Development of a New Electro-thermal Simulation Tool for RF circuits

22 2.2. Chip thermal model - mathematical considerations
The time constants value τi are important parameters due to the fact, a thermal system
can be represented as distributed linear RC system, with wide range of time-constant
values [36]. According to [37], the range of possible time-constant values in thermal
systems can be divided into following ranges:
➤ 10µs-100ms: semiconductor chip / die attach.
➤ 10-50ms: package structures beneath the chip.
➤ 50ms - 1s: further structures of the package.
➤ 1 - 10s: package body.
➤ 10-10000s: cooling assemblies.
In the appendix A, the algorithm for automated identification of thermal impedance
using Foster networks has been developed, based on desired amount of RC pairs N,
which are the input parameter for a routine. The routine has been implemented in the
electro-thermal simulator presented in chapter 3.
Cauer network
Figure 2.11: The simplest Cauer RC network.
The Cauer network is attractive from the thermal point of view, since provides
more physical description with respect to the Foster one [13]. The problem with
Cauer network is that it is easily defined in the Laplace domain using partial fraction
expansion, however the time-domain equation is almost impossible to obtain using
inverse Laplace transform. Concluding, the mathematical representation of Cauer
networks is much more complicated than the Foster one.
For the simplest case, presented in Fig. 2.11, the impedance in the Laplace domain can
be described as follows:
Z(s) =
1
sC + 1
R
In case when N resistors and N capacitors are present, the network is described as:
ZN(s) =
sC1 +
1
1
1
R1+
sCN + 1
RN +...
(2.38)
(2.39)