Scattering Matrices for Nonuniform Multiconductor Transmission Lines
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1 Interaction Notes Note March 2005 Scattering Matrices for Nonuniform Multiconductor ransmission Lines Carl E. Baum Air Force Research Laboratory Directed Energy Directorate Abstract A section of nonuniform multiconductor transmission line of finite length l can be characterized by various forms of scattering matrix. If the transmission line is lossless special properties apply. his work was sponsored in part by the Air Force Office of Scientific Research and in part by the Air Force Research Laboratory Directed Energy Directorate. 1
2 1. Introduction A recent paper [5] addresses some of the properties of the supermatrizant and its blocks based on reciprocity energy and norms for nonuniform multiconductor transmission lines (NMLs. Associated with such NMLs are also scattering matrices. Such is the subject of the present paper. As indicated in Fig. 1.1 we have a section of NML of length l. At z = 0 this connects to a uniform ML with characteristic impedance matrix ( (1 nm. Similarly at z = l another uniform ML with characteristic impedance matrix ( (2 nm. connected. Z of size N x N and assumed lossless and symmetric (reciprocity. Z of size N x N is also have As discussed in [1] one can define waves by combining voltage and current variables. For the left ML we ( V n (1 ( zs ( V n (1 ( zs ( Zn (1 m ( I n (1 ( zs = ± ± ~ = two-sided Laplace transform over time t s = Ω+ jω Laplace-transform variable or complex frequency ( V (1 (0 ( V (1 n (0 s n s wave incident on NML from left (1.1 + wave leaving from NML to left N N-1 (1 (2 Z nm ( Z nm 0 uniform ML z = 0 NML z = l uniform ML Fig. 1.1 Section of NML Connected to wo Lossless MLs. 2
3 Similarly we have for the right ML ( V n (2 ( zs ± = ( V n (2 ( zs ± ( Zn (2 m ( I n (1 ( zs ( (2 n ( s + ( (2 n ( s wave incident on NML from right l wave leaving from NML to right (1.2 he NML can then be described by a scattering matrix with ( n s ( n l s ( n s ( n l s (1 (1 V (0 V (0 ( Snm ( s + = (2 (2 V ( V ( + (1.3 relating incoming incident waves to outgoing waves. For later use we have (1 (1 (1 (1 ( Znm = ( Znm = ( Ynm = ( ynm (2 (2 (2 (2 ( Znm = ( Znm = ( Ynm = ( ynm 1/2 1/2 1/2 1/2 (1.4 his allows us to define a renormalized form of the waves [6] as ( 2 ( 2 ( 2 zs y ( zs v V n n m n ± ± ( 2 ( 2 ( 2 ( 2 = y nm V n z s ± z nm I n ( z s (1.5 Corresponding to this we have a renormalized scattering matrix as 3
4 ( n ( s ( n ( l s ( z ( z ( n ( s ( n ( l s (1 (1 v 0 v 0 ( R + = ( S nm ( s (2 (2 v νν v + ( y ( y (1 (1 0 0 nm nm nm nm ( R ( S nm ( s = ( S nm ( s (2 (2 0 νν nm nm 0nm nm (1.6 See also [2] for some properties of these scattering matrices. For the NML we have the combined telegrapher equations ( V n ( zs d = dz ( Znm ( In( z s ( 0 nm ( Z nm ( z s = ( Ynm ( Znm ( Y nm ( z s ( 0nm ( V n ( zs ( Znm ( In( z s ( ( V s n ( zs + ( s ( Znm ( I n ( z s Z 1 nm Znm Ynm = = normalizing constant impedance matrix (chosen at our convenience (1.7 For present purposes we set the distributed sources to zero. he associated matrizant differential equation is d ( Gnm ( z z0 ; s ( ( ( ( 0; nm z s G nm z z s dz ν ν = Γ νν ν ν ( G nm ( z0 z0 s ( nm ( 0 nm ( Z nm ( z s ( Ynm ( Znm ( Y nm ( z s ( 0nm ( nm ( zs Γ = ; 1 = (boundary condition (1.8 he solution is the product integral 4
