1. ECE 546 – Jose Schutt-Aine 1 ECE 546 Lecture - 06 Multiconductors Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu

2. ECE 546 – Jose Schutt-Aine 2 TELGRAPHER’S EQUATION FOR N COUPLED TRANSMISSION LINES V and I are the line voltage and line current VECTORS respectively (dimension n).

3. ECE 546 – Jose Schutt-Aine 3 N-LINE SYSTEM L and C are the inductance and capacitance MATRICES respectively

4. ECE 546 – Jose Schutt-Aine 4 N-LINE ANALYSIS In general LC ≠ CL and LC and CL are not symmetric matrices. GOAL: Diagonalize LC and CL which will result in a transformation on the variables V and I. Diagonalize LC and CL is equivalent to finding the eigenvalues of LC and CL. Since LC and CL are adjoint, they must have the same eigenvalues.

5. ECE 546 – Jose Schutt-Aine 5 MODAL ANALYSIS LC and CL are adjoint matrices. Find matrices E and H such that is the diagonal eigenvalue matrix whose entries are the inverses of the modal velocities squared.

6. ECE 546 – Jose Schutt-Aine 6 MODAL ANALYSIS Second-order differential equation in modal space

7. ECE 546 – Jose Schutt-Aine 7 V m and I m are the modal voltage and current vectors respectively. EIGENVECTORS

8. ECE 546 – Jose Schutt-Aine 8 EIGEN ANALYSIS *The eigenvalues of A satisfy the relation |A - I| = 0 where I is the unit matrix. *A scalar is an eigenvalue of a matrix A if there exists a vector X such that AX = X. (i.e. for which multiplication by a matrix is equivalent to an elongation of the vector). *The vector X which satisfies the above requirement is an eigenvector of A.

9. ECE 546 – Jose Schutt-Aine 9 EIGEN ANALYSIS Assume A is an n x n matrix with n distinct eigenvalues, then, *A has n linearly independent eigenvectors. *The n eigenvectors can be arranged into an n  n matrix E; the eigenvector matrix. * Finding the eigenvalues of A is equivalent to diagonalizing A. *Diagonalization is achieved by finding the eigenvector matrix E such that EAE -1 is a diagonal matrix.

10. ECE 546 – Jose Schutt-Aine 10 EIGEN ANALYSIS For an n-line system, it can be shown that * Each eigenvalue is associated with a mode; the propagation velocity of that mode is the inverse of the eigenvalue. * LC can be transformed into a diagonal matrix whose entries are the eigenvalues. * LC possesses n distinct eigenvalues (possibly degenerate). * There exist n eigenvectors which are linearly independent.

11. ECE 546 – Jose Schutt-Aine 11 EIGEN ANALYSIS * Each normalized eigenvector represents the relative line excitation required to excite the associated mode. * Each eigenvector is associated to an eigenvalue and therefore to a particular mode.

12. ECE 546 – Jose Schutt-Aine 12 E : Voltage eigenvector matrix H : Current eigenvector matrix EIGENVECTORS

13. ECE 546 – Jose Schutt-Aine 13 Eigenvalues and Eigenvectors gives

14. ECE 546 – Jose Schutt-Aine 14 Modal Voltage Excitation Voltage Eigenvector Matrix

15. ECE 546 – Jose Schutt-Aine 15 Modal Current Excitation Current Eigenvector Matrix

16. ECE 546 – Jose Schutt-Aine 16 Line variables are recovered using V  E  1 V m I  H  1 I m EIGENVECTORS Special case: For a two-line symmetric system, E and H are equal and independent of L and C. The eigenvectors are [1,1] and [1,-1] E : voltage eigenvector matrix H : current eigenvector matrix E and H depend on both L and C

17. ECE 546 – Jose Schutt-Aine 17 MODAL AND LINE IMPEDANCE MATRICES By requiring V m = Z m I m, we arrive at Z m is a diagonal impedance matrix which relates modal voltages to modal currents By requiring V = Z c I, one can define a line impedance matrix Z c. Z c relates line voltages to line currents Z m is diagonal. Z c contains nonzero off-diagonal elements. We can show that

18. ECE 546 – Jose Schutt-Aine 18

19. ECE 546 – Jose Schutt-Aine 19 MATRIX CONSTRUCTION

20. ECE 546 – Jose Schutt-Aine 20 Modal Variables General solution in modal space is: Modal impedance matrix Line impedance matrix

21. ECE 546 – Jose Schutt-Aine 21 N-Line Network Z s : Source impedance matrix Z L : Load impedance matrix V s : Source vector

22. ECE 546 – Jose Schutt-Aine 22 Reflection Coefficients Source reflection coefficient matrix  s  [1 n  EZEZ s L  1 E  1  m ]  1 [1 n  EZEZ s L  1 E  1  m ] Load reflection coefficient matrix  L  [1 n  EZEZ L L  1 E  1  m ]  1 [1 n  EZEZ L L  1 E  1  m ] * The reflection and transmission coefficient are n x n matrices with off-diagonal elements. The off-diagonal elements account for the coupling between modes. * Z s and Z L can be chosen so that the reflection coefficient matrices are zero

