Direct Analysis of Wave Digital Network of Microstrip Structure with Step Discontinuities

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1 Direc Analysis of Wave Digial Nework of Microsrip Srucure wih Sep Disconinuiies BILJANA P. SOŠIĆ Faculy of Elecronic Engineering Universiy of Niš Aleksandra Medvedeva 4, Niš SERBIA MIODRAG V. GMIROVIĆ IMEL-Communicaions Insiue Mihajlo Pupin Boulevard, 65b, Belgrade SERBIA Absrac: - A microsrip srucure, divided ino cascade connecion of uniform secions, can be efficienly modelled by wave digial nework [4]-[7]. Appropriae choice of a secion number in digial model of microsrip srucures is very imporan because of direc influence on he sampling frequency of he digial model, and on accuracy of desired response. Also, effecs of he idenified sep disconinuiies have o be compensaed. In his paper, a choice of minimal number of secions based on given relaive error, and one wave digial elemen for he sep disconinuiy are presened.verificaion of he obained resuls is done on one example of lowpass filer. Key-Words: - Wave digial approach, Microsrip lines, Microwave srucure, Sep disconinuiy. Inroducion Modeling of he planar srucures by wave digial elemens, based on well known heory of wave digial filers []-[3], can be efficienly used for analysis of hese srucures in boh he ime and he frequency domains. Microwave planar srucures can be modeled by one-dimensional [4]-[6] and by wo-dimensional [7] wave digial elemens. A nonuniform srucure has o be divided ino cascade connecion of uniform ransmission lines (UL) where each UL is modeled by uni wave digial elemens. A lossless uniform ransmission line is modeled by a wo-por digial elemen wih a delay appears in forward pah. his wave digial wo-por is called he uni elemen (UE) []. he por resisances of he UE are equal and correspond o he characerisic impedance of UL. he connecion of wo UE wih differen por resisances is achieved by wo-por series/parallel adapors [3]. In he complex microsrip srucures, delays of he ransmission lines vary from one anoher and because of his, each ransmission line has o be represened as a cascade connecion of a cerain number of UE. A goal of his paper is o presen a way of deerminaion a minimal secion numbers in wave digial nework (WDN) of complex microsrip srucure. In he previously published papers [4]-[6], a nonuniform microsrip srucures are observed as cascaded UL segmens, bu effecs of he sep disconinuiies have no been aken under consideraion. Bu, once he sep disconinuiies have been idenified in he srucure, hey mus be correced. In his paper, a wave digial model of asymmerical equivalen -nework of his disconinuiy (WDE_Sep) is given. A nonuniform planar microsrip srucure wih sep disconinuiies is modeled by a WDN composed of hree ypes of building blocks UE, WDE_Sep and adapors. A general WDN is depiced in Figure. he wo-por adapors a he ends are used for maching source and load resisances o he res of he WDN. =Us ADP-S Line_ WDE-Sep_ Line_ WDE-Sep_M- Line_M Fig. Wave digial nework ADP-L =*U ISSN: ISBN:

2 A very simple mehod of analysis of he WDN is a block-diagram mehod. WDN is formed direcly in he Simulink oolbox of he MALAB environmen. Programs dlinmod.m, dimpulse.m and ff.m are employed o find a response. Minimal Secion Numbers. Deerminaion of he secion numbers A real delay of a complex microsrip srucure differs from he delay of he WDN. In he complex microsrip srucures, delays of he ransmission lines are no muliple inegers of he minimum delay. he number of secion n k used for modeling an individual ransmission line is found as he neares ineger of k / min () h raio, where k is a delay on he k ransmission line, min min,,..., M is a minimum delay, k,,..., M and M is a number of ransmission lines in he microsrip srucure, [5]. Delay of he individual UE in WDN is found as / n, () where M (3) k k is he sum of all ransmission line delays, i.e. a oal real delay of he srucure, and M n n (4) i k is he oal number of UE in he WDN.. A choice of minimal secion numbers for known error If a number of cascaded UE for each ransmission line is found in he above described manner, hen he relaive error of he oal delay is high, wha means ha response accuracy is lower. In order o find a lower relaive error, an exra segmenaion of he ransmission lines have o be done (muliple facor q has o be used). he number of secion n k used for modeling an individual ransmission line is found as he neares ineger of q k / min (5) raio. According o hese daa, a oal delay for he digial model of he srucure is n q. (6) min / Relaive error of a oal delay in percens is found as er[%] (7) where is given by (3) and is given by (6). A minimal number of secions needed for modeling of observed srucure is found by using relaive error in percens, n _ er[%], which is already known. A procedure for deerminaion of he minimal number of secions wih he error less han he given one can be done in a few seps. A he beginning, errors er[%] are found for he differen variable values q,,..., qmax, where q max is an arbirary chosen value. hen, firs relaive error wih absolue value less hen previously given error, er[%] n _ er[%], is chosen. he number of h secions for he k ransmission line nk, k,,..., M, a oal number of secions n and a oal delay of a digial model of he srucure, corresponding o ha error are hen used for modeling. 3 Sampling Frequency A sampling frequency of he digial model of he planar srucure is found for he chosen minimal number of secions, and is F s / s, (8) where s / n (9) is a sampling period of he planar srucure modeled by n secions. In order o mach response of digial model wih a real response, a new frequency is defined F /. () sm F s 4 Modeling of he Seps in Line Widh Disconinuiy in he widh of a microsrip line is a very offen used in he microsrip circuis in orded o change characerisic impedance of he line. his is imporan for design of he filers and he impedance maching neworks. In pracice, accuracy of he disconinuiy models depends on heir physical dimensions [8]. For a sep disconinuiy shown in Figure a, equivalen asymmerical - nework has a high series inducance L h placed near he narrow line, a low series inducance L l placed near he wide line, and parallel capaciance. C s ISSN: ISBN:

