Munich07 Foils

543 views

Published on

Published in: Travel, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
543
On SlideShare
0
From Embeds
0
Number of Embeds
7
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Munich07 Foils

  1. 1. EMC Zurich Munich 2007 Circuit Extraction for Transmission Lines Modeling of PCBs Giulio Antonini, Antonio Orlandi UAq EMC Laboratory Department of Electrical Engineering University of L’Aquila, 67040 AQ, Italy e-mail: antonini@ing.univaq.it, orlandi@ing.univaq.it Munich, September 28, 2007 Slide 1 of 88
  2. 2. Introduction High-speed interconnect modeling: SI and EMC issues • Ringing, attenuation, signal delay, distorsion • Crosstalk • EM radiation and susceptibility • Non-linear terminations • Incorporation of frequency dependent phenomena (conductor and dielectric losses) Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 2 of 88
  3. 3. Telegrapher’s equations: a brief review d V (x, s) = − R (s) + sL (s) I (x, s) = −Z (s)I (x, s) dx d I (x, s) = − G (s) + sC (s) V (x, s) = −Y (s)V (x, s) dx Existing methods for time domain solution • Lumped network (closed-form available for HTLN) • Method of characteristics • Matrix Rational macromodeling (with and without delay extraction) • Krylov-based (projection) methods • Vector Fitting techniques Crucial issues • Stability • Passivity (causality) Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 3 of 88
  4. 4. Outline 1st part: the half-T ladder network Development of the transmission line model by using G DFF and DFFz polynomials Two ports representation G Extraction of poles and residues in a closed form G Model order reduction G Stability and passivity G Realization G Numerical results G Conclusions G Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 4 of 88
  5. 5. Development of the transmission-line model Telegrapher’s equations ∂ ∂ v (x, t) = −Ri (x, t) − L i (x, t) ∂x ∂t ∂ ∂ i (x, t) = −Gv (x, t) − C v (x, t) ∂x ∂t R , L , C and G : per unit length parameters, non-negative definite symmetric matrices of order N , being N + 1 the number of conductors. In [1] M.Faccio, G.Ferri, A.D’Amico. A New Fast Method for Ladder Net- works Characterization. IEEE Trans. on Circuits and System, I, 38(11): 1377-1382, September 1991 it was shown that an open-ended half-T lad- der network can be analytically studied. Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 5 of 88
  6. 6. Development of the transmission-line model The voltage at the generic node β in the Laplace-domain can be expressed as: Pbn−β (K (s)) Vβ (s) = n (K (s)) Vin (s) Pb being Z1 (s) K (s) = Z2 (s) n−β and Pn (K(s)) an n − β order polynomial in K (s) with 0 ≤ β ≤ n. More specifically, the voltage at node β is: n−β bj,n−β K j (s) j=0 Vβ (s) = Vin (s) n j (s) j=0 bj,n K Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 6 of 88
  7. 7. Development of the transmission-line model The polynomial coefficients b are generated accordingly to the following recursive expression    j+i j+i  =  bi,j = j−i 2j The general expression of the longitudinal branch current Iβ1 (s) is: n−β+1 cj,n−β+1 K j+1 (s) 1 j=0 Iβ1 (s) = Vin (s) n j (s) Z1 (s) j=0 bj,n K where the polynomial coefficients c are obtained by means of the recursive expression [2] Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 7 of 88
  8. 8. Development of the transmission-line model     i+j+1 i+j+1 = =  ci,j j−i 2j + 1 Longitudinal current can be re-written in a more compact form as: 1 Pcn−β+1 (K(s)) Iβ1 (s) = Vin (s) n (K(s)) Z1 (s) Pb where Pcn−β+1 (K(s)) is a n − β + 1 order polynomial in K (s). Similarly, the shunt branch current Iβ2 (s) can be expressed as: n−β n−β bj,n−β K j (s) 1 Pb (K) 1 j=0 Iβ2 (s) = Vin = n (K) Vin (s) n j (s) Z2 j=0 bj,n K Z2 Pb Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 8 of 88
  9. 9. DFF and DFFz polynomials x x Pbn (x) = Un 1 + − Un−1 1 + 2 2 x n Pc (x) = Un 1 + 2 From the properties of Chebyshev polynomials it has been proved that [2] Pbn (x) and Pcn (x) polynomials are orthogonal in the interval [−4, 0] with respect to the weight functions 1 −1 Pu (x) = − (x + 4) x 2 2 1 1 Pv (x) = − (x + 4) x 2 2 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 9 of 88
  10. 10. DFF and DFFz polynomials Extension to MTLs: MTL equivalent half-T ladder network Z1 Z1 Z1 Z1 I1 b In b-1 1 n 0 Vin Z2 Z2 Z2 Z2 Vout l l Z 1 (s) = (R + sL) = Z l n n l −1 −1 l Z 2 (s) = (G + sC) = Yt n n K (s) = Z 2 (s)−1 Z 1 (s) Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 10 of 88
  11. 11. DFF and DFFz polynomials Extension to MTLs n Pbn (K(s)) = bj,n K j (s) DFF polynomial of order n j=0 n Pbn−1 (K(s)) = bj,n−1 K j (s) DFF polynomial of order n − 1 j=0 n Pcn (K(s)) = cj,n K j+1 (s) DFFz polynomial of order n j=0 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 11 of 88
  12. 12. Closed form zeros of DFF and DFFz polynomials (2j − 1) π 2 for Pbn (x) uj,n = −4sin j = 1···n (2n + 1) 2 j π 2 for Pcn (x) vj,n = −4sin j = 1···n (n + 1) 2 n Pbn (K(s)) = (K(s) − uj,n ) j=1 n−1 Pbn−1 (K(s)) = (K(s) − uj,n−1 ) j=1 n−1 Pcn (K(s)) = K(s) (K(s) − vj,n−1 ) j=1 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 12 of 88
  13. 13. Two port representation ABCD representation n bj,n K j (s) = Pbn (K(s)) G A= j=0 n cj,n K j+1 (s) · Z 2 (s) = Pcn (K(s)) · Z 2 (s) G B= j=0 n · Z −1 (s) = Pcn (K(s)) · Z −1 (s) cj,n K j+1 (s) G C= 1 1 j=0 n−1 = Pbn−1 (K(s)) bj,n−1 K j (s) G D= j=0   (Pcn (K(s)) · Z 2 (s))−1 Pbn−1 (K(s)) (Pcn (K(s)) · Z 2 (s))−1 Y =  (Pcn (K(s)) · Z 2 (s))−1 (Pcn (K(s)) · Z 2 (s))−1 Pbn (K(s)) Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 13 of 88
  14. 14. Two port representation DFF and DFFz polynomials factorization n Pbn (K) = s2 CL + s (GL + CR) + GR − uj,n U j=1 n−1 Pbn−1 (K) = s2 CL + s (GL + CR) + GR − uj,n−1 U j=1 n−1 Pcn (K) = K s2 CL + s (GL + CR) + GR − vj,n−1 U j=1 ⇓ Y matrix entries factorization Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 14 of 88
  15. 15. Y matrix entries factorization   −1 n−1 Y 11 =  s2 CL + s (GL + CR) + GR − vj,n−1 U  (R + sL) · j=1 n−1 s2 CL + s (GL + CR) + GR − uj,n−1 U · j=1   −1 n−1 Y 12 = −  s2 CL + s (GL + CR) + GR − vj,n−1 U  (R + sL) j=1 Y 21 = Y 12 Y 22 = Y 11 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 15 of 88
  16. 16. Two port representation Poles extraction    n−1 det  s2 CL + s (GL + CR) + GR − vj,n−1 U  (R + sL) = 0 j=1 ⇓   n−1  det s2 CL + s (GL + CR) + GR − vj,n−1 U  det (R + sL) = 0 j=1 ⇓ Closed-form evaluation of residues ⇒ Spice equivalent circuit Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 16 of 88
  17. 17. Two port representation Poles extraction for single-conductor transmission lines R s0 =− L v 2 RG − j,n−1 1 RG 1 RG ∆l2 sj,1 = − + + + − 2 LC 4 LC LC v 2 RG − j,n−1 1 RG 1 RG ∆l2 sj,2 =− + − + − 2 LC 4 LC LC for j = 1 · · · P , P being the total number of poles. All the P + 1 poles have a negative real part → stability ensured Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 17 of 88
  18. 18. Model Order Reduction The set of poles for a given order of the half-T ladder network is analytically known Condition 1 |Im (pi ) | < ωmax for i = 1 · · · P , ωmax being the fixed bandwidth of the MOR. A first set P1 of poles is selected. Condition 2 |Residue (pi ) | > th for i = 1 · · · P1 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 18 of 88
  19. 19. Stability, Passivity Stability The proposed model is characterized by poles which strictly satisfy this condition because they represent exactly the poles of a half-T ladder network which is intrinsically stable. Passivity It has been assumed that p.u.l. parameters matrices R, L, C, G are non- negative definite symmetric matrices. This implies that matrices Z 1 (s) and Z −1 (s) are positive real (PR) matrices. This ensures that the half-T 2 ladder network of order n is intrinsically passive (more details in [3]). Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 19 of 88
  20. 20. Realization Realization d x (t) = Ax (t) + Bu (t) dt y (t) = Cx (t) + Du (t) Standard realization procedures can be adopted to obtain the state space representation A, B, C, D. Linear and non-linear terminations are described by additional equations. Circuit synthesis See 3rd part Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 20 of 88
  21. 21. Poles location finding: single conductor transmission line Frequency independent per unit length parameters (FIPUL) order Tanji in [4] DFFLN 0 -1.933837 -1.933837 -2.706589 -2.706589 1 −1.240641e8 + j8.064423e9 −1.2406408e8 + j8.06235e9 −1.240641e8 − j8.064423e9 −1.2406408e8 − j8.06235e9 −1.079620e8 + j8.831359e9 −1.0796194e8 + j8.82908e9 −1.079620e8 − j8.831359e9 −1.0796194e8 − j8.82908e9 2 −1.240495e8 + j1.612970e10 −1.240495e8 + j1.611312e10 −1.240495e8 − j1.612970e10 −1.240495e8 − j1.611312e10 −1.079766e8 + j1.766434e10 −1.079765e8 + j1.764618e10 −1.079766e8 − j1.766434e10 −1.079765e8 − j1.764618e10 5 −1.240454e8 + j4.032485e10 −1.240453e8 + j4.006624e10 −1.240454e8 − j4.032485e10 −1.240453e8 − j4.006624e10 −1.079807e8 + j + 4.416198e10 −1.079806e8 + j4.387875e10 −1.079807e8 − j4.416198e10 −1.079806e8 − j4.387875e10 The proposed method doesn’t use the modal decomposition . Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 21 of 88
  22. 22. Numerical results One-conductor line with linear terminations (50 Ω) [5] R = 1776 Ω/m L = 0.5978 µH/m C = 18.61 pF/m G = 0 S/m 11 x 10 0.07 MOR Order 90 Spice MOR 4 PEEC 0.06 3.5 3 0.05 2.5 Voltage [V] 0.04 2 Im 1.5 0.03 1 0.5 0.02 0 0.01 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −1.4854 −1.4854 −1.4854 −1.4854 −1.4854 −1.4854 Time [s] Re −9 9 x 10 x 10 Dominant poles Voltage at the output port Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 22 of 88
  23. 23. Numerical results Numerical methods used for comparison 1) Transmission Line Theory via IFFT (TLT-IFFT) 2) Half-T Ladder network via IFFT (HTLN-IFFT) 3) DFF and DFFz polynomials via IFFT (DFFLN-IFFT) 4) DFF and DFFz polynomials with MOR via IFFT (DFFLN-MOR-IFFT) 5) DFF and DFFz polynomials without MOR via Pspice (DFFLN-Pspice) 6) DFF and DFFz polynomials with MOR via Pspice (DFFLN-MOR- Pspice) 7) DFF and DFFz polynomials with MOR via ODE solver (DFFLN-MOR- ODE) Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 23 of 88
  24. 24. Numerical results Two-conductor line with linear terminations (50 Ω) (I) 3 1 4 2     4.63 0.74 337 58.4 R=  Ω/m L=  nH/m 0.74 4.63 58.4 337     193 −1.53 00 C=  pF/m G=  S/m −1.53 193 00 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 24 of 88
  25. 25. Numerical results HTLN order 129, MOR 32 11 11 x 10 x 10 4 1.5 Order 129 Order 129 MOR MOR 3 1 2 0.5 1 Im Im 0 0 −1 −0.5 −2 −3 −1 −4 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −7.4 −7.3 −7.2 −7.1 −7 −6.9 −6.8 −6.7 −6.6 −6.5 −6.4 Re Re 7 6 x 10 x 10 1 1 10 10 TLT TLT HTLN HTLN DFFLN DFFLN DFFLN−MOR DFFLN−MOR 0 10 0 10 −1 10 −1 10 Y14 [S] Y11 [S] −2 10 −2 10 −3 10 −3 10 −4 10 −5 −4 10 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Frequency [Hz] Frequency [Hz] 9 9 x 10 x 10 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 25 of 88
  26. 26. Numerical results 1.2 IFFT 1 MOR IFFT MOR 1 0.8 0.8 0.6 Voltage V1 [V] 0.6 Voltage V3 [V] 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time [s] Time [s] −8 −8 x 10 x 10 0.05 0.1 IFFT IFFT MOR MOR 0.04 0.08 0.03 0.06 0.02 0.04 0.01 0.02 Voltage V [V] Voltage V4 [V] 2 0 0 −0.01 −0.02 −0.02 −0.04 −0.03 −0.06 −0.04 −0.08 −0.05 −0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time [s] Time [s] −8 −8 x 10 x 10
  27. 27. Numerical results Two-conductor line with linear terminations (50 Ω) (II) 3 1 4 2     0.2 0 0.28 0.07 R=  Ω/m L=  µH/m 0 0.2 0.07 0.28     122 −50 00 C=  pF/m G=  S/m −50 122 00 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 27 of 88
  28. 28. 160 poles, MOR 68 0.6 0.6 TLT−IFFT TLT−IFFT HTLN−IFFT HTLN−IFFT DFFLN−IFFT DFFLN−IFFT DFFLN−MOR−IFFT DFFLN−MOR−IFFT 0.5 0.5 DFFLN−MOR−ODE DFFLN−MOR−ODE DFFLN−MOR−Pspice DFFLN−MOR−Pspice 0.4 0.4 Voltage V1 [V] Voltage V3 [V] 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Time [s] Time [s] −8 −8 x 10 x 10 0.1 0.03 TLT−IFFT TLT−IFFT HTLN−IFFT HTLN−IFFT DFFLN−IFFT 0.08 DFFLN−IFFT DFFLN−MOR−IFFT DFFLN−MOR−IFFT DFFLN−MOR−ODE DFFLN−MOR−ODE 0.02 DFFLN−MOR−Pspice DFFLN−MOR−Pspice 0.06 0.04 0.01 0.02 Voltage V2 [V] Voltage V [V] 4 0 0 −0.02 −0.01 −0.04 −0.06 −0.02 −0.08 −0.1 −0.03 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Time [s] Time [s] −8 −8 x 10 x 10 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 28 of 88
  29. 29. Numerical results Three-conductor line with linear terminations (resistive loads)     5/12 0 0 87 25 23     1 R=  Ω/cm L = 3600  25 25  µH/cm 0 5/12 0 85     0 0 5/12 23 25 87     68 −40 −13 3/512 0 0     C = 324  −40 1 −40  pF/cm G=  mS/cm 95 0 3/512 0     −13 −40 68 0 0 3/512 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 29 of 88
  30. 30. Numerical results 0.35 0.5 IFFT IFFT IFFT MOR MOR MOR 0.3 0.9 0.4 0.8 0.25 0.3 0.7 0.2 0.2 0.6 0.15 Voltage V3 [V] Voltage V2 [V] Voltage V [V] 0.1 0.5 1 0.1 0.4 0 0.05 0.3 −0.1 0 0.2 −0.2 −0.05 0.1 −0.3 −0.1 0 −0.4 −0.1 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Time [s] Time [s] Time [s] −8 −8 −8 x 10 x 10 x 10 IFFT IFFT IFFT MOR MOR MOR 0.9 0.25 0.2 0.8 0.2 0.7 0.1 0.6 Voltage V6 [V] 0.15 Voltage V5 [V] Voltage V4 [V] 0 0.5 0.4 0.1 −0.1 0.3 0.05 −0.2 0.2 0.1 0 −0.3 0 −0.05 −0.4 −0.1 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Time [s] Time [s] Time [s] −8 −8 −8 x 10 x 10 x 10
  31. 31. Numerical results Lightning over-voltage on a two-conductor line V0 (t/τ1 )n −t/τ2 vs (t) = e n+1 η (t/τ1 ) where V0 = 105 V, η = 1, τ1 = 0.5 µs τ2 = 10 µs and n = 2. It has been considered a 1 m long cable whose p.u.l. parameters are:     0.4 0 1.25265 0.87324 R=  Ω/m L=  µH/m 0 0.4 0.87324 1.25265     17.2799 −12.0461 00 C=  pF/m G=  S/m −12.0461 17.2799 00 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 31 of 88
  32. 32. Numerical results 0.2 TLT−IFFT TLT−IFFT HTLN−IFFT HTLN−IFFT DFFLN−IFFT DFFLN−IFFT DFFLN−MOR−IFFT DFFLN−MOR−IFFT 0 1 DFFLN−MOR−ODE DFFLN−MOR−ODE −0.2 0.8 Voltage V4 [V] −0.4 0.6 Voltage V2 [V] −0.6 0.4 −0.8 0.2 −1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 −5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 −5 x 10 Near-end Far-end Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 32 of 88
  33. 33. FDPUL MTL: 316 poles, MOR 69 11 x 10 1.5 HTLN MOR 1 0.5 Im 0 −0.5 −1 −1.5 −12 −10 −8 −6 −4 −2 0 Re 8 x 10 0.2 TLT−IFFT TLT−IFFT DFFLN−MOR−Pspice DFFLN−MOR−Pspice 2.5 0.15 2 0.1 0.05 1.5 Voltage [V] Voltage [V] 0 1 −0.05 0.5 −0.1 0 −0.15 −0.5 −0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time [s] Time [s] −8 −8 x 10 x 10 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 33 of 88
  34. 34. Numerical results: non linear terminations 1 2 four-ports R24 network 3 R24=1000M , 4 R20=0.01V203 R40 R20 Rg1 Rg2 + R40=0.01V403 Vg1 Vg2 + 6 5 4 Voltage [V] Voltage 3 source 2 1 0 0 1 2 3 4 5 6 7 -8 x 10 Time [s] Full model: 80 half-T sections (160 poles) Model Order Reduction (MOR) model: 24 poles Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 34 of 88
  35. 35. Numerical results: non linear terminations 3 2.5 DFFLN-MOR-Pspice DFFLN-MOR-Pspice 2.5 2 2 1.5 Voltage [V] Voltage [V] 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 -8 -8 x 10 x 10 Time [s] Time [s] 0.2 0.5 DFFLN-MOR-Pspice DFFLN-MOR-Pspice 0.4 0.15 0.3 0.1 0.2 0.1 Voltage [V] Voltage [V] 0.05 0 0 -0.1 -0.2 -0.05 -0.3 -0.1 -0.4 -0.5 -0.15 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 -8 -8 x 10 x 10 Time [s] Time [s] Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 35 of 88
  36. 36. Composite left/right handed MTLs ∂ 1 −1 V (z, s) = − R + sLR + CL I (z, s) ∂z s ∂ 1 −1 I (z, s) = − G + sCR + LL V (z, s) ∂z s • All the transmission lines model can be used. • Since the CRLH TL is realized by cascading unit cells, the closed form half-T ladder network provides exact results. Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 36 of 88
  37. 37. Composite left/right handed MTLs Elementary half-T cell R LR CL CR G LL n R=R LR = LR CL = CL n n n G=G CR = CR LL = LL n n Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 37 of 88
  38. 38. Composite left/right handed MTLs 1 Z1 (s) = R + sLR + = Z (s) sCL n n 1 = R + sLR + sCL 1 Y2 (s) = G + sCR + = Y (s) sLL n n 1 = G + sCR + sLL Computation of poles s2 LR CL + sCL R + 1 = 0 Z1 (s) Y2 (s) − vj,n−1 = 0, for j = 1 · · · n − 1 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 38 of 88
  39. 39. Numerical results: unbalanced CRLH-TL R = 10−3 Ω, LR = 2.45 nH, CL = 0.68 pF, G = 10−3 S, CR = 0.5 pF, LL = 3.38 nH, = 6.1 mm 5 4 10 HTLN HTLN MOR MOR 3 0 10 2 1 −5 10 Phase(Y12) [rad] |Y12| [S] 0 −10 10 −1 −2 −15 10 −3 −20 −4 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency [Hz] Frequency [Hz] 10 10 x 10 x 10 Magnitude and phase spectra of admittance Y12 . Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 39 of 88
  40. 40. Numerical results: unbalanced CRLH-TL R = 10−3 Ω, LR = 2.45 nH, CL = 0.68 pF, G = 10−3 S, CR = 0.5 pF, LL = 3.38 nH, = 6.1 mm 8 12 x 10 x 10 1.5 4 Reference HTLN MOR−GE−SH MOR 3 1 2 0.5 1 Voltage [V] Im 0 0 −1 −0.5 −2 −1 −3 −1.5 −4 0 0.5 1 1.5 −12 −10 −8 −6 −4 −2 0 Time [s] −7 Re 8 x 10 x 10 Location of poles in the complex plane and transient output. Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 40 of 88
  41. 41. Conclusions Advantages of the proposed method G the analytical knowledge of poles and residues of the half-T ladder network allows to obtain a rational representation of MTL Y (s) matrix; G the dominant poles can be extracted thus leading to a reduced order model; G the reduced model is easily synthesized into an equivalent circuit which is suitable for Spice simulations with linear and non-linear terminations; G it can be used to generate macro-models of MTLs to be interfaced with other tools. Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 41 of 88
  42. 42. Outline 2nd part: the spectral model G Introduction Existing techniques to multiconductor transmission lines (MTLs) time domain modeling G Green’s function based MTLs model Sturm-Liouville problem Spectral form of the Green’s function Eigenvalue problem Z matrix computation G Numerical results MTL with frequency independent per-unit-length parameters MTL with frequency dedependent per-unit-length parameters G Conclusions Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 42 of 88
  43. 43. Telegrapher’s equations: a brief review General solution of Telegrapher’s equations:     V ( , s) V (0, s)   = eΨ(s)   I( , s) I(0, s) where Ψ(s) = (α(s) + sβ(s))     0 −R (s) 0 −L (s)  , β(s) =   α(s) = −G (s) 0 −C (s) 0 Closed-form Pad´ rational function is commonly used to approxi- e mate eΨ(s) . Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 43 of 88
  44. 44. Telegrapher’s equations: the Green’s function method Port currents are treated as current sources d V (x, s) = − R (s) + sL (s) I (x, s) = −Z (s)I (x, s) dx d I (x, s) = − G (s) + sC (s) V (x, s) + I s (x, s) dx = −Y (s)V (x, s) + I s (x, s) I s (x, s) = I 0 (s)δ(x) + I (s)δ(x − ) The 2nd differential problem becomes: d2 V (x, s) − γ 2 (s)V (x, s) = −Z (s)I s (x, s) , (γ 2 (s) = Z (s)Y (s)) dx2 with homogeneous boundary conditions: d d I(x, s) |x=0 = I(x, s) |x= = 0 =⇒ V (x, s) |x=0 = V (x, s) |x= = 0 dx dx Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 44 of 88
  45. 45. Telegrapher’s equations: the Green’s function method Telegrapher’s equations as a vector Sturm-Lioville problem [L + λr(x)] y(x, s) = f (x, s) with boundary conditions (Dirichlet or Neumann or mixed type) d d α1 y(x, s) + α2 y(x, s) |x=0 = 0; β 1 y(x, s) + β 2 y(x, s) |x= = 0 dx dx Multiconductor transmission lines: • L = U d2 /dx2 , • λ = −γ 2 (s) = −Z (s)Y (s) • r(x) = U , (U unitary matrix) • f (x, s) = −Z (s)I s (x, s) • y(x, s) = V (x, s) Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 45 of 88
  46. 46. Telegrapher’s equations: the Green’s function method Dyadic Green’s function for a vector Sturm-Liouville problem y(x, s) = G(x, x , s)f (x , s)dx 0 N N N G x, x , s = Gj (x, x , s)uj = Gij (x, x , s)ui uj j=1 j=1 i=1 where Gj (x, x , s), j = 1, · · · , N must satisfy [L + λr(x)] Gj (x, x , s) = δ(x, x )uj j = 1, · · · , N + homogeneous boundary conditions d α1 Gj (x, x , s) + α2 Gj (x, x , s) |x=0 = 0 dx d α1 Gj (x, x , s) + α2 Gj (x, x , s) |x= = 0 dx Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 46 of 88
  47. 47. Telegrapher’s equations: the Green’s function method Self-adjoint problem =⇒ spectral representation of the Green’s function ∞ Gj (x, x , s) = anj (x , s)φn (x) n=0 where [L + λn r(x)] φn (x) = 0 with the same boundary conditions for Gj (x, x , s). Uniform MTLs: L = U d2 /dx2 and r(x) = U =⇒ scalar eigenvalue problem [L + λn ] φn (x) = 0 + homogeneous boundary conditions Eigenfunctions φn (x), n = 1, · · · , ∞ satisfy the orthogonality condition φH (x)r(x)φn (x)dx = φm (x)r(x)φn (x)dx = δmn U m 0 0 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 47 of 88
  48. 48. Telegrapher’s equations: the Green’s function method 1. Enforce Gj (x, x , s) to be the solution of the equation [L + λr(x)] Gj (x, x , s) = δ(x, x )uj + b.c. 2. Use the orthonormality condition 3. Obtain the vector of amplitude coefficients amj (x , s) amj x , s = (λ − λm )−1 φm (x )uj j = 1, · · · , N 4. Obtain the matrix of amplitude coefficients am (x , s) am (x , s) = (λ − λm )−1 φm (x ) 5. Obtain the dyadic Green’s function G (x, x , s) as ∞ ∞ (λ − λn )−1 φn (x )φn (x) G x, x , s = an (x , s)φn (x) = n=0 n=0 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 48 of 88
  49. 49. Computation of the eigenfunctions and eigenvalues d2 2 + kn φn (x) = 0 dx2 2 where kn = λn + homogenous boundary conditions of the Neumann type d d φn (x) |x=0 = φn (x) |x= = 0 dx dx The solution is φn (x) = An cos (kn x) with • kn = nπ/ n = 0, 1, 2, · · · 1 • A0 = 2 • An = , n = 1, · · · , ∞ Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 49 of 88
  50. 50. Series form of the dyadic Green’s function and multiport representation of the MTL ∞ −1 nπ nπ nπ 2 2 A2 cos G(x, x , s) = − γ (s) + U · x cos x n l n=0 V 0 (s) = G(0, x , s) −Z (s)I s (x , s) dx 0 = G(0, 0, s) −Z (s)I 0 (s) + G(0, , s) −Z (s)I (s) ∞ −1 nπ 2 γ 2 (s) + · A2 Z (s)I 0 (s) + A2 cos (nπ) Z (s)I (s) = U n n l n=0 V (s) = G( , x , s) −Z (s)I s (x , s) dx 0 = G( , 0, s) −Z (s)I 0 (s) + G( , , s) −Z (s)I (s) ∞ −1 nπ 2 2 · A2 Z (s)I 0 (s) cos (nπ) + A2 Z (s)I (s) = γ (s) + U n n l n=0 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 50 of 88
  51. 51. Series form of the dyadic Green’s function and multiport representation of the MTL      V 0 (s) Z 11 (s) Z 12 (s) I 0 (s)  = ·  V (s) Z 21 (s) Z 22 (s) I (s) ∞ −1 nπ 2 2 · A2 Z (s) Z 11 (s) = Z 22 (s) = γ (s) + U n l n=0 ∞ −1 nπ 2 2 · A2 Z (s) cos (nπ) Z 12 (s) = Z 21 (s) = γ (s) + U n l n=0 A rational model can be developed provided a rational representa- tion of γ(s) and Z (s). Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 51 of 88
  52. 52. Poles and residues computation FIPUL: MTLs with frequency independent per-unit-length parameters Z (s) = R0 + sL0 Y (s) = G0 + sC 0 Z (s) and Y (s) polynomial matrices FDPUL: MTLs with frequency dependent per-unit-length parameters PZ B p (s) RZ Z (s) = R0 + sL0 + = s − pq,Z Ap (s) q=1 PY D p (s) RY Y (s) = G0 + sC 0 + = s − pq,Y Cp (s) q=1 Z (s) and Y (s) rational matrices Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 52 of 88
  53. 53. Poles and residues computation • The poles of the transmission line can be evaluated as the zeros of the common polynomial at the denominator of impedances nπ 2 2 P n (s) = det γ (s) + U =0 • Additional poles may be generated by the per-unit-length longitudinal impedance Z (s) in the case of FDPUL-MTLs. • FIPUL-MTLs is a special case of FDPUL-MTLs. • For FDPUL-MTLs, poles can be evaluated as the solution of the following equations nπ 2 Qn (s) = det B p (s)D p (s) + Ap (s)Cp (s) U =0 A(s) = 0 Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 53 of 88
  54. 54. Poles and residues computation Modes-poles Each mode n generates several poles, depending on the order of rational approximation of Z (s) and Y (s) and the number of conductors. npoles,n = order [conv(B p , D p )] N = (PZ + PY + 2) N ˜ Global number of poles nmodes npoles = npoles,n = order [conv(B p , D p )] (nmodes + 1) N ˜ n=0 = [PZ + PY + 2] (nmodes + 1) N Residues matrix of pole pn (k)   (−1)n adj (E(s)) A2 B p (s)Cp (s) |s=pn (k) 1 n ·  Rk = npoles (n) (−1)n 1 Qn1 [pn (k) − pn (l)] l=1 l=k Munich, September 28, 2007 EMC Zurich Munich 2007 Slide 54 of 88

×