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In Praise of Dimensional Analysis

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Peter Nachtwey

Jan 1, 1970
0
:
:
: Tim Wescott wrote:
: > So why is dimensional analysis so cool?
:
: Because it is generating so much traffic! It is trivial, so anyone can
: add his two cents.
:
: Vladimir Vassilevsky

Yes, this made me laugh. Everyone has an opinion but few have answers,.
Fewer yet have right ones.

Peter Nachtwey
 
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Jon

Jan 1, 1970
0
After six and a half years, I still get email about my article "PID
Without a PhD"
(http://www.wescottdesign.com/articles/Sampling/pidwophd.html) from
people who are asking for clarifications, or are asking questions that
go beyond what I could say in the 5000 word limit that Embedded
..
You raise some good points.
..
One of the problems with the modern unit system (Pascals, Siemens,
etc) is that the fundamental meaning of the unit is lost. For
example: Pressure X Area = Force.
Pascals X Square meters = Pascal Meter^2. So what? You must first
convert Pascal to its fundamental definition, which is Newtons/Square
Meter. Now dimensional analysis makes sense: (Newtons/square meter) X
(Square meter) = Newtons, which is consistent with a unit of force.
..
Another example is characteristic impedance of a transmission line.
We've all learned that the equation is R = SQRT(L/C), where L =
inductance/unit length, C = capacitance/unit length. Resistance =
(Henries / Farads)^.5? You need to get back to the fundamental
relationships among voltage, current and time for inductance, and the
fundamental relationships among current, voltage and time for
capacitance. v = Ldi/dt.
i = C dv/dt. From these fundamental equations, you can get the
fundamental units of L and C: L=VoltSeconds/Ampere, C= AmpereSeconds/
Volt. Now the equation for characteristic impedance makes sense, in
terms of its fundamental units: R = SQRT([(VoltSeconds/Amps)/
(AmpsSeconds/Volt)]
R = SQRT(Volts^2/Amps^2) = Volts/Amps = Ohms.
 
J

Jerry Avins

Jan 1, 1970
0
Jon wrote:

...
Another example is characteristic impedance of a transmission line.
We've all learned that the equation is R = SQRT(L/C), where L =
inductance/unit length, C = capacitance/unit length. Resistance =
(Henries / Farads)^.5? You need to get back to the fundamental
relationships among voltage, current and time for inductance, and the
fundamental relationships among current, voltage and time for
capacitance. v = Ldi/dt.
i = C dv/dt. From these fundamental equations, you can get the
fundamental units of L and C: L=VoltSeconds/Ampere, C= AmpereSeconds/
Volt. Now the equation for characteristic impedance makes sense, in
terms of its fundamental units: R = SQRT([(VoltSeconds/Amps)/
(AmpsSeconds/Volt)]
R = SQRT(Volts^2/Amps^2) = Volts/Amps = Ohms.

So what is that in meters, kilograms, and seconds? Does it matter?

Jerry
 
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glen herrmannsfeldt

Jan 1, 1970
0
Jon wrote:

(snip)
Another example is characteristic impedance of a transmission line.
We've all learned that the equation is R = SQRT(L/C), where L =
inductance/unit length, C = capacitance/unit length. Resistance =
(Henries / Farads)^.5? You need to get back to the fundamental
relationships among voltage, current and time for inductance, and the
fundamental relationships among current, voltage and time for
capacitance. v = Ldi/dt.

Capacitance is in cm, resistance in s/cm, inductance in s**2/cm.

Capacitance per unit length is dimensionless, inductance per
unit length s**2/cm**2, so sqrt(L/C) is s/cm, just like
resistance!

I used to be able to do it in MKS and CGS units about equally,
and sometimes Heaviside-Lorentz units. (Similar to CGS, but
without so many 4pis around.)

-- glen
 
J

Jerry Avins

Jan 1, 1970
0
glen said:
Jon wrote:

(snip)


Capacitance is in cm, resistance in s/cm, inductance in s**2/cm.