5 z ( Γnm ( zs ( Gnm ( z z0; s e = z0 dz ( nm nm ( 0 tr z s 0 det G z z ; s Γ 1 = = (1.9 We will take z 0 as 0 and l the ends of the NML for our present purposes. 5
6 2. Scattering Matrices Consider an M-port network (lumped and/or distributed elements. Let it be characterized by an M x M impedance matrix ( (M ( (M ( (M Z 1 nm ( s = Znm ( s = Ynm ( s (2.1 his is driven by an external set of sources which we can think of as incident from an M-conductor (plus reference uniform ML of characteristic impedance (ext (ext (ext ( Z 1 nm Znm Ynm ( s = = (2.2 Which we take as real and frequency-independent. he port voltages and currents are related as with ( In ( s (M ( V n( s = ( Zn m ( s = ( In( s (2.3 taken as positive into the network. In turn wave variables are defined [1] as ( (ext ( V n s V n s Zn m ( s In( s ± = ± + incident wave (2.4 reflected (scattered wave A scattering matrix can now be defined via ( M V n s _ = S n m s V n s (2.5 his scattering matrix can take several forms [1]. For convenience define ( M ( M (ext ( M z nm s Z nm s Ynm = ynm s (2.6 Giving various equivalent forms as 6
7 ( M ( M ( M S nm s = z nm s + 1nm z nm s 1nm ( M ( M nm 1nm nm z s z s ( 1nm = + ( M ( M 1nm ynm s 1nm ynm ( s = + ( M ( 1nm ynm ( s = ( 1nm ( M + y nm s (2.7 Note that all the matrices in (2.7 have the same eigenvectors and commute. From [5] for a lossless network we have ( nm( = nm Z s Z s (odd in s (2.8 In (2.7 this gives ( M ( M ( M S nm s znm s 1nm znm ( s ( 1nm = + ( 1 ( s M = S nm ( M ( M nm = S nm s S nm s (2.9 his is suggestive of unitary with s = jω but not quite because ( s M S nm is not symmetric in general. Now consider the power delivered into the M-port. he real power can be expressed in a general form as 1 P ( s = ( Vn( s ( In( s + ( Vn( s ( In( s 2 (2.10 Which for s = jω must have 0 P jω (2.11 As a function of s (2.8 gives the analytic continuation into the s plane. We also note that 7
8 ( = P s P s = even function of s (2.12 his can be cast in terms of the scattering matrix via ( (ext ( (ext _ ( (ext (ext ( ( 8 P s = V n s + V n s _ Y n m V n s V n s _ + + ( = V n s + V n s Y n m V n s V n s _ + + (ext ( (ext 4 P s = V n s Yn m V n s V n s _ Yn m V n s _ + + ( = V n s Yn m Sn m s Yn m Sn m s V n s + + (2.13 his requires that we have the Hermitian matrix (ext ( M ( (ext ( M H nm jω Ynm S nm jω ( Ynm S nm ( jω = ( H nm ( jω (ext ( M (ext ( M Ynm S nm jω ( Ynm S nm ( jω = (2.14 Which is also termed self adjoint. For a lossless network we have ( Hnm ( jω = ( 0 nm (2.15 And by analytic continuation ( Hnm ( s = ( 0 nm (2.16 hroughout the complex s plane. For such a lossless network we can rearrange (2.12 as ( ext ( M ( ext ( M nm Znm Snm s Ynm Snm ( s = (2.17 ( 1 If we choose 8
