Fig. 1

Let’s quote Racine [1667] :

Although these are the words of a forlorn princess wringing her hands in a castle, concerned about uncertainties in her design space¹, they could surely have been expressed by any engineer. As an engineer she’d be wringing her hands in a factory, and she wouldn’t feel much like a princess.  Let’s go through some of her woes.

Lost records are one of the commonest causes of uncertainty, coupled with our inability to remember all the details. But even when we have records, reports, specifications, and documentation, they can contain uncertainties and even amplify them. We need to write, and we need to write clearly. Ideally we need to write in such a way that records are still intelligible even after a long time. We also need to ‘negotiate meaning’, by asking for clarification whenever it is needed. Communication is a dialog, not a one-way street.

Units cause a great deal of uncertainty. They are supposed to make things more definite, and remove uncertainty. But when you say ‘volts’, what exactly do you mean? If the point of measurement, the units, and the form of measurement are not all stated, the value on its own becomes a possible source of uncertainty.

Conventions are supposed to remove uncertainty. Engineering is full of them. A common uncertainty that arises every day is Fleming’s right-hand rule or Fleming’s left-hand rule — which is it? In physics this might be simple enough, but the scope for uncertainty is much wider in engineering — for example, in laying out machine windings with the correct distribution of coils and interconnectors.

The customer is a prime source of uncertainty. We often have no idea what the customer will do to (or with) the product once he or she gets hold of it. It is completely outside the designer’s control. What is the nature of uncertainty? What are its characteristics? Think of it in terms of a road junction. You arrive at a road junction. For a start you are in the wrong lane. The resulting uncertainty is out of all proportion. Your sense of direction is useless, and the road signs are meaningless. You have to make a decision, but the lights give you no time. From previous experience you know the consequences. If you get it right, you will make progress as intended, towards success, fame, wealth, and a happy retirement. But the wrong decision will lead to imperfect results, faulty products, failure, lawsuits, sleepless nights, and even catastrophe. One of the characteristics of this situation is that uncertainty in a small detail can have grossly disproportionate consequences.

Imagine a scrap of paper with an engineering dimension labelled ‘WPM’ and three values ‘9.0 or 9.1’, and ‘9.115’. The label WPM probably means ‘width of permanent magnet’.  The word or expresses uncertainty. The third value 9.115 is expressed with three decimal places as if it were accurate to ± 1 micron. We have no idea why there are 3 values: surely one would be enough? Are these values measured or calculated? — It’s uncertain because it isn’t stated. As a document, this jotting, this scrap of paper, is seriously incomplete because of the lack of important details. Even the units are not stated : they could be millimetres or inches, and we make the tacit assumption that they must be mm, for who in his or her right mind would measure magnets in inches, and who in the world has magnets 9 inches wide?

Lawsuits — documentary scraps such as this (and even worse) have lost cases when presented as evidence in the High Court. When you enter the High Court, you are entering a theatre of profound uncertainty. The lawyers will try to distil certainty out of the kinds of uncertainty we see in our notebooks and on odd scraps of paper, and only the judge (whose engineering judgement may be somewhat uncertain) can save you from an incorrect interpretation.

Tolerances — It may be that the three values reflect the engineer’s consideration of tolerances. In any assembly of components, the parts must fit together. How do we remove any uncertainty as to whether they will fit when the time comes to assemble them? The classical approach is through the use of tolerances, which we see in engineering drawings as ±0.01, for example. Engineering components do not have exact dimensions. They vary, maybe by very small amounts; but even small variations can result in a situation where the parts do not fit, or they don’t fit correctly. So we need to control the variations and try to limit them.

What do you do with a consignment of $100,000 worth of magnetized magnets that won’t fit into the rotor? There’s an element of uncertainty for the purchasing team! This type of uncertainty can develop into a lawsuit where the uncertainties multiply and may become existential.

So the art of tolerancing — and I would say that it is an art — is a very important tool for reducing uncertainty or even eliminating it. Engineers can get one step ahead of the lawyers, by putting the tolerances (the uncertainty) at the beginning and not at the end when it’s too late.

Another common uncertainty arises when test results don’t agree with design calculations. The process of comparison has a fancy name: ‘design verification’. The first reaction is normally a quick process of trying to imagine what might be wrong. Sometimes this is enough. But in serious cases the uncertainty can be resolved only by a systematic process which may involve

All these are relevant techniques for reducing or eliminating uncertainty.

Another huge source of uncertainty in engineering is that we can’t see very far into the future. We can see some way into the future, but how far? This gives rise to the idea of evolution. We can say that evolution gradually eliminates uncertainty. As it does so, many of the uncertainties of a past era disappear completely, and many of them are forgotten.

We’ve all heard about hindsight and foresight. These are important elements in the analysis of failure modes and effects. Safety engineering is very much concerned with the management and elimination of uncertainty. We see this in our safety procedures and in protocols and equipment everywhere.

Comparing past and present, we can also see that the elimination of uncertainty relies on past practice; and is facilitated by engineering theory. Evolution in engineering is also truly international in scope.

Sometimes the design space starts with nothing but a specification : total uncertainty! It’s a jig-saw puzzle! It starts empty. There are no instructions. It’s easy to make mistakes. Often the specification is incomplete — even wrong or unachievable. It may contain requirements that lead to high costs. It often contains redundant information. We can sometimes see uncertainty in the mind of the customer. One of the most important means of eliminating uncertainty is negotiation— for instance, between the designer and the customer; and including colleagues who represent different facets of the project. Manufacturing engineers are critically important, and this really extends to everyone involved in manufacturing.

