## Comparing Preece, Onderdonk, Simulation and Experimentation

#### by Douglas Brooks, PhD and Dr. Johannes Adam

Historically, Dr. Johannes Adam and I (Douglas Brooks) have looked at two sources for guidance here. W. H. Preece developed an equation in 1888 (appropriately known as “Preece’s Equation”) that tried to specify the “fusing current” as a function of conductor cross-sectional area. I. M. Onderdonk developed “Onderdonk’s Equation” that did the same thing (perhaps around 1928) but added an additional variable, time to fusing, into the analysis.

In our book, * PCB Design Guide to Via and Trace Currents and Temperatures *(Note 1), Dr. Adam and I look at both Preece’s Equation and Onderdonk’s Equation as an aid in solving the problem of required trace size, as well as two other approaches, simulation and experimentation. This article uses a specific example to compare the four approaches to see how useful they can be in actual application.

**Preece’s Equation:** In 1888, Sir William Henry Preece was Consulting Engineer (and later Engineer in Chief) for the British General Post Office, which at that time was also responsible for the telegraph system. He was concerned about possible lightning strikes to telegraph lines and how those strikes might travel down the line injuring personnel. He wanted to find a conducting material and size that would perform safely under general operation yet fuse (melt) and protect an operator if there were a lightning strike anywhere along the line. He experimented with a large number of materials by passing current down them and bringing them to just below their fusing (melting) point. He determined that the proper point was reached when the conductor glowed bright red.

He considered a large number of materials and sizes while generating his general equation. If we use the constant he determined for copper and use our US dimensional system, his equation reduces to:

I = 12277*[A^.75] [Eq. 1]

Where I is the fusing current in Amps and A is the conductor cross-sectional area in square inches.

If we assume a 0.5 Oz. PCB trace that is 15 mils wide, this equation predicts a fusing current of 2.14 Amps:

I = 12277 * ((15*0.65*10^{-6})^{.75}) = 2.14 [Eq. 2]

**Onderdonk’s Equation:** Unlike Preece, almost nothing is known about I. M. Onderdonk. When we see his/her equation in print it is offered almost as a given, much like we treat Ohm’s Law. The earliest reference we were able to locate dated to 1928. The 1928 article was written by an engineer (E. R. Stauffacher) at Southern California Edison. At the same time that article was published (coincidence?), SoCal was building a very high voltage transmission line between Los Angeles and Nevada to support the construction of Hoover Dam. The reference seemed to be related to a similar problem that Preece was concerned with. The high voltage line towers were supported by guy wires, which collected dust in the desert and became damp in wet weather. There was a risk of lightning and/or arcing between the power line and the guy wires. The problem was how large did the transmission line conductors and guy wires have to be to withstand the arc while the system went through a controlled shutdown.

While Preece’s Equation was determined experimentally, Onderdonk’s Equation was rigorously derived (Note 2). The equation is quite complicated in its general form but can be simplified if we put in constants for copper and set 20^{o} C as the reference temperature. Under those conditions the equation reduces to three variables, current (I) in Amps, cross-sectional area (A), in mil^{2}, and time (t) in seconds. Using Onderdonk’s Equation to solve for fusing time leads to:

t = 0.0346*(A/I)^{2} [Eq. 3]

This equation is graphed with the black line in Figure 1

Figure 1

Fusing Time vs Current.

Earlier we assumed we had a 15 mil wide, 0.5 Oz PCB trace. We calculated a fusing current of 2.14 Amps for that trace [Eq. 2] using Preece’s Equation. We can calculate an *implied* fusing *time* for that trace by plugging that current and trace area into Onderdonk’s Equation (Equation 3), or simply by locating a current of 2.14 Amps on the Onderdonk curve in Figure 1 (shown by the large black dot.) This suggests that Preece’s approach and Onderdonk’s approach are reasonably consistent, despite the fact that they were developed by totally different means.

