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2 Mirowave Filters 33 6 x Mirowave Filters iafeng Zhou University of Bristol UK. Introdution Filters are two-port networs used to ontrol the frequeny response in a system by permitting good transmission of wanted signal frequenies while rejeting unwanted frequenies. Generally there are four types of filters: low-pass, high-pass, band-pass, and band-stop. Mirowave filter design has been a persistent and produtive field for investigation from the very beginning of mirowave engineering. Nowadays, high performane filters are needed in many mirowave systems. Beause of the importane of mirowave filters, a great deal of material on the theory and design of filters is widely available in the literature. The purpose of this hapter is to introdue the basi theory of mirowave filters, to desribe how to design pratial mirowave filters, and to investigate ways of implementing high performane filters for modern ommuniation systems.. The lowpass prototype. The lowpass prototype filters The lumped-element iruit of an n -order lowpass prototype filter is shown in Fig.. The iruit shown in Fig. (b) is the dual form of that shown in Fig. (a). Both forms give idential responses. In Fig., g to gn are the values of the indutanes or apaitanes of the reative elements. g and gn are the values of terminal immittanes (usually pure resistanes or ondutanes) of the soure and load respetively. The g -values for different types of lowpass filters are given in the following setions.. Butterworth lowpass prototype filter A typial Butterworth, or maximally flat, lowpass response is shown in Fig.. The attenuation harateristi an be expressed by (Matthaei et al. 98) where is the radian frequeny variable, and ut-off frequeny, as defined in Fig.. The value of is given by A n ( ) log[ ( ) ] () is the frequeny of the passband edge, or

3 34 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations Ar () where filters, Ar is the attenuation at the ut-off frequeny, In most ases for Butterworth is defined as the frequeny of the 3-dB passband edge point. That is, = 3 db, and. The parameter n in equation () is the order of the filter, or the number of reative elements in the iruit. Ar (a) (b) Fig.. (a) The prototype of lowpass filters and (b) its dual. Fig.. A typial Butterworth lowpass filter response. For the Butterworth filters with response of the form shown in Fig., the element values, normalized to mae g = and =, an be alulated by g gn ( ) (3) g sin[ ] (,,,n) n The above values an be used to find out the required indutanes and apaitanes in a real filter having a different ut-off frequeny and different terminal impedanes (or admittane / ) by (Matthaei et al. 98) Z g ( even or odd) (4) Z

4 Mirowave Filters 35 Z g ( odd or even) where is the new ut-off frequeny and Z is the new soure and load impedane. Fig. 3. A typial hebyshev lowpass filter response (Matthaei et al. 98)..3 hebyshev lowpass prototype filter A typial hebyshev, or equal-ripple, lowpass response is shown in Fig. 3. The attenuation harateristi an be expressed by (Matthaei et al. 98) where A A ( ) log ( ) log { { os osh [n os [n osh ( )]}, ( )]}, (for ) (for ) (5) Ar (6) In this ase, Ar is the maximum attenuation in the pass band, while is the equal-ripple band edge. The parameter n is the order of the filter. The normalized g -values for an n -order hebyshev low-pass prototype filter an be alulated as follows: g g g a 4a a (,3,4 n ) b g (7)

5 36 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations where and gn oth 4 (n odd) a b (n ln sinh( ) n ( ) sin[ ], n sin ( ), n even) (,,,n) (,,,n) (8) (9) The above values an be used to find the required indutanes and apaitanes in a real filter to realize different ut-off frequeny and soure impedane by Z s Z Z g s g g g n Z s n Z ( even or odd) s ( odd or even) (n even) (n odd) () where is the new ut-off frequeny, Z S is the new soure impedane and Z is the load impedane. It should be notied that, in the prototype filter, the impedanes of the soure and load ould be different for a hebyshev filter, while they are the same for the Butterworth ase. ompared with a Butterworth filter of the same order, a hebyshev response has a muh sharper ut-off rate at the transition region, and usually a worse group delay distortion..4 Quasi-ellipti lowpass prototype filters The synthesis proedures of lowpass prototype filters with Butterworth and hebyshev harateristis are given above. For these two types of responses, all of transmission zeros (attenuation poles) are at infinite frequenies. The inverse hebyshev filters and ellipti filters have all of the transmission zeros at finite frequenies. They have a muh better utoff rate in the transition region than Butterworth and hebyshev filters (Rhodes, 976). However, due to the diffiulties in synthesis and realization of the inverse hebyshev and ellipti filters, it is usually of more interest to develop alternative quasi-ellipti filters, whih have transmission zeros at both finite and infinite frequenies (Rhodes & Alseyab, 98). Espeially quasi-ellipti filters with a single transmission zero (or a pair of transmission

