# A Little Heat Sink Math

Quote of the Day

I know not so many people want to do math, but sometimes it would be easier to clarify many things if you do some math.

— From a WiFi paper on cyclic delay diversity. I cannot imagine designing ANY RF/wireless system without math – it boggles my mind.

## Introduction

Figure 1: Common TO-220 Heat Sink.

I am conducting a seminar next week on cooling electronics. One of the topics I will cover involves basic heat sink usage. Most of the products that are designed by my team do not use heat sinks because we are not allowed to use fans in our designs – fan-based cooling systems generally have air filters that require regular maintenance that is unacceptable for optical hardware deep in the network (example deployment).

I shudder every time I think of fans in an outdoor deployment. Adding fans and filters means that you need to add diagnostic hardware to monitor the health of both the fans and their air filters (e.g. sensors to measure fan speed and air flow). Diagnostics are always a mixed blessing;it is nice to know the health of your hardware, but the diagnostics frequently create a false alarm problem. Also, fans have poor reliability compared to electronic parts – moving parts are likely to stop moving. Generally, this means that you have to add redundant fans so that if one fails you have backup capacity. This costs money in terms of initial outlay and operating costs.

However, my group is now working on a number of designs for the central office that do use fans, and it is time that I provide some heat sink usage guidelines for my team. Let’s start this exercise with a simple heat sink for a TO-220 transistor (Figure 1). I have used this heat sink many times at my previous companies, and I am very comfortable discussing it.

## Background

### Definitions

heat sink
A heat sink  is a passive heat exchanger that transfers the heat generated by an electronic or a mechanical device into a coolant fluid in motion. The transferred heat leaves the device with the fluid in motion, therefore allowing the regulation of the device temperature at physically feasible levels (Source).
A heat spreader is a heat exchanger that moves heat between a heat source and a secondary heat exchanger whose surface area and geometry are more favorable than the source. Such a spreader is most often simply a plate made of copper, which has a high thermal conductivity. By definition, heat is “spread out” over this geometry, so that the secondary heat exchanger may be more fully utilized (Source).
This definition defines the heat spreader as an interface between a hot object and the its heat exchanger. In industry, we often use the term “heat spreader” when discussing the use of a heat sink without a fan. This improves the thermal coupling between the integrated circuit being cooled and the air.
Sink-to-Ambient Thermal Resistance (θSA)
θSA is a parameter selected to make the relationship $T_{HS} = T_A + P\cdot\theta_{SA}$ true, where THS is the heat sink temperature, TA is the ambient temperature, and P is the power to be dissipated.

### Effect of Moving Air on Heat Sink Performance

The efficiency of a heat sink is strongly related to the air flow across the heat sink. We say that ${{\theta }_{{SA}}}\propto \frac{1}{{{{h}_{{Air}}}}}$, where hAir is the thermal conductivity of air. Figure 2 shows the thermal conductivity of both still and moving air (Source).

Figure 2: Thermal Conductivity of Air (h) vs  Sink-to-Ambient Temperature (ΔTSA).

Figure 2 tells us several things:

• h increases with increasing air speed.
• Since θSA ∝ 1/h, θSA will decrease ⇨ better heat transfer from the sink to the air.
• θSA is only approximately constant because h increases with increasing ΔTSA.
• Observe the strong non-linearity in the 0 m/s air speed curve at low ΔTSA.

### Heat Sink Specifications

If you want to understand all the nuances of heat sink specifications, read this paper from Aavid. For this post, we are going to focus on this chart for the heat sink in Figure 1 – the call-out bubbles are my annotations.

Figure 3: Specification Graph for Heat Sink of Figure 1.

## Analysis

### Objective

In Figure 1, the heat sink’s θSA is given as 24 Ω. However, if you look at closely Figure 3, you will see that θSA varies with air speed (curve labeled “Moving Air”) and θSA is always less than 24 Ω. Where does the 24 Ω value come from? It represents the thermal resistance in still air curve at low power levels, which I will demonstrate below. I also want to determine the thermal resistance for still air at various power levels.

### Analysis

Figure 4 show the my analysis, which consists of:

• Digitizing the still air curve (i.e. ΔTSA vs P) from Figure 3.
• Interpolating and smoothing my digitized still air curve.
• Computing ${{\theta }_{{SA}}}\left( P \right)=\frac{{d\Delta {{T}_{{SA}}}}}{{dP}}$

Figure 4: Generation of Thermal Resistance for Still Air.

We can see that θSA varies from ~28 Ω for power dissipations near zero and decreases to ~12 Ω at 5 W. Thus, the 24 Ω value shown in Figure 1 represents the approximate θSA in still air and at low power levels.

## Conclusion

I derived the effective thermal resistance of this heat sink in still air for power levels from 0 to 5 W. I was able to show that the 24 Ω value stated in the specification reflects the thermal resistance of the heat sink at low power levels and under still air conditions. The actual thermal resistance is a function of the allowed ΔTSA value and the air speed.

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### 1 Response to A Little Heat Sink Math

1. Alexander729 says:

This calculator ought to not be made use of in circumstances where the warm resource is much smaller sized than the base of the heat sink. The concentration of the warm resource over an area a lot smaller than the warm sink base is not taken into consideration in this calculator.