I'm putting the finishing touches to a proposal that I hope allows me to teach a short course at the local high school this summer or autumn for seniors who plan on entering the sciences. It's a course I've wanted to do for 20 years, drawing on info I've pulled together over 40 years, and I'm confident it'll provide the typical student with new perspectives.

That's because I know from observation that even some with years in the technical fields and academia have long forgotten how to address basic topics ranging from technical terminology to helping their students understand a problem in the simplest way.

Who understands the everyday words we use? I'll find that out first, providing some terms that are appropriate to analog design. One key term is “linear.” The graph of the equation for a straight line is y=mx + b. When that graph is representing a linear system such as an amplifier, we usually use the x-axis for the input signal and the y-axis for the output signal. We usually consider the system to be linear only if the line passes through the origin. Or put another way, y=mx + b is linear only if b = 0.

Now, 25 years ago, I explained this to a class of graduate students and their professor, none of whom accepted it. Even after I drew a line that crossed the y axis at 5, thus defining a machine that delivered an output (an offset) with no input!

I'll be providing a few more math examples. How about symbolic division? Everyone divides a numerator by a denominator to secure a quotient expressed by a number. But most don't realize that the quotient can be letters or symbols (i.e., resulting from a direct division of letters by letters), which sometimes comes in quite handy.

Another important area for students is “dimensional analysis.” Your dimensional analysis may be correct but yet the equation you've derived might not be. I'll give numerous examples.

I'll cover some circuit applications. Few people seem to notice that a power supply rectifier/filter and an AM diode detector (for recovering the audio from an RF carrier) are basically the same circuits and can be analyzed similarly. Moreover, they say they know oscillators, but couldn't fashion a simple proof to show that oscillation initiates from the system's inherent noise.

I'll be asking students to bring meaning to an audio-amp spec that has 0.1 percent THD in the context of our inability to hear distortion in the music below 5 percent. I'll also be asking why efforts to match an RF power amp to its load (to secure efficiencies up to 90 percent) is not at odds with the maximum power-transfer theorem (50 percent efficient). Surprisingly, this topic was a subject of intense debate among engineers in a respected radio magazine a few years back.

I haven't yet read a clear, simple explanation in an elementary textbook of the counter-EMF phenomenon in a resistor-inductor circuit. The words most often used, “equal and opposite,” tend to lead to true confusion at some point relating to the “instantaneous” applied-minus-back voltages. Some authors are just beginning to address it, and contributors to the various web fora are still creating ingenious constructs to explain what they think happens.

Taking a step back, there's the matter of “convention,” which apparently runs afoul of energy conservation principles when electrons “flow” from the negative terminal of a battery through a resistor, do work, and yet come out of the resistor at a higher potential. Now the so-called “field-dynamics” explanations say energy flow is opposite in direction to that of electron migration — but it's a bit of obfuscation, I think. Some recent texts imply electron flow is from the plus terminal through the resistor and then to the minus terminal. But plus or minus may be more than just about declaring a “convention.” Let's illuminate.

I could provide many more examples. But my initial goal this summer will be getting young people to inspect the ground floor before they even think of “high technology.”

I've gotta side with the grad students and prof on the question of what constitutes linear. Why should “b” have to be zero? y=3x+5 is a linear line.

Is an adder circuit therefore nonlinear? Or a level-shifter? Is any real opamp circuit, even one with only 1nV of output offset, nonlinear?

Maybe it's just semantics, but I've never heard anyone else claim a linear circuit has to have zero output for zero input.

I've gotta side with the grad students and prof on the question of what constitutes linear. Why should “b” have to be zero? y=3x+5 is a linear line.I agree.

There is a rigourous process for determining if a function is linear or not using tests like communtativity and associativity, if I remember my course on linear algebra (and since I only remember two of the tests, obviously I don't remember much). If it meets the criteria a function is linear, otherwise it isn't. It's like being pregnant, either you are or you aren't. And from what I remember the constant can be any value and not restricted to 0.

Hi Vincent and guests. I can see the logic of saying am amp with a significant offset isn't linear. I think it makes sense to teach students to think about what the specs and the curves mean, but not get overly pedantic about it. In most cases, the question is “is it linear over the needed range” (and maybe with some margin).

Hi Vincent–I'm interested in an example of a proper dimensional analysis resulting in an incorrect equation. Are you saying they just had the wrong equation but the dimensions happened to work out “right”, or something else?

One area this was a problem for many are equations involving gravity in other than SI units. The equations involve terms (g/Gc) where Gc has units of

ft * lb

_{f}lb

_{m}*s^{2 }and Gc has the value 32.174, which people mistakenly take to be the gravitational constant g. Then there is that whole Newtons and slugs thing.

“Now, 25 years ago, I explained this to a class of graduate students and their professor, none of whom accepted it.”

