Civil Engineering Reference
In-Depth Information
follows:
where the latter expression corresponds to a partial fraction expansion. In the
time domain‚ this corresponds to the following sum of exponentials
The transient here is dominated by the time constant of the first exponential
term‚ which corresponds to the first few exponential terms‚ and the pole at
is referred to as a
dominant pole
2
.
While such a distinction is qualitative
rather than quantitative‚ if one were to approximate the exact system by a
lower-order system that captures the dominant pole(s) of the original system‚ a
very good approximation to the waveform could be obtained. Figure 3.4 show
the results of applying the AWE technique (to be discussed in Section 3.5) to the
above system and it is apparent that even a first or second order approximation
is adequate to capture the most important characteristics of this waveform.
This chapter will discuss techniques for the reduced order modeling of lin-
ear systems‚ starting with the asymptotic waveform evaluation method‚ and
then progressing into Krylov subspace-based methods. For a more in-depth
description‚ the reader is referred to [CN94‚ CPO02].
3.2
INTERCONNECT MODELING
Interconnect wires may be modeled at various levels of abstraction‚ ranging
from relatively simple models to those that arise from a full 3-dimensional
extraction. We will outline a set of low-frequency models from the former cat-
egory here‚ which will consider resistance and capacitance‚ but not inductance‚
and for inductance extraction approaches‚ the reader is referred to‚ for exam-
ple‚ [BP01‚ ]. The analysis methods described later in this chapter‚
however‚ are valid for RLC systems.
Given a wire of length
and width
its resistance
R
may be modeled as