6 Mirowave Filters 37 zeros for the transformed bandpass filters) at finite frequeny are very attrative. This is not only beause these filters an be exatly synthesized and physially realized with little pratial diffiulty, but also beause they give important improvements ompared with onventional Butterworth and hebyshev filters (evy, 976). The typial response of quasiellipti filters is shown in Fig. 4. The response of suh filters has a generalized hebyshev equal-ripple passband, and a different stopband with transmission zero(s). In filter realization, the transmission zero(s) are usually implemented by ross oupling a pair of nonadjaent elements of the filters. Fig. 4. A typial quasi-ellipti lowpass filter response (Rhodes & Alseyab, 98). The attenuation harateristi of an n -order lowpass prototype filter with a single transmission zero at finite an be expressed by (Rhodes & Alseyab, 98) A n F ( ) () where is given by equation (6) and the transfer funtion is given by F ( ) osh[osh n (n )osh ( a a )] () where n is an odd number, and the transmission zeros are at a and infinity. The transmission zero at a is shown in Fig. 4, while the other one a is the mirror image of the positive one. Another similar form of the transfer funtion is given by (Hong & anaster, ), F ( ) osh[(n )osh n osh a ( ) osh a a ( )] a (3) where the transmission zeros are at a and infinity. Some useful quasi-ellipti filters have been synthesized in the referene (Hong & anaster, ). The general synthesis proess to find the values of elements in the prototype quasi-ellipti lowpass filter has been given in the referenes (Rhodes & Alseyab, 98; evy, 976).

7 38 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations However, the exat synthesis proedure is quite ompliated, and there does not exist losed-form formulas for the element values of quasi-ellipti filters as those for Butterworth and hebyshev filters. Instead, a set of data is tabulated in the referene (Hong & anaster, ) for pratial design of quasi-ellipti filters up to eighth-order. 3. The lowpass to bandpass transformation In this setion, only hebyshev lowpass and bandpass filters are disussed. Butterworth and quasi-ellipti filters an be studied in a similar manner. The lowpass prototype shown in Fig. (a) is used for the disussion, while the result will be the same if Fig. (b) were used instead. The transformation from lowpass to highpass and bandstop filters an be aomplished similarly. A typial hebyshev bandpass filter response is shown in Fig. 5. The response an be related exatly to a orresponding lowpass prototype filter by the lowpass to bandpass mapping (Matthaei et al. 98; Hong & anaster, ; ollin, ) ( ) (4) where, are the frequeny variable and the ut-off frequeny of the lowpass prototype filter defined in the above setions respetively. and are the ut-off frequenies of the passband, and the entre frequeny is given by (5) Equation (4) may be solved for, giving ( ) ( ) 4 (6) The point = of the lowpass filter is seen to map into the points of the transformed filter, and map into four points and. Thus the lowpass band of the prototype filter maps into the passband extending from to and to, whih represents bandpass filters with ut-off frequenies at and. The entre frequeny is the geometri mean of the ut-off frequenies. The transformed response for is shown in Fig., and the response for is the mirror image of the 5 positive one.

8 Mirowave Filters 39 Fig. 5. A typial hebyshev bandpass filter response. Aording to equation (4), to obtain the transformed bandpass response, the shunt apaitane in the prototype lowpass filter needs to be hanged to a new suseptane jb j[ g j( ( )]g ) j ( g ) ( odd) (7) whih is equivalent to a shunt tuned resonator with a shunt apaitane indutane given by g g ( odd) and a shunt Similarly, the series indutane in the prototype filter should to be hanged to a new reatane jx j[ ( )]g g j( ) j ( g ) ( odd) whih is equivalent to a series tuned resonator with a series apaitane indutane given by (8) (9) and a series

9 4 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations g g ( even) The equivalent iruit of the transformed bandpass filter is shown in Fig. 6. () Fig. 6. The equivalent iruit of a bandpass filter transformed from a lowpass filter. 4. Transformation of bandpass filter using K- or -inverters The filter shown in Fig. 6 onsists of series tuned resonators alternating with shunt-tuned resonators. Aording to equation (8) and equation (), suh a filter is diffiult to implement, beause the values of the omponents are very different in the shunt and series tuned resonators. A way to modify the iruit is to use - (admittane) or K - (impedane) inverters, so that all resonators an be of the same type. 4. Impedane and admittane inverters An idealized impedane inverter operates lie a quarter-wavelength line of harateristi impedane K at all frequenies. As shown in Fig. 7(a), if an impedane inverter is loaded with an impedane of Z at one end, the impedane Z seen from the other end is (Matthaei et al. 98) K K ZK () Z An idealized admittane inverter, whih operates lie a quarter-wavelength line with a harateristi admittane at all frequenies, is the admittane representation of the same thing. As shown in Fig. 7(b), if the admittane inverter is loaded with an admittane of at one end, the admittane seen from the other end is (Matthaei et al. 98) () It is obvious that the loaded admittane an be onverted to an arbitrary admittane by hoosing an appropriate value. Similarly, the loaded impedane Z an be onverted to an arbitrary impedane by hoosing an appropriate K value. Fig. 7. Definition of K- (impedane) and - (admittane) inverters.

10 Mirowave Filters 4 As indiated above, both the impedane and admittane inverters are lie ideal quarterwave transformers. While K denotes the harateristi impedane of an inverter and denotes the harateristi admittane of an inverter, there are no oneptual differenes in their inverting properties. An impedane inverter with harateristi impedane K is idential to an admittane inverter with harateristi admittane = /K. Espeially for a unity inverter, with a harateristi impedane of K = and a harateristi admittane of =. Besides a quarter-wavelength line, there are some other iruits that operate as inverters. Some useful - and K - inverters are shown in Fig. 8 and Fig. 9. It should be notied that some of the indutors and apaitors have negative values. Although it is not pratial to realize suh omponents, they will be absorbed by adjaent resonant elements in the filter, as disussed in the following setions. It should also be noted that, sine the inverters shown here are frequeny sensitive, these inverters are best suitable for narrowband filters. It is shown in the referene (Matthaei et al. 98) that, using suh inverters, filters with bandwidths as great as perent are ahievable using half-wavelength resonators, or up to 4 perent by using quarter-wavelength resonators. (a) = /() Fig. 8. Some iruits useful as -Inverters. (b) = (a) K = Fig. 9. Some iruits useful as K-Inverters. (b) K = /() 4. onversion of shunt tuned resonators to series tuned resonators Beause of the inverting harateristi indiated by equation (), a shunt apaitane with a -inverter on eah side ats lie a series indutane (Matthaei et al. 98). iewise, a shunt tuned resonator with a -inverter on eah side ats lie a series tuned resonator. To verify this, a shunt tuned resonator, onsisting of a apaitor and an indutor, with a - inverter on eah side is shown in Fig. (a). Both -inverters have a value of. If the iruit is loaded with admittane of an arbitrary value at one end, from equation (), the admittane looing in at the other end is given by j j (3)

11 4 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations The impedane is, therefore, j Z (4) j This impedane is equivalent to a series tuned resonator loaded with an impedane of Z /, as shown in Fig. (b). The apaitor of and indutor of the equivalent iruit are given by (5) Beause the above equations are orret regardless the value of the load, the two iruits shown in Fig. are equivalent to eah other. (a) (b) Fig.. A shunt tuned resonator with -inverters on both sides and its equivalent iruit. It is very useful for the disussion in the following setions to point out that, from equation (5), the transformed resonator an have an arbitrary impedane level / tuned at the same frequeny. That is, the shunt tuned iruit with -inverters shown in Fig. (a) an be onverted to a series tuned resonator with an arbitrary or, as long as =, by hoosing the inverter (6) Thus, the bandpass filter shown in Fig. 6 an be onverted to a iruit with only shunt resonators by using -inverters, as shown in Fig.. The dual ase of a series tuned resonator with a K -inverter on eah side an be derived in a similar manner. Fig.. The bandpass filter using only shunt resonators and -inverters.

12 Mirowave Filters onversion of shunt resonators with different -inverters In the above setion, the shunt-tuned resonator is onverted into a series tuned resonator by -inverters of the same value at both ends. More generally, the inverters may have different values. Fig. (a) shows a shunt-tuned iruit with -inverters at both ends. The resonator onsists of a apaitor and an indutor. The -inverters have a value of on one end and on the other. This iruit an be transformed to an equivalent iruit shown in Fig. (b), where the shunt tuned resonator has a apaitor and an indutor, whereas =, and the -inverters have values of and respetively. (a) (b) Fig.. (a) A shunt tuned resonator with -inverters of different values, and (b) its equivalent iruit. The iruit shown in Fig. (a) is not symmetrial. If the iruit is loaded with an admittane of R at the right-hand-side end, the admittane and impedane Z looing in at the lefthand-side end are given by Z j j R j j R (7) Similarly, if the iruit is loaded with an admittane of at the left-hand-side end, the impedane Z R looing in at the other end are given by Z R R j (8) j In a similar manner, with a load at the one end, the impedane from the other end of the iruit shown in Fig. (b) an be given by And Z' Z' R ' R j' (9) ' ' j'' ' R ' j' (3) ' ' j'' '

13 44 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations If the two iruits shown in Fig. are equivalent, Z Z and ZR ZR, from equation (7) to equation (3), it an be obtained ' ' ' ' ' ' (3) This transformation is very useful in a sense that the bandpass the filter shown in Fig. an be further onverted to a iruit where all of the resonators have the same indutane and apaitane. Suh onversion will be shown in the next setion. 4.4 Filter using the same resonators and terminal admittanes In filter design, it is usually desirable to use the same resonators in a filter, and have the same harateristi impedanes or admittanes at the soure and load. In this setion, an n - th order bandpass filter will be transformed to use the same shunt resonators tuned at the same frequeny, with an indutane of and a apaitane of, and the same terminal admittanes at both ends. The equivalent iruit of a bandpass filter using only shunt resonators and -inverters is shown in Fig.. As disussed in setion, the admittane of the soure and the load an be onverted to the same value by adding a -inverter, or hanging the value of the - inverter if there is a -inverter diretly onneted to the soure or load. By the transformation disussed in setion, the iruit shown in Fig. an be transformed to Fig. 3, where all resonators have the same indutanes and apaitane. The values of the inverters are given by:,, n,n g g ' n g g ' n (,,,n ) ' g g (3) where is the frational bandwidth of the bandpass filter given by (33) where, and are the ut-off frequenies, and is the entre frequeny of the filter as defined in equation (5). The values of g,g,g gn and ' are defined in the low-pass prototype filter disussed above.

14 Mirowave Filters 45 Fig. 3. A transformed bandpass filter using the same resonators. The above equations are based on the lumped-element equivalent iruit of the filter. More generalized form of these equations will be given in setion. This transformation is very useful beause all the resonators in the filter have the same harateristis, whih maes the design and fabriation of the filter muh easier. The above transformation an also be implemented by using series tuned resonators and K -inverters in a similar manner. 5. oupled-resonator filter The -inverters in the filter shown in Fig. 3 an be replaed by any of the equivalent iruits shown in Fig. 8 or other equivalent iruits. One form of suh filters is shown in Fig. 4, using the equivalent iruit shown in Fig. 8(b) for those -inverters. The results of this setion would still hold if other equivalent iruit were hosen for the inverters. Fig. 4. The transformed filter using the same resonators with apaitive ouplings between resonators. In the filter shown in Fig. 4, the equivalent iruit of eah -inverter onsists of one positive series apaitor and two negative shunt ones. In filter design, the positive apaitane represents the mutual apaitanes between resonators, while the negative apaitors an be absorbed into the positive shunt apaitors in the resonators. It should be noted that the negative apaitanes adjaent to the soure and load annot be absorbed this way. Further disussion about these negative apaitanes will be given below in setion. From equation (3), it is obvious that the nowledge of the equivalent iruit of the resonators will be needed to find out the values of the required -inverters, whist the g- values an be obtained from the low-pass prototype filter. One the values of the - inverters are determined, the required mutual apaitanes between resonators an then be alulated by the equation shown in Fig. 8(b). It should be noted that, as indiated in setion, the inverters shown in Fig. 8 are atually frequeny dependent. However, in the narrowbandwidth near the entre frequeny, the inverters an be regarded as frequeny insensitive by approximating, ( ) (34),, where /, and,, and, are defined in Fig.. 4

15 46 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations 5. Internal and external oupling oeffiients Due to the distributed-element nature of mirowave iruits, it is usually diffiult to find out the equivalent iruit of the resonators diretly. It is therefore diffiult to determine the required the values of the -inverters, or mutual apaitanes between resonators. However, from equation (3), it is possible to obtain the required ratio of the mutual apaitane to the shunt apaitane of eah resonator without the nowledge of the equivalent iruit. For example, the ratio of the required mutual apaitane between resonators to the apaitane of eah resonator is, from equation (3) and equation (34), M,,,, (,,,n ) (35) ' g g where is the frational bandwidth of the bandpass filter, and ', g, g are defined in the prototype lowpass filter. M, is the strength of the internal oupling, or the oupling oeffiient, between resonators. The external ouplings between the terminal resonators and the soure and load are defined in a similar manner by, with the approximation of equation (34), Q en,n Q e,, n,n, n,n gg' gngn ' (a) (b) (36a) (36b) The values of Q e, and Qe n,n are the strength of the external ouplings, or the external quality fators, between the terminal resonators and the soure/load. It an be seen from equation (35) and equation (36) that these required internal and external ouplings an be obtained diretly from the prototype low-pass filter and the passband details of the transformed bandpass filter, without speifi nowledge of the equivalent iruit of the resonators. From equation (3), it an be proved that fixing the internal and external ouplings as presribed by equation (35) and equation (36) is adequate to fix the response of the filter shown in Fig. 4 (Matthaei et al. 98). The following two setions will onentrate on experimentally determining these ouplings. 5. Determination of internal ouplings by simulation After finding the required oupling oeffiients and external quality fators for the desired filtering harateristis as disussed above, it is essential to experimentally determine these ouplings in a pratial iruit so as to find the dimensions of the filter for fabriation. This setion desribes the determination of the oupling oeffiients between resonators by the use of full wave simulation. The details about the external ouplings between the terminal resonators and the soure and load are given in the next setion. As disussed above, the same resonators are usually used in a filter. The equivalent iruit of a pair of oupled idential resonators is shown in Fig. 5, whih an be regarded as part

