There are two kinds of structure functions: cumulative structure function and differential structure function. Hereinafter we describe how these functions are derived and give some hints how to use them.
Basic features of the structure functions
The structure function shown in Figure 3‑14 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 3-14: The "structure function" of an IC package
In Figure 3‑15 one can see the so called cumulative structure function where the x-axis is the cumulative thermal resistance - starting from the chip while the y-axis is the cumulative heat capacitance.
Figure 3-15: Cumulative structure function of the heat-flow path. The left-hand side corresponds to the chip, the right-hand side to the ambience.
Figure 3-16: Cauer-network for driving-point models
The structure function is theoretically derived from the one-dimensional heat-flow equation. (In this case one-dimensional 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 3‑16. The thermal resistance between the n-th element of the model network and the heat source is
and the cumulative thermal capacitance is
where Ki and Ri denote the element values of the i-th stage of the Cauer-type model network. It can be proved that the derivative of Kn, the differential structure function
is proportional to the square of the cross-sectional area of the conducting path.
The 
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 
and 
is
.
The physical distance between points of the structure characterized by parameters and is
,
where 
and 
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 see e.g. reference [3].) Values of k for different materials are given in Table 3‑1.
As it can be seen in Figure 3‑15, 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 3‑15).
Table 3-1: k values for different materials; A denotes the cross-sectional area.
The features characterized by the above two relations show 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 if its geometrical dimensions are known; but in any case, the total chip-to-ambience thermal resistance of the structure can be identified with its help.
Simple case studies for using structure functions
The first test chip encapsulated into a ceramic DIL package was used in various case studies. In one of our experiments we investigated the effect of an IC socket.
We had two setups: the only difference between them was that the test chip was either soldered directly into the test board or was plugged into an IC socket soldered into a similar PCB having the same geometrical and thermal properties. The final results obtained from thermal transient measurement are shown in Figure 3‑17. The increased thermal resistance towards the ambience introduced by the socket is clearly visible on the plots. This value is about
= 25 K/W (see the difference of the singularities of the curves).
Figure 3-17: Effect of a chip socket, demonstrated by the differential structure function.
In the second experiment the test chip was plugged into a socket and in one of the measurements a cooling mount was attached to the top of the IC package. The obtained structure functions are plotted in the same diagram in Figure 3‑18. This clearly shows that the cooling mount introduces an excess thermal capacitance (the corresponding “bump” in the structure function is marked by an arrow) as well as it reduces the thermal resistance by a great extent, compensating for the excess thermal resistance introduced by the IC socket.
A third experiment was performed in order to demonstrate the sensitivity of our tester. The same setup was used: the test chip plugged into an IC socket that was soldered into a printed circuit board. The difference between the two measurements shown in Figure 3‑19 was, that in one case the test IC was fully plugged into the socket, while during the other measurement it was only slightly plugged in: to realize the electrical connections only. In this second case the heat removing path was extended by the extra pin-length that remained outside the socket. This excess heat resistance can be identified as the difference between the singularities of the two curves, which is about = 5 K/W, corresponding to 1.5 mm excess pin length of the IC.
Figure 3-18: Effect of a cooling mount, demonstrated by the differential structure function.
Figure 3-19: Effect of the pin length.
