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If people are asked what determines PCB trace temperature, the most common response would probably be current, or I^{2}R power dissipation. While these answers are not necessarily wrong, they are woefully incomplete.

The units of I^{2}R are joules/sec; it is the rate at which energy is supplied to the trace. If we keep applying this energy to a trace indefinitely, the temperature of the trace would continue to increase indefinitely. It doesn’t happen because there are corresponding cooling effects that work to cool the trace. These effects include conduction through the dielectric, convection through the air, and radiation away from the trace.

In this article, we are going to pretty much ignore convection and radiation, and consider them to be constants. We will focus on things that conduct heat away.

The formula for conductive heat transfer is shown in Equation 1:

Q/t = kA(ΔT)/d (1)

Where:

Q/t = rate of heat transfer (watts or joule/sec)

k = thermal conductivity coefficient (W/mK)

about 0.5 for FR4 and about 350 for copper

ΔT = change in temperature (°C = °K)

In our case, between the trace and the dielectric

A = overlapping area

d = distance between the overlapping areas

A constant temperature occurs when the rate of heating (I^{2}R) equals the rate of cooling (Equation 1).

The situation is further complicated by the fact that I^{2}R and Q/t are point concepts. That is, they vary by (a) points in time and also by (b) points along the trace. They can vary by points in time because several of the variables—current and ΔT, for example—can change with time. They can vary along the trace because several of the variables—resistivity, thermal conductivity, and ΔT, for example—can vary along the trace.

Below is a sneak peek at a few of the less obvious influences on trace temperature.

**Thermal modeling**

In our recent book, PCB Design Guide to Via and Trace Currents and Temperatures, we explore these concepts in detail. In this article, we will use a simple thermal simulation model based on the thermal simulation software Thermal Risk Management (TRM) to illustrate the concepts. The model consists of a 50×200-mm board with a single 6-inch-long trace down the middle. The setting is a normal laboratory environment with ambient temperature of 20°C. Important parameters are:

Trace width 100 mil

Trace thickness 1.3 mils (approx. 1.0 Oz.)

Current 8 A

Resistivity (ρ) 1.72 μΩ-cm (annealed copper)

Tc (in plane) 0.7

Tc (through plane) 0.5

Board thickness 63 mils

Convection and radiation effects are assumed constant.

**Time transient**

As mentioned above, thermal effects exist at a point in time. When current is first applied to a trace, the trace takes some time to reach a thermal equilibrium. The time frame is typically 5 to 10 minutes or so. **Figure 1** illustrates the thermal response time for our model. The time frame is determined principally by how quickly the heat can be conducted through the board material.

**Figure 1** Thermal response transient of our model determines how quickly the heat can be conducted through the board material.

So, what is the temperature of our trace? It’s a function of when we measure it.

**Thermal gradient**

Also, as mentioned above, the thermal effects along a trace are a function of the point along the trace where the temperature is measured. **Figure 2** is a thermal image of our model after the temperature has stabilized. The graph in **Figure 3** shows the temperature as a function of the distance along the trace from the left edge. The trace is clearly and significantly cooler at the ends than at the middle. This is characteristic of most PCB traces.

**Figure 2** The thermal gradient shows cooling at the ends of a trace.

**Figure 3** The thermal gradient is seen along the trace.

The reason is as follows. The cooling—heat flow path—at the middle of the trace is pretty much confined to be perpendicular to the trace. But the cooling path at the ends of the trace covers more than 180 degrees. The heat has a much wider area to “conduct into.” So, the cooling at the ends of the trace is much more efficient than it is at the midpoint of the trace; therefore, the ends of the trace are cooler.

So, what is the temperature of our trace? It depends on where we measure it.

**Board thickness**

Our model assumes a 63-mil thick board. The temperature of a trace is surprisingly dependent on the board thickness, up to a point. A trace on top of a thin board gets hotter than one on a thicker board. That is because a thicker board has more material for the heat to conduct into. Thus, a thicker board cools more efficiently. But there are diminishing returns. At some point, there is more material under the trace than the trace can efficiently utilize.

**Figure 4** illustrates this effect. Our base temperature of 66.4°C is equated to a 63-mil thick board. If the board is only 32-mil thick, the trace temperature rises to 78.9°C. But if it’s 126-mil thick, the trace temperature falls to 60°C. Additional thickness beyond that point does not buy us much.

**Figure 4** The trace temperature depends on board thickness.

So, what is the temperature of our trace? It depends on the board thickness.

