S-Parameters for Three and Four Two- Port Networks

Transcription

1 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi -Prmeters for hree nd Four wo- Port Networks rhn M. Mus, Mtthew N.O. diku, nd Cjetn M. Akujuoi Center of Exellene for Communition ystems ehnology Reserh (CECR) Pririe View A&M University, Pririe View, exs Astrt: he sttering prmeters re fundmentl in the hrteriztion of eletril devies t high frequenies. hey re prtiulrly useful for nlyzing nd designing multiport high frequeny nd mirowve networks. his pper presents expliit mthemtil formuls for the resultnt -prmeters for prllel, series, nd sde of three nd four two-port networks in terms of the -prmeters of the individul two-port networks. he mthemtil formuls re derived using the signl flow grph (FG) method. I. Introdution he sttering prmeters formlism is one of the most useful sujets tht mirowve engineers need to fully omprehend due to its hrteriztion of mirowve networks. Elementry iruit nlysis provides mny methods for desriing n-port networks. hese methods inlude impedne prmeters, dmittne prmeters, trnsmission line, nd hyrid prmeters []-[3]. hese methods est desrie d nd low-frequeny iruits. Prtiulrly, they re diffiult to pply to mirowve networks. o hrterize suh high-frequeny networks, one n employ sttering prmeters (or -prmeters) in ple of impedne (Z) or dmittne (Y). hey re used in defining the performne of mny eletril devies, suh s mirowve devies, filters, trnsformers, nd mplifiers. he - prmeters re defined in terms of wve vriles, whih re more esily mesured t high frequenies thn re voltge nd urrent. However, the -prmeters re not diretly suitle for the nlysis of prllel nd series network of two or more two-port networks. uh prllel, series, nd sde networks re usully nlyzed y multiplying the individul mtries using the Y-prmeters, Z-prmeters, - prmeters, respetively. It should e noted tht there re different definitions of other prmeters in the literture [4]-[6]. ine mny mirowve networks onsist of prllel, series, nd sde onnetions of oth pssive nd tive elements, it is desirle to hve expliit formuls diretly involving the -prmeters. uh formuls re useful for nlyzing wveguide nd mirostrip disontinuities. In this pper, we provide the formuls for three nd four of prllel, series, nd sde

2 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi two-port networks. he prolem of determining the -prmeters for prllel, series, nd sde two-ports n e solved in using the signl flow grph (FG). Using the Y nd Z mtries will involve knowing in dvne the referene hrteristi of dmittne Y 0 nd impedne Z 0, respetively. herefore, we hoose to use the FG pproh. he -prmeters re tught in ourses in mirowve engineering t the undergrdute nd grdute levels. However, textooks on mirowve engineering do not over the nlysis of three or four two-port networks. herefore, this pper will supplement suh mirowve engineering textook nd help in overing three nd four two-port networks. his pper presents expliit formuls for the resultnt -prmeter for three nd four two-port networks onneted of prllel, series, nd sde in terms of the - prmeters of the individul two-ports. he formuls re derived y FG nd verified using simultion. In setion 2, we present the kground informtion on -prmeters for two-port networks. etion 3 dels with the derivtion for the - prmeter nlysis, while setion 4 is on onlusion. II. Bkground he sttering (or -) prmeters re fixed properties of the liner iruit. hey desrie how the energy ouples etween eh pir of ports (or trnsmission lines) onneted to the iruit. For two-port network (with one input nd one output), the mtrix equtions re = = 2+ 2 () (2) where nd 2 re the inident wves, while nd 2 re the sttered (refleted) wves, s illustrted in Fig.. In mtrix form, equtions () nd (2) n e written s where [ ] = [ ][ ] (3) [ ] 2 = 2,[ ] = 2, [ ] = 2 (4) Figure 2 shows the flow grph for two-port network. From equtions () nd (2), the - prmeter re otined s

3 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi 2 2 = = Input refletion oeffiient t port (5) = = = Diret trnsfer rtio t port to port 2 (6) = 0 2 = = Output refletion oeffiient t port 2 (7) 2 = 0 2 = 0 nd = = Reverse trnsfer rtio t port 2 to port (8) III. Derivtions he most ommon nd suitle method used for the nlysis of prllel two-port networks is Y-prmeters (dmittne prmeters) suh tht [ ] = [ Y][ ] (9) Or = Y + Y22 2 = Y2 + Y2 he Y-prmeters of overll networks re the sum of the individul Y prmeters of the individul networks. his mens tht (0) () [ Y] [ Y ] [ Y ]... [ Y ] = (2) n where n is the numer of two-port networks. An exmple for n =3 is shown in Fig. 3. As mentioned erlier, the FG is method of writing set of equtions, wherey the relted vriles re represented y points nd the interreltions y direted lines giving diret piture of signl flow. he min dvntge of suh grphil tehnique in solving prllel or series networks re the onvenient pitoril representtion nd the pinless method of proeeding diretly to the solution from the grph. With tht preliminry kground, we now onsider the prllel twoport networks s in Fig. 4, tht shows the -prmeter FG for three prllel twoport networks. We find the following -prmeters:

