In part 2 of this blog series we looked at the three different modes of heat transfer and related them to equivalent thermal resistances. In this blog, we’ll use the concept of thermal resistors to develop a thermal equivalent network of a system and determine its equivalent junction to ambient thermal resistance.

As an electrical engineer, one of the most basic thermal requirements we must guarantee in our systems is to ensure the maximum junction temperature of our integrated circuits (ICs) is not exceeded. With the concept of thermal resistance, we can estimate the maximum power dissipation our IC can have if we know the equivalent thermal resistance seen from the IC junction to the ambient environment, ,the maximum junction temperature of the IC, and the maximum ambient temperaturethe IC is placed in with the following equation:

However, in many cases as system designers we cannot control the maximum power dissipation of the integrated circuit, but we can control the equivalent junction to ambient thermal resistance by designing the environment that the IC is placed in. If we rearrange the above equation, we can find the maximum junction to ambient thermal resistance for our system as:

Thus, our goal is to make sure our equivalent junction to ambient resistance is less than the maximum junction to ambient resistance to ensure our IC never exceeds the maximum junction temperature with the following equation:

Now, let’s take a look at how we can estimate with a simple example shown in figure 2 consisting of an IC in a package on top of a printed circuit board (PCB). In this example, the IC Die is the source of heat (power) and we will analyze how heat transfers from the IC die through the package and PCB to the ambient environment.

*Figure 2. A simple system consisting of an IC in a package on a PCB.*

Recall from our earlier blogs that conduction heat transfer, unlike electrical current conduction, is not well constrained and flows in all directions from the heat source. Technically speaking, electrical current also flows in all directions but electrical components are designed to constrain current flow with conductors and insulators that have very high isolation (>> 10^{8}) whereas the isolation of heat flow is much less, typically in the 1000s to 10,000s range. Thus, heat from the IC die will conductively flow in all three dimensions through the solid package and PCB and can be modeled with their respective thermal conductive resistances.

When the heat flow reaches the surfaces of the package and PCB, the mode of heat transfer will change from conduction to convection and radiation. Notice that both convection and radiation heat transfer occur at the surface to the environment and thus appear in parallel. In general, this parallel combination will always occur at the surface of an object to an environment in a medium such as air. As electrical engineers, we know that if we parallel two resistors and if one resistor is significantly smaller than the other resistor, then the parallel resistor combination can be approximated by the much smaller resistor. The same concept also applies to thermal resistors where heat always flow through the path of least thermal resistance. In many cases heat convection dominates radiation and thus the heat transfer mechanism from the surface to the environment can be approximated with the smaller thermal convection resistor.

*Figure 3. A simplified 2D model of our system with thermal resistors.*

For simplicity, lets analyze our system in 2D but the techniques can be readily applied to 3D. Figure 3 shows a simplified 2D model of our system with thermal resistors. The number of resistors used to model the system can vary depending on how complicated and accurate you want to model the system. In this example, we model the solid components to allow heat propagation to all the surfaces or sides of the object. The package has four thermal conduction resistors that allows heat to travel from the IC die to the top of the package, , to the sides of the package, and , and to the bottom of the package, . The PCB is modeled with ten conduction resistors to give a more distributed heat transfer effect since the PCB has a much larger area than the package. Heat is transferred from the package to the PCB through to two top sides of the PCB, and , and then internal to the sides of the PCB , and to the bottom of the PCB. As mentioned in the earlier paragraph, all the solid surfaces will then have parallel convection and radiation resistors to model heat transfer from the solid surface to the environment. Again, the PCB surface is modeled with multiple parallel convection and radiation resistors for a distributed effect.

*Table 1. Equivalent resistors for thermal network shown in Figure 4.*

*Figure 4. Equivalent thermal network of our 2D system shown in Figure 3.*

As electrical engineers, we can simplify the 2D model of our system with an equivalent network by grouping resistors as shown in table 1 and figure 4. The advantage of this type of resistive grouping is that each equivalent resistor still maintains a physical interpretation with its 2D model. For example, represents the equivalent resistor from the IC die to the ambient environment through the package top and represents the contact resistance from the package bottom to the PCB top interface. So, if we wanted to include the thermal resistance of solder connecting the package to the PCB, we can add it to . Furthermore, by examining figure 4, we see heat transfers directly from the IC die to the ambient environment through the package top and package side resistors. The other propagation path is through the package bottom and into the PCB via resistor and then eventually to the PCB top, side, and bottom surfaces through the respective resistive paths.

*Figure 5. Thermal network used to find the equivalent junction to ambient thermal resistance.*

The network in figure 4 can be further simplified by observing the bottom resistors and can be added in series and then paralleled with resistor . This pattern continues up the resistor ladder until we can come to the equivalent thermal network as shown in figure 5. Here our desired equivalent junction to ambient thermal resistance of our system is found as:

As we can see from the equation above (and also from the intuition we gained from figure 4), is equal to the parallel resistance seen from the package top( )with the resistance from the package sides (), and with the equivalent resistance from the package bottom through the PCB to the ambient environment .

Using the information we learned about thermal resistances being inversely proportional to the either the cross-sectional area for conductance or surface area for both convection and radiation resistances, we can further simplify the by neglecting the large thermal resistances due to the package and PCB sides (since the area is small) to get the following simplification:

Using these simplifications, we can see the main paths for heat transfer are either through the top of the package or through the top and bottom surfaces of the PCB as we would intuitively expect. However, if we look closely at the above equation and the definitions for each of the equivalent resistances, we can gain a little more insight.

- The package top resistance, , consists of a conduction resistance from the package (which is a function of the package material and top surface area) and convection and radiation resistances which are also a function of the package top surface area. Its not likely we can change the package our IC is using but we can use a heat sink to combat the package’s convection and radiation resistances. This will be discussed in our next blog.
- The next main path for heat flow is through the conduction resistance between the package and PCB, , and then out through the top surface of the PCB, . With a large PCB top surface, it is important to reduce the contact conduction resistance between the package and PCB and also reduce the internal conduction resistances of the PCB.
- The final main path of heat flow is through the bottom surface of the PCB, . Here it is even more critical to reduce the conduction resistances of the PCB since this is a longer path to the bottom surface of the PCB.

In our next blog, we’ll discuss some techniques to cool an electronic system and utilize our knowledge of thermal resistors and networks to get a better understanding of how they work.