9 ( ext Z nm = constant times ( 1 nm (2.18 then ( nm S jω = unitary matrix (2.19 It is this last form that is often used in circuit theory [8]. More generally let us try ( M (ext ( M (ext nm = nm nm ( nm (ext (ext 1/2 (ext ( znm ( Znm ( ynm S s y S s z (2.20 Which in (2.14 gives ( M ( M ( M ( M nm Snm s Snm s Snm s Snm ( s = = ( 1 ( ( jω M S nm = unitary matrix (2.21 his is a more general result than (2.18 (2.19 allowing various choices for ( (ext nm Z. Revisiting (2.5 we now have ( M ( ext ( M ( ext S nm s = znm S nm s ynm (ext ( M (ext ynm V n s _ = S nm s ynm Vn s + (2.22 Where the bracketed terms are the renormalized combined-voltage waves. his has the same form as (1.5 and following allowing us to identify ( ( s M S nm with ( s R S nm in Section 1. Going back to (2.7 we have (with some manipulation 9
10 ( M ( ext ( M ( ext S nm s = ynm S nm s znm (ext (M (ext ( ynm Z nm y ( 1 nm nm (ext (M (ext ( ynm Z nm ynm ( 1 nm (ext (M (ext ( y ( Z nm ( s ( ynm ( 1 nm (ext (M ( ynm ( Z (ext nm ( s ( yn m + ( 1 n m = + = = + (ext (M ( ext ( 1nm ( znm ( Y nm ( s znm (ext (M ( ext ( 1nm ( znm ( Y nm ( s znm (ext (M ( ext ( 1nm ( znm ( Y nm ( s znm = (ext (M ( ext ( 1nm + ( znm ( Y nm ( s znm (2.23 Noting the commuting of the matrix combinations we also have ( M ( M Snm s Snm ( s = (symmetric (2.24 As discussed in Appendix A a lossless unitary symmetric matrix also has a complete set of M real eigenvectors with eigenvalues all of magnitude one. he passive nature of the scattering Babanian matrix also gives [8] ( M ( M 1 nm S nm jω S nm ( jω = = positive semidefinite 2 ( M Snm ( jω for each m ( n ( ( jω S nm M 1 For a lossless network we also have 10
11 ( M ( M S nm jω S nm ( jω = 1nn (2.26 n Which extends into the complex plane as S M nm ( s S nm ( s = 1nn (2.27 m As discussed in Appendix A a lossless unitary symmetric matrix also has a complete set of M real eigenvectors with eigenvalues all of magnitude one. 11
12 3. Scattering Supermatrix for Section of NML As discussed in Section 2 we need the scattering supermatirx in the form (ext ( Snm ( s ( y nm Snm s z νν = nm νν (ext ( znm (1 ( ynm ( 0nm (2 ( 0nm ( ynm (1 ( znm ( 0n m ( 0nm ( znm (ext 1/2 (ext (ext (ext ( ynm = = ( Ynm = ( znm ν ν ν ν ν ν = ( ν ( ν ( ν Znm = Znm = ynm ν ynm Ynm ν = 1/2 (ext = ( Znm (2 1/2 1/2 (3.1 Since ( S nm ( s ( S nm s νν = (3.2 we have for the submatrices (each N x N ( S nm ( s = S nm ( s ( S nm ( s = S nm ( s ( S nm ( s = S nm ( s (3.3 hese are related to the other form of the scattering matrix as (1 (1 (1 (2 ( ynm ( S nm ( s ( z ( nm ynm S nm s ( z 12 nm ( ynm ( S nm ( s ( znm ( ynm ( S nm ( s ( znm ( Snm ( s = νν (2 (1 (2 (2 21 (3.4 12
13 So the blocks are still relatively simple in form and have the symmetries given in (3.3. his relates renormalized wave variables as ( n ( s ( n ( l s (1 ( vn ( 0 s ( (1 ( v n ( 0 s (2 ( v n ( l s (1 ( 1 ( ynm V n ( 0 s (2 ( (1 ( 1 ( ynm V n ( 0 s (2 ( 2 ( ynm V n ( l s ( n ( s ( n ( l s (1 (1 v 0 v 0 ( Snm ( s + = (2 (2 v v + = (2 v 2 n s l ynm Vn l s = (3.5 he impedance supermatrix for the NML takes the form ( n ( n ( l ( n V 0 s I ( 0 ( N n s Znm ( s = = V s νν I ( l s ( N ( N ( N Z nm s = Z nm s = Y nm s (3.6 νν = impedance supermatrix for the NML section Note the use of ( I n ( l s since we need currents into the section for impedance purposes. here are various ways to calculate the ( S nm ( s ν ν submatrices. For present illustration let us calculate the impedance or admittance supermatrix for (1.9 which gives 13