That's CGS absolute, with volts in abvolts (1 abvolt = .01 microvolt)
and current in abamps (1 abamp = 10 ampere) I'd rather use MKS units; so
would you! :)
Capacitance per unit length is dimensionless, inductance per
unit length s**2/cm**2, so sqrt(L/C) is s/cm, just like
resistance!

I used to be able to do it in MKS and CGS units about equally,
and sometimes Heaviside-Lorentz units. (Similar to CGS, but
without so many 4pis around.)

Maybe you wouldn't rather. :)

Jerry
 
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glen herrmannsfeldt

Jan 1, 1970
0
Jerry Avins wrote:

(snip)
That's CGS absolute, with volts in abvolts (1 abvolt = .01 microvolt)
and current in abamps (1 abamp = 10 ampere) I'd rather use MKS units; so
would you! :)

One professor, when explaining the unit system he expected in
answers, explained that his house power line was 1/3 of a statvolt.

-- glen
 
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Peter Nachtwey

Jan 1, 1970
0
["Followup-To:" header set to sci.electronics.design.]

John said:
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.

Yes.

What if my output is 4-20ma or +/- 50 ma like a servo valve?

Peter Nachtwey
 
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joseph2k

Jan 1, 1970
0
Peter said:
["Followup-To:" header set to sci.electronics.design.]

John said:
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.

Yes.

What if my output is 4-20ma or +/- 50 ma like a servo valve?

Peter Nachtwey
And just what part of it is a simple scalar units transform is so beyond
you.
 
H

Heinrich Wolf

Jan 1, 1970
0
Vladimir Vassilevsky said:
Tim Wescott wrote:

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

e = c = h = 1

How about that?

I knew someone who claimed to be a professor of theoretical nuclear
physics--- oh no, he really was --- who wildely objected
against this.

My scientific origins are in theoretical physics but I mostly worked
as a softwareengineer--- now lurking here to learn about DSP. What
baffled me again and again is how engineers (in Germany) approach a
problem: their question is ``where is the applicable formula''. I was
trained to ask: ``what is the mathematical modell that allows me to
handle this kind of problem.''

From studying mathematical modells, the physicist is probably just
much more trained to handle long symbolic calculations as the average
engineer and from this he has a stronger ability to analyze such
formulas ``at a glance''--- w/o inserting units. To him the essential
thing is the physical quality that is measured, like length or time,
not the units that someone happens to use.

Measured values just tell us how often some well defined object will
fit ``into'' the measured object; the sole purpose of units is to pass
along what object the one who measured happened to use for reference.
Thus I like to claim: all those units were completely superflous,
hadn't things been hopelessly messed up, starting at beginning of the
world, by merchants and engineers.

Comparing any speed to the speed of light in vacuum is perfect---
might even have some advantages, when on speed-limit signs a non-zero
digit does not appeare until the 9-th (or so) position after the
decimal point ...
...

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.

Strongly typed languages were designed to support this style of
programming. They come in two flavours: ``static typing'' and
``dynamic typing''. If you want to see powerfull representatatives of
both paradigms, just look at ML and Scheme.

Static typing gives you strict compile-time type check at no run-time
overhead; dynamic typing gives you strict run-time type-check .
 
R

robert bristow-johnson

Jan 1, 1970
0
Tim Wescott wrote:

(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.

i disagree, i guess. when i was teaching EE classes (circuits) and as
a grad student in EE, i would be pretty hard on any engineering
student that did it the latter way. i think Jerry's little proverb
said it best: "Mathematicians routinely ignore units, but engineers do
so at their peril."

the cumbersome method you ascribed to engineers, Glen, is almost as
bad as ignoring the unit. (because your root algebraic equation is
dimentionless. in the latter case, F = ma is actually expressed as
the dimensionless F is the product of the dimensionless m and the
dimensionless a, as long as you express all three in their base SI
units.

r b-j
 
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robert bristow-johnson

Jan 1, 1970
0
I knew someone who claimed to be a professor of theoretical nuclear
physics--- oh no, he really was --- who wildely objected
against this.

