# Performing Transient Thermal Analysis With a 3D Field Solver

### Key Takeaways

• When a component begins generating heat, the temperature in regions surrounding the component will start rising.

• This temperature rise to a new equilibrium temperature is not instant; there is a transient response that occurs in the system.

• In systems with a thermostat, the transient response can be oscillatory, which appears as relaxation oscillations in the system over time. Even your air conditioning system can use some transient thermal analysis

When you turn on your air conditioning in your home, it can take hours for the temperature in your house to reach a new equilibrium. Similarly, turn on your electric stove, and it takes time for the burner to reach its final temperature. Examining how a system reaches its final equilibrium temperature is the cornerstone of transient thermal analysis.

Any system with complex geometry and feedback can be difficult to analyze using equations. Some important insights can be gained in simplified systems, but mission critical systems require very accurate results that can be easily obtained with the right field solver.

## Analytical Techniques for Transient Thermal Analysis

When we refer to using “analytical techniques” for transient thermal analysis, we’re referring to using closed-form equations to understand something about the transient thermal response in a system. Any heat source (such as a switching IC) or heat sink (such as a fan) might be switched on at different times and be located in different regions of the system. As a result, the temperature can rise and fall due to feedback in the system. Many systems use a thermostat to control a heat sink, meaning the heat source/sink depends on the temperature in the system.

As a result, there is a complex coupled relationship between the sources and sinks in a system. When we consider the heat equation describing the temperature in a system, we have a system of partial differential equations that defines how the temperature evolves in space and time. One way to model this set of complex relationships is using the heat equation with sources/sinks, where each source is a function of space, time, and temperature. This set of equations describes coupling between sources, sinks, and temperature in a complex system for performing transient thermal analysis.

The Laplacian term in the top equation is normally reduced to the system eigenmodes using the Helmholtz equation, giving a set of coupled first-order partial differential equations. Transient analysis in these systems, particularly where a heat sink is also a function of temperature and time, is quite complicated and typically only some high-level quantitative results can be obtained.

Each source/sink term can be defined using its own partial differential equation in time and/or space, which creates a system of coupled partial differential equations. The consequence of this is that the eigenvalues of the system are really functions of space and can be functions of each other. In other words, the transient thermal response of the system can be very complicated and varies throughout space and time, depending on where sources are located and when they operate.

The result is that the system can exhibit a complex transient response or mixed transient responses in different locations in the system. Some example results obtained for a PCB with a single switching IC are shown below. Each curve is gathered from a different location in the system. Measurements taken farther from the IC exhibit very little variation in the transient response, while those taken closer to the IC exhibit very large changes in the temperature curve as the overall system rises to its final temperature. Very long into the simulation, the transient response tends to vary around the system’s equilibrium temperature. Example transient thermal analysis results for different regions in a PCB. The switching IC acts as a periodic source of heat due to power dissipation. The transient response varies in time as the IC switches and adds heat to the system.

This area of research involving the transient response of different systems in space and time is very active and mathematically intense. It takes many of the important skills from coupled partial differential equations to analyze a system of coupled equations. Similar techniques are used in continuous wave lasers to examine how the power output saturates to its final value. Because these systems are analytically very complicated and often can’t be solved in closed form, a 3D field solver is generally used to perform transient thermal analysis in complex systems like PCBs.

## Creating a Transient Thermal Simulation in Your 3D Field Solver

When you need to perform transient thermal analysis in your PCB, you need to define some important quantities in your system. In addition, if you plan to consider airflow in your board, you’re now creating a complex multiphysics problem involving heat generation, heat conduction, and heat transfer via airflow. The important quantities in the heat equation, Navier-Stokes equation, and continuity equation are:

• Material thermal constants. These include the thermal conductivity, mass density, and volumetric specific heat of all materials in the system. If you’re looking at structural changes due to thermal expansion, you’ll also need to enter the coefficient of thermal expansion.

• Boundary conditions. Regions in the system that have defined temperature or no-slip boundaries need to be defined in your simulation.

• Initial conditions. Because transient thermal analysis is a time-domain simulation, you need to define some initial conditions in the system.

• Sources and sinks in space and time. The locations of any sources and when they operate need to be defined. This includes your components and any fans or other active cooling equipment (e.g., refrigeration) in your system.

The simulation you build needs to take your PCB layout data and create a spatial mesh for your system, which is then used in a standard numerical algorithm for solving the coupled heat and CFD equations. For transient thermal analysis, you’ll use an FDTD simulation for this type of complex system. A great field solver will use your PCB design data to create a multiphysics simulation.

The goal in this simulation is to understand something about how the system reaches its equilibrium temperature once the components in the system are turned on. Mathematically, the system takes an infinite amount of time to reach equilibrium, but in reality, you only need to wait ~3 time constants (see the solution to Newton’s form of the heat equation) for the temperature to rise to ~95% of its final value. You can estimate the final temperature and the time constant by fitting data from a short time period to an exponentially increasing function: Transient response without modulated sources that can be used for fitting and determining the time constant associated with heat transfer and approach to an equilibrium temperature.

Similarly, if the system starts off at a high temperature, you can examine how long it takes for the system’s temperature to drop back to a lower temperature or ambient temperature when components are switched off. To determine a time constant, simply fit the temperature data over time to an exponential decay equation.

The goal in determining time constants is to understand how long the simulation should run in order to examine the transient thermal response in the system. You should run the simulation for 3 or 4 time constants to determine how long it takes for the system to transition between different temperature levels. Setting the simulation time too long means the simulation requires a longer computation time to complete.

Note that because these simulations are defined in time and space, the temperature variation is different in various locations in the system. This is due to the location of different sources and the variation in material constants in space. With the right simulator, you can generate temperature maps in your system that can be used to calculate spatial averages, which will consider the entire system. You could also examine the temperature at specific points in the system over time.

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