I'm curious. Have you found anyone who has agreed with you?

When reviewing datasheet, normal range is with linear, but if systems go to saturation point, behavior is nonlinear high order differential function. It is not easy to analyze this range. For example, let us look voltage versus junction temperature in the IC chips or some components. As junction temperature is increased, voltage varies exponentially. So, I think that most system has nonlinear behavior in real world.

Hello,

The expression y = 3x + 5 describes a

straightline, not alinearone. The rigorous test for linearity is by way of the so-called “scaling” property, i.e.,If the function y = f(x) is such that f(Ax) = Af(x), where we define

Aas a real number, then the function is linear. Otherwise, we have what is called anaffinefunction.Could be semantics. But let's consider if we had a very high power audio amplifier or servo amplifier. Like the the old Crown DC300 that was used for theater sound systems and shaker tables. You would (i expect) prefer the transfer function to pass thru the origin. Otherwise, your speakers/shaker table would be distended/moved a bit off of center point whilst you awaited the music to start.

Certainly, in real-world systems, there will always be a little offset. But that's not quite where Vincent was going with this (as he is noting in a separate comment).

@Vincent

“When that graph is representing a linear system such as an amplifier”

If I were in your class when you made the above claims, my response would be that then there can never be a linear amplifier system because there is no such thing as zero offset voltage.

So what is the point you are making to the class when discussing an analog system in the context of something that is impossible to build?

Hi Scott,

Yes, although not many in the engineering and science fields. But if you're a mathematician, it's a fundamental piece of knowledge.

OK, so you're on a mission to rename all…of

thosecircuits to “affine”? Best of luck.Engineering is not mathematics, even if there's a strong connection. And different fields often use the same terms to mean different things. Personally, I have bigger battles to fight.

May as well say that a resistor shouldn't be called a resistor, because it isn't perfect. Ditto all other components. Now excuse me while I go misuse my ohmeter to measure my resistinducnoisesourcethermistcapacitor.

Michael,

Again, the issue is not about redefining any terms, nor is it about “offset” or “designing the perfect system.” The issue is defining “linearity,” whether in op amps or kilowatt systems. In mathematics, which has its basis in precise definitions and rigorous proofs, the lesson is that the engineer or scientist cannot use the terms “straight-line equation” and “linear equation” interchangeably.

@Michale engineering is all about mathematics, its concerned more on mathematical method and techniques used primarily in solving problems and in engineering works.

Hi eafpres,

I'll be asking students to use their imagination to try to create new laws based on consistent dimensional analysis, and then explain the significance of each relation. If the student grasps the challenge, he/she might well create equations that are dimensionally consistent but yet unrealistic. Thus he'll realize the advantages and possible perils of his depending on dimensional analysis.

Say, for instance, I ask the student to develop a relationship between pressure and density. The student then posits that pressure is directly proportional to density, a possible scenario. If the student follows through correctly on the dimensional analysis, he'll find that he can interpret the result to mean that the pressure exerted on an envelope passing through a vacuum-mail system is directly proportional to the area of the envelope and the density of its contents, and inversely proportional to the square of the time it takes the envelope to travel down the mail chute. He'll have done the dimensional analysis correctly, but what about his conclusion in the real world?

I'm wondering which mathematician came up with the name “linear equation” for describing y=mx+b when as we're told, that equation is not linear.

This is a very interesting discussion.

I hesitate to use Wikipedia as a source, but nevertheless Wikipedia differentiates linear equations from linear functions. y=mx+b is a linear equation as virtually all engineers and scientists would agree. For the narrow case where b = 0, that is a linear FUNCTION. Linear functions are a concern for mathematicians whose first focus is not on physically realizeable systems-something engineers/scientists MUST do or they don't eat.

FYI…if the objective is to teach with the intent to attract young minds to science, I don't think semantic/pedantic tricks are a good teaching method. Better to encourage than to demonstrate to young minds they don't know something. You know…the old adage of attracting bees with honey versus a fly swatter approach.

Sorry you don't agree. But even Wikipedia carefully couches its terms and discussion on “Linearity” under the section “Linear Polynomials,” where the section begins (and also ends) with a statement that their use of the word “linear” is a different one to the initial definition that has gone before. See http://en.wikipedia.org/wiki/Linearity

That implies they recognize a formal definition of linear, and there is also a definition that is sometimes called the “rebellious” definition. As to your “differentiation” of y = f(x) as an equation versus f(x) as a “function,” it's actually of no import in the context of this discussion on linearity.

To me, defining a term such as “linear” is not a question of semantics or pedantics or making students happy. It's a question of defining it formally the first time so students don't become confused later, independent of a discussion of electronics or chemistry or physics.

The most positive thing that has happened in these discussions is we all learned something. It was enjoyable. Thanks for engaging with me.

