Accurate broadband modelling of multiconductor line RLGC

Accurate broadband modelling of
multiconductor line RLGC-parameters in
the presence of good conductors
and semiconducting substrates
Daniël De Zutter (Fellow IEEE)
Ghent University
Dept. of Information Technology
Electromagnetics Group
1
Overview
Accurate broadband modelling of multiconductor line RLGC-parameters
in the presence of good conductors and semiconducting substrates

Introduction

RLGC-parameters modelling approach

Examples

Further research and challenges

Questions & discussion
2
Introduction
How to model signal integrity?
 full 3D numerical tools
 direct access to multiport S-parameters
PlayStation 3 motherboard
and time-domain data
 holistic but time-consuming
 (some times too easily) believed to be accurate
 divide and conquer
 multiconductor lines, vias, connectors, packages, chips, ….
 model each of them with a dedicated tool
 derive a circuit model for each part
 obtain the S-parameters and time-domain data from
overall circuit representation
 gives more insight to the designer (optimisation)
 overall accuracy might be difficult to assess
3
Introduction
Multiconductor Transmission Lines
simplify (idealize) to a 2D problem
2D fields, charges, currents
RLGC
PlayStation 3 motherboard
3D fields, charges,
currents
Transmission lines
voltages & currents
4
Multiconductor TML
Telegrapher’s equations (RLGC)
1
2
…..
N
reference
N+1 conductors – one reference conductor
i : Nx1 current vector
v : Nx1 voltage vector
C : NxN capacitance matrix
L : NxN inductance matrix
G : NxN conductance matrix
R : NxN resistance matrix
5
Multiconductor TML
wish list
reference

broadband results (time-domain)

many regions

some semi-conducting

good conductors (e.g. copper)

small details

exact current crowding and
skin effect modelling
on-chip interconnect example:
• 4 differential line pairs
• semi-conducting region
• unusual reference conductor

modelling approach?
6
Model building
vector potential

How to “correctly” define voltages?
scalar potential
Step 1: define the voltages on the good conductors
z
c
 is constant on c = Telegrapher’s v(z)
good cond.
 >> 
v(z)
i.e. we only take the effect of longitudinal currents into account
(quasi – Transverse Magnetic approximation; hz-field is negligible)
7
Model building
Step 2: quasi-TM approximation in lossy dielectrics and semi-conductors
lossy dielectric (  << )
and
tranversal ø
i.e. as we neglect transversal currents, we also neglect possible charges
 sufficiently large w.r.t. 
semi-conductor
behaves as a good conductor
c
vsc
semi-cond.
 smaller than or comparable to 
behaves as a “lossy” dielectric but
8
Model building
How do we keep a link between the (v,i)-description and the modal fields?
1V
1V
0V
differential pair – even mode
-1 V
1V
0V
differential pair – odd mode
Step 3: judicious choice of model to properly link fields and MTL representation
field eigenmodes (Em, Hm)
voltage-current TL eigenmodes (vm, im)
9
Model building
use field reciprocity and carry it over from the fields to voltages and currents
=0

important consequence:
the symmetry of the RLGC-matrices is
automatically guaranteed
10
Model building
What about the currents? Is our choice still free?
Step 4: current interpretation

classical: integrated current density
through cross-section
i
v1,i1

extension to semi-conducting substrates
c
semicond.
diel.
c
v2,i2
Maxwell + quasi-TM approx. + mode orthogonality leads to:
circuit current in is actual current through conductor n
+ suitably weighted sum of currents through semiconductors!
11
Model building
Step 5: determine complex capacitance matrix with a boundary integral equation
+ consider equivalent polarisation charges eq in free space
+ express the potential everywhere in terms of eq
sc
c
diel.
+
c
~
Q=Cv
+ for each piece of material we need a relationship between  and
its normal derivative /n on the boundary of that material
12
Model building
Step 6: determine complex inductance matrix with a boundary integral equation
+ consider equivalent contrast currents
note: these currents spread out over all cross-sections
of conductors and semi-conductors
in free space
sc
c
c
diel.
integral equation for ez
~
yielding L
but current distribution must
be accounted for from DC to
high skin-effect !
13
Model building
Suppose that we can replace current distribution in cross-section
by equivalent one on the boundary and valid from DC to skin effect
sc
c
c
diel.
sc
c
diel.
c
admittance operator
14
Model building

~
determination of C = C + G/j
+ eq: equivalent charges residing
in free (half)-space
+ to be determined by solving an
semi-cond.
v1
integral equation for eq
+ we need a relation between  and /n
on each boundary c

cond.
cond.
c
qc
~
determination of L = L + R/j
+ jc,z: equivalent currents residing
in free (half)-space
+ to be determined by solving an
integral equation for jc,z
+ we need a relation between jc,z and ez
on each boundary c
v2
diel.
semi-cond.
i1
cond.
cond.
diel.
i2
c
jc,z
15
Model building
Admittance operator Y
B
in S:

(f =  or f = ez)
determine relationship between f and
f/n on c
c
S
r
r’
n
A
Y(A,B): how f at A contributes to f/n at B

does this not generate more difficulties than it solves???
can be solved analyticallly for rectangles and triangles (also circles)
and these shapes can be combined to more general ones!
16
Model building

broadband results (time-domain)
+ admittance operator works fine

from DC to beyond 100GHz

many regions and small details
+ all of them replaced by equivalent quantities

on boundaries placed in homogeneous space
+ integral equations are highly accurate in modelling these details


some semi-conducting

good conductors (e.g. copper) and exact current crowding and skin effect
+ admittance operator works fine even in copper

+ only the boundary is needed (no internal currents – no problem
with exponentially decaying internal fields near boundary at high
frequencies)
17
Admittance operator
45
B
26
5 m
copper
A
50
1
20 m
20
B
A
79.1 MHz - skin depth  = 7.43m
10 GHz - skin depth  = 0.66m
18
Examples
Metal Insulator Semiconductor (MIS) line
good
dielectric
good
conductor
19
Examples
Coated submicron conductor
3117 nm
500 nm
238 nm
450 nm
500 nm
450 nm
copper: 1.7 cm
chromium: 12.9 cm
coating thickness : 10 nm
20
Examples
L
R
inductance and resistance p.u.l
as a function of frequency
21
Examples
4 differential pairs on chip interconnect
+ all dimensions
in m
+ sig = 40MS/m
+ sub = 2S/m
+ dop = 0.03MS/m
22
Examples
eight quasi-TM modes
quasi-even
the modal voltages V = V0exp(-j)
are displayed (V0 = bb ) @ 10GHz
quasi-odd
slow wave factor:
mode prop. velocity v = c/SWF
23
Examples
complex capacitance matrix @10GHz
24
Examples
complex inductance matrix @10GHz
25
Examples
Pair of coupled inverted embedded on-chip lines
26
Examples
Discretisation
27
Examples
L and R results
28
Examples
C and G results
29
Examples
Cross-section of a new high-speed connector (courtesy of FCI)
dielectric 1
mm
copper
1
2
dielectric 2
mm
30
Examples
L and R results
L (nH/m)
L11 & L22
L12 = L21
frequency
R (Ohm/m)
R11 & R22
R12 = R21
frequency
31
Examples
C and G results
C (pF/m)
C11
C22
C12 = C21
frequency
G (S/m or Mho/m)
G22
G11
G12G=12G21
frequency
32
Examples
Fibre weave (in progress)
differential
stripline pair
cross-section
33
Examples
x 10
8
-4
Fibre weave (in progress) - discretisation
6
mm/10
4
2
0
-2
-4
0
0.5
mm
1
1.5
x 10
-3
34
Ongoing research
Starting point: fast and accurate determination of RLGC-data

sensitivity and tolerance analysis of multiconductor lines with respect
to geometrical and material data

full stochastic analysis (probability density functions)

time-domain modelling with non-linear (stochastic) loads

combination with macromodelling for RLGC-data to tackle
the problem of many stochastic variables
35
Challenges

to account for surface roughness (several empirical formulas do exist)

to go beyond quasi-TM approximations

a full-wave approach is needed

voltages and currents become difficult to define

many models in literature of past decades:
current/power – voltage/power – current/reciprocity
voltage/reciprocity – causal - ......


computationally very demanding
to drop “divide and conquer”? or find a smart way out?
36
Acknowledgement
Thanks to all PhD students and colleagues of the EM group I have
been working with on these topics over very many years:
Niels Faché (now with Agilent Technologies)
 Jan Van Hese (now with Agilent Technologies)
 F. Olyslager (full-time professor at INTEC, UGent – deceased)
 Thomas Demeester (post-doc at INTEC, UGent)
 Luc Knockaert (part-time professor at INTEC, UGent)
 Tom Dhaene (full-time professor at INTEC, UGent)
 Dries Vande Ginste (full-time professor at INTEC, UGent)

37
Questions
and
Discussion?
38