3 UL UL UL 3 UL 4 UL 5 UL 6 UL 7 ef Zc, ef r Zc, r Zc C s Z c a) b) Fig. a) Sep disconinuiy and b) Equivalen -nework of he disconinuiy z - R Ll R Cs Zc Zc RL Z Z l c RL h c L l L h z - z - R Lh Fig.3 WDE for seps in line widh (WDE_Sep) Once he disconinuiies in he microsrip srucure have been idenified, hey mus be correced. Analog -secion shown in Figure b, is modeled by wave digial elemen shown in Figure 3. WDE_Sep elemen is realized as cascade connecion of wo hree-por series adapors for inducances L h and L l (adapors wih reflecion free pors) and one hree-por parallel adapor for capaciance C s, []-[3]. For he coefficiens of he hee-por adapors shown in Figure 3, can be wrien Zc /( Zc RL ), /( ) l k G k G G G3, k, and Zc /( Zc RL ), where is h G /( RL Z, G /( l c ) RL Zc ) and h G 3 / R C. Por r esisances are R s L al h h, RL al l and R C / l s acs, where a fc / an( fc / Fsm ) is a scaling parameer and f c is he cuoff frequency for lowpass and highpass filers and he cener frequency for bandpass and bandsop filers. 5 5 Fig.4 Layou of lowpass filer Here, 5 leader lines a he ends of he srucure are no included during he analysis of he srucure modeled by UE. hese leader lines affec o he oal delay in he WDN and have effec of shifing response characerisics. So, he microsrip lowpass filer is observed as a cascade connecion of seven UL segmens. heir delays vary from one anoher because of heir dependance on he effecive dielecric consan, and are: 7.7 ns, ns, ns and ns. In order o have delays in he wave digial models as possible equal o hese delays, each ransmission line has o be represened as a cascade connecion of a cerain number of UE. In order o find he minimal number of secion for he given error, esing is done for he nex values q,,..., qmax, and q max. A oal number of secions n and he couned errors er [%], for q,,..., 37, are given in he able. able A oal Number of Secions and Relaive Error q u n u er [%] q u n u er [%] Analysis Example A microsrip sepped-impedance 7 h order Chebyshev lowpass filer wih passband ripple of.37 db and cuoff frequency of 9 MHz [9], is used for verificaion of he proposed mehod. he layou is shown in Figure 4, and h is circui is also analyzed in [5]. As shown in able, relaiv e error of he oal delay does no have convergence effec wih increasing oal secion number. his error vary from one anoher, and because of ha a way of deerminaion of minimal number of secion for given error is presened in his paper. Also, exra segmenaion of he ransmission lines leads o he bes soluion in he modeling procedure, i.e. o he minimal chosen relaive error. ISSN: ISBN:

4 Parameers of elemens of equivalen disconinuiy circui according o relaion from [8], are C s.383 pf, Lh.85 nh and L l.79 nh. Furhe r, in order o give a beer explanaion of he deerminaion of a minimal number of secion for a given error, a few cases from able will be exraced and explained. If he number of secions is found for q, hen UL segmens assigned as UL, UL4 and UL6 are modeled wih one UE (blocks _, _ and _3), segmens assigned as UL and UL7 wih wo cascaded UE (blocks _ and _), and segmens assigned as UL3 and UL5 wih four cascaded UE (blocks 4_ and 4_). Formed WDN is depiced in Figure 5 and a block for cascade connecion of 4 UE is shown in Figure 6. Figure 7 shows WDE_Sep formed in Simulink oolbox in MALAB enviromen. Delay of he individual UE in WDN, according o relaion () is ps, where ps is he sum of all ransmission line delays, and n 5 is he oal number of UE in he WDN. According o relaion (8), a sampling frequency is F s 7.47 GHz. A oal delay of he digial model of he srucure is ps. Relaive error of delay found by (7) is er.756 %. - ADP-S ADP-L Sep_ Sep_6 3 Sep_ Sep_5 4_ 4_ Sep_3 Sep_4 Fig.5 Wave digial nework for n 5 /z ADP-S /z /z /z /z Fig.6 Block 4_ /z ADP-P - /z ADP-S Fig.7 WDE_Sep in Simulink oolbox _ For n 5, agreemen beween he resuls obained by WDN and hose ones obained in ADS (Advanced Design Sofware) is bad, Figure 8. [db] S Fig.8 Response comparison for 5 From able for given error n _ er.%, firs posiive error is for q 37, and firs negaive error is for q 3. For q 3, a oal secion number in he WDN is n 474. For individual ransm ission lines, number of secions is 67, 3, 8, 4, 8, 3 and 67, respecively. A sampling frequency is F s GHz. A oal delay of he srucure digial model is ps. Relaive error of delay is er %. For q 37, a oal secion number in he WDN is n 548. For individual UL segmens, number of secions is 77, 37, 37, 46, 37, 37 and 77, respecively. A sampling frequency is F s 69.9GHz. A oal delay of he srucure digial model is ps. Relaive error of delay is er.747 %. Response comparison are shown in Figures 9 and. In he region below.5 GHz, he agreemen beween he resuls calculaed by using WDN wih modeled disconinuiies and he resuls obained in ADS is very good. For he region above.5 GHz he curve is shifed slighly o he lef. A curve for WDN wihou modeled disconinuiies is shifed o he righ in whole frequency band. Sandard values for a ccuarcy conrol, AE (Average es Error), WCE (Wors Case Error), and Pearson-Producd Momen correlaion coeficien r, of he response S [ db ] obained b y wave digial approach and ha one obained in ADS, are shown in able. I is clear ha he agreemen n ISSN: ISBN:

5 beween he responses is very good in boh cases, for ( n 474 ) and for ( n 548 ). - secions. A very simple algorihm for deerminaion of he minimal number of secion for given error is described here. Also, a wave digial model for sep disconinuiy is given. One applicaion example, proving he response accuracy of he new mehod, is given. AE, WCE and correlaion coeficien r are used o prove accuracy of he response obained by wave digial approach. S [db] S [db] Fig.9 Response comparison for n Fig. Response comparision for able Sandard Values for Accuracy Conrol in Case of Given Error n _ er.% n 548 n 548 n 474 AE [%] WCE [%] r Conclusions Planar microwave srucures wih is sep disconinuiies can be compleely analyzed and modeled by wave digial elemens. Improved accuracy of he analysis resuls is obained by choosing an appropriae minimal number of References: [] A. Feweis, Digial Circuis and Sysems, IEEE ransacions on Circuis and Sysems, Vol. CAS-3, No., January, 984, pp [] A. Feweis, Wave Digial Filers: heory and Pracice, Proceedings of IEEE, Vol. 74, 986, pp [3] W. K. Chen, he Circuis and Filers Handbook, CRC Press, 995. (Wave Digial Filers, pp ). [4] M. V. Gmirović i B. P. Sošić, Analysis of Planar Srucures Modeled by Wave D Digial Elemens, 4h elecommunicaions forum ELFOR 6, Serbia, Belgrade, November - 3, 6, pp [5] B. P. Sošić and M. V. Gmirović, Implemenaion of Wave Digial Model in Analysis of Arbirary Nonuniform ransmission Lines, Microwave and Opical echnology Leers, 7, Vol. 49, No. 9, Sepember 7, pp. 53. [6] B. P. Sošić and M. V. Gmirović Equivalen hevenin Source Mehod as ool for Response Compuaion of Wave Digial Srucures, 8h Inernaional Conference on elecomm. in Modern Cable, Saellie and Broadcasing Services - ELSIKS 7, Serbia, Niš, Sepember 6-8, 7, Volume, pp [7] B. P. Sošić and M. V. Gmirović, Generaing of Basic Wave Digial Elemens for Modeling of wo-dimensional Planar srucures, XLII Inernaional Scienific Conference - ICES 7, Macedonia, Ochrid, June 4-7, 7, pp [8] P. F. Combes, J. Graffeuil and J.-F. Sauereau, Microwave Componens, Devices and Acive Circuis, John Wiley & Sons, New York, 987. [9] R. W. Rhea, HF Filer Design and Compuer Simulaion, Noble Publishing Corporaion, USA, 994, Secion 7.. ISSN: ISBN: