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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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