Example a lossless reciprocal network
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This document describes finding the remaining scattering parameters of a 3-port lossless reciprocal network where S11 = 1/2, S31 = 1/2, and S33 = 0. Using the properties of losslessness and reciprocity, the document derives equations relating the known and unknown scattering parameters and solves them to find the full scattering matrix.
Example a lossless reciprocal network
- 1. 2/23/2007 Example A Lossless Reciprocal Network 1/4 Example: A Lossless, Reciprocal Network A lossless, reciprocal 3-port device has S-parameters of S11 = 1 2 , S31 = 1 2 , and S33 = 0 . It is likewise known that all scattering parameters are real. Find the remaining 6 scattering parameters. Q: This problem is clearly impossible—you have not provided us with sufficient information! A: Yes I have! Note I said the device was lossless and reciprocal! Start with what we currently know: ⎡ 1 2 S12 S13 ⎤ S = ⎢S21 S22 S23 ⎥ ⎢ ⎥ 1 ⎢ 2 S32 0 ⎥ ⎣ ⎦ Because the device is reciprocal, we then also know: S21 = S12 Jim Stiles S13 = S31 = 1 The Univ. of Kansas 2 S32 = S23 Dept. of EECS
- 2. 2/23/2007 Example A Lossless Reciprocal Network 2/4 And therefore: ⎡ 1 2 S21 1 2 ⎤ S = ⎢S21 S22 S32 ⎥ ⎢ ⎥ 1 ⎢ 2 S32 0 ⎥ ⎣ ⎦ Now, since the device is lossless, we know that: 1 = S11 + S21 + S31 2 = ( 1 2 ) + S21 + ( 1 ) 2 2 2 2 2 2 2 2 2 2 2 1 = S13 + S23 + S33 2 2 2 = ( 1 2 ) + S32 + ( 1 Columns have unit magnitude. 2 ) 1 = S12 + S22 + S32 = S21 + S22 + S32 2 2 2 2 and: ∗ ∗ ∗ 0 = S11S12 + S21S22 + S31S32 = 1 2 ∗ ∗ S21 + S21S22 + 1 2 ∗ S32 ∗ ∗ ∗ 0 = S11S13 + S21S23 + S31S33 = 1 2 ( )+S 1 2 ∗ S32 + 21 1 2 Columns are orthogonal. (0) ∗ ∗ ∗ 0 = S12S13 + S22S23 + S32S33 = S21 ( 1 Jim Stiles 2 )+S ∗ S32 + S32 ( 0 ) 22 The Univ. of Kansas Dept. of EECS
- 3. 2/23/2007 Example A Lossless Reciprocal Network 3/4 These six expressions simplify to: S21 = 1 2 2 2 1 = S21 + S22 + S32 S32 = 0= 1 2 1 2 S21 + S21S22 + 0= 1 2 1 2 S32 (2 2 ) + S21S32 0 = S21 ( 1 2 )+S S32 22 where we have used the fact that since the elements are all ∗ real, then S21 = S21 (etc.). Q: I count the expressions and find 6 equations yet Q only a paltry 3 unknowns. Your typical buffoonery appears to have led to an over-constrained condition for which there is no solution! A: Actually, we have six real equations and six real unknowns, since scattering element has a magnitude and phase. In this case we know the values are real, and thus the phase is either 0 or 180 (i.e., e j 0 = 1 or e j π = −1 ); however, we do not know which one! From the first three equations, we can find the magnitudes: Jim Stiles The Univ. of Kansas Dept. of EECS
- 4. 2/23/2007 Example A Lossless Reciprocal Network S21 = 1 2 S22 = 1 2 4/4 S32 = 1 2 and from the last three equations we find the phase: S21 = 1 2 S22 = 1 2 S32 = − 1 2 Thus, the scattering matrix for this lossless, reciprocal device is: 1 1 ⎡ 12 ⎤ 2 2 S = ⎢ 1 2 1 2 −1 2 ⎥ ⎢ ⎥ 1 −1 ⎢ 2 0 ⎥ 2 ⎣ ⎦ Jim Stiles The Univ. of Kansas Dept. of EECS