J
Jon Kirwan
Guest
On Sat, 18 Jul 2009 19:57:05 -0700 (PDT), fungus
<openglMYSOCKS@artlum.com> wrote:
where it is titled "Energy Storage in Inductor." But above that, as
well. In that titled area, they will take note of the fact that the
permeability factor of the iron core (or other material) actually is
in the divisor of the iron core term in the energy equation, which
with any large permeability means it gets divided by a pretty large
number. Note that in the example they analyze, about 2% of the energy
is in the core and the rest is in the air gap of the gapped core.
Also, inductors work just fine in vacuum.. so it's not the molecules
-- it is space itself. The atoms (those able to align their spin
states, anyway) actually are more like dead-shorts where energy isn't
much stored. They align up and then bridge over between bits of
vacuum where the energy gets mostly placed. In air, which is more a
very thin liquid, there aren't so many 'dead shorts' (they don't even
align that much where they exist) and the effective permeability is
much the same as vacuum. With a chunk of iron, and many many more
atoms present which are each quite willing to align with the field and
become dead shorts for the field, the energy that is present gets
stored again in the interstitial areas between these aligned atoms.
However, the atoms themselves, because they align so well, in effect
shorten the magnetic path length from what we humans on the outside
imagine.
Okay, now I'm going to make both our heads hurt for a moment.
Inductance is really just a bag holding loose constants laying around
in the equations. There's a whole bunch:
(1) L = mu_0 * mu_r * N^2 * A_e / l_e
mu_0 is the magnetic constant for a vacuum. mu_r is some arbitrary
multiplier for the core material, in cases where it isn't a vacuum.
We'll get to that one in a moment. N is just the number of windings.
A_e is the effective cross-section area of the core and l_e is the
length of the magnetic loop or circle that the magnetic field must go
through. Think of A_e times l_e as the total volume that the magnetic
field's energy occupies. With an iron core, this is easier to figure
out as you can pretty much measure it with a tape measure. Very
little energy leaks beyond the volume of the core itself (in
well-designed operation anyway) because all those iron atoms line up
and allow the field to remain mostly contained due to their very low
reluctance, dead-shorting effect. The magnetic energy 'wants' to take
the path of least resistance and the iron atoms practically beg to be
used by the field. Since vacuum is more 'difficult' for the field, it
channels itself through all those dead-shorts. So the volume is
pretty easy to measure since the magnetic field has no 'interest' in
going beyond the core (unless the core becomes 'saturated.') If the
inductor is an air core (or vacuum, in effect) then the magnetic field
concentrates in the interior of the coil and then, as it reaches
either the north or south end of the wire coil it then 'blooms' out
into the space around the coil and tries to find the way to the other
pole through a path of least resistance. This volume is harder to
measure with a tape, obviously. It's hard to 'see' exactly how much
volume is occupied by most of the field. So designers measure what
they can, which is the area of the coil cross-section, and then fudge
things by guessing at the length of the loop, instead. The area is
easy to measure, the magnetic path length is harder.. for an air core.
So experimental data becomes the basis for educated guesses about that
path length and we get these funky air core equations with what may
appear to be arbitrary constants tacked into them. (Neither of these
cases are a 'gapped core' case, but careful thinking about the two
I've already talked about lead to an easy understanding of gapped
cores, as well.)
Now let's return to the nifty inductance equation and address
ourselves to the 'dead-short' aspect of iron atoms:
(1) L = mu_0 * mu_r * N^2 * A_e / l_e
For an air core, just remove mu_r:
(2) L = mu_0 * N^2 * A_e / l_e
This won't actually look like Wheeler's equation (see: Wheeler, H.A.
'Simple Inductance Formulas for Radio Coils', Proc. I.R.E., Vol 16,
p.1398, Oct.1928), which is:
(3) L = 0.001 * N^2 * r^2 / (228*r + 254*l)
Where L is in Henrys, r is the coil radius in meters, l is the coil
length in meters (and must be greater than 0.8*r to work well) and N
is the number of turns used.