16 Mirowave Filters 47 of the filter shown in Fig. 4. As the iruit is symmetrial, the admittane looing in at either side is, in j j(, ) j, j(, ) j (37) At resonane, in =. By equating the right-hand side of equation (37) to zero, four eigenvalues of the frequeny an be obtained. The two positive frequenies are given by ( (,, ) ) (38) The other two negative frequenies are the mirror image of these positive ones. Fig. 5. The equivalent iruit of a pair of oupled idential resonators. If this iruit is wealy oupled to the exterior ports for measurement or simulation, the typial measured or simulated response for the sattering parameter S is as shown in Fig.. More details of the measurement or simulation will be given in the next setion. The two 6 resonant frequenies as expressed in equation (38) are speified in Fig. 6. By inspeting equation (35) and equation (38), the oupling oeffiient an be determined by, M,, (,,,n ) (39)

17 48 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations Fig. 6. A typial response of the oupled resonators shown in Fig Determination of external ouplings by simulation The proedure to determine the strength of the external oupling, the external quality fator Q e, is somewhat different from determining the internal oupling oeffiient between resonators. It is possible to tae the orresponding part of the iruit, for example, the load, the last resonator and the inverter between them, from Fig. 4, and determine the external quality fator by measuring the phase shift of group delay of the seleted iruit (Hong & anaster, ). More onveniently, a doubly loaded resonator shown in Fig. 7 is onsidered. One end of the iruit is the same as in Fig. 4, while another load and inverter of the same values are added symmetrially at the other end. The ABD matrix of the whole iruit, exept the two loads, is given by A Q BQ j j D e j e (4) Q Q je j je where e =,, or n,n+ as defined in Fig. 4. The sattering parameter S an be alulated by S A Q BQ Q D Q j e ( ) (4) By substituting equation (36) and /, this equation an be rewritten as S j ( e ) jqe ( ) where Q e = Q, or Q n,n+ is the external quality fator for the soure or load as defined in equation (36). At a narrow bandwidth around the resonant frequeny, / / / with. The magnitude of S is given by (4)

18 Mirowave Filters 49 S (Q / ) e (43) Fig. 7. The equivalent iruit of a doubly loaded resonator. If this iruit is onneted to the exterior ports for measurement or simulation, the typial measured or simulated response of the doubly oupled resonator is shown in Fig. 8. It an be found from equation (43) that S has a maximum value S = (or db) at, and the value falls to.77(or 3 db) at Qe (44) The two solutions of equation (44) are given by (45) Q The two orresponding frequenies / Qe and / Qe an be easily found by simulation or measurement as shown in Fig. 8. The external quality fator therefore an be given by Qe (46) ( ) As indiated above, the external quality fator Q e is atually defined for a singly loaded resonator. One possible way to determine Q e of a singly oupled resonator is to measure the phase shift of group delay of the refletion oeffiient (S ) of a singly loaded resonator, and the external quality fator is given by (Hong & anaster, ) e where o 9 and 9 o Q e (47) o o 9 9 are the frequenies at whih the phase shifts are The external qualify fator an also be given by (Hong & anaster, ) 9 respetively. ( ) (48) 4 Qe where ( ) is the group delay of S at the entre frequeny.

19 5 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations Fig. 8. The typial response of a doubly oupled resonator. Another more pratial way to determine the external quality fator of a singly loaded resonator is to use an equivalent iruit shown in Fig. 9. The iruit is similar to Fig. 4, exept that one end of the iruit has the external oupling to be measured, while the other end has a relatively muh weaer oupling, namely w «e. The ABD matrix of the iruit an be expressed as, similar to equation (4), A Q Q B D Q Q j e j e j j j w j w (49) Fig. 9. The equivalent iruit of a singly loaded resonator. It is alled singly loaded beause the oupling at one end, represented by e s, is muh stronger than the oupling at the other end, represented by w s. The sattering parameter S an be obtained, similarly to equation (4), by S ( e w w e ) j It is obvious that if w = e, this equation is the same as equation (4). Here as w «e, equation (5) an be rewritten as, w S e ( jqe ) (5) The typial response of the iruit is very similar to Fig. 8, exept that the value of S has a maximum value of w / e [or log( w / e ) db] at. The value is 3 db lower at frequenies where is given by, e w Q e (5)

20 Mirowave Filters 5 Qe The two orresponding frequenies are /( Qe) and /( Qe) external quality fator, therefore, an be determined by (5). The Q e (53) 5.4 Equivalent iruit of the inverters at the soure and load In the above disussion, some negative shunt apaitanes are used to realize the inverters. Most of these negative apaitanes an be absorbed by the adjaent resonators. However, this absorption proedure does not wor for the inverters between the end resonators and the terminations (soure and load), as the terminations usually have pure resistanes or ondutanes. This diffiulty an be avoided if another equivalent iruit, shown in Fig., is used for the -inverter. As indiated above, by using any equivalent iruits to realize the required inverters, the filter response will be the same. All the methods to determine the external quality fator as desribed by equations (46), (47), (48) and (53) are still valid. In the iruit shown in Fig., at the resonant frequeny, the admittane looing in from the resonator towards the soure is given by in j a j b b j( a b b ) (54) Beause the required value of the -inverter is e,, or n,n as defined in Fig. 3, the required admittane is therefore e /. By equating this value to the real part of in, two solutions of b an be obtained, and the positive one is given by e b e (55) ( ) By equating the imaginary part of in to, a an be found by a b ( This negative shunt apaitane an be absorbed by the resonator. b ) (56)

21 5 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations Fig.. Another equivalent iruit to realize the inverter between the end resonator and the termination. 5.5 More generalized equations Sine purely lumped elements are diffiult to realize at mirowave frequenies, it is usually more desirable to onstrut the resonators in a distributed-element form. Suh a resonator an be haraterized by its entre frequeny and its suseptane slope parameter (Matthaei et al. 98) db( ) (57) d where B is the suseptane of the resonator. For a shunt tuned lumped-element resonator, equation (57) an be simplified as /( ). The values of -inverters for filters using distributed-element resonators an be alulated by replaing with in equation (3), where is the suseptane slope parameter of the distributed-element resonators. More generally, if the slope parameter of eah resonator is different from the others, equation (3) an be rewritten as (Matthaei et al. 98), ', n,n g g g g ' n g g n n ' (,,,n ) (58a) (58b) (58) where is the suseptane slope parameter of -th resonator, is given in equation (33), and the values of g,g,g gn and ' are defined in the low-pass prototype filter. The definition of the oupling oeffiient equation (35) an be modified to (Matthaei et al. 98), M,, ' g g (,,,n ) (59) If it is possible to find the equivalent apaitanes, + for the -th and (+)-resonators, and the equivalent mutual apaitane,+ in the viinity of the entre frequeny, the oupling oeffiient M,+ an be expressed by

22 Mirowave Filters 53 M,,,, (,,,n ) (6) In a similar manner, equation (36) an be modified to Q Q e, en,n, n n,n g g n g' gn ' (6a) (6b) Or, if the equivalent apaitanes in the viinity of the entre frequeny of the terminal resonators an be found, Qe, (6a) Q en,n, n n,n, n n,n (6b) For the ase when a filter uses resonators tuned at different frequenies, the determination of the oupling oeffiients are desribed in hapter 8 of the referene (Hong & anaster, ). 6. Design example of a hebyshev filter At mirowave and millimetre wave frequenies, filters are not usually built by using the lumped-element omponents as disussed above, but by utilizing transmission lines, usually alled distributed-element omponents. The omplex behaviour of the distributed-element omponents maes it very diffiult to develop a omplete synthesis proedure for mirowave filters. It is, however, possible to approximate the behaviour of ideal apaitors and indutors by using appropriate mirowave omponents in a limited frequeny range. Thus the mirowave filter is realized by replaing apaitors and indutors in the lumpedelement filters by suitable mirowave omponents with similar frequeny harateristis in the frequeny band of interest. The mirowave filter design proedure is further simplified by the aid of AD program. 6. Filter synthesis In this setion, a three-pole hebyshev bandpass filter with a frational bandwidth of.46% entred at 6 MHz, and a ripple of.db in the passband, will be designed by simulation (Sonnet Software, 9) using the above theory. Firstly, the g -values of the three-pole hebyshev prototype lowpass filter, with a ripple of. db, an be alulated by equation (7): g =, g =.69, g =.97, g 3 =.69 and g 4 =. Substituting these values with = and the frational bandwidth.46% into equation (35) and (36) results in, M, = M,3 =.59 (63)

