A. Thermal Equivalents

Solving the one-dimensional static lattice heat flow equation

one can derive a thermal equivalent model for a device with a geometry as shown in Fig. A.1a. With the boundary conditions

T_{L}(x) = ^{ . }1 -
- ^{ . }
+ |
(A3) |

with

T |
= T_{1} - T_{2}, |
= | (A4) | |

P_{E} |
= H^{ . }A^{ . }2^{ . }a, |
G_{th} |
= . | (A5) |

It consists of a linear term arising from the boundary condition and a
quadratic term arising from heat generation inside the device due to the
dissipated electrical power *P*_{E}. The temperature distribution is shown in
Fig. A.1b.

To derive a thermal equivalent circuit the discretized lattice heat flow
equation as solved by *MINIMOS-NT* can be used. Assuming constant electrical power
dissipation *P*_{E} and a constant thermal heat capacity *c*_{L} the
expression for a grid point *i* reads [36]

P_{i, j} |
= |
+ C_{th}^{ . } |
(A6) |

P_{i, j} |
= | G_{th}^{ . }T_{i} - T_{j} |
(A7) |

C_{th} |
= | V_{i}^{ . }^{ . }c_{L} . |
(A8) |

The sum in (A.6) considers the contribution of all neighbor points

On the other hand, for the electrical active region the thermal resistances and capacitances are of minor importance due to their small size. This is the reason why power dissipation in electrical devices is normally modeled by a power source alone which can be considered as reducing the power source to a point source.

It is worthwhile to point out the simplifying assumptions made in the derivations above:

- A constant thermal conductivity
has been assumed which is a very
crude approximation. Using the exponential expression from [56]
= 1.5486 ^{ . }(A9)

gives and error in the solution of the linear heat diffusion equation in the order of 30 % [8]. - Heat generation has been assumed constant inside the device. This gives
a good approximation only when the temperature rise due to
*H*is small compared to the temperature rise induced by the boundary condition.

1999-05-31