Tim Wang Lee

Tim’s Blackboard #1: The Melting Trace Paradox

Blog Post created by Tim Wang Lee Employee on May 24, 2017






Unlike other famous paradoxes such as the Zeno’s paradox, where Achilles and the Tortoise are involved, the melting trace paradox is one with a segment of copper trace and a current source, see Fig. 1. 


Fig. 1: The circuit setup for the melting trace paradox.


If you find the equations for power of the trace and the required energy to change the trace temperature, you will find the temperature change of a trace expressed as follows:



whereis the temperature change, and  is the specific heat and mass of the copper and  is the time elapsed.


The equation states that the temperature increases with time. That is, the longer I leave the current source on, the hotter the trace gets. As the temperature reaches the melting temperature of copper, the trace melts.


There is clearly a paradox here. In labs, we know that the temperature of the trace does not increase with no bound, and the trace does not come with a warning label that says “don’t leave current on for too long, trace will melt.” At some point, the temperature of the trace reaches a steady state value.


But how come our prediction is not consistent with our expectation?

Is our math wrong? Or is the world we know broken?

We will now look at the melting trace paradox and find an explanation and solution for it. 


Power and Energy of the Trace

To reconstruct the melting trace paradox, let’s first look the power dissipated by the trace. Assume the trace has resistance  and the current source is delivering  Amps. We can write: 




Since power has the unit Joule per second, we know that the longer we leave the current source on, the more energy,, is consumed:



Next, we will look at how much energy it takes to increase the temperature of the trace.


Increasing the Trace Temperature

Let’s calculate the energy required to increase the temperature of the trace. Recall the definition of specific heat: the amount of heat per unit mass required to raise the temperature by one degree Celsius. Let be the specific heat of the trace and the mass of the trace, we write:



where  is the energy required to increase the trace temperature bydegree Celsius.

Because we have an expression for the energy of the trace, we can re-write the temperature change in terms of energy: 


Melting Trace Paradox     

Here is the energy delivered to the trace:


Here is the temperature change of the trace with given energy:


Replacing  in first equation with  in the second equation , we get:



Putting in the numbers, given 1 A current source and a 1 inch long 20 mil wide copper trace, leaving the current source on for 1 minute results in 1100 oC temperature change, which exceeds the melting point of copper at 1085 oC.

We end up with a melted trace and a frowny face.

What Is Going On?

The assumption we made in calculating the energy generated is that ALL the electric energy goes to heat up the trace. The assumption is wrong for a lab environment.


In a lab environment, we have air to provide heat transfer through convection. However, if we are in space (vacuum) where there is no air for convective heat transfer, the lack of heat transfer traps the heat in the trace and increase the trace temperature with time and finally, the copper trace melts.   


Fig. 2: Air should be included in the setup to correctly represent the lab environment.


As shown in Fig. 2, in most labs, air is present to provide heat transfer through convection, allowing the trace to reach a steady state temperature.


It would be hard to calculate the steady state temperature with pencil and paper. I’ll show you how to find the temperature by performing an electro-thermal simulation with ADS PIPro.  


Electro-Thermal Simulation of the Trace

Entering the same parameters in the ADS PIPro electro-thermal simulator, we can setup two experiments: one with a vacuum and another where air and convection is present.


We would expect the trace in the experiment with a vacuum to have an extremely high temperature, and expect the one with air to reach an equilibrium temperature.


Moreover, with air surrounding the trace, we expect the surrounding of the trace to heat up and have a higher temperature. On the other hand, if the trace is in a vacuum, the surrounding of the trace should stay at the ambient temperature.   


Fig. 3: ADS PIPro electro-thermal simulation of a trace in a vacuum. Since there is no air to allow convective heat transfer, the trace reaches a very high temperature, ~35000 oC.


Shown in Fig. 3, as expected, the trace in vacuum reaches a high temperature, with no heat spreading to the trace’s surroundings. (But if there is no convective heat transfer in vacuum, how does the warmth of the sun gets to the earth!?)



Fig. 4: ADS PIPro electro-thermal simulation of a trace in a lab environment. Since air is present to provide convective heat transfer, the simulated temperature reaches a reasonable value, 56.8 oC


Shown in Fig. 4 is the analysis done with air around the trace. As expected, the air surrounding the trace provides means of heat transfer and allows some energy to escape instead of being all trapped in the trace. Reading from the result plot, we find the final temperature of the trace to be about 56.8 oC.   


The World is Not Broken and Math is Good

After the analysis, we are now sure the world is not broken and our calculations are correct. It is our assumption that needs to be improved. We took the air around us for granted and forgot to include it in our initial analysis.


By performing the consistency tests, we found the solution to the melting trace paradox and now have a better understanding of the thermal aspect of the current source and the trace. 


That's this week's Tim's Blackboard. See you in two weeks!


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