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### The undergraduate electrical engineering experience

Electrical engineering begins with electromagnetics, the beautiful theory embodied in Maxwell's equations. And it is beautiful. But I don't like it. To a first-year engineering student, you can't even understand the concepts, really. Take electric field. Wikipedia says: “In physics, an electric field is the region of space surrounding electrically charged particles …” This is false, of course: Electric field is not a region; it’s a vector-valued function of space and time. The U of T course ECE110 Electrical Fundamentals uses the text Fundamentals of Physics, by Halladay. For Halladay, electric field is too difficult to approach head-on, so he tries to tiptoe up to it via temperature, which is supposed to be easier because it's a scalar field. He begins thus: “The temperature at every point in a room has a definite value.” I guess everyone knows intuitively what that means. But in a physics course there should be precise, not vague, definitions. What is temperature? Temperature is a concept defined in the subject of thermodynamics. Coincidentally, in 1871 Maxwell himself wrote a book called Theory of Heat. His definition is this: "The temperature of a body is its thermal state considered with reference to its power of communicating heat to other bodies." Thermal state is just another vague term. Let’s try a different tack. Let’s say that temperature is the reading on a thermometer. Of course, two thermometers will give slightly different readings, but let’s ignore that point. From the viewpoint of what a thermometer reads, temperature is a measurement of the average kinetic energy of the molecules in an object or system. But what is average kinetic energy? I believe it’s something like this: Temperature at a point is a limit as the radius goes to zero of the average kinetic energy of molecules in a spherical volume centred at that point. But how can there be a limit? As the radius of the sphere gets smaller, eventually it’s smaller than the size of one molecule. It may be that the limit is not really a limit, but rather an extrapolation on a curve. Again, it’s vague.

So that’s what a first-year engineering student is fed: Electric field is approached via the simpler concept of a scalar temperature field, but temperature is an abstract, mathematical concept based on a dubious limit. So physics, which is supposed to help us understand the real world, isn't real at all -- its method is to invent a model: molecules, kinetic energy, the limit of an average, etc. How is a young person supposed to understand this? And so far I've been discussing only electric field. Maxwell's equations represent a much more complex fantasy of a model. It seems to me that physicists are not as forthcoming as they might be in emphasizing that physics is a subject of model building and verification by experiment. An experiment does not provide truth; it relates to the non-falsification of a model. It seems there is no absolute truth; we will never arrive at the “true” physics model. Why is that?

I am not knocking physics or its contributions. Without physics we would still be living in the dark ages. I am only saying that I don't like physics as an academic subject because I don't understand it, and the reason I don't understand it is because it is abstract.

Over the years as a professor I occasionally would ask a fourth-year undergraduate, “What is electric field?” I got various answers. Most would say they don’t know. Some would say it’s force per unit charge, to which I would then ask, “What is electric field in a region of space where there are no charges -- presumably Maxwell’s equations still hold?” They were stuck. Not one student gave the answer I would have liked to hear: Electric field is a vector-valued function of space and time that appears in Maxwell’s equations. My conclusion is that, even after several courses on electromagnetics and fields and waves, students have only a vague understanding of the subject.

### Electromagnetics vs control theory

Control theory results are in the definition/theorem/proof format. Thus in principle a result is either true or not, depending on whether the proof is correct. To see the contrast with electromagnetics, what follows is an email dialogue I had with a specialist in that subject.

Dear K,
I didn’t get an opportunity to look at your paper until now. On skimming through it, I don’t see any formal theorem statements, with proofs. So may I ask, are there mathematical results? That would help an outsider like me (of course, you didn’t write the paper for outsiders).
Cheers,
Bruce

Dear Bruce,
The paper is not written in the formal language of “theorems” and “proofs.” Certainly, though, there are mathematical results reported that describe the physical properties of complex modes. … In general, though, in the physics literature papers are not written as mathematical papers in the form of “theorems,” “lemmas,” and “proofs.”
Best wishes,
K

Dear K,
I wonder how physicists check each other’s work without explicit assumptions, mathematical statements, and proofs. For example, your paper talks about “casual suggestions in the literature.” Had they instead been rigorous statements, one could have checked them to be true or not. ... Your model in the appendices involves a limit (homogeneous limit), so I guess something converges and can be proved to converge in a precise way. Or is it that one doesn’t actually prove convergence but instead verifies it by experiment? You don’t have to bother replying if my questions are irrelevant. It’s opening a can of worms for a control theorist to try to read physics papers.
Bruce

Dear Bruce,
This is a philosophical question really; for example one cannot prove Maxwell’s equations mathematically. Mathematics is a tool, it cannot describe physical reality. So in the narrow area of electromagnetics, whether something is true or not boils down to whether it satisfies Maxwell’s equations.
Best wishes,
K

K’s statement seems to yield the following argument: Mathematics is not real; truth lies in Maxwell’s equations; Maxwell’s equations are mathematics; therefore, truth is not real.

