1. Historical brochure on DWM simulation of Multiconductor Transmission Lines (1990)
Piero Belforte copyright 1990-2012
DWN MODELS FOR
MULTICONDUCTOR TRANSMISSION
LINES
This application note refers to
SPRINT&SIGHTS utilization for
dealing with multiconductor
transmission lines (MTL). Different
methods are available from
literature for both homogeneous
and nonhomogeneous dielectrics, as
well as symmetrical and
asymmetrical structures (Fig. 1).
Simple situations are simulated in
order to compare SPRINT with
SPICE.
Thanks to the availability of
bimodal and multimodal adaptors,
SPRINT implementation of modal
method produces simple netlists.
The applicability of these methods
are summarized in Fig. 2.
GENERAL MODAL
METHOD
The general modal method
applies for nonhomogeneous and
asymmetrical MTL structures. It is
based on the theory that each
physical voltage 'v' and current 'i' at
the MTL points can be represented
by means of a linear combination of
modal voltages and currents, each
of them associated to the related
propagation mode generated within
the MTL structure considered (Fig.
3).
All the parameters needed for the
analytical solution of the MTL
problem can be extracted from [L]
and [C] matrices obtained by means
of analysis of the structure
crossection. This analysis can be
performed analytically only for a
modal method
(general)
Marx method
(homogeneous)
two-line bimodal adaptors
(balanced and symmetrical
structure)
tridiagonal
(symmetrical: coupling
between adjacent lines only)
Chang transformers multimodal
adaptor
SPRINT
Fig. 2: Different methods for modeling MTL.
a) symmetrical and not
homogeneous structure
b) symmetrical and
homogeneous structure
c) not symmetrical and
homogeneous structure
Fig. 1: Different MTL structures.

2. | 2
limitated set of structures, while
general configurations can be
afforded with the help of 2D
Electromagnetic Field Solver only.
In particular, the modal parameters
that define a system of n coupled
lines are represented by the
eigenvalues of [C] matrix, while the
[X] block is a matrix of
eigenvectors that implements the
linear transformation.
The physical meaning of [X] is to
specify what is the contribute of
each propagation mode to define
physical voltages and currents.
From a simulation point of view, it
is necessary to define electrical
blocks implementing the analytical
relationship expressed by [X].
Chang defined a method based on
ideal transformers (Fig. 4) to
implement the translation, where
the ratios of transformers refer to
elements of [X].[1]
Because the number of transformers
to define in the model grows
quadratic with the number of lines
(2*n2 transformers are needed for a
n lines configuration) this method
generates a lot of elements with the
increase of n. This can be a problem
for conventional SPICE-derived
simulators. SPRINT can easily
simulate large nets with
transformers, but offers the user
also the possibility to describe the
block [X] by means of a dedicated
.MODEL card with a more efficient
and compact description of the
eigenvectors.
Another semplification is available
for simple symmetrical two-lines
situations. Infact for this case, the
eigenvectors are very simple and
independent from the [L] and [C]
matrices. As result, a dedicated
SPRINT primitive (the bimodal
adaptor) allows a compact
description of [X] block (for two-
line, symmetrical structures) within
one single instruction, without
specifing any .MODEL card. This
situation is very common in
practice, especially to model
balanced situations.
TRIDIAGONAL METHOD
This method is a semplification of
the general modal method and
applies for symmetrical MTL
structures showing weak coupling,
assuming that coupling is
significant between adjacent lines
only.
For example, we can assume that in
a MTL system composed by n lines
equally spaced and having same
crossection and distance from
reference conductor the i-line is
coupled only with the closest i-1
and i+1 lines.
As result, the [L] and [C] matrices
are tridiagonal and symmetrical.
This means also that the
characteristic parameters
(eigenvalues and eigenvectors) can
be calculated very easily by means
of sine and cosine functions.[2]
Physical voltage
and current
Modal voltage
and current
K11
K21
K31
K12
K22
K32
K13
K23
K33
Fig. 4: Chang modal translator.
[x] [x]
Physical Modal Physical
Modal translators
Physical
current and
voltage
i
v
Fig. 3: MTL model using modal method.

3. | 3
Because the particular
characteristics of the [L] and [C]
matrices, this method is the only
one that allows to model losses on
the physical conductors as
combination of losses on the modal
lines.
MARX METHOD
In case of MTL embedded in a
homogeneous dielectric, like the
coupled striplines of the inner layer
of multilayer PCB, the Marx
method [3] can be implemented
very efficiently by SPRINT. It can
be used also for nonhomogeneous
structures like microstrips of the
outer layers of PCBs but, in this
case, the far end crosstalk is
neglected.
With homogeneous dielectric
assumption, the relation between
matrices [L] and [C] is very simple
and expressed in terms of the
square of the propagation velocity
in the material (v2). Thanks to this
semplification, it is possible to
model the structure composed by n
coupled lines (including ground) by
means of m = n * (n - 1) / 2
uncoupled lines whose parameters
can be easily calculated analytically
(Fig. 5).
SPRINT-SPICE3
COMPARISON
A very simple structure composed
by three-conductors shown in Fig. 6
is simulated in order to compare
SPRINT with SPICE using both
Marx and modal decomposition
method [3].
As can be pointed out looking at the
correponding SPRINT netlists (Fig.
8), while the two-wire line method
produces a netlist very similar to
the SPICE counterpart, the modal
decomposition netlist is simpler,
thanks to the availability of bimodal
adaptors (AM) among SPRINT
primitives.
Both cases lead to extremely simple
nets with respect SPRINT
capabilities, so that running times
on a typical workstation are
fractions of second.
Fig. 7a shows the simulation
results: SPICE3 outputs match
SPRINT responses within machine
precision errors, with simulation
times one order of magnitude
longer.
Using SPRINT, modal
decomposition runs faster than two-
wire equivalent, so that modal
decomposition is more convenient
at least in the case of three
conductor (two-mode) symmetrical
lines.
The observed small differences
between outputs coming from the
two methods (Fig.7b) are due to the
truncation errors of the numerical
values of line impedances as they
are assigned in the netlist file. Due
to SPRINT performance, Marx
2 cm
2 cm
20 Gauge Wire
(rw = 0.41 mm)


GROUND PLANE
a)
Rne=50 ohm
Rs=50 ohm
VS(t)
0
1
2
0
1
2
Rfe=50 ohm
Rl=50 ohm
MTL
z = 0 z = 4.67 m
b)
1
12.5 20.0 32.5 t (ns)
VS (volts)
c)
1 2
Fig. 6: Three-conductor line and electrical configuration used to compare Marx and modal
methods. (a) Crossection of MTL. (b) Electrical scheme. (c) Input voltage waveform.
Fig. 5: MTL model using two-wire lines (Marx method).

4. | 4
method can be conveniently applied
to deal with n-conductor lines with
n>3 despite the quadratic growth of
the number of lines versus n,
without incurring in SPICE
simulation time and memory
limitations mentioned in ref. [3].
LOSSES
Losses effects in MTL structures
are very difficult to simulate due to
the frequency dependance of
resistance and conductance
matrices ([R] and [G] respectively).
Infact, the procedure adopted for
modal decomposition leads
frequency dependent to physical-
modal transformer blocks [X] very
difficult to implement by means of
circuital equivalents.
Only for tridiagonal method it is
possible to obtain transformer
blocks [X] with real, frequency-
independent parameters and use it
to transform the [R] matrix in
physical domain to [R'] matrix in
modal domain. In this way it is
possible to simulate physical losses
by means of losses applied to the
modal lines without errors.
----------------
[1] F.Y.Chang, "Transient Analysis
of Lossless Coupled Transmission
Lines in a Nonhomogeneous
Dielectric Medium" IEEE
Trans.Microwave Theory Tech.,
vol. MTT-18, sep. 1970, pp.616-
626.
[2] F.ROMEO and M.Santomauro
"Time-Domain Simulation of n
Coupled Transmission Lines" IEEE
Trans. Microwave Theory Tech.,
vol. MTT-35, feb. 1987, pp. 131-
137.
[3] K.D.Marx and R.I.Eastin, "A
Configuration-oriented Spice
Model for Multiconductor
Transmission Lines with
Homogeneous Dielectrics" IEEE
0.00 40.00 80.00 120.00 160.00 200.00
TIME[nS]
-12.5
-8.75mV
-5.00mV
-1.25mV
2.50mV
6.25mV
10.00mV
13.75mV
17.50mV
21.25mV
25.00mV
V(17)
mV
CROSSTALK ON NODE 17
Fig. 7a:Simulation result using two-wires lines (Marx method)
********************************************************************
* Ref. "A Configuration-oriented SPICE Model for MTL with Homogeneous
* Dielectrics"
* IEEE Trans Microwave Theory and Techniques vol.38 n.8 Aug. 1990 p.1123-1129
********************************************************************
* MTL MODEL USING TWO-WIRE DELAY LINES
*
VS 1 0 PULSE( 0 1 0 12.5NS 12.5NS 7.5NS 1000 )
RS 1 2 50
RL 9 0 50
RFE 10 0 50
RNE 17 0 50
*
T01 2 0 9 0 Z0=323.6 TD=15.58NS
T12 2 17 9 10 Z0=1522 TD=15.58NS
T02 17 0 10 0 Z0=323.6 TD=15.58NS
*
.TRAN TSTEP=.5NS TSTOP=200NS V(1) V(2) V(9) V(17) V(10)
.END
**********************************************
* MTL MODEL USING MODAL DECOMPOSITION
*
VS 1 0 PULSE( 0 1 0 12.5NS 12.5NS 7.5NS 1000 )
RS 1 2 50
RL 9 0 50
RFE 10 0 50
RNE 17 0 50
*
AM1 17 2 5 14
TC 5 0 6 0 Z0=323.60 TD=15.58NS
TD 14 0 16 0 Z0=227.07 TD=15.58NS
AM2 10 9 6 16
*
.TRAN TSTEP=.5NS TSTOP=200NS V(1) V(2) V(9) V(17) V(10)
.END
Fig.8: MTL models using Marx method and modal method.
0.00 40.00 80.00 120.00 160.00 200.00
TIME[nS]
0.0u#
816.7u#
#3$
-6.16uV
2.41uV
V2$
RELATIVE ERROR
ABSOLUTE ERROR
Fig. 7b: Relative and absolute errors of modal decomposition vs Marx
method

5. | 5
Trans. Microwave Theory Tech.,
vol. MTT-38, aug. 1990, pp. 1123-
1129.