4 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi For the prllel onnetion of three two-port networks: = = = = + + (3) (4) (5) (6) For the prllel onnetion of four two-port networks: = + d = 2 + d = 2 + d = + d + + (7) (8) (9) (20) we n oserve tht the generl rule for the prllel onnetion of n two-port networks: ij n k = k ij = (2) k is,,...,k two-port networks, i j = or 2. he most ommon nd suitle method used for the nlysis of series two-port networks is Z-prmeters (impedne prmeters) suh tht [ ] = [ Z][ ] () or = Z + Z 2 2 (23) 2 = Z2+ Z2 (24) he Z-prmeters of overll networks re the sum of the individul Z prmeters of the individul networks. his mens tht

5 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi [ Z ] [ Z ] [ Z ]... [ Z ] = (25) n where n is the numer of two-ports networks suh s in Fig. 5. We now onsider the series of two-port networks s in Fig. 6, tht shows the -prmeter FG for three series-onneted two-port networks. We find the following -prmeters: For the series onnetion of three two-port networks: = (26) = (27) 2 2 = (28) = (29) + For the series onnetion of four two-port networks: d d d d d = (30) d = (3) 2 2 = (32) d d d d d = (33) We now relte the -prmeters to the hin trnsfer prmeters (lso known s the trnsfer sttering prmeters) or simply -prmeters, whih re suitle for the nlysis of sded two-ports. It should e noted tht there re different definitions of the - prmeter in the literture [4-5]. Here we follow the type desried y Hewlett Prkrd [5] sine it is the most ommon. 2 2 = (34) 2 2 By mnipulting the mtries in equtions (), (2), nd (34) we n redily show tht

6 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi 2 2 =, = (35) 2 where = 2 2, the determinnt of the -mtrix, nd = 22, the determinnt of the -mtrix. With tht preliminry kground, we now onsider three sded two-ports s shown in Fig. 7. he hin mtrix of the overll sde onnetion n e written s 2 = (36) 2 where = (36) or 2 2 = (36) For exmple, = (37) ustituting for the individul -mtries using eq. (34) yields 2 = = = = [ + ] (38) where = 2 2, the determinnt of the -mtrix. By tking similr pproh for other elements of the overll -mtrix, we otin 2 2( ) = + (39) + [ ]

7 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi 2 = (40) [ + ] 2 2 ( ) = + (4) [ + ] As mentioned erlier, the signl flow grph (FG) n e used to derive the sme formuls. FG is method of writing set of equtions, wherey the relted vriles re represented y points nd the interreltions y direted lines giving diret piture of signl flow. he min dvntge of suh grphil tehnique in solving sded networks is the onvenient pitoril representtion nd the method of proeeding diretly to the solution from the grph. he sded onnetion of three two-ports in Fig. 7. is represented y the flow grph of Fig. 8. With Fig. 8, we n derive eq. (38)-(4). imilr steps n e tken to otin the -prmeters for four sded two-ports. he results re 2 2 d d 22( + ) d d d d [ ] = + (42) 2 d 2 d 2 2 = (43) 2 2 d d d d [ ] 2 = (44) 2 d d d d [ ] d d d 2 2( + ) d d d d [ ] = + (45) where = 2 2, the determinnt of the -mtrix Port wo-port network 2 Port 2 Figure. Inoming nd outgoing wves for two-port network.

8 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi Figure 2. FG for two-port network. [Y ] 2 [Y ] 2 [Y ] Figure 3. Prllel onnetion of three two-port networks Figure 4. FG for three prllel two-port networks. 2

9 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi 2 [Z ] [Z ] 2 [Z ] Figure 5. eries onnetion of three two-port networks Figure 6. FG for three series onnetion of two-port networks. 2 2 Figure 7. Csded onnetion of three two-port networks.

10 the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi Figure 8. FG for three sded two-port networks. IV. Conlusion Expliit mthemtil formuls for finding the omposite -prmeters for three nd four two-port networks onneted in prllel, series, nd sde hve een derived in this pper. he mthemtil formuls re derived using FG. V. Referenes []. C.K. Alexnder nd M.N.O. diku, Fundmentls of Eletri Ciruit. MGrw-Hill, New York, 2007, [2]. R. Dorf, Introdution to Eletri Ciruits. Wiley, New York, 993, [3]. J. D. Irwin, Bsi Engineering Ciruit Anlysis. Mmilln, New York, 993, [4]. G. Gonzlez, Mirowve rnsistor Amplifiers Anlysis nd Design: Prentie- Hll, Englewood Cliffs, N J, 984. [5]. K. C. Gupt, R. Grg, nd R. Ghdh, Computer-Aided Design of Mirowve Ciruits: Arteh House, Dedhm, MA, 98. [6]. G.A. Deshmps, nd J. D. Dyson, ttering Mtries, in Jordn, E.C. (ed.): Referene Dt for Engineering: Rdio, Eletronis, Computer, nd ommunitions (Howrd W. ms, Indinpolis, IN, 7 th ed., ).