23. ECE 546 – Jose Schutt-Aine 23 TERMINATION NETWORK I 1 = y p11 V 1 +y p12 (V 1 -V 2 ) I 2 = y p22 V 2 +y p12 (V 2 -V 1 ) In general for a multiline system Note : y ii  y pii, y ij = -y ji, (i≠j)

24. ECE 546 – Jose Schutt-Aine 24 Termination Network Construction To get impedance matrix Z Get physical impedance values y pij Calculate y ij 's from y pij 's Construct Y matrix Invert Y matrix to obtain Z matrix Remark: If y pij = 0 for all i≠j, then Y = Z -1 and z ii = 1/y ii

25. ECE 546 – Jose Schutt-Aine 25 1)Get L and C matrices and calculate LC product 2)Get square root of eigenvalues and eigenvectors of LC matrix   m 3)Arrange eigenvectors into the voltage eigenvector matrix E 4)Get square root of eigenvalues and eigenvectors of CL matrix  m 5)Arrange eigenvectors into the current eigenvector matrix H 6) Invert matrices E, H,  m. 7) Calculate the line impedance matrix Z c  Z c = E -1  m -1 EL Procedure for Multiconductor Solution

26. ECE 546 – Jose Schutt-Aine 26 8)Construct source and load impedance matrices Z s (t) and Z L (t) 9) Construct source and load reflection coefficient matrices  1 (t) and  2 (t). 10) Construct source and load transmission coefficient matrices T  (t), T 2 (t) 11) Calculate modal voltage sources g 1 (t) and g 2 (t) 12) Calculate modal voltage waves: a 1 (t) = T 1 (t)g 1 (t) +  1 (t)a 2 (t -  m) a 2 (t) = T 2 (t)g 2 (t) +  2 (t)a 1 (t -  m) b 1 (t) = a 2 (t -  m ) b 2 (t) = a 1 (t -  m ) Procedure for Multiconductor Solution

27. ECE 546 – Jose Schutt-Aine 27  mi is the delay associated with mode i.  mi = length/velocity of mode i. The modal volage wave vectors a 1 (t) and a 2 (t) need to be stored since they contain the history of the system. 13) Calculate total modal voltage vectors: V m1 (t) = a 1 (t) + b 1 (t) V m2 (t) = a 2 (t) + b 2 (t) 14) Calculate line voltage vectors: V 1 (t) = E -1 V m1 (t) V 2 (t) = E -1 V m2 (t) Wave-Shifting Solution

28. ECE 546 – Jose Schutt-Aine 28 7-Line Coupled-Microstrip System L s = 312 nH/m; C s = 100 pF/m; L m = 85 nH/m; C m = 12 pF/m.

29. ECE 546 – Jose Schutt-Aine 29 Drive Line 3

30. ECE 546 – Jose Schutt-Aine 30 Sense Line

31. ECE 546 – Jose Schutt-Aine 31 Multiconductor Simulation

32. ECE 546 – Jose Schutt-Aine 32 LOSSY COUPLED TRANSMISSION LINES Solution is best found using a numerical approach (See References)

33. ECE 546 – Jose Schutt-Aine 33 Three-Line Microstrip

34. ECE 546 – Jose Schutt-Aine 34 Subtract (1c) from (1a) and (2c) from (2a), we get This defines the Alpha mode with: and The wave impedance of that mode is: and the velocity is Three-Line – Alpha Mode

35. ECE 546 – Jose Schutt-Aine 35 In order to determine the next mode, assume that By reciprocity L 32 = L 23, L 21 = L 12, L 13 = L 31 By symmetry, L 12 = L 23 Also by approximation, L 22  L 11, L 11 +L 13  L 11 Three-Line – Modal Decomposition

36. ECE 546 – Jose Schutt-Aine 36 In order to balance the right-hand side into I , we need to have or Therefore the other two modes are defined as The Beta mode with Three-Line – Modal Decomposition

37. ECE 546 – Jose Schutt-Aine 37 The Beta mode with The characteristic impedance of the Beta mode is: and propagation velocity of the Beta mode is Three-Line – Beta Mode

38. ECE 546 – Jose Schutt-Aine 38 The Delta mode is defined such that The characteristic impedance of the Delta mode is The propagation velocity of the Delta mode is: Three-Line – Delta Mode

39. ECE 546 – Jose Schutt-Aine 39 Alpha mode Beta mode* Delta mode* Symmetric 3-Line Microstrip In summary: we have 3 modes for the 3-line system *neglecting coupling between nonadjacent lines

40. ECE 546 – Jose Schutt-Aine 40 Coplanar Waveguide

41. ECE 546 – Jose Schutt-Aine 41 Coplanar Waveguide

42. ECE 546 – Jose Schutt-Aine 42 Coplanar Waveguide

43. ECE 546 – Jose Schutt-Aine 43 Coplanar Strips

44. ECE 546 – Jose Schutt-Aine 44 CharacteristicMicrostripCoplanar WguideCoplanar strips e eff *~6.5 ~5 ~5 Power handlingHigh Medium Medium Radiation lossLow Medium Medium Unloaded QHigh Medium Low or High DispersionSmall Medium Medium Mounting (shunt)Hard Easy Easy Mounting (series)Easy Easy Easy * Assuming  r =10 and h=0.025 inch Qualitative Comparison