The infinite

At one point I was interested in very large dynamical systems, such as those in crystal lattices. One may try to approximate a very large dynamical system by an infinite one. As a very simple example, I studied a 2D resistive lattice; this is posted elsewhere. I corresponded with Professor Atkinson of the Physics Department, Groningen, who had written a prior article on the subject. This is his reply to one of my enquiries:

"I completely agree with you that the infinite network of resistors yields a system of equations that possess an infinity of solutions. As we noted in the introduction of our paper, (p. 486), “we tacitly obtain uniqueness by requiring that the currents at infinity vanish, ...”. In your note you object that an infinite network is not physical, so appeal to physics is disallowed. Indeed, if you stick to the mathematics, the equations do admit multiple solutions, unless you add the requirement at infinity, in which case there is only one solution, namely the one we published. If you further press the point as to why we should be interested in this particular solution, then physics can be brought in via the Aristotelian distinction between actual and virtual infinity. In an actually infinite network (without a boundary condition at infinity), there is an infinity of solutions of the equations. In a virtually infinite network, which means that you consider a finite network of size s, and then let s tend to infinity, there is only one solution, and moreover it is one in which the currents at the periphery do indeed vanish in the limit. This is the “physical” solution, according to the Aristotelian canon."

Thus, again, physical reality is defined by an abstract mathematical property: a limit as the size goes to infinity. Along this line, I like the quote, "Nothing is as abstract as reality."

### Classical control

Classical control is a course that follows electric circuit theory. Circuit theory introduces phasors, i.e., sinusoidal steady-state analysis, Laplace transforms, transfer functions, poles and zeros, and Bode plots. Classical control should build on that base, not be something totally new. An electric circuit diagram is a graphical representation of physical components (resistors, capacitors, etc.) connected together. Getting to appreciate a circuit diagram, and to think in the language of circuit diagrams, is absolutely essential. Classical control is built on block diagrams, usually with feedback. It is essential that students come to think in the language of block diagrams. Draw a feedback loop with a reference input and plant output. We want the output to follow the reference input. Students don’t get this initially; they don’t know what’s the big deal about that problem, why you can’t just make the output BE the input. It is the teacher’s responsibility to make this problem clear: Why there’s the requirement of stability of the feedback loop, and why this requirement means you can’t just, willy-nilly, make the output equal to the reference input. At first I thought, surely it doesn’t take a whole course to teach just that. But, yes, it does, because the tools you need are non-trivial. The main tool is the Nyquist criterion. Nothing is more revealing about where you stand than the Nyquist plot. Classical control IS the Nyquist criterion.

### The EE curriculum

An interesting question is how the electrical engineering undergraduate curriculum should be structured. There are only four years, so not everything can be covered. Take for example signals and systems. This in itself is huge: Fourier series, discrete Fourier transforms, discrete-time Fourier transforms, continuous-time Fourier transforms, z-transforms, Laplace transforms, linear systems modeled by difference equations or differential equations or convolution equations or state equations, causality, time invariance, finite-dimensionality, stability, transfer functions, perhaps sampling, perhaps modulation, and so on. Certainly a bad way to structure the curriculum is to try to include everything. Another bad way is to include a topic that is never used again, or is not used until much later. A good example of this is to put linear algebra in first year and not use it again until fourth year. Another bad way is to put topics in the wrong order, for example putting electromagnetics before the required calculus. I haven’t seen an undergraduate curriculum that is really sensible. Consequently, when they reach the end of fourth year, most students are confused even about the fundamentals.

There is no feasible way to fix this situation, because no group of professors could agree what should be in the curriculum.

One thing that does work, but helps only a few students, is to have fundamental courses in graduate school taught by really good instructors. Imagine a graduate course on signals and systems taught by Frank Kschischang.