**Thermal conductivities**

Board materials or dielectrics, and indeed almost all elements, have a thermal conductivity coefficient. This relates to how well the material conducts heat. Its unit is W/mK. For most PCB dielectrics, this coefficient ranges from about 0.3 to about 0.8—for copper it’s about 350. But there are newer board materials emerging with significantly-higher conductivity coefficients. Higher thermal conductivity coefficients lead to lower trace temperatures.

There are typically two such coefficients for PCB materials: “in-plane” parallel to the traces and “through-plane” perpendicular to the traces. Board materials typically have higher in-plane coefficients than through-plane coefficients because of, we believe, the directions in which the fiberglass is laid. A frustration to us is the fact that these coefficients are often not published by the material manufacturers—although this situation is improving—or they are published in an incomplete manner.

PCB trace temperatures are very sensitive to thermal conductivity coefficients. If we lower the coefficients slightly, trace temperatures significantly increase. In our model, if we lower the in-plane coefficient from 0.7 to 0.6, the trace temperature increases from 66.4°C to 70.7°C. If we lower the through-plane coefficient from 0.5 to 0.4, the trace temperature increases from 66.4°C to 67.2°C. Clearly, the in-plane coefficient is the more important of the two.

So, what is the temperature of our trace? It depends on the thermal conductivity coefficients of the board material.

**Trace dimensions**

When we are concerned about trace temperatures, we are usually dealing with relatively wide traces. In such cases, there is not much uncertainty about trace width, relatively speaking. But the same is not true regarding trace thickness. Trace thickness is relatively small and thickness along a trace can routinely vary by several tenths of a mil. As a result, trace temperatures are not uniform along a trace. We cannot safely assume that trace thickness is what is nominally specified. Indeed, plated copper on a top layer can vary by 0.4 to 0.5 mils at different points around the board. Currently, there is no practical way to know with certainty what the actual thickness of our trace is.

Trace temperature is very sensitive to trace thickness. If, for example, we reduce trace thickness from 1.3 mils to 1.2 mils in our model, the trace temperature increases from 66.4°C to 70.8°C, a 6.6% increase.

So, what is the temperature of our trace? It depends on the actual trace thickness, which unfortunately is usually uncertain.

**Copper**

In general, there are two types of copper on our boards: plated (ED, or electro-deposited) and rolled (drawn). Plated copper is very close to “pure” copper. It has a resistivity of about 1.64 μΩ-cm. Rolled copper is rolled from a copper ingot, which is typically a copper alloy or annealed copper. Its resistivity varies, but is around 1.72 μΩ-cm (which we assumed in our model). The resistivity of the copper is, of course, directly related to the resistance of the trace—and therefore the I^{2}R term. So, if we change from rolled copper to ED copper, the trace temperature would go down. That change in our model reduced the trace temperature from 66.4°C to 63.4°C.

So, what is the temperature of our trace? It depends on whether we are using ED or rolled copper.

**Presence of a plane**

Most of our boards today include the presence of planes, both for power-distribution reasons and for signal-integrity reasons. The presence of a plane has a major impact on trace temperature. The reason is that the thermal conductivity of a copper plane is so much higher than for the board material—350 vs 0.7 W/mK. The heat can be conducted to a plane, then it can very efficiently be conducted all over where the plane spreads.

If we place a plane on the opposite side of the board for our model, the trace temperature falls from 66.4°C to 45.2°C. If we place the plane 12 mils below the trace, the trace temperature falls to 38.1°C. This latter case is illustrated in **Figure 5**. Compare this figure with Figure 2. Note in particular how much wider the thermal “plume” is above and below the trace. This illustrates the effect of the plane distributing the heat in a greater area around the board, and why the trace is cooler.

**Figure 5** The plane distributes the heat in a greater area around the board.

So, what is the temperature of our trace? It depends on whether there are planes present and where they are.

**The need for simulation models**

The temperature of a trace is dependent on much more than just the I^{2}R power dissipation along the trace. Some of the more important variables include the presence or absence of planes (and their size), the thermal properties of the board dielectric, the thickness of the board, and the actual thickness (variations) of the trace along the trace length.

It’s no longer practical to determine trace temperature using charts and equations; we need computer simulation models. And we have been here before. In the 1990s, we began to worry about controlled impedance traces. Back then, we could use impedance equations that were found in various standards and publications. Today, such equations are not sufficient and we need field effect solutions. We are at that same point with trace thermal considerations.

*This article was originally published on EDN.*

*Douglas Brooks has written two books and **numerous technical articles on PCB design. He gives seminars on PCB designs around the world.*

*Johannes Adam has worked on numerical simulations of electronics cooling at companies like Cisi Ingenierie, Flomerics and Mentor Graphics. He currently works as a technical consultant.*

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