@Vincent, you are right at some point. Defining linearity is purely mathematics and defining linear systems in the context of engineering or science is totally different. Practical systems can be considered linear with exceptions like

“system is linear over desired range”which is engineering. An emplifier will never be able to satisfy the mathematical equation over infinite range. While theorical calculations for linear equation can. So, we should keep both saperate.When taking one of automatic control subject very long time ago, all contents were math formulation including a variety of function type. During the class, all students were wondering if this class is pure math (application math) subject. Teacher has a lot of experience in automatic industry with a very strong math background. After completed this class, for example, we knew how to approach robotic behavior as math formulation. After understanding this type behavior, all students can better design control analogy circuit for each components including passive components in detail. I think that Math is the basic foundation course for current engineers and the future engineers.

I understand linearity of any equation by its slope. If any two adjacent points on a graph of the equation are pointing to same direction, then equation is linear at those two points. This is what is done in small signal analysis where nonlinear function is assumed to be linear at infinitely small section of graph.

It's kind of interesting that in the heyday of analog, the three jobs that required great attention to the issue of linearity were 1) designing data converters, 2) designing CRT sweep circuits and 3) designing audio equipment. Of the three it was likely the audio design folks who developed the most precise measurements of nonlinearity, but they were also the LEAST sensitive to “non-zero offset” since that was generally discarded by the next non-DC-coupled stage! In that field it was customary to ignore an offset of 1% or even higher while chasing nonlinearity at levels that were 80 decibels or less under “reference” where maximum undistorted output was probably 20 decibels higher still. I believe data converters on the other hand were specified for distortion by the greatest displacement from a straight line between the endpoints, and that spec obviously considered offset to be important. In other words individual fields have entirely unique working definitions of “linearity” and how they measure deviations from ideal, and trying to force a particular definition down some poor student's throat (without knowing what field they might be inclined to work in) seems to me to be emphasizing some of the WORST, not best, of what I remember from my own days in school!

It's quite simple – the word “linear” has several meanings. It is pointless trying to force /your/ particular choice of meaning on others without explaining which one you mean. “Linear” can mean “degree 1 polynomial” – i.e., a straight line. It can also mean “linear operator”, which requires that f(ax) = af(x) and f(x + y) = f(x) + f(y), and in simple terms often means “proportional”. In electronics, it is often this second meaning that is relevant. But your job as an educator is to tell people to use the right meaning in the right context – not to try and tell them other meanings don't exist.

Back when I was in Trig, a few decades ago, I got tired of plotting the parabolic equations by picking X and calculating Y. After several equation manipulations, I discovered a form of y=m(x+a)^2 + b that allowed me to plot a parabola as quick as a straight line; plot one point and then plot the slope. My teacher through I was conforming to what I had already plotted, and gave me increasingly more difficult problems to the point that she would not give the students on a test. When I showed here that my method worked and was able to explain it, she was quite impressed. That was never taught to the class.

When my kids started plotting parabolas, they were shown the 'pick X, calculate Y' method and I had fun helping them. I took my tutoring a bit further and tried to show them what would be later coming in math with the equation form and challenged them to try and plot a parabola as fast as a straight line. Later, the teacher did teach the form, but still had them do the old method for plotting.

What I find math avoids is to show students a 'why' things are done. Teachers seem to show students what the teacher was taught and do not encourage the students how to expand their thoughts and ask the 'why' questions. Maybe it is because the teacher does not know.

If I can figure out the best way to write my quick-plot method, I will post later.

That's the part I left out: Stages were typically AC coupled, so the offset was automatically eliminated.

I had thought, when you started that sentence it was going to end differently:

>>Back when I was in Trig, a few decades ago,

we only had sine and cosine – tangent hadn't been discovered yet. And we had to walk to school, barefoot, in two feet of hyperbolic functions.But seriously, I look forward to your more detailed explanation.

Vicent,

I agree with you, the first step is to say straighten mathematics for applications in electronic circuits, no matter where you are going, there will always be the math and calculations correctly associate the solution of problems.

Our analog world and or digital, there are no ways to solve a problem, matrices, complex numbers, trigonometric functions will not be able to miss …

But what will really make the difference is the application of all calculations to a feature, be it with more complex circuits, or more simple …

Good luck Vince!

About RF matching and not meeting 50% matched load efficiency…

RF amplifier output is more like a current source seeing a resistive load. Current source is has infinite impedance (in real world higher than the load) and therefore is not limited by 50% efficiency of voltage source with series internal resistance.

It took me a while to understand it.

Dear Vincent,

My book 'An Analog Electronics Companion' (ISBN 978-0-521-68780-5), which includes maths, physics, simulation and electronics may be of assistance for your proposed course.

Scott Hamilton.

