ECE 598 JS Lecture 06 Multiconductors

ECE 598 JS Lecture 06 Multiconductors Spring 2012 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu 1

TELGRAPHER S EQUATION FOR N COUPLED TRANSMISSION LINES V 1 (z) V 2 (z) z=0 z=l... V 3 (z) L, C V = L z I = C z V and I are the line voltage and line current VECTORS respectively (dimension n). I t V t 2

V V V = V 1 2 n V 1 (z) V 2 (z) V 3 (z) N-LINE SYSTEM z=0 z=l I I = I L and C are the inductance and capacitance MATRICES respectively L, C... I 1 2 n L L L L 11 12 L 21 22 = L nn C C C C 11 12 C 21 22 = C nn 3

N-LINE ANALYSIS V z V = LC t 2 2 I z 2 2 I = CL t 2 2 2 2 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. 4

MODAL ANALYSIS EV z EV t 2 2 1 = ELCE 2 2 HI z HI t 2 2 1 = HCLH 2 2 LC and CL are adjoint matrices. Find matrices E and H such that ELCE = HCLH =Λ 1 1 2 m is the diagonal eigenvalue matrix whose entries are the inverses of the modal velocities squared. 5

MODAL ANALYSIS V 1 (z) V 2 (z) z=0 z=l... V 3 (z) L, C I z 2 2 m 2 m =Λ 2 m 2 V z I t V t 2 2 m 2 m =Λ 2 m 2 V m = EV I m = HI Second-order differential equation in modal space 6

7 V m and I m are the modal voltage and current vectors respectively. EIGENVECTORS 1 2 m m m mn V V V V = 1 2 m m m mn I I I I = I m = HI V m = EV

EIGEN ANALYSIS * A scalar λ is an eigenvalue of a matrix A if there exists a vector X such that AX = λx. (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. * The eigenvalues λ of A satisfy the relation A - λi = 0 where I is the unit matrix. 8

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. 9

EIGEN ANALYSIS For an n-line system, it can be shown that * 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. * Each eigenvalue is associated with a mode; the propagation velocity of that mode is the inverse of the eigenvalue. 10

EIGEN ANALYSIS * Each eigenvector is associated to an eigenvalue and therefore to a particular mode. * Each normalized eigenvector represents the relative line excitation required to excite the associated mode. 11

EIGENVECTORS E : Voltage eigenvector matrix 1 2 ELCE = Λ m e e e E e e e 11 12 13 = 21 22 23 e e e 31 32 33 H : Current eigenvector matrix 1 2 HCLH = Λ m h h h H h h h 11 12 13 = 21 22 23 h h h 31 32 33 12

1 2 ELCE =Λ m gives 1 2 HCLH =Λ m gives Eigenvalues and Eigenvectors e e e E e e e e e e 11 12 13 = 21 22 23 31 32 33 h h h H h h h h h h 11 12 13 = 21 22 23 31 32 33 1 0 0 v m1 1 Λ m = 0 0 v m2 1 0 0 v m3 1 0 0 v m1 1 Λ m = 0 0 v m2 1 0 0 v m3 13

Modal Voltage Excitation Voltage Eigenvector Matrix e e e E e e e e e e 11 12 13 = 21 22 23 31 32 33 e 11 e 12 + + + e 13 - - - MODE A e 21 e 22 + + + e 23 - - - MODE B MATCHING NETWORK MATCHING NETWORK e 32 + + + e 31 e 33 - - MODE C - MATCHING NETWORK 14

Modal Current Excitation Current Eigenvector Matrix h 11 h 12 h 13 MODE A MATCHING NETWORK h h h H h h h 11 12 13 = 21 22 23 h h h 31 32 33 h 21 h 22 h 23 MODE B MATCHING NETWORK MATCHING NETWORK h 31 h 32 h 33 MODE C 15

EIGENVECTORS E : voltage eigenvector matrix H : current eigenvector matrix E and H depend on both L and C 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] Line variables are recovered using V = E 1 V m I = H 1 I m 16

MODAL AND LINE IMPEDANCE MATRICES By requiring V m = Z m I m, we arrive at Z m = Λ m 1 ELH 1 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 Z c = E 1 Z m H = E 1 Λ m 1 EL 17

18 1 2 ( ) m m mn j u v j u v j u v e e Wu e ω ω ω = 1 2 1 1 1 m m m mn v v v Λ =

MATRIX CONSTRUCTION [ L] = L ( m) s [ ] ii L ij = L ij n ( m) ii = + s ij j= 1 [ C] C C [ C] ij = C ( m) ij [ R] ii = R s [ R] ij = R m ij n ( m) ii s ij j= 1 [ G] G G = + ( m) [ G] ij = G ij 19

