In Praise of Dimensional Analysis

T

Tim Wescott

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
go beyond what I could say in the 5000 word limit that Embedded Systems
Programming magazine imposed.

The one I got today made me think.

It didn't make me think because it was hard to figure out the answer, or
because it was hard to explain the answer so it would be understood. It
the gains in a PID controller, and I was impressed that someone who was
obviously just approaching the subject was paying enough attention to
want to do a correct job of dimensional analysis.

It made me think of the innumerable times when I've done some complex
calculation and ended up proving something absurd such as measures of
length in units of gallons, and the subsequent discovery of my error.
It made me think of the Mars Climate Orbiter which crashed because one
team specified a motor in pound-seconds while another one used
Newton-seconds (http://mars.jpl.nasa.gov/msp98/news/mco990930.html). It
made me think of how much I like MathCad, because I can set variable
values with the correct units, and how inconvenient nearly every other
programming language is because numbers are just that -- anonymous numbers.

So I thought I'd write a little bit about dimensional analysis, why I
like it, and why you should use it, even if you think it is tedious and
trite.

Dimensional analysis is a method used by engineers and scientists that
extends the notion that "you can't compare apples to oranges" to the nth
degree. Dimensional analysis says that every variable in any physical
problem has units, and that you ignore these units at your peril.

The basic rules of dimensional analysis are these:

1. There are very few naked (i.e. dimensionless) numbers.
2. You can only add (or compare) two numbers of like dimension --
you can't compare feet with pounds, and if you're being strict
you can't even compare pounds (which is strictly a measure of
force) with kilograms (which is strictly a measure of mass).
3. You can multiply (or divide) numbers of any dimension you want;
the result is a new dimension. So if I pour water into a pan, the
water at the base of the pan exerts a certain force on each bit
of area, the resulting pressure is measured in pounds/square inch
(PSI), or in Newtons per square meter (N/m^2, or Pascal).
3a. You can honor famous people by naming dimensions after them --
Pascal is the metric unit of pressure, Ampere of current. Avins,
Grise, and Wescott have yet to be defined.

So why is dimensional analysis so cool?

If you are doing a long calculation and you track your dimensions, some
mistakes will show up as incorrect dimensions.

It can give you insight into the operation of a system -- most
explanations of fluid dynamics that I have seen rely heavily on
dimensional analysis in their arguments, and when they do so such
aerodynamically important quantities as Reynold's numbers and Mach
numbers fall out.

When you do design with physical systems, careful dimensional analysis
keeps you out of trouble, even when you're getting the math right.
Remember that Mars Lander? Had someone been carefully checking
dimensions instead of looking at numbers and making assumptions, it
would have been a successful mission instead of a famous crash.

How do you use dimensional analysis?

For working forward, such as finding the relationship between an
aircraft's speed and it's lift, or finding the relationship between
mass, speed and energy (remember E = mC^2?) you find out the dimensions
that your answer _must_ have, then you hunt down candidates in that field.

For working backward, you do all of your calculations with dimensions,
then make sure that, for example, if you're commanding a system in feet
it's moving in feet, and not apples/minute. If your answer ends up
being in gallons/inch^2, you know you're close but may have to multiply
by a constant.

For a more full exposition of dimensional analysis I refer you to the
Wikipedia article (http://en.wikipedia.org/wiki/Dimensional_analysis),
or whatever your own web searches turn up.

In the mean time, the next time you're dealing with a knotty problem --
make sure to track those dimensions. The Martian lander you save may be
yours!

J

Jan 1, 1970
0
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!"

M

Mike Monett

Jan 1, 1970
0
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

C

Charles

Jan 1, 1970
0
Essential in electro-mechanical system design and analysis!

R

Randy Yates

Jan 1, 1970
0
Tim Wescott said:

Perhaps you can say dimensional analysis is where theory
and reality meet.
--
% Randy Yates % "And all that I can do
%% Fuquay-Varina, NC % is say I'm sorry,
%%% 919-577-9882 % that's the way it goes..."
%%%% <[email protected]> % Getting To The Point', *Balance of Power*, ELO

J

John Larkin

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
go beyond what I could say in the 5000 word limit that Embedded Systems
Programming magazine imposed.