What I'm bedeviled by are the number of conflations of “linear” with “constant”. So many electronics folk throw the word linear around and think they understand what it means. So I have recently read that the input capacitance of a tube is very linear, and there is the long-running near-universal denotation of a transistor with constant beta versus collector current as “linear beta” ! I sometimes try to get the person using that terminology to see that a graph of beta versus collector current could be a line with a substantial slope and still be properly described as having linear beta, but this usually is not appreciated, the attitude often being that “We all know what we mean so why be so perverse? “

Now one could argue that a linear function in slope/intercept form y = mx + b has a sewt of solutions when m = 0, i.e. with y = b, and hence that it is still proper to call it linear. But surely this is not helpful to the student.

When one looks at mechanisms for distortion again we hear it said that a capacitance varies nonlinearly and hence is a mechanism. But although the distortion spectrum will likely be different, it is sufficient most of the time merely that the capacitance /changes/ with voltage (say), and can do so adhering to a linear dependence. Even though most if not all such C /does/ vary in a nonlinear way, again the student is not helped by having that term appear /as if/ it's the nonlinearity of the dependence that is responsible.

Brad Wood

A function can certainly be linear and not pass through zero if you recognize and identify the offset. That makes it linear with an offset. We see that in many amplifier circuits, and wind up needing to derive a work-around.

The very excellent description and derivation of counter emf is found in a very interesting book, “Mathmatics and the Physical world”, by Morris Kline, published by Thomas Crowell company. It goes through the discovery of how mathmatics relates to just about everything physical, from planetary motion through light, gravity, and magnetics. A very interesting book. It delivers great insight from a somewhat different point of view.

I couldn't help but notice that most of the respondents have taken it upon themselves to try to “correct” the various misconceptions that this article talks about, and in doing so, many of them (many of YOU) only further those misconceptions. Guys, you are only making examples for Vincent to use in his course. I'm sure he knows a thing or two about electronics, and doesn't need to be “taught”.

So this is how all those arguments get started.

I wonder how far this will snowball before it dies down.

Hi, a RF PA output is indeed more often nearly a current source, but only as long as your Vce or Vds is above the saturation voltage. So there is no single best point you may take the impedance from!! This causes that there is a mismatch between the points best for small-signal gain (S21), best output power (Pout) or best efficiency (like PAE). So you need a compromize, e.g non of these points might be the best overall, like also for low IM3.

The best starting point to find it is for design is to 1st look at the load line, mainly defined by supply voltage-Vsat and by max current – much more essential & 1st semester engineering and well before any RF teaching class!!

Just keep your head down and no one will get hurt (sounds like we're in the middle of a bank robbery).

Maciel: “there are no ways to solve a problem”

Might as well go do something other than engineering if there are no ways to solve a problem. Maybe go into politics where problems are created, but never solved.

Probably not expressed myself very well, what I meant and that there is no other way to solutions of problems in our world without a prior knowledge of mathematics …

Vince, I applaud your effort to persuade potential engineering students to concentrate on understanding the fundamentals. I consider myself a pretty clever analog circuit designer, but I must admit I have a certain disdain for “math snobs”. I recently read a biography of Michael Faraday and I felt an immediate kinship. He was a brilliant mind who thought in concepts, which I tend to do … visualizing the interactions between various components. Unfortuately, Faraday lived in a time of intense intellectual snobbery and, because Maxwell developed equations to describe Faraday's work, Maxwell is the name everyone remembers and gives credit to for describing the fundamental relationships between electricity and magnetism. I do a fair amount of technical writing and deliver seminars to any receptive audience on the subject of system grounding and interfacing, with an emphasis on high performance audio (where dynamic ranges of 100+ dB require attention to a lot of pesky details). My audiences are most often sound system installers, sometimes equipment designers, and less often universities (including an invited lecture at MIT in 2007). I find that by explaining concepts in simple terms, and skipping the intimidating math, most students can even understand inductance and counter-EMF. I hope your talks to high schoolers sticks to the beauty of simple concepts and doesn't scare them away with fancy equations. In the end, most circuit components have relatively loose tolerances anyway, so why get carried away with equations that include every possible contribution to the answer when most of them turn out to be insignificant. As I approach semi-retirement (I'll be 70 next year), I've decided that teaching is my real passion and I'd like to do something similar to Vinces venture.

@GuruOfGrounding—Agreed, simplest is best, as I stressed during the latter part of an interview a few years back (see for example, http://www.hbagency.com/podcast, find B2B Technology section, 3.12.09, click play button).

As for the math, one should endeavor to get by with the simplest math possible. It seems to me when an author suddenly jumps from a conceptual technique to a chapter that fills the textbook (or blackboard) with a never-ending run of vector calculus equations, he has departed from teacher to one who wants to impress you with how much

heknows. The reader/student thus loses out.VB – still looking forward to more blogs from you on topics relating to education, math, analog design, engineers, and a blend of all of these in any combination.