Frequency-Domain Analysis of Transmission Line Circuits (Part 1)

Transcription

1 Frequency-Doman Analyss of Transmsson Lne Crcuts (Part ) Outlne -port networs mpedance matrx representaton Admttance matrx representaton catterng matrx representaton eanng of the -parameters Generalzed -parameters

2 An -port Hgh-peed etwor o For the -th port: o o -port Lnear etwor o 3 Examples of -port etwors (R. Ludwg and P. Bretcho, RF Crcut Desgn, Prentce Hall, 000) 4

3 3 5 mpedance atrx Representaton () Each element of matrx s gven by for 0 6 -Parameters for -Port etwors Equvalent crcut:

4 4 7 Admttance atrx Representaton () Each element of matrx s gven by for 0 8 -Parameters for -Port etwors Equvalent crcut:

5 ome Propertes of the an atrces f the networ s recprocal (not contanng any nonrecprocal meda such as ferrtes or plasmas), then and are symmetrc, f the networ s lossless, all the elements n and are purely magnary n most practcal cases, hgh-speed nterconnects can be consdered as recprocal networs, and n many cases, lossless networs 9 The catterng atrx Representaton () They can be more easly obtaned at hgh frequences: ncdent and reflected waves can be measured usng a ector etwor Analyzer (A) They do not requre shorts or opens (actve devces mght oscllate or self-destroy) They are more drectly related to hgh-frequency effects (Γ, T, L, RL, WR, etc.) We can convert bac and forth between, and parameters 0 5

6 The catterng atrx () o o o -port Lnear etwor o 0 for 0 f we termnate port wth a matched load ( L o ) eanng of the -parameters represents the reflecton coeffcent at port when all the remanng ports are termnated wth matched loads represents the transmsson coeffcent from port to port when all the other ports are termnated wth matched loads 6

7 eanng of the -parameters (cont) forward reflecton coeffcent (nput match) reverse reflecton coeffcent (output match) forward transmsson coeffcent (gan or loss) reverse transmsson coeffcent (solaton) Γ Γ 0 for 0 for T 0 for T 0 for 3 Converson between and and atrces ( ) ( ) ( ) ( ) (normalzed, assumng all ports have the same o ) where s the dentty matrx ( by ) 4 7

8 Converson between and and (cont) ( o ) ( o ) ( ) ( ) o where s the dentty matrx ( by ) and o dag( / o / o / o ) 5 ome Propertes of the atrx f the networ s recprocal (not contanng any nonrecprocal meda such as ferrtes or plasmas), then s symmetrc, f the networ s lossless, matrx s untary (orthonormal complex) * * 0 n most practcal cases, hgh-speed nterconnects can be consdered as recprocal networs, and n some cases, lossless networs 6 8

9 Generalzed catterng Parameters n many practcal cases, o o... o o 50Ω f o s dfferent for each port, we defne the ncdent and reflected waves at the -th port as a o b o The total voltage and current at port -th are ( a b ) ( a b ) / o o o o 7 Generalzed catterng Parameters (cont) ) ( a b o ( a b ) / o The average power delvered to the -th port s P Re{ a b (ncdent power mnus reflected power) * P Re{ ( b a * } a b )} * a b 8 9

10 Generalzed catterng Parameters - ummary The Generalzed catterng atrx relates the ncdent and reflected waves b a a b a a b b o a a b 0 for b To convert from the orgnal -parameters to generalzed -parameters o o 0 for o 9 0