Modal Variables V m = EV I m = HI General solution in modal space is: V m (z) = W(z)A +W(- z)b - I m (z) = Z 1 m [ W(z)A- W(- z)b] Modal impedance matrix Z m = Λ ELH 1 1 m Line impedance matrix Z = E Z H = E Λ EL 1 1 1 c m m 20

N-Line Network Z s : Source impedance matrix Z L : Load impedance matrix V s : Source vector 21

Reflection Coefficients Source reflection coefficient matrix Γ s = [1 n + EZ s L 1 E 1 Λ m ] 1 [1n EZ s L 1 E 1 Λ m ] Load reflection coefficient matrix Γ L = [1 n + EZ L L 1 E 1 Λ m ] 1 [1n EZ L L 1 E 1 Λ 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 22

TERMINATION NETWORK V 1 y p11 y p12 V 2 y p22 I1 = yp11v1 +yp12(v1-v2) I2 = yp22v2 +yp12(v2-v1) In general for a multiline system I = YV V = ZI Z = Y 1 Note : yii ypii, yij = -yji, (i j) Also, z ij 1 y ij 23

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 24

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

Procedure for Multiconductor Solution 8) Construct source and load impedance matrices Z s (t) and Z L (t) 9) Construct source and load reflection coefficient matrices Γ 1 (t) and Γ 2 (t). 10) Construct source and load transmission coefficient matrices T 1 (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) + Γ 1 (t)a 2 (t - τm) a 2 (t) = T 2 (t)g 2 (t) + Γ 2 (t)a 1 (t - τm) b 1 (t) = a 2 (t - τ m ) b 2 (t) = a 1 (t - τ m ) 26

ai ai a i( t τ m ) = ai mode 1 mode mode n ( t τ 2( t τ ( t τ τ mi is the delay associated with mode i. τ 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 m1 m2 mn ) ) ) 27

7-Line Coupled-Microstrip System ALS04 Drive Line 1 ALS240 ALS04 Drive Line 2 ALS240 ALS04 Drive Line 3 ALS240 Sense Line 4 ALS04 Drive Line 5 ALS240 ALS04 Drive Line 6 ALS240 ALS04 Drive Line 7 ALS240 z=0 z=l L s = 312 nh/m; C s = 100 pf/m; L m = 85 nh/m; C m = 12 pf/m. 28

5 Drive line 3 at Near End Drive Line 3 5 Drive Line 3 at Far End 4 4 3 2 1 3 2 1 0 0 0 100 Time (ns) 200-1 0 100 Time (ns) 200 29

This image cannot currently be displayed. Sense Line 2 Sense Line at Near End 2 Sense Line at Far End 1 1 0 0-1 0 100 Time (ns) 200-1 0 100 Time (ns) 200 30

Multiconductor Simulation 31

LOSSY COUPLED TRANSMISSION LINES V 1 (z) V 2 (z) z=0 z=l... V 3 (z) L, C V = RI + L z I = GV + C z Solution is best found using a numerical approach (See References) I t V t 32

Three Line Microstrip 1 2 3 V1 I1 I2 I = L11 + L12 + L13 z t t t 3 I1 V1 V2 V = C11 + C12 + C13 z t t t 3 V2 I1 I2 I = L21 + L22 + L23 z t t t 3 I2 V1 V2 V = C21 + C22 + C23 z t t t 3 V I I I = L + L + L z t t t 3 1 2 3 31 32 33 I V V V = C + C + C z t t t 3 1 2 3 31 32 33 33

Three Line Alpha Mode Subtract (1c) from (1a) and (2c) from (2a), we get Vα I = ( L11 L13) z t Iα V = ( C11 C13) z t This defines the Alpha mode with: α α and I Vα = V1 V = I α 1 I3 3 The wave impedance of that mode is: Z α = L C L C 11 13 11 13 and the velocity is u α = 1 ( L L )( C C ) 11 13 11 13 34

Three Line Modal Decomposition In order to determine the next mode, assume that V = V + βv + V β 1 2 3 I = I + β I + I β 1 2 3 Vβ I I I = 11 + 21 + 31 + 12 + 22 + 32 + 13 + 23 + 33 z t t t 1 2 3 ( L βl L ) ( L βl L ) ( L βl L ) I β V V V = 11 + 21 + 31 + 12 + 22 + 32 + 13 + 23 + 33 z t t t 1 2 3 ( C βc C ) ( C βc C ) ( C βc C ) 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 35

