Fig. 1

In a recent session of the Biggar Science Café ¹ entitled Mathematics in Action, I found it very easy to borrow some examples of applied mathematics from the world of modern automotive engineering, illustrated with one or two published images from seminars presented by J.R. Hendershot ². Fig. 1 is a composite diagram for the discussion, and it can be supplemented with an Internet search for EV Drive-Train, that will produce a rich collection of images and diagrams from the real world of electric vehicles. I have to say that when I tried that search myself, I was shocked at the extent of development that has taken place in just the last few years.

The basic idea was to focus on one or two images and think about the mathematics behind them. It is quite humbling to contemplate the extent of the mathematics involved not only in automotive engineering, but in all engineering. The Science Café talk began with a light-hearted discussion of number theory in relation to simple systems of counting, and the mathematics of scales in music; and although these topics might seem far from the daily work of JMAG, they quickly show the enormous influence of mathematics in ordinary life. In contrast, automotive engineering is not at all far from the daily work of JMAG!

It can be argued that the digital computer is limited to the arithmetic processing of binary numbers: to put it colloquially, computers don’t do math, they only do arithmetic. They don’t work with irrational numbers or complex numbers or transcendental functions, or any functions for that matter: they only work with binary approximations to these things. They don’t work with logic, but only with symbolic logic expressed in the form of binary arithmetic. They have no innate intelligence, but only algorithms implemented in fixed microelectronic circuits. A computer is a machine: it does not express itself in concepts, hypotheses, proofs, or caring, or moral judgement, or any other higher sentiment. Without a continuous uninterrupted supply of electrical power, and lots of it, the computer is nothing (or worse than nothing). For most of the technical elements missing from the working of the digital computer, we need mathematics with its theorems, its hypotheses, its proofs, its formulations, and other elements from its vast resource of ideas conceived by the human mind, often working with a pencil and a sheet of paper, or even with a sharp stick on a patch of damp sand by the banks of the River Nile.

For many, our experience of serious mathematics is often through the lens of advanced modern software — through the computer screen, if you like. Looking through this lens, we immediately lose sight of the classical mathematics: we are looking at the output of binary-arithmetic algorithms which approximate the solution of the mathematical equations; we do not see the equations themselves. However precise this data may be, it is generally only an approximation to the total reality of the physical world, while in engineering and in other disciplines the classical theory and the functional forms remain at the heart of the process, and that means mathematics, not Fortran or Python or C++ : mathematics.

Back to the motor-car. There is so much mathematics in the motor-car (and particularly in the EV) that we can only discuss a few pointers.

The motor-car itself has a shape, an envelope, which is defined mathematically in engineering CAD drawings that would surely make Euclid’s hair stand on end. One of the most sophisticated mathematical processes is to work out the air-flow and the aerodynamic characteristics, using equations associated with names such as Navier, Stokes and Bernoulli, discretized and solved numerically on enormous computers. Some of the solutions take weeks. That might seem expensive: but it says something about the financial value of the solutions. Think how much time and effort is saved in prototype modifications and wind-tunnel testing.

Inside we can see the battery, which again is the object of extensive mathematical analysis & simulation, going to its chemical, electrical and thermal behaviour.

Then we have the drive-train, with the electric motor & gearbox; and the inverter which is a box of power-electronic components and digital controls. The inverter converts the DC battery voltage to variable-frequency AC for the traction motor, and it operates with DC voltage levels anywhere from 300 to as high as 1200 V. That, by the way, is a serious safety concern for maintenance technicians, [3].

Inside the motor is a stator with complex coils or windings of copper, and a rotor whose structure may include stacks of laminated steel punchings. The geometry of both the stator and the rotor may be complex, involving interconnectors, permanent magnets, insulators, sensors, and other components provided to enhance heat transfer. Rare-earth magnets are expensive and they are generally too strong and too dangerous to handle manually, so that special insertion tooling is necessary.

We can see the circuit diagram of the pulse-width modulated current-regulated inverter, using insulated-gate bipolar transistors made from silicon or silicon carbide. Engineering jargon is much worse than mathematical jargon!

The switching states of the six transistors are represented in the complex plane (the so-called Argand diagram) by complex numbers called space-vectors. As the transistors switch on and off (at typically 10 − 20 kHz), the voltage applied to the motor windings is represented by a succession of these vectors. The circuit diagram at the bottom shows the six possible states of the switches.

The space-vector diagram looks a bit like the circle of fifths so beloved of musicians, but it is much more complex, and it doesn’t stand still. This diagram is the link between the digital on/off switching states of the transistors and the analogue or continuous currents in the motor windings. It is at the heart of all power-electronic control of all AC motor drives and a great many wind-turbines and other devices used in electric power systems. You can sometimes find this technology even in your dishwasher drain pump.

The control of all this complex electronics is done digitally in a microcontroller, a small but powerful computer operating in real time, and we can see a schematic diagram of a mathematical simulation in software — another sophisticated process which is built on the solution of ordinary but non-linear simultaneous differential circuit equations in the time domain together with what is called a state machine which accounts for the on/off states of the switches (transistors) in binary logic (0/1).

