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Re: Impedance &c.



good show Leigh!

one of MY random questions on impedance, mulled over for some the years.

Electrical resistance of course is V / i ( or Z = E / j for the
sophisticates )

Power is (V) x (i.).

My question (suspicion) is:
are other impedances also going to be expressed as the ratio of an
intensive variable to an extensive variable - with the product of
these two variables being power???

a 'simple' system to analyze would be a spring loaded gun shooting a
projectile of mass m.

and what IS the impedance of a simple lever system?


At 11:01 AM -0800 1/5/04, Leigh Palmer wrote:
Happy New Year! The lake is frozen over, and despite the thin layer of
granular remelted snow on its surface, it is skateable, so I will be
out on it later today. It is -10 degrees C here in Burnaby, very cold
for this location. There is much interesting physics to be observed at
this time of year, including my old favorite, the dispersive chirp
heard when a distant hockey stick strikes the ice.

How about a new topic? A list member who read the gyrogenerator series
asked me: "Could you say some more about what you mean by impedance
matched loads?" It happens that I have a great long standing interest
in this topic, so I'm going to try to interest the rest of you.

Let me start with another example. Consider the problem of transferring
one's mechanical output power to the performance of a task. Consider
the old hand powered egg beater. Running the eggbeater in air will
scarcely require any effort and will raise only a puny, low power
breeze, even at high speeds. On the other hand, applying the eggbeater
to cool honey for the purpose of warming it up will be discouraging.
The honey is so thick that it is difficult to transfer much power to it
with this device. In a medium of intermediate viscosity, like eggs, it
is possible to transfer a much higher power than was the case in the
previous examples. While one probably cannot cook the eggs with this
device, it is admirably suited to the task of scrambling them by
transferring one's personal power output to the load, a better coupling
mechanism than, say, a table fork applied to the same task.

The eggbeater is an impedance matching device! perhaps you have never
thought of it in that way; if not, let me sensitize you to it. There
are many, many other examples of impedance matching devices we
encounter in our mundane experience. The hydraulic jack is one that is
easy to recognize, as are the "simple machines" that occupy all too
prominent* a niche in our early science education curricula. So is
another favorite of mine, the differential hoist. There are so many
that enumerating them is a distraction, but I will mention two more of
my favorites, the n-speed bicycle and the atlatl.

This is a shadowy topic about which I have little formal knowledge, but
if I were to write an introductory textbook, I would certainly develop
it formally and give it some needed exposure. I won't ever write a
textbook, but perhaps this group will help me develop the concept
formally anyway. (A radical textbook won't sell, and selling it is the
only sane reason for writing a textbook.)

In the gyrogenerator problem the impedance matched load to which I
referred is that load that extracts the maximum power from the system.
Note that the power generated was given by an expression that varied
from zero power at no load to zero power at shorted output, through a
maximum at a load that permitted the gyro to precess at half Earth's
rotational angular speed. The load is said to be matched to the power
source when it has the value permitting maximum power dissipation.

In this case the narrow electrical meaning of "impedance" can be
inferred, though from the data given one could scarcely calculate the
requisite load impedance. One needs also some detail regarding the
generator itself, for the generator is an impedance matching device,
matching a mechanical power source to an electrical load.

The key concepts here seem to be impedance, power, matching, source (of
power) and sink (load). My goal would be to clarify and precisely
define these concepts and relate them formally. Since others have also
done so, it appears that there is some variety in the systematics of
others' treatments. I note in passing that while impedance has
implicitly meant a lumped quantity in my discussion so far, there are
also the related concepts of distributed and intrinsic impedance. For
example, cool honey is a high viscosity, therefore high intrinsic
impedance medium.

Of course the most common use of the term "impedance" is the electrical
kind. The notion of impedance is much more widely applicable than this,
however, and we use the term much more broadly in mechanical,
hydrodynamic, and acoustical contexts, to name but a few. I checked ten
mechanics textbooks on my home bookshelf. Not a single one has the term
in its index. My "Basic Acoustics" by Donald C. Hall gives considerable
attention to this topic, to my great relief. I would like citations of
other texts that treat the question of mechanical impedance.

Impedance as a quantity will relate to the ratios of force to velocity,
or of torque to angular speed, just as electrical impedance is the
ratio of electromotive force to electric current. Note that in all
these cases power (input and/or output) is the product of the
quantities named. Note also that, unlike the case of power, impedance
takes on many different sorts of units depending upon its type. Perhaps
someone will be clever enough to devise a dimensionless framework so
that it will become unnecessary to have "acoustical ohms" and
"mechanical ohms" that differ in dimension from electrical ohms. That
would be truly glorious!

My thanks to all who have endured these unformed thoughts. I hope
something good will emerge from this discussion. I'm off to GR class
before skating. Retirement can be glorious.

Leigh

* I feel they should be introduced *en passant* when treating the many
other devices to which they are related. I don't know myself what a
"class III lever" is (though of course I could Google it if I wanted
to), and I feel that has never been an impediment to my teaching of
physics.


--
Chuck Britton Education is what is left when
britton@ncssm.edu you have forgotten everything
North Carolina School of Science & Math you learned in school.
(919) 416-2762 Albert Einstein, 1936