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

12 2.1. Assumptions for analytical thermal model
2. The image function method (e.g. [35]). In this approach the temperature
solution is first determined by neglecting the boundary conditions at the
lateral and bottom boundaries . This implies that in this approximation
one treats the heat source as if it were in a semi-infinite domain. It has
been shown that it is possible to finish a simple closed-form solution for
rectangular heat sources (see section 2.3.1 and 2.3.2). The second step is to
incorporate the effect of boundary conditions. This is done by introducing
an infinite number of fictitious heat sources. As a result, the temperature
solution is expressed as an infinite sum of terms, each accounting for a
fictitious image source. Note that if the heat source-to-chip area ratio is
small, so the effect of boundary conditions will be negligible in practice and
a small number of terms (or even just one one term) need to be included.
Therefore this method is very efficient for small devices.
In section 2.2, the problem of solution of the heat-flow equation will be described
from the mathematical point of view, introducing possible approximations. The
idea of thermal resistance and thermal impedance will be demonstrated.
2.1.
Assumptions for analytical thermal model
The steady-state heat-flow equation is described as follows:
▽ [k(T ) ▽ T ] + g(x, y, z) = 0 (2.1)
where k(T ) is the thermal conductivity, T is the temperature and g(x, y, z) is the power
density (which depends on a position) per unit volume [W/cm 3 ].
To solve the heat-flow equation (Eq. 2.2) one needs to specify:
➤ How the heat is exchanged with the environment (boundary conditions).
➤ The generation of the heat (power density g function).
Usually this problem is resolved by simplifying both:
Power Density. The active area where the heat is generated, is assumed of a simple
geometry (parallelepiped or rectangle), with a power density g function, that is
(1) constant and uniform in the heat source area (2) zero outside the heat source
area.
Boundary Conditions (B.C.). The boundary conditions are simplified as follows: