Modeling, Simulation, and Measurement - Computer Architecture Techniques for Power-Efficiency - page 15

In this equation V ds is the voltage across the drain and the source and V gs , the voltage
across the gate and the source terminal. V off is an empirically determined model parameter
and

t is a physical parameter proportional to temperature—the exponential dependence to
temperature is immediately obvious. The term n encapsulates various device parameters. The
term I s0 depends on transistor geometry and can be written as I s0

v

W / L . The Butts and Sohi
model examines and simplifies the above equation for a single device in its normal “off ” state,
where V ds

×

= V CC and V gs = 0. This makes the factor

1

t

e − V ds −

v

approximately 1 since V ds = V dd V T . By grouping more terms together, Butts and Sohi

simplify the formula to

W
L

10 − V T

I Dsub = × k Tech × .

S t

The simplified formula exposes only the relationship of leakage to transistor geometry, to
threshold voltage ( V T ), and indirectly to temperature via the s t term. Accounting for the many
( N ) similarly sized transistors with similar characteristics in larger structures further simplifies
the formulas by encapsulating the various ( W

L ) terms of each transistor into a new parameter
k design . The formula for the static power consumption for the N transistors can then be expressed
as:

/

10 − V T P leakage = V dd × N × k design × k Tech × .

S t

The parameters k design and k Tech can be looked up from tables such as Table 2.2 for k design .The
number of devices, and the supply and threshold voltages are the only concerns of the architect
for estimating the leakage power of a design.

2.2.3 Thermal models
A remaining important piece of the modeling landscape is the ability to model thermal behavior.
A cyclic relationship exists between power and thermal modeling. On one hand, thermal
behavior depends on power dissipation and density, since temperature is essentially a function
of how much power is dissipated in a region versus how that region is cooled. On the other
hand, power also depends on temperature. This is particularly true due to the exponential
dependence of leakage energy on temperature as indicated in the equations in Section 2.2.2.
In addition to this cyclic dependence of power and temperature, another wrinkle in
thermal modeling concerns the timescale of interest for the model. For example, in terms of
provisioning the cooling capacity of a chip, a system, or a data center, long-term “steady-state”
temperature may be the metric of interest. In terms of either designing a microprocessor or