14 ( n ( l ( n ( n V s V 0 s = ( Gnm ( 0; s l ( Znm I n( l s ( Znm I n( 0 s ( n V 0 s V l s = ( Gnm ( 0 ; s l ( Znm I n( 0 s ( Znm I n( l s ( Gnm ( 0 ; s ( G ( 0 l = nm l (3.7 Writing out the blocks we have ( V n( l s = ( G n m( l0; s ( V ( ( n 0 s + G n m l0; s Z 12 n m I n 0 s ( Z nm ( I n( l s = ( G nm ( l0; s ( V ( 0 ( ( 0; ( ( ( 0 21 n s + G nm l s Z 22 nm I n s ( V n( 0 s = ( G n m( 0 l; s ( V ( ( ( 0 ; ( ( ( n l s + G n m l s Z 12 n m I n l s ( Znm ( I n( 0 s = ( G nm ( 0 l; s ( V n( l s + ( G nm ( 0 l; s ( Znm ( I n( l s (3.8 From the first and third of these we have ( V n( 0 s = ( G n m( 0 l; s ( G ( n m l0; s V n 0 s + ( G nm ( 0 l; s ( G ( nm l0; s Z 12 nm I n 0 s + ( G nm ( 0 l; s ( Z ( ( 12 nm I n l s ( V n( l s = ( G n m( l0; s ( G ( 0 ; ( ( n m l s V n l s + ( G nm ( l0; s ( Z ( ( 0 12 nm I n s + ( G nm ( l0; s ( G nm ( 0 l; s ( Z nm ( I n( l s 12 (3.9 Defining ( 0 ( ( 1 ( ( 0 ; ( nm nm nm nm ( 0; D l s G l s G l s ( l ( ( 1 ( ( 0; ( nm nm nm nm ( 0 ; D l s G l s G l s (3.10 we have the blocks of the impedance matrix as 14
15 ( N ( 0 Z nm s = D nm l s Gnm ( 0 l; s G nm l0; s Z 12 nm ( ( ( N ( 0 Z nm s = D nm l s G nm ( 0 l; s Z 12 nm 12 ( N Z nm s = D l nm l s G nm ( l0; s Z 12 nm 21 ( N Z nm s = D l nm ( l s ( G nm ( l0; s G ( 0 ; 22 nm l s Z 12 nm (3.11 Note that the submatrices with arguments (0 l can be found in terms of those with arguments (0 l (and conversely from the formulae in Appendix B. From the formulae in (2.7 this is converted to (( S nm ( s ν ν. From (3.4 this is converted to (( S nm ( s ν ν. It is this last form which is unitary for s = jω in the case of a lossless NML. 15
16 4. Junction of wo MLs If the length of the NML shrinks to zero the formulae in (3.11 are not well behaved. For convenience first let the NML be a uniform ML so that ( nm ( 0 ( G ( 0; s z nm z s Γ νν = e (4.1 And all derivatives of (( Γ nm ( zs ν ν are zero. he matrizant differential equation is d ( Gnm ( z 0; s ( ( nm 0 s G nm z 0; s dz ν ν = Γ νν ν ν (4.2 From which we can evaluate derivatives recursively as d p 1 ( ( 0; ( ( 1 0 d p G p nm z s nm s G p nm z 0; s dz ν ν = Γ νν dz ν ν (4.3 Expanding the exponential matrix we have 1 p ( G ( 0; ( 1 ( ( 0 p nm z s nm nm s z νν = νν = Γ p! (4.4 p= 1 Noting that (( Γ nm (0 s ν ν has zero blocks on the diagonal then this property holds for all odd p. Hence we can write 1 ( Gnm ( z0; s = ( 1 O nm + γ nm s z = z 2 ( 2 2 ( 4 ( 0 ( ( 0 ( ( 0 ( ( 0 2 γ 2 nm s = Znm s Ynm s = s Lnm Ynm for usual lossless NML (4.5 We also have for the off-diagonal block ( 3 G nm z0; s = Z 12 nm 0 s Ynm z + O z (4.6 16