But that is because, as I said earlier, while we have an easy time
measuring A_e if we assume it is simply the cross-section area of the
coil's winding, we have a VERY hard time estimating the effective path
length of the magnetic field, which is l_e. And because of that, we
reach for experimental evidence to guide us and come up with rather
more pragmatic ways of estimating values than the pure theory approach
gives.
Refer back to equation (1) and compare it again with equation (2). Now
imagine all those dead-short iron atoms in the case of equation (1).
Since the iron atoms bridge over between bits of vacuum in the
material, and act like magnetic dead-shorts, the distance that these
atoms account for are, in effect, ignorable. The magnetic field gets
a free ride with each iron atom and then faces that nasty vacuum
before it can hop over to the next iron atom for some more of that
wonderful free ride, again. It takes energy to hop across the vacuum
and that is where the energy sits. Not in the iron atoms, which give
a free ride. But in the vacuum spaces in between the atoms because
that is where it takes energy to make the hop. So all the energy gets
stored in the vacuum gaps, not the atoms themselves. (Not if they
align easily, anyway. Many atoms don't 'help' the field any, don't
line up, and so the magnetic field doesn't get a free ride from them.
But iron works really good this way.)
All these iron atoms, giving the magnetic field a free ride over to
the next bit of vacuum, in effect short out or bypass that much of the
distance. So that distance doesn't count in the magnetic path length.
Really, the actual magnetic path length is just the vacuum parts that
the magnetic field must punch through and consume energy bridging. So
if we could just somehow only add up the vacuum parts and ignore the
iron atom parts of the loop length (l_e), we'd be able to figure out
just how much of the iron core loop length is actually just vacuum
(from the magnetic field point of view.) If we knew that, we'd just
use equation (2). That value for l_e would be much, much shorter than
what our tape measure would say about it, obviously.
However, we have tape measures and iron is easy to hold and see and
measuring the bits of vacuum isn't easy to measure. So the other
alternative, since the measurable l_e is so much longer due to all
those iron atoms in there, is to come up with something in the
numerator of the equation to compensate. In other words, add another
factor. This is mu_r seen in equation (1). It is there to compensate
l_e in the denominator for the error due to the fact that we measure
the total length using a tape measure instead of using some magical
means of adding up all those bits of vacuum hops that the magnetic
field actually must make and ignoring all those iron atoms. Since the
loop length of the iron core measured with common measuring means is
so much longer than that of the vacuum (which is what we really want
to know) hops the magnetic field will make, we'd calculate far too
little an inductance figure if we didn't add in this new factor.
In effect, __all__ of the energy gets stored in vacuum. Always has
been that way, always will be. For atoms that don't align at all, no
bridging takes place and the vacuum space they occupy gets filled with
it's part of the magetic energy hopping past them. For atoms that do
align easily, they act as dead-shorts so that the magnetic field
expends very little energy bridging across the tiny bit of space they
occupy along the path.
Jon
<openglMYSOCKS@artlum.com> wrote:
Go here and start near Figure 5 and read down from there. Particularly
where it is titled "Energy Storage in Inductor." But above that, as
well. In that titled area, they will take note of the fact that the
permeability factor of the iron core (or other material) actually is
in the divisor of the iron core term in the energy equation, which
with any large permeability means it gets divided by a pretty large
number. Note that in the example they analyze, about 2% of the energy
is in the core and the rest is in the air gap of the gapped core.
Also, inductors work just fine in vacuum.. so it's not the molecules
-- it is space itself. The atoms (those able to align their spin
states, anyway) actually are more like dead-shorts where energy isn't
much stored. They align up and then bridge over between bits of
vacuum where the energy gets mostly placed. In air, which is more a
very thin liquid, there aren't so many 'dead shorts' (they don't even
align that much where they exist) and the effective permeability is
much the same as vacuum. With a chunk of iron, and many many more
atoms present which are each quite willing to align with the field and
become dead shorts for the field, the energy that is present gets
stored again in the interstitial areas between these aligned atoms.
