fgure 14. Confguraton of the quarter-wavelength resonator bandpass flter realzed n rectangular coaxal structures.
Progress In Electromagnetcs Research B, Vol. 17, 29 227 From (41) and (42) we know that t s very easy and fast to calculate the or J parameters for the element wth a gven dmenson by employng the full-wave smulaton or mode-matchng program, however, we usually need to calculate the element dmenson for the requred or J parameters, whch wll cost much tme f a search or matchng program s employed. Here we use the cubc splne data nterpolaton to calculate the element dmenson and other parameters A,+1 (Ω 1 ) and φ (Ω ). Frst we employ a full-wave smulaton to calculate the scatterng parameters of the nductve rs wth a seres of wdths [w] and the capactve gap wth a seres of wdths [v]. Then the samplng data of [(Ω )], [(Ω 1 )] and [φ (Ω )] can be calculated usng (41). The samplng data of [J(Ω )], [J(Ω 1 )] and [φ J (Ω )] can be also calculated usng (42). Wth these samplng data, we can buld the cubc splne functons. The wdths of the nductve rs and the capactve gap can be calculated as w,+1 S ([(Ω )], [w], A,+1(Ω )), 2,..., n v,+1 S ([J(Ω )], [v], A,+1 (Ω )) 1, 3,..., n 1, (43) where the functon S([a], [b], x) denotes the cubc splne nterpolaton functon based on the samplng data [a] and [b] and x s the nterpolaton varable. The calculatons of A,+1 (Ω 1 ) and ϕ (Ω ) can be also expressed as S ([(Ω )], [(Ω A,+1 (Ω 1 ) 1 )], A,+1 (Ω )), 2,..., n S ([J(Ω )], [J(Ω 1 )], A,+1 (Ω )) 1, 3,..., n 1, (44) S ([(Ω )], [φ ϕ (Ω ) (Ω )], A,+1 (Ω )), 2,..., n S ([J(Ω )], [φ J (Ω )], A,+1 (Ω )) 1, 3,..., n 1, (45) After the compensated transmsson lne phase ϕ (Ω ) s calculated, the transmsson lne resonator lengths are gven by l λ g(ω ) [ π 2π 2 + φ 1(Ω ) + φ (Ω )]. (46) Wth ths approach, the element parameters extracton usng the full-wave smulaton s performed only one tme and the samplng data can be calculated. The approach, therefore, presents a calculaton procedure wth the advantage that the soluton s always possble and occurs rapdly. 3.4. Desgn Procedure Fgure 15 shows the flow dagram of the desgn procedure, whch comprses the followng steps.
228 Zhang and Lu Determne the center frequency and lower edge frequency Element parameters extracton usng full-wave smulaton Intal synthess Ideal nverter model Intal parameters Improved nverter model Flter synthess Iteratons Corrected parameters Converge? N Y Full-wave analyss of the desgned flter Fgure 15. Flow dagram of the desgn procedure. Step 1) Frst we should determne the center frequency and lower edge frequency accordng to the flter desgn requrement. And then dfferent nductve rs and capactve gap elements wth a seres of wdths are chosen and the full-wave smulaton s employed to calculate the scatterng parameters at the center frequency and lower edge frequency. By applyng (41) and (42), the samplng data of the and J parameters at the center frequency and the lower edge frequency and the compensated transmsson lne phase are calculated.
Progress In Electromagnetcs Research B, Vol. 17, 29 229 Step 2) The deal nverter model s employed to calculate the ntal parameters A (),+1 (Ω ) by applyng (39) and (4). The other two ntal parameters A (),+1 (Ω 1) and φ () (Ω ) are obtaned usng the cubc splne data nterpolaton n (44) and (45) together wth the samplng data n Step 1). Step 3) Based on the ntal parameters n Step 2), the mproved frequency-dependent nverter model s then establshed. And the corrected and J parameters at the center frequency A (1),+1 (Ω ) are calculated accordng to (37) and (38). Usng the cubc splne data nterpolaton n (44) and (45) together wth the samplng data n Step 1), the other two corrected parameters A (1),+1 (Ω 1) and φ (1) (Ω ) are obtaned. Step 4) A further mproved nverter model s then establshed based on the corrected parameters. By repeatng Step 3), all the parameters A (),+1 (Ω ), A (),+1 (Ω 1) and φ () (Ω ) wll be further corrected. Step 5) The Step 3) and Step 4) are repeated untl the and J parameters after N teratons A (N),+1 (Ω ) converge. Wth the converged and J parameters, the nductve rs and the capactve gap wdths and the resonator lengths are calculated usng (43) and (46). Fnally, a full-wave analyss of the computed flter s carred out before the desgned flter s fabrcated and measured. 4. DESIGN EXAMPLE AND RESULTS The flters here are desgned wthout global optmzatons and are expected to have good n-band equal rpple performance n a wde bandwdth. Snce no excessve global optmzatons are needed, the tme and complexty of the desgn procedure can be mproved sgnfcantly. One example of rectangular coaxal quarter-wavelength resonator bandpass flters wth a center frequency of 5 GHz s desgned and presented here. The dmensons for the flter are lsted n Table 1. The dmenson of the rectangular coaxal cable s chosen as a 16 mm, b 6.5 mm, c 3 mm and d 3 mm. Table 1. g w,1, w 4,5 v 1,2, v 3,4 w 2,3 l 1, l 4 l 2, l 3 l 1.44.53 4.244 7.15 9.72