The one I got today made me think.

It didn't make me think because it was hard to figure out the answer, or
because it was hard to explain the answer so it would be understood. It
the gains in a PID controller, and I was impressed that someone who was
obviously just approaching the subject was paying enough attention to
want to do a correct job of dimensional analysis.

It made me think of the innumerable times when I've done some complex
calculation and ended up proving something absurd such as measures of
length in units of gallons, and the subsequent discovery of my error.
It made me think of the Mars Climate Orbiter which crashed because one
team specified a motor in pound-seconds while another one used
Newton-seconds (http://mars.jpl.nasa.gov/msp98/news/mco990930.html). It
made me think of how much I like MathCad, because I can set variable
values with the correct units, and how inconvenient nearly every other
programming language is because numbers are just that -- anonymous numbers.

So I thought I'd write a little bit about dimensional analysis, why I
like it, and why you should use it, even if you think it is tedious and
trite.

Dimensional analysis is a method used by engineers and scientists that
extends the notion that "you can't compare apples to oranges" to the nth
degree. Dimensional analysis says that every variable in any physical
problem has units, and that you ignore these units at your peril.

The basic rules of dimensional analysis are these:

1. There are very few naked (i.e. dimensionless) numbers.
2. You can only add (or compare) two numbers of like dimension --
you can't compare feet with pounds, and if you're being strict
you can't even compare pounds (which is strictly a measure of
force) with kilograms (which is strictly a measure of mass).
3. You can multiply (or divide) numbers of any dimension you want;
the result is a new dimension. So if I pour water into a pan, the
water at the base of the pan exerts a certain force on each bit
of area, the resulting pressure is measured in pounds/square inch
(PSI), or in Newtons per square meter (N/m^2, or Pascal).
3a. You can honor famous people by naming dimensions after them --
Pascal is the metric unit of pressure, Ampere of current. Avins,
Grise, and Wescott have yet to be defined.

So why is dimensional analysis so cool?

If you are doing a long calculation and you track your dimensions, some
mistakes will show up as incorrect dimensions.

It can give you insight into the operation of a system -- most
explanations of fluid dynamics that I have seen rely heavily on
dimensional analysis in their arguments, and when they do so such
aerodynamically important quantities as Reynold's numbers and Mach
numbers fall out.

When you do design with physical systems, careful dimensional analysis
keeps you out of trouble, even when you're getting the math right.
Remember that Mars Lander? Had someone been carefully checking
dimensions instead of looking at numbers and making assumptions, it
would have been a successful mission instead of a famous crash.

How do you use dimensional analysis?

For working forward, such as finding the relationship between an
aircraft's speed and it's lift, or finding the relationship between
mass, speed and energy (remember E = mC^2?) you find out the dimensions
that your answer _must_ have, then you hunt down candidates in that field.

For working backward, you do all of your calculations with dimensions,
then make sure that, for example, if you're commanding a system in feet
it's moving in feet, and not apples/minute. If your answer ends up
being in gallons/inch^2, you know you're close but may have to multiply
by a constant.

For a more full exposition of dimensional analysis I refer you to the
Wikipedia article (http://en.wikipedia.org/wiki/Dimensional_analysis),
or whatever your own web searches turn up.

In the mean time, the next time you're dealing with a knotty problem --
make sure to track those dimensions. The Martian lander you save may be
yours!

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

J

Jerry Avins

Jan 1, 1970
0
Tim Wescott wrote:

...
of how much I like MathCad, because I can set variable values with the
correct units, and how inconvenient nearly every other programming
language is because numbers are just that -- anonymous numbers.

So I thought I'd write a little bit about dimensional analysis, why I
like it, and why you should use it, even if you think it is tedious and
trite.