**Simulation:** Dr. Adam and I go into great detail in our book about using his software, TRM, (Note 3) to solve PCB trace thermal issues. In Section 12.5 we simulate a fuse using Onderdonk’ conditions with almost perfect results. When we simulate a normal PCB trace (like the one we assumed above) the results differ considerably.

A major assumption in the Preece and Onderdonk equations is that there are no cooling effects — i.e. the conductors continue to heat without any corresponding cooling (dissipation of heat). The Onderdonk derivation assumes that explicitly. Preece’s reporting is silent on that point, but his procedure ignores any cooling impacts as well. In reality, a conductor in still air cools somewhat slowly. So everywhere you see his equation there is an accompanying warning that it is valid for up to 10 seconds or so. But a PCB trace on a dielectric cools much more rapidly, i.e. the thermal conduction of the dielectric is typically much stronger than the convection and radiation cooling effects in air. Therefore, a conductor heats more slowly when current is applied than would be predicted by Onderdonk’s Equation.

Given the trace we assumed above, we can simulate the fusing time for each level of current. The result is the red curve shown in Figure 1. There is very close agreement between the two curves (Onderdonk and the simulation) at very high currents where the fusing time is very short. But the curves quickly separate as the times lengthen. This is because the thermal conduction of the dielectric allows for heat spreading and slows the heating of the trace. Indeed, at some current between 5 and 6 Amps the trace does not reach the melting temperature at all.

Suppose we tried to predict (by simulation) the fusing time of our assumed trace with a 6 Amp current. When we try that, the results are EXTREMELY sensitive to the model assumptions we make, especially regarding trace thickness. Starting just below 7 Amps (and lower) the curve gets VERY steep. For example, the difference in fusing time between 2 seconds and 20 seconds is just a tiny fraction of the current or of the trace area. This limits the precision of simulation for predicting fusing times just to cases where the fusing time will be very short (Note 4). Longer fusing times are subject to too many uncontrollable variables.

**Experimentation:** We benefitted from the support of a board fabricator (Note 5) who gave us numerous boards for experimentation. One such board was suitable for some fusing experiments. There was a trace on that board that was nominally 0.5 Oz. and 15 mils wide, but these dimensions varied slightly across the board (due to normal tolerances.)

Figure 2

Fusing time for experimental trace.

We applied 6 Amps to this trace and measured the voltage across the trace with an oscilloscope (Figure 2, Note 6). The voltage is approximately related to temperature, but there are some lags (including chemical decomposition and board delamination) and non-linearities that occur during an experiment like this. The total time from the start of the current pulse to the moment of fusing, however, is quite precise, and, as shown, was 2.75 seconds. This experimental result is shown by the large red dot in Figure 1.

**Conclusion**: Onderdonk’s and Preece’s Equations, as well as simulations, are useful in a practical sense (on PCB traces) for fusing estimates with very short fusing times, say 0.5 seconds or less. But the impact of conductive cooling through the PCB dielectric becomes a dominant factor slowing down (increasing) the fusing time fairly quickly.

**Caution:** This blog article does not relate to “trace fuses” and we do not endorse the use of “trace fuses” except in very special situations. An overheated or fused trace is a destructive event. Boards where traces have “fused” should be considered as “destroyed” and should not be repaired and/or reused.

**Notes:**

- “PCB Design Guide to Via and Trace Currents and Temperatures,” Douglas Brooks, PhD and Dr. Johannes Adam, Artech House, 2021, Chapters 11, 12 and Appendix G. https://us.artechhouse.com/PCB-Design-Guide-to-Via-and-Trace-Currents-and-Temperatures-P2191.aspx
- Dr. Adam provides a rigorous derivation of Onderdonk’s Equation in Appendix G of our book.
- Thermal Risk Management, see https://www.adam-research.com
- In Section 12.9 of our book we suggest that fusing times less than 0.5 seconds are modestly predictable.
- Prototron Circuits, Inc., Tucson Az.
- Figure 2 is a copy of Figure 12.10 in our book.

Previous article from the same author in Artech House Blog.