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Simulation and modelling underlying radio frequencies
P. Heres
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Introduction
 The continuing trend of higher frequencies and
smaller feature sizes force the designer of electronic circuits to take
into account electromagnetic effects caused by the interconnect
structures. Straightforward coupling of circuit simulators with Maxwell
solvers is prohibitive because of the large computational effort
associated with this. Also, it seems unneccessary to use a detailed
simulation of the electromagnetic e ects for current problems.
Therefore, one often attempts to capture the main electromagnetic
effects into a compact model for the interconnect structure, and
couples this to the circuit simulation programme. In this, numerical
mathematics plays an important role, since most reduced order modelling
techniques are based on Krylov subspace techniques. On the other hand,
it is important to use simplifications which can be motivated using
arguments from the theory of electromagnetics. Combining these two
aspects, the SMURF project attempts to generate reduced order models
for electronic interconnect structures.
System formulation
The interconnect can be modelled as a lumped-RLC-model with n ports and
therefore can be described by a set of Kirchhoff's Current Laws and
branch equations for the current controlled branches:
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(1)
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where x are the internal node voltages and currents,  is a set of voltages at the ports. This leads to a unique
set of currents over the ports  . After Laplace
transforming the system to the frequency domain, a transfer function
can be formulated. This function gives a direct relation between
input ,and output in the frequency domain and is therefore used as a measure
of approximation for a reduction technique. For a multiport system as
given in (1) the transfer function is a matrix valued function, given
by:
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(2)
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Reduced Order Modelling
There are many techniques available in Reduced Order Modelling. The
basis of our research is the SVD-Laguerre reduction technique proposed
by
Knockaert and De Zutter in 1999 [a]. In this method a Krylov-space is
generated,
based on the observation that the transfer function can be expanded
into
Laguerre functions:
The matrices  are used to generate the Krylov subspace.
In the first year of the project we modified this algorithm in such a
way that more accurate models are obtained. This was achieved by
orthogonalizing the columns of the Krylov-space during the process,
rather than at the
end of the iterative process. In addition, an efficient implementation
for multiple input columns was developed. Via the Laguerre expansion we
also found a new way to represent the reduced models in form of an
electronic circuit, thus enabling a direct coupling with circuit
simulations programmes.
Results
The example is a PCB of a lowpass lter, see the rst
gure below. On the board two spirals are printed which represent the
two inductors. The two capacitors are added later, in order to be able
to change the characteristics of the filter. We therefore want a
circuit, with 6 ports, of which 4 ports are needed to connect 2
capacitors to it later. The printed board and the pins are shown in the
second figure.
With the new version of the SVD-Laguerre algorithm this system can
be reduced. The example, derived from a layout simulator, originally
consisted of an RLC model, of which the MNA formulated matrix has size
465 x 465. The system is reduced to the size 84 x 84. After reduction
the passivity of
the system is preserved and we are still able to connect components at
the
ports of the reduced model. In the last picture the transfer function
of
the reduced system is compared with the results of a layout simulator.
Footnotes
[a] Luc Knockaert and Daniel De Zutter, "Passive Reduced Order
Multiport Modeling: The Padé-Arnoldi-SVD Connection", Int.
J. Electronics and Communications, nr. 53, 1999, pp. 254-260.
[b] This work is done by means of a grant of NWO (Dutch Organisation for Scientific
Research, le nr. 635.000.010)
and in corporation with Philips.
[c] Check http://www.win.tue.nl/smurf
for a detailed project description.
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