Hear! hear! We need more of this. (T as the dimension of gain is absurd.)
Dimensional analysis is a method used by engineers and scientists that
extends the notion that "you can't compare apples to oranges" to the nth
degree. Dimensional analysis says that every variable in any physical
problem has units, and that you ignore these units at your peril.

The basic rules of dimensional analysis are these:

1. There are very few naked (i.e. dimensionless) numbers.

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.
2. You can only add (or compare) two numbers of like dimension --
you can't compare feet with pounds, and if you're being strict
you can't even compare pounds (which is strictly a measure of
force) with kilograms (which is strictly a measure of mass).

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.)

...

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

Jerry

T

Tim Wescott

Jan 1, 1970
0
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. The integral is in volt-seconds. But we sum
them, and nothing explodes!

John
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.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html

R

Robert Latest

Jan 1, 1970
0
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.

The integral is in volt-seconds.

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

R

Robert Baer

Jan 1, 1970
0
Tim said:
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
go beyond what I could say in the 5000 word limit that Embedded Systems
Programming magazine imposed.

The one I got today made me think.

It didn't make me think because it was hard to figure out the answer, or
because it was hard to explain the answer so it would be understood. It
the gains in a PID controller, and I was impressed that someone who was
obviously just approaching the subject was paying enough attention to
want to do a correct job of dimensional analysis.

It made me think of the innumerable times when I've done some complex
calculation and ended up proving something absurd such as measures of
length in units of gallons, and the subsequent discovery of my error. It
made me think of the Mars Climate Orbiter which crashed because one team
specified a motor in pound-seconds while another one used Newton-seconds
of how much I like MathCad, because I can set variable values with the
correct units, and how inconvenient nearly every other programming
language is because numbers are just that -- anonymous numbers.

So I thought I'd write a little bit about dimensional analysis, why I
like it, and why you should use it, even if you think it is tedious and
trite.

Dimensional analysis is a method used by engineers and scientists that
extends the notion that "you can't compare apples to oranges" to the nth
degree. Dimensional analysis says that every variable in any physical
problem has units, and that you ignore these units at your peril.

The basic rules of dimensional analysis are these:

1. There are very few naked (i.e. dimensionless) numbers.
2. You can only add (or compare) two numbers of like dimension --
you can't compare feet with pounds, and if you're being strict
you can't even compare pounds (which is strictly a measure of
force) with kilograms (which is strictly a measure of mass).
3. You can multiply (or divide) numbers of any dimension you want;
the result is a new dimension. So if I pour water into a pan, the
water at the base of the pan exerts a certain force on each bit
of area, the resulting pressure is measured in pounds/square inch
(PSI), or in Newtons per square meter (N/m^2, or Pascal).
3a. You can honor famous people by naming dimensions after them --
Pascal is the metric unit of pressure, Ampere of current. Avins,
Grise, and Wescott have yet to be defined.

So why is dimensional analysis so cool?

If you are doing a long calculation and you track your dimensions, some
mistakes will show up as incorrect dimensions.

It can give you insight into the operation of a system -- most
explanations of fluid dynamics that I have seen rely heavily on
dimensional analysis in their arguments, and when they do so such
aerodynamically important quantities as Reynold's numbers and Mach
numbers fall out.

When you do design with physical systems, careful dimensional analysis
keeps you out of trouble, even when you're getting the math right.
Remember that Mars Lander? Had someone been carefully checking
dimensions instead of looking at numbers and making assumptions, it
would have been a successful mission instead of a famous crash.

How do you use dimensional analysis?

For working forward, such as finding the relationship between an
aircraft's speed and it's lift, or finding the relationship between
mass, speed and energy (remember E = mC^2?) you find out the dimensions
that your answer _must_ have, then you hunt down candidates in that field.

For working backward, you do all of your calculations with dimensions,
then make sure that, for example, if you're commanding a system in feet
it's moving in feet, and not apples/minute. If your answer ends up
being in gallons/inch^2, you know you're close but may have to multiply
by a constant.