We can also see several reference-frame transformations or mappings into different frames of reference — a sophisticated matrix algorithm that is executed thousands of times per second as an integral part of the control-system mathematics. Cosmologists may baffle us with 4-dimensional space, but here we have multi-dimensional spaces (mathematically), and we’re using them to drive around and shop for groceries.

Where is the maths? It’s everywhere, but I’ve included this famous small equation of Euler, which includes all the five basic elements of mathematics — 0, 1, j [= √(−1)], π and e [= 2.7183 . . .] Euler’s constant, the base of natural logarithms (invented by Napier, another Scotsman). This iconic equation reminds us of the comprehensive power and presence of mathematics in what we do. If you want to know the meaning of Euler’s equation, don’t ask a mathematician: ask an electrical engineer. It should be obvious! (See Engineer’s Diary No. 43).

In a trace of the currents and voltages at the motor terminals, we will see that the three line currents are not bad approximations to sinewaves. We can see some high-order harmonics (what musicians call overtones or partials). The fundamental frequency determines the speed of the motor, and the amplitude determines the torque. The phase of the currents can be adjusted to control the voltage needed to drive them into the motor; this is a process known as flux-weakening, and it’s important for maximizing the utilization of the limited battery voltage at high speed.

In the terminal voltages and we will see the pulse-width modulation which is what causes the currents to be sinusoidal. We may also see a secondary ringing on every single pulse, which is a parasitic oscillation or resonance. The high-frequency ripple is inherent in power-electronic systems, and it causes losses, acoustic noise, and electromagnetic interference, all of which have to be analysed and controlled. More maths — lots of it!

For engineering students to learn the workings of such systems, they need a specialised course lasting 2−4 years, and maybe a period of postgraduate work as well, and ideally an industrial apprenticeship to help keep them sane — and employable. Throughout this course they need a parallel course of study in several relevant branches of mathematics.

But so far we haven’t seen much real mathematics, have we? In the familiar world of electromagnetic finite-element analysis, we see the solution of some of the equations that constitute the great foundations of mathematical physics. We often refer to these in connection with James Clerk Maxwell, to whom we attribute the four partial differential equations deduced from the observed physical laws of Faraday, Ampère, Oersted, and Gauss in the early 19^th century. Many people put Maxwell above Einstein and alongside Galileo and Newton, and I certainly do; (but this is a purely personal opinion). The equations that need to be solved in electromagnetic design problems are more precisely specified as the Laplace equation and the Poisson equation for magnetostatics; the diffusion equation and its complex form the Helmholtz equation for eddy-current and skin-effect calculations; and the wave equation for electromagnetic wave propagation. All of these are applicable to various aspects of the electromagnetic design and analysis of all motors and generators, and much else besides.

To solve the field equations in software, we discretize the geometry by dividing it in to very small elements: sometimes as many as several million. After transformation into a huge set of non-linear algebraic equations, the solution is executed iteratively and concurrently in each and every element, by a process known as Gauss-Seidel and Newton-Raphson successive approximation, until it converges to a satisfactory result.

The rate of convergence is a mathematical problem of enormous importance, because it affects the solution time and cost, as well as the precision. The finest mathematical minds in a generation work on this type of problem, just as Euler improved the convergence of infinite series in the 18^th century for calculating trigonometric functions, and for many other purposes, including refinements in approximations to π.

We can see a result in the form of a colour-map of the magnetic flux-density over the cross-section of a motor. This result is not the most sophisticated example of mathematics in action, but it gives the idea in the form of a pretty picture with symmetry and almost mystical patterns, and the trained eye of the design engineer can abstract a wealth of detailed information at a glance. It’s a nice thought, to ask yourself where is the nearest factory where you can see this software being used. In my case, it’s about 60 km from here. For anyone reading this, it might well be in the same building. I hope so!

I hope I’ve made a convincing argument that mathematics is full of action, and is an integral and essential part of our daily lives, even when we can’t see it; or if we can only see its results approximated (however precisely) in software.

Notes and References

¹ https://www.biggarsciencefestival.co.uk/cafes

² Publisher and co-author of the Green and Blue Books [1] and [2]

[1] Hendershot J.R. and Miller T.J.E., Design of Brushless Permanent-Magnet Machines, Motor Design Books LLC, ISBN 978-0-9840687-0-8, 2010, sales@motordesignbooks.com (Green Book)

[2] Hendershot J.R. and Miller T.J.E., Design Studies in Electric Machines, Motor Design Books LLC, ISBN 978-0-9840687-4-6, 2022, sales@motordesignbooks.com (Blue Book)

[3] Vesa Linja-aho, Assessing the Electrical Risks in Electric Vehicle Repair, IEEE Industry Applications Magazine, September/October 2024, pp. 32−41(DOI 10.1109/MIAS.2024.3387142)