Impedance characterization of Interconnects and intentional Inductors in
the broad frequency domain that extends from near DC to 100’s of GHz in
integrated circuits is full of unachieved goals. Existing computational methods are
near the end of their usefulness, since accurate characterization of the Impedance
matrix Z(ω) at high-frequencies with existing methods can only be applied to
small structures. We present a computationally inexpensive approach that extends
the ability for accurate characterization to problem sizes that are between one and
two orders of magnitude larger, opening the door to the validation of high
frequency wireless circuits in terms of real time simulation, rather than the less
desirable alter-native of validation by manufacturing and testing. The starting
point in our approach is an integral representation of the Green’s function for the
magnetic vector potential in classical Electromagnetic theory. The intermediate
computation involves a least-square fit to refection coefficients in terms of linear
combinations of complex exponentials, so as to render integrable the coordinate
space representation of the Green’s function. The end result is an analytical
description of derivative quantities, including the matrix elements of the serial
Impedance matrix of the interconnect configuration, for all frequencies of interest.
We study the problem in two and three dimensions. Among the alternative least
square fits, we found that those utilizing VARPRO in the complex domain – an
extension of VARPRO created specifically to attack this problem – give the best
results. The levels of accuracy (errors less than 3%) and efficiency (better than an
order of magnitude lower computational cost than existing methods) have a major
impact on nano-electronic circuit design.
Keywords: High-frequency impedance matrix; VLSI interconnects; VLSI inductors;
multi-layer substrate; high-frequency wireless circuits; real time simulation;
complex exponentials
