Scattering Parameters. Scattering Parameters

Transcription

1 Motivatio cattrig Paramtrs Difficult to implmt op ad short circuit coditios i high frqucis masurmts du to parasitic s ad Cs Pottial stability problms for activ dvics wh masurd i oopratig coditios Difficult to masur ad I at microwav frqucis Dirct masurmt of amplituds/ powr ad phass of icidt ad rflctd travlig wavs Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm Motivatio cattrig Paramtrs Difficult to implmt op ad short circuit coditios i high frqucis masurmts du to parasitic s ad Cs Pottial stability problms for activ dvics wh masurd i oopratig coditios Difficult to masur ad I at microwav frqucis Dirct masurmt of amplituds/ powr ad phass of icidt ad rflctd travlig wavs Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm

2 Gral Ntwor Formulatio Port oltags ad Currts Charactristic (Port) Impdacs, I I I, I I I I I port port N port Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm Nport Ntwor I I N N N I N N I N Not: all currt compots ar dfid positiv with dirctio ito th positiv trmial at ach port I,, N I 3 Impdac Matrix I N I I N Port Port Nport Ntwor Port N N N I I I N Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm I OpCircuit Impdac Paramtrs i I i, oc I for N I i,oc N N NN Port i Port Nport Ntwor Port N 4

3 dmittac Matrix I N I I N Port Port Nport Ntwor Port N I Y I Y I N YN Y Y Y N I Y I i,sc Y Y Y N N NN Port i N hortcircuit dmittac Paramtrs Y i I i, sc for Port Nport Ntwor Port N Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm _ 5 Th cattrig Matrix Th scattrig matrix rlats icidt ad rflctd voltag wavs at th twor ports as (assum, ): N N N N N NN with voltag ad currt at port : I I I ( ) N I, I or I port Nport Ntwor I port N I, port N I N I N I N Not: paramtrs dpd o port impdacs, I N N Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm, N 6 3

4 Th cattrig Matrix Th scattrig matrix rlats icidt ad rflctd voltag wavs at th twor ports as (assum, ): N N N N N N NN Itrprtatio? with voltag ad currt at port : I Powr rlatioships?, I I I ( ) I or I port Nport Ntwor I port N I, port N I N I N I N Not: paramtrs dpd o port impdacs, I N N Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm, N 7 Trasmissio i Basics osslss Trasmissio i _,, "l Port Port z, _ I I I (z) z z I(z) I z I z I I I Phas Costat: " C Charactristic Impdac: C I I Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 8 4

5 Nt Powr Flow o osslss i (ral) γ β I(z') ) (z' ) Γ z ' ( z') I( z') βz ( ' βz Γ ' ) βz ( ' βz Γ ' ) P av (z) { } (z) ( I(z) ) # $ " % & cost. Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 9 Gralizd cattrig Paramtrs _ I cosidratios ad dfiitios I Γ I I I I ( Γ) ( I ) ( I ) _ Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm (assumig is ral ) I P max { ( I } ) P 5

6 Normalizd Wav Quatitis It is usful to xprss powr P without charactristic impdac (port impdac) (but P still dpds o ) P { ( I } ) a P { ( I } ) b _ a b I P P { a } b Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm P (assumig ral ),, I I cattrig Matrix Port Nport Ntwor Port b b b N N N N NN a a a N N _,N I N N Port N i bi a a for i i, i i, i ( ) ( ), for all for all ( i ) Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm, 6

7 cattrig Paramtrs Physical maig of i ( ral) i Pi P max, actual powr lavigport i maximumpowr fromport ( i ) Physical maig of b,,, a a,,, P P P P { }, max _, I, I Port Nport Ntwor Port Port N 3 Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm,N I N N, Impdac matrix: / latio to Matrix I xprss,i i trms of a ad b ( a b) / ( a b) with ormalizd impdac matrix ( ral) / b Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm, ( U) ( U)a N, N N ( U) ( U) ( U)( ) U with / NN / / / ad port impdac matrix ( ) 4 7