why would a theoretical nuclear physicist have a problem with that?
these are one example of "Natural units" (look that up in Wikipedia)
and i think these are called "electronic units" if the "h" is replaced
with "hbar". but qualitatively, the are electronic units. they can be
compared to Stoney units (but i think Stoney normalizes G instead of
hbar), Planck units (but Planck normalizes G instead of e), or Atomic
units (which normalize the mass of the electron instead of c).
personally i like Planck units the best because they have no prototype
particle, object, or "thing" that needs to be sorta arbitrarily chosen
as a base unit. it's better, in my opinion, to normalize these
parameters of free space, before introducing any special particles or
objects, to define your base Natural units. i which Planck would have
normalized 4*pi*G instead, that would have been more natural.

other good Wikipedia articles that are related is

Planck units
Physical constant
Dimensionless physical constant
Fundamental unit

i had something to do with those articles, including Dimensional
analysis.

r b-j
 
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Clay

Jan 1, 1970
0
Our educators should make more of a big deal of this than they do.
Thanks, Tim.

Jerry

Hello Jerry et al,
I can assure you that my students are forced to show units all of the
way through their physics calculations. I also show the utility of
expressing things as ratios, so the units cancel out. Such as given a
pendulum clock that runs normally on Earth, how much faster/slower
will it be on the Moon? (g_moon approx 1/6 that of the Earth's)

Clay
 
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redbelly

Jan 1, 1970
0
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

That was one of Gary Larson's classic Far Side cartoons.
"Everything is squared away. Yep, squa-a-a-a-a-ared away ..."

Mark
 
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redbelly

Jan 1, 1970
0
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

Somewhere, often in the form of an R*C, those volt-seconds are divided
by seconds to get back to volts.

Mark
 
C

Clay

Jan 1, 1970
0
why would a theoretical nuclear physicist have a problem with that?
these are one example of "Natural units" (look that up in Wikipedia)
and i think these are called "electronic units" if the "h" is replaced
with "hbar". but qualitatively, the are electronic units. they can be
compared to Stoney units (but i think Stoney normalizes G instead of
hbar), Planck units (but Planck normalizes G instead of e), or Atomic
units (which normalize the mass of the electron instead of c).
personally i like Planck units the best because they have no prototype
particle, object, or "thing" that needs to be sorta arbitrarily chosen
as a base unit. it's better, in my opinion, to normalize these
parameters of free space, before introducing any special particles or
objects, to define your base Natural units. i which Planck would have
normalized 4*pi*G instead, that would have been more natural.

other good Wikipedia articles that are related is

Planck units
Physical constant
Dimensionless physical constant
Fundamental unit

i had something to do with those articles, including Dimensional
analysis.

r b-j

Hello Robert,

That is cool you had some input on those articles. If you recall I use
atomic units all of the time. There are two variations. One uses a
Rydburg and the other uses a Hartree for the fundamental energy unit.

The main reason for using these is the Schrodinger equation becomes
quite simple in appearance. A factor of two shows up in the energy
term. So that is why there are two variations. The distance becomes
scaled to Bohr radii - the mean distance to the electron in the ground
state of Hydrogen. And 1 Ry is the binding energy.

Clay
 
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robert bristow-johnson

Jan 1, 1970
0
... If you recall I use atomic units all of the time.

what i recall Clay, is that it was YOU that first told me the name for
Planck Units:
http://groups.google.com/group/sci.physics/msg/e1d9352d0a6b64b3
and i thank you profusely. because you did that, i was able to do web
searches, found papers/web_sites/books by Michael Duff, Gabriele
Veneziano (pioneers of string theory), Lev Okun, John Baez, and John
Barrow and i've had several really neat email conversations with ALL 5
of these guys about the nature of fundamental physical constants (the
only ones that count are the dimensionless ones, any notions of a
"varying c" or "varying G" are not even wrong, they're meaningless,
and for those who can't see that, just think of everything measured in
Planck units). and then later i got into Gravito-electro-magnetism
(GEM) a little and some interest in the Gravity Probe B (which*still*
hasn't been able to conclusive say that frame-draggin or
gravitomagnetism or gravity waves can be measured, they are behind
schedule.)

none of that fun would have happened if you hadn't done that for me
nearly a decade ago. thanks.
There are two variations. One uses a
Rydburg and the other uses a Hartree for the fundamental energy unit.

