# The history and mystery of the Neher-McGrath formula

## In its abbreviated form, the Neher-McGrath formula appears straightforward. For electrical engineers, it must be carefully reviewed and used.

**Learning objectives**

1. Learn the scientific basis for the Neher-McGrath formula.

2. Understand how to use the Neher-McGrath formula to find conductor ampacity.

3. Appreciate factors that can impact accuracy.

A while back, a colleague brought up a problem that had occurred on a $100 million industrial project. A few months after plant turnover, several feeder breakers started tripping. There are few things that will cost an electrical engineer more sleep than random breaker operations. After considerable consternation, the under-slab feeders were pulled out and found to have heat damaged insulation. The cables had been sized per National Electrical Code (NEC) Article 310 and there wasn’t any overloading, so what was the problem?

The cause is an object lesson we could all learn from: You can not blindly apply the NEC tables to determine cable ampacity. The installation and operating conditions must be considered.

Located above the slab were a series of industrial furnaces. The cables’ Joule (*I ^{2}R*) losses, combined with heat radiating through the slab, elevated the temperature above the insulation’s thermal limit. Overheating degraded the insulation; add a little moisture, and breakers start opening.

In hindsight the cable ratings should have been adjusted for the actual temperature in which they were expected to operate. Article 310.15 of the NEC points this out and provides the Neher-McGrath formula as the engineered solution for doing so.

In its abbreviated form, Neher-McGrath appears straightforward (see Figure 2).

The devil is in the details. *T _{c}* is readily available from product data; an

*R*value (not necessarily the correct value) is available from published sources, but what about the other factors? Is the

_{dc}*ΔT*significant? What if there is an external heat source affecting the installation? What if a complex cable type is used? While the NEC version of Neher-McGrath appears straightforward, finding the right parameters that accompany it is not.

_{d}

If a cable of complex construction is used (see Figure 10), the rating equation takes the form of Figure 3.

In practice the terms “conductor” and “cable” are used interchangeably. For clarity the term “conductor,” as used here, indicates the current-carrying part of a cable. The term “cable” refers to a complete assembly, for example, conductors, filler, insulation, jacket, armor, serving, and so on.

Adapted from Ander’s, Rating of Electric Power Cables , *ΔT* is the temperature rise over ambient, *W _{d}* the dielectric heating,

*R*the thermal resistances, n the number of conductors,

_{i}*R*the ac resistance, and

_{ac}*λ*the loss ratios for armor and shielding.

_{i}Labeling Neher and McGrath’s work a “formula” implies ampacity problems can be solved algorithmically; provide a few inputs, crank through some calculations, and out pops an answer. Their paper does not provide an algorithm; ampacity problems cannot be solved by plug-and-chug.

Neher and McGrath’s work is a body of knowledge, some aspects of which are applicable to an installation, some are not. Equations like those above must be considered case-specific; as presented they include assumptions and simplifications that affect the validity of any calculated result.

**History of Neher-McGrath**

John Hutchins Neher (1899–1973) and Martin Hager McGrath (1902–1980) did not have an epiphany when they wrote their 1957 paper, “The Calculation of the Temperature Rise and Load Capability of Cable Systems.” They had worked with a cadre of engineers on the problem for decades; the significance of their contribution was in compiling the collected knowledge as it stood at the time.

Neher, a 1921 graduate of Princeton, was a senior engineer with the Philadelphia Electric Co. A Navy Commander during WWII, he served in London working on the use of radar to find U-boats. He was made a Fellow of the American Institute of Electrical Engineers (AIEE), now the Institute of Electrical and Electronics Engineers (IEEE), in 1957 for his contributions in both cable heating and protective relaying.

McGrath, a 1924 graduate of the Carnegie Institute and a founding member of the AIEE Insulated Conductors Committee, was vice president and chief engineer of the General Cable Corp. In 1962 he arranged for then competitor Anaconda Wire and Cable (General Cable acquired Anaconda Wire and Cable in 1999) to develop the first set of ampacity tables on an IBM 650—tables that became AIEE Special Publication S-135, Power Cable Ampacities, a forerunner of IEEE 835, Standard Power Cable Ampacity Tables.

Neher and McGrath’s goal was to develop something the practicing engineering could use with the computing tools available at the time. For most engineers that was the slide rule. This was no easy task considering that in their lifetimes published ampacities varied widely. AWG #1/0 copper, for example, had values ranging from 71 to 372 amps.

(In full disclosure, Neher and McGrath also dealt with conductors in air, a topic which will not be discussed here. Interested readers are directed to IEEE Std 738-2006, IEEE Standard for Calculating the Current-Temperature of Bare Overhead Conductors.)

**The Kennelly hypothesis**

Arthur E. Kennelly (1861-1939) was one contemporary Neher and McGrath drew on. His research on ampacity was a de facto standard until 1938 when Samuel J. Rosch, an employee of Anaconda Wire and Cable working under the auspices of National Electrical Manufacturers Association (NEMA), produced what became the NEC ampacity tables, which were in use until the 1980s.

Aside from working on other topics, Kennelly developed the hypothesis that underlies Neher-McGrath: The movement of heat through a cable system’s thermal resistances will cause a temperature rise. Treat the cable system as an infinitely long, cylindrical heat source buried at some depth in a uniform medium, and the temperature rise at any point—be it the surface of a cable, interior of a duct, or any other point—can be predicted. This is shown graphically and mathematically in Figure 4.

A heat source, *+q _{c}*, is buried at a depth

*L*in uniform soil (constant ambient temperature, constant thermal resistivity, etc.). Heat will migrate from the source to all points of lower temperature by conduction (and by convection and radiation if enclosed in a pipe or conduit containing air). If the heat sink is treated as a single point,

_{b}*-q*, represented by the reflected image of the source, the temperature rise can be found from the difference in the two heat flows.

_{c}A relatively simple design problem (Figure 5) helps to put Neher-McGrath’s work in context; the objective being to find the cable system’s current carrying capacity. (This example was not chosen randomly; NEC Annex B and IEEE 835 values are used so the results can be compared to published values, an important verification and validation step.)

For underground cable systems, determining ampacity involves five elements:

1. Defining the installation

2. Finding the conductor’s ac resistance at the desired operating temperature

3. Determining the thermal resistances of each element

4. Calculating the total effective thermal resistance

5. Computing the cable’s ampacity.

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