Three Line Modal Decomposition Vβ I I3 I = ( L + βl + L ) + + ( 2L + βl ) z t t t 1 2 11 12 13 12 11 In order to balance the right-hand side into I β, we need to have ( + β ) = β ( + β + ) β ( + β ) 2L L I L L L I L L I 12 11 2 11 12 13 2 11 12 2 2L = β L 2 12 12 or β =± 2 Therefore the other two modes are defined as The Beta mode with 36

The Beta mode with Three Line Beta Mode Vβ = V + 2V + V 1 2 3 Iβ = I + 2I + I 1 2 3 The characteristic impedance of the Beta mode is: Z β = L + 2L + L 11 12 13 C + 2C + C 11 12 13 and propagation velocity of the Beta mode is 1 u = β + + + + ( L11 2L12 L13 )( C11 2C12 C13 ) 37

Three Line Delta Mode The Delta mode is defined such that V = V δ 2V + V 1 2 3 I = I δ 2I + I 1 2 3 The characteristic impedance of the Delta mode is Z δ = L 2L + L 11 12 13 C 2C + C 11 12 13 The propagation velocity of the Delta mode is: 1 u = δ + + ( L11 2L12 L13 )( C11 2C12 C13 ) 38

Symmetric 3 Line Microstrip 1 2 3 In summary: we have 3 modes for the 3-line system E 1 0 1 = 1 2 1 1 2 1 Alpha mode Beta mode* Delta mode* *neglecting coupling between nonadjacent lines 39

Coplanar Waveguide 1 2 3 ε r = 4.3 346 162 67 LnH ( / m) = 152 683 152 67 162 346 113 17 5 C( pf/ m) = 16 53 16 5 17 113 E 0.45 0.12 0.45 = 0.5 0 0.5 0.45 0.87 0.45 H 0.44 0.49 0.44 = 0.5 0 0.5 0.10 0.88 0.10 40

Coplanar Waveguide 1 2 3 ε r = 4.3 Z m 73 0 0 ( Ω ) = 0 48 0 0 0 94 Z c 56 23 8 ( Ω ) = 22 119 22 8 23 56 vp 0.15 0 0 ( m/ ns) = 0 0.17 0 0 0 0.18 41

Coplanar Waveguide S' W S W S' h ε r Kk ( ) : Complete Elliptic Integral of the first kind S k = 30 π K'( k) S + 2W Zocp = (ohm) ε ( ) r + 1 Kk K'( k) = K( k') 2 k' = (1 k ) 2 1/2 v cp 2 = εr + 1 1/2 c 42

Coplanar Strips W S W h ε r Z ocs = 120 π K'( k) (ohm) ε 1 Kk ( ) r + 2 43

Qualitative Comparison Characteristic Microstrip Coplanar Wguide Coplanar strips ε eff * ~6.5 ~5 ~5 Power handling High Medium Medium Radiation loss Low Medium Medium Unloaded Q High Medium Low or High Dispersion Small Medium Medium Mounting (shunt) Hard Easy Easy Mounting (series) Easy Easy Easy * Assuming ε r =10 and h=0.025 inch 44

3-Line Matching Network Extend matching network synthesis to multiconductor transmission line (MTL) systems, specifically coupled microstrip Characterize MTL systems by their normal mode parameters Mode configurations Propagation constants Characteristic impedance matrices 45

Longitudinal Immittance Matrix Functions (LIMFs) Given terminations, the immittance matrices are expressed as input longitudinal functions looking toward the load S Y m,c m,c in,out, Γ,, Z m,c in,out m,c in,out 46

Calibration Methods 1st order calibration: 6 distinct TRL calibrations were required to deembed the fan-in and fan-out sections 2nd order calibration: 1 multimode TRL calibration required to deembed all fan-ins and fan-outs Multiconductor transmission line matching network device under test 47

Bias Networks and Multiline/Multimode TRL Self-biasing for DC power done using air bridges Regulator circuit may be built on board with surface mount/chip components Calibration must include all modes and allow for the 6- port discontinuity S- parameter measurement Proposed calibration fixture for transistor amplifier device discontinuity with self-biasing network 48

3-line Transistor Amplifier The general amplifier structure with generic matching networks Note that a transistor seating hole may be present for convenience, but is not necessary Photograph of a three-line transistor amplifier autobiased w/o matching networks 49

Transistor Amplifier Matching Matching 2-0.8 pf chip cap. 2-0.4 pf chip cap. Gains 3.38 db @ 3.1 GHz m=5.14 / um= 1.76 6.22 db @ 3.22 GHz m=6.54 / um= 0.32 S-parameter comparison between unmatched and matched three-line amplifier. Dotted line - unmatched; dashed line - matched. Design frequency: 3.1 GHz. 50

V Transmission Line w signal strip h dielectric α ground 2p 51

V Line Characteristic Impedance 250 CHARACTERISTIC IMPEDANCE Zo (ohms) 200 150 100 30º 37.5º 45º 52.5º 60º Microstrip 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 w/h 0.7 0.8 0.9 1 Calculated values of the characteristic impedance for a single-line v-strip structure as a function of width-to-height ratio w/h. The relative dielectric constant is εr = 2.55. 52

V Line Effective Permittivity 2.2 2.15 30º 37.5º EFFECTIVE PERMITTIVITY 2.1 45º 52.5º Effective Relative Dielectric Constant 2.05 2 1.95 1.9 1.85 1.8 60º Microstrip 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w/h Calculated values of the effective relative dielectric constant for a single-line v-strip structure as a function of width-to-height ratio w/h. The relative dielectric constant is εr = 2.55 53

Three Line V Transmission Line y signal strip w s 2p h ground surface dielectric x Region 1 Region 2 Region 3 Region 4 Region 5 Region 6 54

V Line and Microstrip Microstrip 52.49-5.90-0.57 C (pf/m) = -5.90 53.27-5.90-0.57-5.90 52.49 609.74 113.46 41.79 L (nh/m) = 113.54 607.67 113.54 41.79 113.46 609.74 V-line 74.00-0.97-0.23 C (pf/m) = -0.97 74.07-0.97-0.23-0.97 74.00 425.84 20.35 7.00 L (nh/m) = 20.36 422.85 20.36 7.00 20.35 425.84 y 1 2 3 w s signal strip 2p h ground surface dielectric x Region 1 Region 2 Region 3 Region 4 Region 5 Region 6 Comparison of the inductance and capacitance matrices between a three-line v-line and microstrip structures. The parameters are p/h = 0.8 mils, w/h = 0.6 and ε r = 4.0. 55

V Line: Coupling Coefficients 350 Mutual Inductance 40 Mutual Capacitance L m (nh/m) 300 250 200 150 100 V-line microstrip C m (pf/m) 35 30 25 20 15 10 V-line microstrip 50 5 0 0 0.2 0.4 0.6 0.8 1 1.2 s/h 0 0 0.2 0.4 0.6 0.8 1 1.2 s/h Plot of mutual inductance (top) and mutual capacitance (bottom) versus spacing-to-height ratio for v-line and microstrip configurations. The parameters are w/h = 0.24, εr = 4.0. 56

V Line vs Microstrip: Coupling Coefficients 0.6 0.5 0.4 Coupling Coefficient V-line microstrip K v 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 s/h Plot of the coupling coefficient versus spacing-to-height ratio for v-line and microstrip configurations. The parameters are w/h = 0.24, εr = 4.0. 57

Advantages of V Line ground reference signal strip w signal strip dielectric substrate h dielectric α ground ground reference 2p Propagation Function Advantages of V Line * Higher bandwidth 1 0.9 microstrip V-60 V-30 * Lower crosstalk * Better transition Linear Attenuation 0.8 0.7 0.6 0.5 1 3 5 7 9 11 13 15 17 Frequency (GHz) 58

Advantages of V Line ground reference signal strip w signal strip dielectric substrate h dielectric α ground ground reference 2p Propagation Function Advantages of V Line * Higher bandwidth 1 0.9 microstrip V-60 V-30 * Lower crosstalk * Better transition Linear Attenuation 0.8 0.7 0.6 0.5 1 3 5 7 9 11 13 15 17 Frequency (GHz) 59

V Line vs Microstrip: Insertion Loss 0 Insertion Loss 20*log10* S21-1 -2-3 -4-5 -6-7 -8 Vline Microstrip 0 5 10 15 20 25 Frequency (GHz) 60

References J. E. Schutt-Aine and R. Mittra, "Transient analysis of coupled lossy transmission lines with nonlinear terminations," IEEE Trans. Circuit Syst., vol. CAS-36, pp. 959-967, July 1989. J. E. Schutt-Aine and D. B. Kuznetsov, "Efficient transient simulation of distributed interconnects," Int. Journ. Computation Math. Electr. Eng. (COMPEL), vol. 13, no. 4, pp. 795-806, Dec. 1994. D. B. Kuznetsov and J. E. Schutt-Ainé, "The optimal transient simulation of transmission lines," IEEE Trans. Circuits Syst.-I., vol. cas-43, pp. 110-121, February 1996. W. T. Beyene and J. E. Schutt-Ainé, "Efficient Transient Simulation of High-Speed Interconnects Characterized by Sampled Data," IEEE Trans. Comp., Hybrids, Manufacturing Tech. vol. 21, pp. 105-114, February 1998. 61