23 Zhang and Lu The flters are desgned for Chebyshev response wth equal rpple performance n the passband. Because the bandwdth of the flter s very wde, the mnmum passband return loss s chosen as -1 db so that the dmenson s not too small and the flter s easer to be fabrcated. The desgned flter s analyzed by full-wave smulatons usng the commercal software Ansoft HFSS [11] before t s fabrcated wthout tunng screws. Fg. 16 shows the fabrcaton photo of the four-pole flter and Fg. 17 shows the measured and smulated return loss, nserton loss and group delay. It s noted from Fg. 17 that the measured results agree well wth the smulaton results. The passband rpple of the flter s nearly equal durng the 63% fractonal bandwdth (3.84 6.97 GHz). It can be also notced from Fg. 17 that the measured nserton loss above 6.5 GHz does not agree very well wth the smulated results. It may be caused by the fabrcaton tolerance or calbraton tolerance durng the measurement. In the practcal applcatons, the flter can be desgned for 2 db return loss and the nserton loss can be mproved. Fgure 16. Fabrcaton photo of the desgned flter. Fgure 17. Smulated and measured return loss, nserton loss and group delay of the four-pole rectangular coaxal bandpass flter. 5. CONCLUSION In ths paper we have presented a dmensonal synthess method for desgnng wde-band quarter-wavelength resonator bandpass flters. In ths synthess method, the alternatve lowpass prototype flter and the edge frequency mappng method were proposed and appled. The mproved - and J-nverter model wth the exponent-weghted turns rato was also proposed n order to ncorporate the frequency
Progress In Electromagnetcs Research B, Vol. 17, 29 231 dependence of nverters. Based on the edge frequency mappng method and the mproved nverter model, an teratve dmensonal synthess procedure has been presented. As desgn examples, a four-pole rectangular coaxal bandpass flter wth 63% fractonal bandwdth was desgned and fabrcated and the measured results agree well wth the smulated results. The smulaton and measurement results show good equal rpple performance n the passband. The proposed synthess method s expected to fnd more applcatons n desgnng wde-band bandpass flters. REFERENCES 1. Matthae, G. L., L. Young, and E. M. T. Jones, Mcrowave Flters, Impedance-matchng Networks and Couplng Structures, McGraw- Hll, New York, 1964. 2. Hong, J.-S. and M. J. Laucaster, Mcrostrp Flter for RF/Mcrowave Applcatons, John Wely & Sons, 21. 3. Levy, R., Theory of drect coupled cavty flters, IEEE Trans. Mcrowave Theory Tech., Vol. 15, 34 348, Jun. 1967. 4. Cohn, S. B., Drect-coupled-resonator flters, Proc. IRE, Vol. 45, 187 196, Feb. 1957. 5. Matthae, G., Drect-coupled, band-pass wth λ /4 resonators, IRE Natonal Conventon Record, Part 1, 98 111, 1958. 6. Lancaster, M. J., J. Zhou, M. e, Y. Wang, and. Jang, Desgn and hgh performance of a mcromachned -band rectangular coaxal cable, IEEE Trans. Mcrowave Theory Tech., Vol. 55, No. 7, Jul. 27. 7. Llamas-Garro, I., M. J. Lancaster, and P. S. Hall, A low loss wdeband suspended coaxal transmsson lne, Mcrow. Opt. Technol. Lett., Vol. 43, No. 1, 93 95, Jan. 24. 8. Chen, R. T., E. R. Brown, and C. A. Bang, A compact lowloss a-band flter usng 3-dmensonal mcromachned ntegrated coax, 17th IEEE Int. MEMS Conf., 81 84, Maastrcht, The Netherlands, Jan. 25 29, 24. 9. Chen, T.-S., Determnaton of the capactance, nductance, and characterstcs mpedance of rectangular lnes, IEEE Trans. Mcrow. Theory Tech., Vol. 8, No. 5, 51 519, Sep. 196. 1. Gruner, L., Hgher order modes n rectangular coaxal wavegudes, IEEE Trans. Mcrow. Theory Tech., Vol. 15, No. 8, 483 485, Aug. 1967. 11. ANSOFT HFSS, http://www.ansoft.com.