Structure functions: ultimate results of evaluation

The structure function

The structure function shown in Figure 4 is a thermal resistance / capacitance map along the heat flow path in an IC package. The x axis is the thermal resistance, left-hand side: chip, right-hand side: ambience. The y axis is proportional to the squared cross-sectional area of the heat flow path.

Irregularities of the heat removing (as faulty die attach etc) can be easily determined and localized with the help of this diagram. Thermal quality of the packages can be determined easy.

Figure 4: The differential "structure function" of an IC package

In Figure 5 one can see the so called cumulative (or integral) structure function, where the x-axis is the cumulative thermal resistance - starting from the chip, the y-axis is the cumulative heat capacitance.

The structure function is theoretically derived from the one-dimensional heat-flow equation. (One-dimensional here means essentially one dimensional, including cylindrical and spherical propagation, too.) In practice, it can be easily constructed from the Cauer-type model network shown in Figure 2. The thermal resistance between the n-th element of the model network and the heat source is

(a)

and the cumulative thermal capacitance is

(b)

where R_i and C_i denote the element values of the i-th stage of the Cauer-type model network. It can be proved, that the derivative of K_n, the differential structure function

(c)

is proportional to the square of the cross-sectional area of the conducting path.

The k(r) differential structure function has some interesting features. For example, its integral is characteristic for the heat capacitance of a given section of the heat conduction path. More precisely, the total heat capacitance between two points of the structure characterized by parameters r₁ and r₂ is

	(d)

Figure 5: Cumulative structure function of the heat-flow path. The left-hand side corresponds to the chip, the right-hand side to the ambience.

The physical distance between points of the structure characterized by parameters r1 and r₂ is

(e)

where l and c denote the thermal conductivity and the specific heat per unit volume of the material. (For further features and for the analytical form of the structure function in special cases refer to e.g. [8].) Values of k for different materials are given in Table 2

As it can be seen in Figure 5, the structure function tends to infinity, corresponding to the fact, that the "universe" as a general thermal environment has an infinite thermal capacitance. The distance between the origin and the location of this singularity of the structure function is exactly the thermal resistance between the heat source (in our case the test chip) and the ambience (which is 45 K/W in Figure 5).

The features characterized by relations (d) and (e) suggest, that the structure function is a good tool to identify geometrical dimensions of the heat conduction path if the material is known. The structure function can also be used to guess the type of the material of the heat conduction path in the knowledge of its geometrical dimensions, but in any case, the total chip-to-ambience thermal resistance of the structure can be identified with its help, see our simple case studies.