The main reason for using these is the Schrodinger equation becomes
quite simple in appearance.

discovered a word for that, too: "Nondimensionalization." there's a
wikipedia article on that also. i may have done a minor edit to that.
but dunno.
A factor of two shows up in the energy
term. So that is why there are two variations. The distance becomes
scaled to Bohr radii - the mean distance to the electron in the ground
state of Hydrogen. And 1 Ry is the binding energy.

isn't normalizing the Rydberg constant have similar effect as fixing
the Bohr radius (with another dimensionless alpha tossed in)?

just curious.



r b-j
 
G

glen herrmannsfeldt

Jan 1, 1970
0
robert bristow-johnson wrote:

(snip)
i disagree, i guess. when i was teaching EE classes (circuits) and as
a grad student in EE, i would be pretty hard on any engineering
student that did it the latter way. i think Jerry's little proverb
said it best: "Mathematicians routinely ignore units, but engineers do
so at their peril."

It isn't ignoring units, but more like tunneling then. One must
convert given quantities to the appropriate units before applying
the equation. One can then use the numbers with a slide rule or
calculator, and apply the appropriate units to the result.

When writing down the process, what is usually called "show your work"
all units will still be shown.
the cumbersome method you ascribed to engineers, Glen, is almost as
bad as ignoring the unit. (because your root algebraic equation is
dimentionless. in the latter case, F = ma is actually expressed as
the dimensionless F is the product of the dimensionless m and the
dimensionless a, as long as you express all three in their base SI
units.

I probably overemphasized the difference, but yes, the algebraic
equation is dimensionless, but so are all calculators that I know of.

The problem with the way I called the physics way is that it can
result in unusual or inconsistent units. One might end up with
an acceleration in meters/second/hour, for example, or worse.

For the engineer way, one converts all given units to those
specified, does the algebra keeping the units (except when
entering them into a calculator), and then converts the result
to the desired unit.

Though the division probably isn't all that strict, and it wouldn't
surprise me if many EE's used the physics way.

-- glen
 
R

robert bristow-johnson

Jan 1, 1970
0
robert bristow-johnson wrote:

(snip)


It isn't ignoring units, but more like tunneling then.

dunno what that means.
One must
convert given quantities to the appropriate units before applying
the equation. One can then use the numbers with a slide rule or
calculator, and apply the appropriate units to the result.

When writing down the process, what is usually called "show your work"
all units will still be shown.


I probably overemphasized the difference, but yes, the algebraic
equation is dimensionless, but so are all calculators that I know of.

sure. as noted by sombuddy else, you can program computers
(especially with C++ or some other OOP) to attach a unit from a known
list to the numerical quantity. that way we conceptually have a
dimensionful quantity in the computer.
The problem with the way I called the physics way is that it can
result in unusual or inconsistent units. One might end up with
an acceleration in meters/second/hour, for example, or worse.

this is what conversion factors are for. these conversion factors,
like (0.3048 m/ft) are dimensionless (even though they have units
inside the expression), in fact should be the dimensionless 1 so that
multiplying or dividing by it changes nothing.
For the engineer way, one converts all given units to those
specified, does the algebra keeping the units (except when
entering them into a calculator), and then converts the result
to the desired unit.

but the requirement is that all units chosen must be consistent. the
so-called "physics" way you can actually accelerate a pound mass by
(mi/hr)/s and have a meaningful equation (and an unusual unit of
force) naturally pop out.
Though the division probably isn't all that strict, and it wouldn't
surprise me if many EE's used the physics way.

i guess i'm a partisan for that method. almost to the point of
facsism.

i'm getting less tolerant in my middle age. (Jerry, does it get worse
or better as we get older?)

r b-j
 
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Robert Adsett

Jan 1, 1970
0
glen said:
The problem with the way I called the physics way is that it can
result in unusual or inconsistent units. One might end up with
an acceleration in meters/second/hour, for example, or worse.

I can imagine circumstances in which that would quite a useful unit for
acceleration.

Robert
 
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