Unlike many engineering problems, the jig-saw puzzle has a unique solution, even when there are missing pieces. Engineers are usually content with a solution, and don’t worry about uniqueness. This often gives rise to uncertainties.

The question of uniqueness or unique solutions is often overlooked or even disregarded by engineers. But if your solutions are not unique they may be uncertain. For example, we sometimes choose one root of a quadratic equation because it seems to be the right one, and simply ignore the other one.

It is not often pointed out that the wave equation has two solutions, one going forwards in time and the other going backwards. Universally we pick the first and ignore the second. Are we missing something? It’s uncertain!

Closer to home, in any electric machine containing more than one winding, including magnets, if the steel is saturable (which it always is), then the various inductances cannot always be defined or measured uniquely. For example, the dq equations used to extract the synchronous inductances from calculated flux-linkage components are non-linear and they are not independent, yet we often use them as though they were simple simultaneous linear equations from a school textbook. It seems we ignore or misuse the principle of superposition. The fact is that inductance is an uncertain quantity in practical machines, and it can be argued that we would be better off if we didn’t use it, but used flux-linkage and Faraday’s law directly, instead. Maybe this seems pedantic, but beware: mathematical uncertainty and ambiguity can weaken the foundations of some of our most cherished engineering precepts.

Uncertainty can arise from failure to understand the difference between accuracy and precision. A 10-in slide-rule has a basic precision of about 3 significant figures, or 1 part in 1000. For many engineering parameters this is sufficient. Many test instruments claim to be accurate to about the same level — between 0.1% and 1%. So there is a nice consistency here. Precision is the number of digits on the display, or how closely you can read your slide-rule. Accuracy is different. It’s how close your calculations are to your measurements.

Years ago, when calculators were relatively new, I had a set of class homework papers with all the results quoted to 8 or 9 places of decimals. To explain the concept of engineering precision, I decided to do an example in class using my old slide-rule. Unfortunately, after I had started writing on the blackboard, I realised that I had forgotten how to use it. Worse : the example required the calculation of an inverse tangent. Fortunately, I could estimate the result with some level of precision (maybe 1 decimal place), and so I escaped total humiliation. But the following week, all the homework papers came in with results quoted to only one or two places of decimals.

What does this story have to do with uncertainty? — A great deal. It suggests that certainty and uncertainty are not ‘black-and-white’ quantities, but come in levels or degrees. Like the weather forecast : 60% chance of rain, not 60.0978672411%. Precision does not guarantee accuracy.

And if two numerical solutions differ by 1 part in 10⁹, which one is right? Maybe neither. The subject of precision is a tricky one — even before we go into the test-house or the laboratory to check the accuracy. We know nothing until we measure.

In another direction, extrapolation creates uncertainty by drawing inferences from a limited sphere of experience and using them beyond the bounds of that experience. For example, cosmologists extrapolate all the way back to the Big Bang — far, far beyond what any sane engineer would risk! In business planning, the value of money is variable, and that’s a source of huge uncertainty in engineering : what will things cost once we get into production?

Does a big software budget reduce uncertainty? It is perfectly possible to have certainty (or sufficient certainty) with simple tools, especially when the data is verified by test and measurement. It is also possible to have uncertainty with expensive tools, especially if there is no test data. The more complex the analytical tools, the more opportunities exist for uncertainties to creep in. That’s why sound engineering theory coupled with discipline and rigour in the software engineering (and its documentation) are so vital in our analytical design schema, together with continuous validation by test and measurement.

Material properties are a major source of uncertainty. In electromagnetic analysis a common example is the need for BH data for electrical steel, going up to high levels of the order of 2.4 T. Measured data is rare, and it is often DC data; yet we use these materials under AC conditions, often at frequencies of several kHz. Epstein test data is often limited to 50/60 Hz and it goes up to only about 1.7 T. Can we extrapolate up to 2.5 T? People do it. I do it. I have to do it. I have my methods! Justification comes from testing the prototypes in the laboratory and the products in service, as is always necessary to eliminate uncertainty.

We often need data that is not readily available even for common materials, for example BH data for shaft steel. It helps to have a collection of data measured by different specialists. My own collection goes back to the 1890s. While I trust the old measurements, I am often uncertain as to which one to use!

So much is unknown or uncertain in engineering that we have special factors called X-factors. X means unknown or uncertain, right? An X-factor defined like this : X = 4.18/2.4 = 1.74, comparing a measured result with a calculated result, is healthy, because it shows how we are quantifying uncertainty and learning from experience. 74% error is certainly a learning experience! In other cases the use of X-factors can be dangerous and risky; it may even indicate cheating or negligence. But an X-factor may sometimes be a safety factor: for example, a factor of 2 or 5 on the safe loading of a crane or a bridge. This is necessary because the actual load may vary, and it may well be unknown. X-factors can be perfectly acceptable if they are used to make reasoned allowances or to quantify our uncertainties, or to ask ‘what-if’ questions (lateral thinking). Engineers do that, a lot. Use them right!

¹ This Engineer’s Diary and Video No. 53 are adapted from a keynote address prepared for the UK Magnetics Society seminar on Uncertainty in the Design Space of Electric Machines, Coventry, U.K., 13 November 2024.