17 Substituting we find from (3.10 and (3.11 ( 0 ( ( 1 ( ( 0 ; ( nm nm nm nm ( 0; D l s = G l s G l s 2 ( 2 ( 4 l γ nm s l O l D nm ( l s = + = 2 ( 2 ( 4 l O l ( ( 0 l O( l 3 ( N 1 Znm s = γ nm ( s + Znm s = l ( Y nm ( 0 s + O( l 1 (4.7 As we can see the leading term is a singular supermatrix [4 (Appendix A] introducing some difficulties in taking the limit as l 0. Fortunately we can directly solve for the scattering matrix for a zero-length section by imposing continuity of voltage and current through this junction as ( 1 ( 2 ( 1 ( 2 0 = 0 0 = ( 0 V n s V n s I n s I n s (4.8 he scattering matrix is defined by ( ( ( ( ( ( V n 0 s Zn m I n 0 s V n 0 s + Zn m I n 0 s = ( S nm ( 2 ( 2 ( 2 νν ( 2 ( 2 ( 2 Vn 0 s Zn m In 0 s Vn 0 s Zn m In ( 0 s + (4.9 where it is now frequency independent. By setting the wave incident from the right to zero we find ( 2 ( 1 ( 2 ( 1 Snm = Z nm Znm Znm + Znm ( 2 ( 2 ( 1 Snm = 2 Z 21 nm Znm + Znm (4.10 Similarly by setting the wave incident from the left to zero we find ( 1 ( 2 ( 1 Snm = 2 Z 12 nm Znm + Znm ( 1 ( 2 ( 2 ( 1 Snm = Z 22 nm Znm Znm + Znm (
18 From (3.4 we have ( 1 ( 2 ( 1 ( 1 ( 2 ( 1 Snm = y ( 1 ( 1 nm Znm ynm nm ynm Znm ynm+ nm ( 1 ( 2 ( 2 ( 1 ( 2 ( 1 = ynm Znm ynm + ( 1nm ynm Znm ynm ( 1nm ( 2 ( 1 ( 2 ( 1 Snm = 2 z 12 nm ynm + ynm znm ( 1 ( 2 ( 1 ( 2 Snm = 2 y 21 nm znm + znm ynm ( 2 ( 1 ( 2 ( 2 Snm ( 1 ( 2 = y ( 1 22 nm Znm ynm nm ynm Zn y + ( 1 m n m n m ( 2 ( 1 ( 2 ( 2 ( 1 ( 2 = ynm Znm ynm + ( 1nm ynm Znm ynm ( 1nm (4.12 Evidently (( Snm ν ν is symmetric and real. From previous discussion it is also unitary. 18
19 5. Concluding Remarks Scattering matrices can be formed in various ways depending on the definitions of the incoming and outgoing waves. he traditional way involving volts ± current is not convenient when dealing with a length of NML. In this case it is convenient to normalize the current by an appropriate characteristic impedance matrix. However the resulting scattering matrix for a lossless section of NML is not unitary. By a renormalization procedure involving the square root of characteristic admittance and impedance matrices the resulting scattering matrix (( S nm ( s ν ν a lossless NML section is again unitary. As a special case we find that an NML of zero length has a real symmetric unitary form. So now we have some general results for lossless NMLs which are independent of the variation of ( Lnm ( z and ( Cnm ( z in the section of interest. 19
20 Appendix A. Unitary Matrices A.1. General Case Summarizing from [9] unitary N x N matrices have the equivalent statements: (a ( U nm is unitary (b ( U nm is nonsingular and ( U nm = ( U 1 nm ( U nm ( Unm = 1 nm = ( Unm ( Unm (c (d ( U nm is unitary (adjoint = (e columns of ( U nm are biorthonormal (f rows of ( U nm are biorthonormal (A.1 (g ( y ( U ( x ( y ( x n = n m n n = n (length preserving Stated in another way we have r n ( m mth row as a vector n n m1 m2 ( m1 ( m2 r r = 1 ( n c m nth column as a vector (biorthonormal (A.2 ( n1 ( n2 c c = 1 m m n1 n2 (biorthonormal Write an eigenvector equation as = uβ xn Un m x β n β uβ ( yn = ( yn ( Unm = ( Unm ( y β β n β ( x ( y n n = 1 (biorthonormal β β ββ u β ( yn = ( Un m ( y β n β ( nm ( n = ( n u β U y y β β (A.3 20