23 54 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations Q e, = Q e 3,4 = 36.5 where M, and M,3 are the oupling oeffiients between resonators, and Q e, and Q e 3,4 are the external fators between the end resonators and the terminations (soure and load). 6. Determination of the ouplings by simulation The shape and dimensions of a mirostrip resonator entred at 6 MHz are shown in Fig.. The entre frequeny an be tuned in a small range by hanging the lengths of the stubs A and B. The resonator is designed on a.5 mm thi MgO substrate. More details on the design of this resonator an be found in the referene (Zhou et al., 5). To determine the oupling strength between resonators, the struture shown in Fig. (a) is used for simulation. The ouplings between the resonators and the feed lines are muh weaer than that between the two resonators. As disussed in setion, two resonant frequenies will be obtained from the simulation as shown in Fig. (b), similar to Fig. 6. The oupling oeffiient an be extrated by using equation (39). The oupling oeffiient is a funtion of the distane d between the resonators, and the relationship between the oupling strength and the distane d is shown in Fig. 3. It an be found in Fig. 3 that two resonators with a distane of.6 mm have a oupling oeffiient.59, whih is very lose to the required value of.59. Fig.. ayout of the resonator entred at 6 MHz. The minimum line and gap widths are.5 mm. Other detailed dimensions are shown in the figure (unit: mm). (a) (b) Fig.. (a) The struture to determine the oupling strength between resonators in the simulation, and (b) the simulated response for d =.6 mm. The external oupling between the end resonator and the termination is realized by a tapped line, as shown in Fig. 4(a). The length t along the signal line of the resonator, from the tapped line to the middle of the resonator, ontrols the strength of the external oupling. The

24 Mirowave Filters 55 resonator is wealy oupled to the other feed line, so that the iruit an be regarded as a singly loaded resonator as disussed in setion. The wide mirostrip line onneted to port has a harateristi impedane of 5 ohm, the length of whih does not affet the response of the iruit. oupling oeffiient Distane (d) between resonators (mm) Fig. 3. The oupling oeffiient against the distane between the resonators. The simulated response is shown in Fig. 4(b), similar to Fig. 8, and the external quality fator an be extrated by using equation (53). The relationship between the external quality fator and the length t is shown in Fig. 5. It an be found that t = 4. mm gives an external Q of 35, whih is lose to the required value of (a) (b) Fig. 4. (a) The struture to determine the external oupling between the end resonator and the termination in the simulation (unit: mm) and (b) the simulated response for t = 3.9 mm. External Q Distane (t) between the external tapped-line and the middle of the terminal resonator (mm) Fig. 5. The external oupling strength against the length from the tapped line to the middle of the resonator. It should be noted that the position of the tapped line also affets the entre frequeny of the end resonator. Therefore the dimensions of the end resonator need to be hanged slightly to eep the desired entre frequeny. This is done by hanging the length of the stubs A and B as shown in Fig. 4.

25 56 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations Fig. 6 ayout and dimensions of the three-pole hebyshev filter (unit: mm), where t = 4.3 and d =.6 after optimisation. More detailed dimensions of the resonators an be found in Fig. and Fig. 4(a). For the required external Q for this filter, the length of A and B is found to be.4 mm, whih ompares to the length of.65 mm in the original resonator shown in Fig.. The filter is formed in a shape as shown in Fig. 6. It will be disussed blow that further optimization of the filter is required to ahieve optimal performane. 6.3 iruit optimisation and simulated response The theoretial response of the three-pole hebyshev filter designed is shown in Fig. 7. The theoretial response is obtained by alulation using the oupling oeffiients given in equation (63). More details on the alulation are given in hapter 8 of the referene (Hong & anaster, ). S & S (db) S simulated S simulated S theoretial S theoretial Frequeny (MHz) Fig. 7. The theoretial and simulated responses of the three-pole hebyshev filter. The dimensions obtained in setion are used by the full-wave simulator Sonnet (Sonnet Software, 9). The simulated response of the filter is shown in Fig. 7. However, the simulated response using these dimensions is lose to, but does not meet the theoretial response very well. Generally, there are two major reasons. One reason is that, in the simulator, the dimensions of the iruit are disrete rather than ontinuous, so that the

