Thermal Resistance Circuit Ideas to Help Explain Heat Flow

Key Takeaways

• One way to think about heat transfer in any system is to consider the resistance to heat flow between hot and cold regions.

• Thermal resistance becomes part of a circuit model for understanding and predicting how heat will transfer throughout a system.

• Thermal resistance can be calculated using the thermal conductivity of different materials in the system.

Thermal resistance will influence how heat travels through this airgel.

Heat transfer is inevitable when there is a temperature difference between two regions in a system. The temperature difference drives heat to flow between hot and cold regions, but this heat transfer does not happen instantaneously. Some thermal engineers have developed a useful analogy for thinking about heat transfer, known as a thermal resistance circuit. If you’re designing a PCB, IC, or complete system, then it helps to design the thermal resistance in different portions of the system to help manage the temperature distribution.

Thankfully, thinking about heat transfer in the same way as electric current gives a convenient way to see how heat moves between different areas of a PCB, IC, or other system type. Every object has some thermal conductivity, which can be used to calculate a thermal resistance and to predict how heat transfers between different regions of your system. Let’s look at how this works mathematically and conceptually.

What Is a Thermal Resistance Circuit?

A thermal resistance circuit is one way to schematically predict how heat will flow in different systems. The thermal resistance of an object depends on its thermal conductivity and dimensions, as defined in the equation shown below. The circuit diagram shows a thermal resistance circuit, which provides a thermal analog for an electrical circuit.

Thermal resistance circuit and its analog to heat transfer.

In the top graph, we have a situation where one side of an object is hotter than the other side, which causes heat to begin transferring through the system at a rate q. The equivalent circuit model for this system is shown in the bottom half, where the thermal resistance is defined for this system. Just like a regular resistor, we can define the thermal resistance in this circuit in terms of the system’s thermal conductivity, just like you would define electrical resistance from the electrical conductivity.

Analyzing Heat Flow in Thermal Circuits

Just like an electrical circuit, thermal resistances of different elements in your system combine to create an equivalent thermal resistance. Different amounts of heat will flow through different legs of the circuit, which describes how heat flow is distributed throughout your system. Thermal circuits can be analyzed very easily using Ohm’s law, which gives the equivalent thermal resistance and total heat flow for a given temperature gradient across the system.

If you can identify areas of a PCB with low heat flow (such as areas of high thermal resistance) or excessively high heat flow (areas of low thermal resistance), there are some factors that you can control to adjust heat flow:

• PCB substrate selection: A substrate with larger thermal conductivity will have lower thermal resistance, and vice versa. You could swap the substrate for a different material (e.g., a ceramic substrate).

• Presence of vias, copper power, and planes: These elements have high thermal conductivity, which means they have low thermal resistance. Adding or removing these elements in a PCB adjusts heat transfer as needed.

• Component grouping: This does not affect thermal resistance of other areas in the board, but you can spread out components with high power consumption rather than grouping them in one area. Doing so leads to a more evenly distributed temperature in your PCB.

Thinking About Equilibrium

If we take an isolated system and set up a temperature gradient between two regions, heat will begin transferring from the hot to cold regions in the system. As heat starts leaving the hot region and approaching the cold region, the temperature in each region will begin to change. The temperature throughout the system will approach an equilibrium temperature that is even throughout the system.

In an electrical system, we would have something of a change in the voltage back to some equilibrium value in the system. Consider what happens in a discharging capacitor; the voltage across the capacitor is like the temperature difference between two points on a PCB. As charges leave the capacitor and recombine elsewhere in the circuit, the voltage across the capacitor drops exponentially over time, just as is seen in the time-dependent behavior predicted with Newton’s law of cooling.

As cooling proceeds, the temperature approaches some equilibrium which can be predicted if you know the thermal conductivity, density, and specific heat of the system (as functions of space). Much like temperature, the potential difference across our theoretical capacitor will settle to zero as measured across the two plates. This transition to an equilibrium from an initial temperature is shown in the graph below:

Transition to thermal equilibrium in a thermal resistance circuit with exponential decay.

In other words, the voltage dropped across the fully discharged capacitor is uniformly distributed between the two plates. If you were to measure the voltage between the plates, the potential you’d measure, with respect to some ground reference plane, would be uniformly distributed throughout the capacitor’s body. Similarly, if we measure the temperature of a system in equilibrium, the temperature is also uniformly distributed throughout the system.

In the presence of heat sources, like an IC in a PCB, the temperature of the IC creates a new equilibrium temperature distribution that is not flat, which can be most easily visualized with a 3D field solver. This type of tool treats heat generation and transfer (including via airflow) as a multiphysics problem and presents the results numerically and visually. You can then use numerical data to calculate thermal resistance in your system and make design changes as needed.

When you need to visualize how heat flows in different regions of an electronic system via a thermal resistance circuit model, you can use a 3D field solver to build a map of the temperature distribution in your system. When you add this tool to your workflow, you can determine the equilibrium temperature in your system and extract thermal resistance values as part of heat management.