For a more full exposition of dimensional analysis I refer you to the
Wikipedia article (http://en.wikipedia.org/wiki/Dimensional_analysis),
or whatever your own web searches turn up.

In the mean time, the next time you're dealing with a knotty problem --
make sure to track those dimensions. The Martian lander you save may be
yours!
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.

R

Robert Baer

Jan 1, 1970
0
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. The integral is in volt-seconds. But we sum
them, and nothing explodes!

John
Hmmm... X double dot = - X

R

Robert Baer

Jan 1, 1970
0
Jerry said:
Tim Wescott wrote:

...

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.)

...

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

Jerry
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.

J

Jan 1, 1970
0
Robert Baer said:
Actually, my science teacher in High School emphasized units...

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.!

V

Jan 1, 1970
0
Tim said:
After six and a half years, I still get email about my article "PID
Without a PhD"

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

It made me think of the innumerable times when I've done some complex
calculation and ended up proving something absurd such as measures of
length in units of gallons, and the subsequent discovery of my error.

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

e = c = h = 1

It
made me think of the Mars Climate Orbiter which crashed because one team
specified a motor in pound-seconds while another one used Newton-seconds

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

of how much I like MathCad, because I can set variable values with the
correct units, and how inconvenient nearly every other programming
language is because numbers are just that -- anonymous numbers.

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
So I thought I'd write a little bit about dimensional analysis, why I
like it, and why you should use it, even if you think it is tedious and
trite.

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 a method used by engineers and scientists that
extends the notion that "you can't compare apples to oranges" to the nth
degree.

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

3a. You can honor famous people by naming dimensions after them --
Pascal is the metric unit of pressure, Ampere of current. Avins,
Grise, and Wescott have yet to be defined.

"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" ?
So why is dimensional analysis so cool?

Because it is generating so much traffic! It is trivial, so anyone can

It can give you insight into the operation of a system -- most
explanations of fluid dynamics that I have seen rely heavily on
dimensional analysis in their arguments, and when they do so such
aerodynamically important quantities as Reynold's numbers and Mach
numbers fall out.

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.

When you do design with physical systems, careful dimensional analysis
keeps you out of trouble, even when you're getting the math right.

This is the basic thing not worth mentioning which every professional
should do automatically.
Remember that Mars Lander? Had someone been carefully checking
dimensions instead of looking at numbers and making assumptions, it
would have been a successful mission instead of a famous crash.

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.

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

J

Jerry Avins

Jan 1, 1970
0

...
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.

Spoilsport!

Jerry

G

glen herrmannsfeldt

Jan 1, 1970
0
Tim Wescott wrote:

(snip)
So I thought I'd write a little bit about dimensional analysis, why I
like it, and why you should use it, even if you think it is tedious and
trite.

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

P

Phil Hobbs

Jan 1, 1970
0
glen said:
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.

-- 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

G

glen herrmannsfeldt

Jan 1, 1970
0
Joel said:
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!"

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

T

Tim Williams

Jan 1, 1970
0
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

R

RRogers

Jan 1, 1970
0
As usual Tim is right. I would like to be more explicit about it's
use in control systems. It's what allows you to "close the loop" on
the block diagram; for instance the plant is something like degC1/
Volt, the transducer is something like Volts/degC2 and the controller
is Volts/Volts. When multiplied this give degC1/degC2 as the gain and
the units to measure during verification. Of course a real diagram
has a lot more terms and different units; but the loop product must
balance. Inserting the open loop process variables in proper units
degC1/degC2 into the equations answers many questions.
One additional point is that the form is not as specialized as it
seems, but carries over to generalized Differential Geometry as vector/
covector spaces. Applying dimensional analysis in this realm has
allowed me to make sense of a lot of formulas. The units changes
(volts->degC) correspond to mapping from one vector space to another
(and back); the reduction to a gain scalar is the contraction of a
covector and vector; and so forth.

RayRogers

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