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Re: emf, potential, voltage



At 05:42 PM 3/17/01 -0500, Bob Sciamanda wrote:
I can get a simple linear model if I make the "reasonable" assumptions:
1) e=C1*omega: back emf is proportional to angular velocity, and
2) omega=C2*I: angular velocity is prop to current.

Then the terminal voltage equation: V=e - Ir becomes

V = C1*C2*I - IR, or [equation 3]

V=C*I, a straight line in V/I space.

How realistic is this (and for what type motors)?
(Obviously, I also assume a constant mechanical load.)

Equation 3 is more realistic than the assumptions that went into the
derivation given here.

To first order, this system is linear(*). Something of the general form of
equation 3 must hold over some range near the operating point, assuming the
mechanical load is not pathological. We can consider equation 3 to be a
two-parameter fit to the data, with phenomenological parameters R and (C1*C2).

So the interesting questions are:
*) Does equation 3 hold over a relatively narrow range or a relatively
broad range?
*) What if anything does this have to do with physics?

The physics is that the canonical motor looks rather similar to a
generator. It is generator-like not just when you are driving the shaft
and applying a load to the electrical terminals; it is _at least_ as
generator-like when you are driving the electrical terminals and applying a
load to the shaft. It is "generating" the back emf.

The following things all work nicely as generators, over a very wide range
of operating conditions:
a) Rotor = permanent magnet, AC connected to stator coil [your typical
synchronous motor (e.g. clock motor)].
b) Stator = permanent magnet, AC connected to rotor coil via slip rings.
c) Stator = permanent magnet, DC connected to rotor coil via commutator.
d, e, f) Any of the above with the permanent magnet is replaced (or
supplemented) by an electromagnet with constant (regulated) DC current.

These stand in contrast with the following things, which make good
generators only under a more-limited range of operating conditions:
x) a "universal AC/DC" motor, where there is a conspicuously
non-constant current in both the stator and rotor.
y) an induction motor.

Even these last two are good generators in the normal motor operating
regime. Even if you think (on physical grounds) that their back emf should
be proportional to the square of the rotation speed, you can (on
phenomenological grounds) expand the parabola to first order and get a
serviceable linear relationship. Linear but not proportional.

===============

As to assuming "constant mechanical load" and "angular velocity
proportional to current".... I don't see how that helps, conceptually or
otherwise. The load is rarely constant in this way. Usually we apply a
voltage to the leads of our motor, and let the current be a dependent
variable. You can't independently dictate the values of the voltage and
the current.

From a pedagogical and conceptual point of view,
== For DC motors I prefer to take the applied voltage and the rotation
rate as my independent parameters. Rotation rate can be kept constant
(over periods long enough to be interesting) by using a flywheel. Data
points for various load conditions can be obtained by applying a brake.

== For synchronous AC motors, only one rotation rate makes
sense. Therefore the magnitude of the back emf is determined by the
_frequency_ of the AC applied to the motor. But notice I said
"magnitude". The I-V relationship for the motor depends on the _phase_ of
the back emf, as well as its magnitude. As you apply more load to the
clock motor, it shifts the phase of its back emf, whereupon more power is
drawn from the electrical supply.

================================================

NOTE (*): I love saying "to first order, such-and-such is linear". People
say I have a keen grasp of the obvious.

But we should not get carried away: Although it's true that motors are
linear to first order (under anything approaching normal operating
conditions), it's not strictly true that "everything" is linear to first
order. There are things, even real-world physical things, that cannot be
expanded in a Taylor series over any useful range.

http://www.nobel.se/physics/laureates/1982/press.html