The Basics of Steady-State Heat Transfer Analysis

Key Takeaways

• Heat flow is driven entirely by thermal gradients, and the goal in cooling system design is to control heat flow.

• In a PCB, components heat up during operation, and heat transfers around the board, eventually reaching a steady state.

• Steady-state heat transfer analysis with a field solver can help you quickly identify areas where a heat sink might be added, cooling needs to be directed, or ventilation in the enclosure is needed. 

This cooling fan is just one element that determines how your PCBs and components approach a thermal steady-state.

Once you turn on your circuit board and it starts consuming power, components will start generating heat and your system’s temperature will start increasing. When designing your system for reliability, it helps to see where heat is flowing and how this relates to the location of components around the board. Eventually, your board’s temperature will reach a steady-state, where the temperature stops changing as long as the heat sources in the board are steady.

What happens in the steady-state is very important. Once the temperature distribution in the board reaches the steady-state for a given set of sources and sinks, you’ll know whether components or other portions of the board are running above your desired temperature limits as the temperature stops changing. When you use a field solver to determine the temperature distribution in your board, you can spot how heat moves around your system and determine whether a redesign will decrease the system’s temperature.

Temperature in the Steady-State

If you want to take greater control over the temperature distribution in the board, you really need to control how heat flows in the steady-state in your PCB. Sure, you could control heat flow by placing the board into an icebox or refrigerator, but a more elegant strategy requires tracking heat flow throughout the board as it runs. To do this, you need to look at the temperature distribution in the steady-state.

The term “steady-state temperature” is often used interchangeably with the idea of thermal equilibrium, but the two ideas are not the same. When a system is in thermal equilibrium with its surroundings, everything (system + surroundings) are at the same temperature. In the steady-state, the temperature field throughout the system has simply stopped changing. However, temperature differences still exist in the system, as shown in the example one-dimensional system below. The exact shape of the steady-state temperature distribution depends on the source function, S(x), and the value of the heat constant, k.

Heat flow in due to a heat source, S(x), in a system with heat constant, k. The heat flow rate is proportional to the gradient of the steady-state temperature distribution.

The first equation shown above gives the double integral used to calculate the steady-state temperature distribution for any source function, S(x), and heat constant, k. Mathematically, the equation for T(x) is found by setting all time derivatives in the heat equation to zero and integrating the result. In a real system, such as a PCB with a large number of components, k will be a complex function of space and is included in the integral. In any case, the resulting heat transfer rate Q through an area A will be a function of space. This formalism can be found in almost any partial differential equations textbook.

Once the system enters the steady-state, the amount of heat that leaves the system through its boundaries at any instant is equal to the heat generated within the system at the same instant. When these two quantities are in balance, the system is in the steady-state. As shown in the graph above, the temperature distribution can have a complex shape, it’s just that the temperature field does not change in time. Heat flow in the steady-state through a defined area A with normal vector n is calculated with the lower equation in the graph above.

Steady-State Heat Transfer Analysis from Field Solver Results

A multiphysics field solver can be used to determine the steady-state temperature distribution in your PCB or IC package quite easily. These types of simulation utilities can include airflow, due to a fan or natural convection, in addition to conduction. A steady-state thermal simulation proceeds quite quickly compared to a transient thermal simulation in the same system. To determine the distribution of the heat transfer rate throughout the system, you need to use the following high-level procedure:

1. Import your PCB or IC layout into your field solver utility. Make sure to define the relevant thermal conductivity values in different materials in the system.

2. Define the static (i.e., time-independent) boundary conditions in the system.

3. Run a steady-state thermal simulation to get the temperature distribution.

4. Use the gradient equation shown above to get the heat flow rate distribution.

5. Calculate an area integral of the resulting gradient (don’t forget the dot product with n) to get the heat transfer rate through the chosen area.

The image below shows an example temperature distribution for a power MOSFET determined in a 2D steady-state simulation. The gradient can then be calculated in this image numerically.

In this power MOSFET simulation, the core temperature reaches well above the typical CMOS temperature rating. 

This type of post-simulation analysis (steps 4 and 5) is simple enough to run with numerical temperature data in 3D. The goal here is to map out how heat moves through the system and identify locations where heat might be removed. The image below shows the thermal gradient in the above IC package calculated with Image.

Thermal gradient of the MOSFET package in the above image.

The white box shows the package outline. We see high gradients across the boundaries of the conductors in this system. This data can then be exported as numerical data and multiplied by an image showing the heat constant distribution in the system. The same process can be used with other IC packages or a 3D simulation in a PCB.

Once you identify areas with high gradient, you can take steps to increase the thermal conductivity in this region (such as adding metals or coatings) to more quickly remove heat. You can then use your layout tools to make the required design changes and run the simulations again for verification. Integrated simulation tools can help you visualize temperature and heat flow in your system, as well as perform steady-state heat transfer analysis.