26 Mirowave Filters 57 required oupling oeffiients an usually be realized proximately, rather than preisely. This is beause in the simulator a ell is the basi building blo of the iruit. Thus any part of the iruit may be as small as one ell or may be multiple ells long or wide. For example, a typial iruit drawn in the simulator is shown in Fig. 8, whih has a ell size of.5 mm.5 mm. The bla dots are the grid points. The dimensions of the iruit in the horizontal diretion, suh as w and d, an only be the multiple of.5 mm; while the dimensions in the vertial diretion, suh as h, an only be the multiple of.5 mm. If d =.75 mm would give the required performane, in the simulator, only the proximate value d =.5 or. mm ould be used. The dimensions an be more preise if the ell size is smaller. But, on the other hand, the simulation time inreases exponentially as the ell size dereases. Another reason is that the unwanted ross ouplings, among non-neighbouring resonators and between input and output ports, are not onsidered in the design. These ross ouplings annot be easily determined before the design as they are not independent to other ouplings, and they beome muh more ompliated in a filter having more resonators. Alternatively, the simulator (Sonnet Software, 9) has an optimisation funtion, whih an be used to optimise the dimensions of a iruit to get an optimised performane. Fig. 8. An example iruit drawn in the EM simulator (Sonnet Software, 9). By using this funtion, the user may selet dimensions of the iruit and define them as a parameter. In the analysis, the simulator ontrols the parameter value, within a user defined range, in an attempt to reah a user defined goal. More than one parameters an be used simultaneously in the simulation if neessary. More detailed information about the optimisation an be found in the referene (Sonnet Software, 9). The dimensions of the three-pole filter after optimisation by the simulator are shown in Fig. 6, where t = 4.3 mm and d =.6 mm. The optimised response is shown in Fig. 9, whih agrees very well with the theoretial result. The optimised passband has a ripple of.4 db, very lose to the target of. db; and the minimum return loss is better than 5 db in the passband, also lose to the theoretial value of 6.3 db. S (db) S optimised S theoretial Frequeny (MHz) (a)

27 58 Mirowave and Millimeter Wave Tehnologies: from Photoni Bandgap Devies to Antenna and Appliations S & S (db) S optimised S optimised S theoretial S theoretial Frequeny (MHz) (b) Fig. 9. The theoretial and optimised performanes of the 3-pole hebyshev filter. 7. Summary The general theory of mirowave filter design based on lumped-element iruit is desribed in this hapter. The lowpass prototype filters with Butterworth, hebyshev and quasiellipti harateristis are synthesized, and the prototype filters are then transformed to bandpass filters by lowpass to bandpass frequeny mapping. By using immitane inverters ( - or K -inverters), the bandpass filters an be realized by the same type of resonators. One design example is given to verify the theory on how to design mirowave filters. 8. Referene ollin R. E. (). Foundation for Mirowave Engineering, ohn Wiley & Sons, In. ISBN: ISBN New ersey. Hong,. S. & anaster, M.. (). Design of Highly Seletive Mirostrip Bandpass Filters with a Single Pair of Attenuation Poles at Finite Frequenies, IEEE Transations on Mirowave Theory and Tehnology, vol. 48, uly. pp Hong,. S. & anaster, M.. (). Mirostrip Filters for RF/Mirowave Appliations, ohn Wiley & Sons, IN. ISBN: , New or. evy, R. (976). Filters with single transmission zeros at real and imaginary frequenies, IEEE Transations on Mirowave Theory and Tehnology, vol. 4, Apr pp Matthaei, G.; oung,. & ones, E.M.T. (98). Miorwave Filters, Impedane-mathing Networs and oupling Struture, Arteh House, IN. 685 anton Street, Norwood, MA 6. Rhodes,. D. (976). Theory of Eletrial Filters, Willey. ISBN: , New or. Rhodes,. D. & Alseyab, S. A. (98). The generalized hebyshev low-pass prototype filter. iruit Theory Appliation, vol.8, 98. pp.3-5. Sonnet Software (9), EM User s Manual, Sonnet Software, In. Elwood Davis Road North Syrause, N 3 Zhou,.; anaster, M..; Huang, F.; Roddis, N. & Glynn, D. (5) HTS narrow band filters at UHF band for radio astronomy appliations, IEEE Transations on Applied Superondutivity, vol.5, une 5. pp

28 Mirowave and Millimeter Wave Tehnologies from Photoni Bandgap Devies to Antenna and Appliations Edited by Igor Minin ISBN Hard over, 468 pages Publisher InTeh Published online, Marh, Published in print edition Marh, The boo deals with modern developments in mirowave and millimeter wave tehnologies, presenting a wide seletion of different topis within this interesting area. From a desription of the evolution of tehnologial proesses for the design of passive funtions in milimetre-wave frequeny range, to different appliations and different materials evaluation, the boo offers an extensive view of the urrent trends in the field. Hopefully the boo will attrat more interest in mirowave and millimeter wave tehnologies and simulate new ideas on this fasinating subjet. How to referene In order to orretly referene this sholarly wor, feel free to opy and paste the following: iafeng Zhou (). Mirowave Filters, Mirowave and Millimeter Wave Tehnologies from Photoni Bandgap Devies to Antenna and Appliations, Igor Minin (Ed.), ISBN: , InTeh, Available from: InTeh Europe University ampus STeP Ri Slava Krautzea 83/A 5 Rijea, roatia Phone: +385 (5) Fax: +385 (5) InTeh hina Unit 45, Offie Blo, Hotel Equatorial Shanghai No.65, an An Road (West), Shanghai, 4, hina Phone: Fax: