Civil Engineering Reference
In-Depth Information
The Elmore metric can be understood in terms of a simple and intuitively
appealing description. Consider‚ for example‚ an RC mesh that is initially
rested‚ i.e.‚ the initial voltage at any node is zero. If a unit step excitation
is applied to this structure‚ then the voltage at each node will monotonically
increase from 0 to 1. Figure 3.9(a) illustrates such a waveform‚ at some
non-source node of this RC network. We will refer to the time derivative of
this waveform as
In [Elm48]‚ Elmore made the observation that the time coordinate of the
center of area of the region under the curve (illustrated by the shaded
region in Figure 3.9(b)) would serve as a reasonable estimate of the delay. In
other words‚ an approximation to the 50% delay‚
may be expressed as
3
From Figure 3.9(a)‚ such a metric can be seen to have great intuitive appeal.
From elementary linear system theory‚ we observe that since is the step
response of the rested system‚ must be the response to an excitation that is
the derivative of the step response‚ namely‚ the unit impulse. Hence‚ Equation
(3.7) also has the interpretation of being the
first time moment of the impulse
response.
An interesting observation is that the quantity is also the area above
the step response‚ as shown in Figure 3.9(c). This can be seen by performing
integration by parts‚ as shown below:
Here‚ we make use of the fact that
since
as
and the latter term decreases exponentially with
while the former
term only increases linearly.
3.4.1 An expression for the Elmore delay through RC networks
Although the above expressions provide formal definitions of the Elmore delay
metric‚ they are rather cumbersome and difficult to use. In this section‚ we will
derive a more usable form for the Elmore delays in a general RC tree or mesh
structure (with no floating capacitors)‚ and derive a closed-form expression for
the Elmore delays in RC trees and RC meshes.