21 From this we identify ( y = ( x n β u 1 β = u u β β uβ = 1 uβ = 1 n β (A.4 So that all eigenvalues have unit magnitude (a not-usually-stated property. We can now write a dyadic expansion as N = β (A.5 β β β = 1 ( Unm u ( xn ( yn Noting that a unitary matrix is a special case of a normal matrix (one which commutes with its adjoint then it also has a complex set of N independent eigenvectors [7] making (A.5 always possible. A.2. Symmetric Unitary Matrices Now let ( Unm = ( Unm ( Unm = ( Unm (A.6 From the eigenvector equation (A.3 we have ( n = ( n m ( n = ( n ( n m uβ x U x x U β β β ( x = ( y n β n β ( xn ( yn = β β β1 β (orthonormal (A.7 the left and right eigenvectors being the same (with scaling constant taken as unity. Now from (A.4 we have 21
22 n = β n = β n (A.8 β ( y ( x ( x We have that x n β = real vector (A.9 Consistent with our choice of scaling constant (the orthonormal requirement. he dyadic expansion now takes the form N = β (A.10 β β β = 1 ( Unm u ( xn ( xn Note that the u β are still in general complex. 22
23 Appendix B. Supermatrix Inverse he inverse of a supermatrix is given in terms of its submatrices (blocks. his can be found in various places [3 7 9]. We have ( B nm ( A 12 νν nm νν = = (B.1 where these matrices are in symmetric compatable order (same division of both rows and columns. he diagonal blocks ( and 22 are square. he submatrices of the inverse supermatrix are given by = ( Anm ( A ( ( nm A 12 nm A 22 nm ( A 21 nm A nm 12 1 Bnm = 21 Anm A 22 nm A 21 nm A nm A 12 nm A 21 nm 1 = ( Anm ( A ( ( ( ( 22 nm A 21 nm A nm A 12 nm A 22 nm 21 ( Bnm = ( Anm ( Anm ( Anm ( Anm ( Bnm = ( Anm ( Anm ( Anm ( Anm ( Bnm = ( Anm ( Anm ( Anm ( Anm ( Anm ( Anm (B.2 If in addition the supermatrix is symmetric we have ( A nm ( A = nm ( B nm ( B = nm (B.3 Which in terms of the submatrices gives ( Anm = ( Anm ( Anm = ( Anm ( Anm = ( Anm ( Bnm = ( Bnm ( Bnm = ( Bnm ( Bnm = ( Bnm (B.4 he reader can take the transposes in (B.1 to verify these relations. 23
24 References 1. C. E. Baum. K. Liu and F. M. esche On he Analysis of General Multiconductor ransmission-line Networks ; also in C. E. Baum Electromagnetic opology for the Analysis and Design of Complex Electromagnetic Systems pp in J. E. hompson and L. H. Luessen (eds. Fast Electrical and Optical Measurements Martinus Nijhoff Dordrecht C. E. Baum Bounds on Norms of Scattering Matrices Interaction Note 432 June C. E. Baum On he Use of Electromagnetic opology for the Decomposition of Scattering Matrices for Complex Physical Structures Interaction Note 454 July C. E. Baum Symmetric Renormalization of the Nonuniform-Multiconductor-ransmission-Line Equations with a Single Modal Speed for Analytically Solvable Sections Interaction Note 537 January C. E. Baum Reciprocity Energy and Norms for Propagation on Nonuniform Multiconductor ransmission Lines Interaction Note 583 April C. E. Baum High-Frequency Propagation on Nonuniform Multiconductor ransmission Lines Interaction Note 589 October F. R. Gantmacher he heory of Matrices Chelsea Publishing Co. New York N. Balabanian. A. Bickart and S. Seshu Electrical Network heory Wiley R. A. Horn and C. R. Johnson Matrix Analysis Cambridge U. Press
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