8 cattrig Paramtrs Port is said to b matchd wh it is trmiatd with a load havig th sam impdac as th port impdac,. Oft, all port impdacs ar chos to b qual ad, 5 Ω. Th valus of th scattrig () paramtrs dpd o th chos port impdacs. paramtrs ca b algbraically rormalizd to diffrt ad uqual port impdacs. (s latr) Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 5 TwoPort Ntwors Isrtio ad tur oss, _, I tur oss idicats th xtd of mismatch i a twor i db port : log i db Isrtio oss masur of trasmittd fractio of powr i db from port to port : I log i db Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 6 8

9 9 Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 7 xampl Port Port _, _, osslss Trasmissio i: If,,, th scattrig paramtrs ca b asily obtaid by ispctio: ± ± U ( ) U ( )( ) U U cot si si cot Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 8 5Ω Trasmissio i 5Ω frcs frq, G Hz mag( (,)) mag( (,))

10 5Ω Trasmissio i Ω frcs. mag( (,)) mag( (,)) i ( 5 ) 5Ω Ω frq, G Hz Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 9 Proprtis of Paramtrs ciprocal twors: i i ymmtrical twors: ii osslss twors: or ad For a losslss passiv twor th scattrig matrix is uitary: traspos i T U i T complxcougat Matrix symmtry lctrical ymmtry ad Matrix symmtry xampl: twoport twor T? Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm

11 Proprtis of Paramtrs ciprocal twors: i i ymmtrical twors: ii osslss twors: or ad For a losslss passiv twor th scattrig matrix is uitary: traspos i T U i T complxcougat Matrix symmtry lctrical ymmtry ad Matrix symmtry xampl: twoport twor T? Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm Th T osslss TwoPort Ntwors T From uitary coditio follows: xampl: losslss T Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm

12 Th T osslss TwoPort Ntwors T From for passiv uitary (lossy) coditio twors follows: Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 3 Trmiatd Port: _ Γ i pplicatios port oport twor port Γ Γ Γ ll port (rfrc) impdacs ar, rf Γ Γ Γ Γ i Γ Γ ( s,rf ) spcial cas: è Γ Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm Γ i 4

13 B hift i frc Pla βδl TwoPort βδl Ntwor B B B B B B B B B Δ l Δl B B shift i shift i rfrc pla rfrc pla B B B B B B B Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 5 Matrix ormalizatio ( U) ( U) ( U)( ) U ( U )( U ) w / /,w,w F ormalizatio matrix old F ( ral) F old, w, old, N w, N Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 6 3

14 xampl.5 Masurmt.75.5 Iitial CM Prturbd & ugmtd CM Circuit Modl Ω.5 55Ω Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 7 Compariso of Diffrt Port Normalizatios 5Ω, 55Ω, 5Ω,, Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 8 4

15 oltag Trasfr Fuctio from cattrig Paramtrs _ I I Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm ( ) ( I ) I i I i I I ( ) 9 xampl Matchd 3dB attuator ( 5 Ω) (f: Pozar pp. 75/6) 5Ω _ 5Ω i, i, i, Ω 8.56Ω 5Ω.777 I log 3dB Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 3 5

16 xampl usig Matrix 5Ω _ 5Ω 5Ω,, / / Ω , 5Ω, Ω / / Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm i _ xampl usig Matrix, ttuator trmiatd i Ω ad Paramtrs wrt 5Ω,, 5Ω 5Ω, Γ, Γ Γ / / Γ ttuator trmiatd i Ω ad Paramtrs wrt, 5Ω, Ω Γ i Γ Γ, 5Ω, Ω / / Ω Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm Γ.67 Γ,,, Ω

17 Covrsio btw Ntwor Paramtrs Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 33 Proprtis of Ntwor Paramtrs ymmtric TwoPort Ntwor Y Y D assumig th sam port impdacs ciprocal Ntwor i i Y Y i i D BC i i osslss Ntwor { } i { Y i }.g. T I { B, C} {, D} Im Prof. dras Wisshaar C58 Ntwor Thory Gust ctur Fall Trm 34 7