In Praise of Dimensional Analysis

My high-school physics book had a cartoon in it where Einstein was standing in front of a chalkboard, having crossed out E=mc and E=mc^3. The caption was something along the lines of, "Aha! So the units *do* work out in that one!"

Very good post! We tend to get lazy doing plain ohm's law calculations, since they are so simple. But this creates the bad habit of ignoring the dimensions on more complex problems where errors are more likely.

We need to break these old bad habits and reinforce new ones. Your article is a very good place to start.

Thanks!

Mike Monett

Essential in electro-mechanical system design and analysis!

Perhaps you can say dimensional analysis is where theory and reality meet.

In a PID controller, we are summing voltages (which is fine) but they also represent an error, the time integral of an error, and the derivative of an error.

The error is in volts. The integral is in volt-seconds. But we sum them, and nothing explodes!

John

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Hear! hear! We need more of this. (T as the dimension of gain is absurd.)

But those are vitally important. The arguments to transcendental functions had darn well be dimensionless, even if you need to do extra work (normalize) to make them so. The sine if two meters is a mistake.

Sometimes you can't even compare numbers of like dimension. For example, the dimensions of torque and work are the same, but they are inherently incommensurable. The MKS dimensions of volts are obscure, but electromagnetic dimensions hang together even if they stand apart from the more common ones. (The unit of flux is a volt-second; the unit of inductance is volt-second/ampere-turn. "Turn", like radian. is dimensionless.)

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Our educators should make more of a big deal of this than they do. Thanks, Tim.

Jerry

If the output of the integrator is in volts, then it's gain must be in volt/volt-second.

If the integrator is buried in software, then it's gain is in counts/count-tick, though you'll often see integrator gain expressed as (something)/(something-seconds) -- because someone has taken it on themselves to obscure the sampled nature of the controller by scaling the integrator (and derivative) gain.

Yes.

No. It's volts. The gain of the integrator has the unit V/(Vs), and the integration is volts over time, so the seconds cancel out and you get volts again. A similar arguments holds for the differential term.

robert

BUT. It is *very* useful to specify "dimensionless" numbers with their underlying dimensions, eg voltage regulation in volts per volt or if really good, in millivolts per volt; the latter giving a clue to a sensitivity or gain in the system. Also rather useful in error analysis.

Hmmm... X double dot = - X

Actually, my science teacher in High School emphasized units; minimal steps that must be shown in solving a given problem was 1) present the equation in standard form (E = I * R), 2) substitute the knowns WITH UNITS (22 Volts = I Amps * 11 Ohms), 3) solve for the unknown AND GIVE UNITS (I = 2 Amps); draw a box around the answer WITH UNITS so it can be found. Any of these criteria found missing will result in a grade of ZERO for that question (therefore, since there is no box, i get a ZERO). It was acceptable to add any number of intervening steps either for clarity or ease of calculation.

Saved my butt in college as i was able to derive an equation from the units involved in the question; with that, i solved the problem and passed the test.

He deserves kudos. I'm amazed how many IEEE papers you see where units are left off of graphs, equations containing "magic constants" are presented without specifying the units assumes that are required to make the constant correct, etc.!

No wonder. The PhDs without the PIDs are much more common, then the PIDs without the PhDs.

BTW, in the theoretical physics, they use the dimensionless units to avoid the heavyweight dimension constants:

e = c = h = 1

How about that?

No. It crashed because it is impossible to be perfect in all but in the very small projects. This problem is mich wider then just the agreement about the dimensions.

The beloved Microsoft style is using the so-called 'hungarian notation' to avoid that sort of mistakes. lpSTR, HANDLE, DWORD and such.

Charles Simonai, who is the inventor of this style, just recently made it safely to the space and back :)

C++ allows you defining the explicit types like "VOLTAGE", "CURRENT" and such, so the dumb mistakes are avoided. However, this approach is seen by many as the counter productive and resulting in the inefficient code.

Dimensional analysis is just a trivial check to avoid a class of a simple mistakes at the low level.

"Wescott" would better be reserved as the name of the not yet discovered radioactive chemical element of the halogen group. "Avins" is a parameter of a Markov source. What could be "Grise" ?

Because it is generating so much traffic! It is trivial, so anyone can add his two cents.

There is a method of thermodynamic potentials, which derives a useful formulae from the dimension considerations by means of differentiation and integration. However if you divide your phone number by your SSN, it is not going to be very usefull.

This is the basic thing not worth mentioning which every professional should do automatically.

It would crashed of some other trivial or non-trivial reason. Or at some other time. Somebody else would be sacrificed as a scapegoat. That's the only difference.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

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Spoilsport! :-)

Jerry

(snip)

This reminds me of something that has occurred to me in the past, and that I would like to see if people here agree.

It seems to me that in calculations physicists usually give variables quantities with dimensions, where engineers usually factor out the dimensions. For example,

A physicist might say:

F = m a, where the variable m might have the value 3kg or 5g.

An engineer might say:

F(Newtons) = m(kg) * a (m s**-2), such that m has the value 3 or 0.005.

That is, the dimensions belong to the equation, but not to variables.

It might be because most programming languages don't keep units with variables, so that one must factor them out before assigning a value to a variable.

I would be interested to see if others agree or disagree.

-- glen

I think that's true--engineers are typically taught to pull out all those fundamental factors of hbar, 4*pi/c**2, and so on, crunching them all into some anonymous constant to save labour and blunders. When you do that, you have to keep track of the units by hand, whereas in the physicists' method, the units get carried along automatically.

In fact, people doing relativistic field theory usually use c=1 style, in which all the calculations are done with just numbers, and the proper conversion factors get figured out at the end, by reverse dimensional analysis--sticking in the right powers of c, hbar, G and so on to make the units come out right. It turns out that this is a well-defined procedure, so it saves labour and you still get the right answer at the end. (I'm not a field theorist, so I've never done this.)

Cheers,

Phil Hobbs

The one I remember has E=mc and E=mc**3 written down, and someone straightening up the books on a nearby bookshelf saying. "Well now that's all squared away."

-- glen

In that example, you showed kilograms or grams -- I grew up with thousands prefixes, so I change them on the fly as it suits me. A 1k resistor has always been, and always will be, a 1000 ohm resistor as well, and not a 1 * (1000 / 1k) resistor. :^P

Tim