However, the atoms themselves, because they align so well, in effect
shorten the magnetic path length from what we humans on the outside
imagine.
Okay, now I'm going to make both our heads hurt for a moment.
Inductance is really just a bag holding loose constants laying around
in the equations. There's a whole bunch:
(1) L = mu_0 * mu_r * N^2 * A_e / l_e
mu_0 is the magnetic constant for a vacuum. mu_r is some arbitrary
multiplier for the core material, in cases where it isn't a vacuum.
We'll get to that one in a moment. N is just the number of windings.
A_e is the effective cross-section area of the core and l_e is the
length of the magnetic loop or circle that the magnetic field must go
through. Think of A_e times l_e as the total volume that the magnetic
field's energy occupies. With an iron core, this is easier to figure
out as you can pretty much measure it with a tape measure. Very
little energy leaks beyond the volume of the core itself (in
well-designed operation anyway) because all those iron atoms line up
and allow the field to remain mostly contained due to their very low
reluctance, dead-shorting effect. The magnetic energy 'wants' to take
the path of least resistance and the iron atoms practically beg to be
used by the field. Since vacuum is more 'difficult' for the field, it
channels itself through all those dead-shorts. So the volume is
pretty easy to measure since the magnetic field has no 'interest' in
going beyond the core (unless the core becomes 'saturated.') If the
inductor is an air core (or vacuum, in effect) then the magnetic field
concentrates in the interior of the coil and then, as it reaches
either the north or south end of the wire coil it then 'blooms' out
into the space around the coil and tries to find the way to the other
pole through a path of least resistance. This volume is harder to
measure with a tape, obviously. It's hard to 'see' exactly how much
volume is occupied by most of the field. So designers measure what
they can, which is the area of the coil cross-section, and then fudge
things by guessing at the length of the loop, instead. The area is
easy to measure, the magnetic path length is harder.. for an air core.
So experimental data becomes the basis for educated guesses about that
path length and we get these funky air core equations with what may
appear to be arbitrary constants tacked into them. (Neither of these
cases are a 'gapped core' case, but careful thinking about the two
I've already talked about lead to an easy understanding of gapped
cores, as well.)
Now let's return to the nifty inductance equation and address
ourselves to the 'dead-short' aspect of iron atoms:
(1) L = mu_0 * mu_r * N^2 * A_e / l_e
For an air core, just remove mu_r:
(2) L = mu_0 * N^2 * A_e / l_e
This won't actually look like Wheeler's equation (see: Wheeler, H.A.
'Simple Inductance Formulas for Radio Coils', Proc. I.R.E., Vol 16,
p.1398, Oct.1928), which is:
(3) L = 0.001 * N^2 * r^2 / (228*r + 254*l)
Where L is in Henrys, r is the coil radius in meters, l is the coil
length in meters (and must be greater than 0.8*r to work well) and N
is the number of turns used.
But that is because, as I said earlier, while we have an easy time
measuring A_e if we assume it is simply the cross-section area of the
coil's winding, we have a VERY hard time estimating the effective path
length of the magnetic field, which is l_e. And because of that, we
reach for experimental evidence to guide us and come up with rather
more pragmatic ways of estimating values than the pure theory approach
gives.
Refer back to equation (1) and compare it again with equation (2). Now
imagine all those dead-short iron atoms in the case of equation (1).
Since the iron atoms bridge over between bits of vacuum in the
material, and act like magnetic dead-shorts, the distance that these
atoms account for are, in effect, ignorable. The magnetic field gets
a free ride with each iron atom and then faces that nasty vacuum
before it can hop over to the next iron atom for some more of that
wonderful free ride, again. It takes energy to hop across the vacuum
and that is where the energy sits. Not in the iron atoms, which give
a free ride. But in the vacuum spaces in between the atoms because
that is where it takes energy to make the hop. So all the energy gets
stored in the vacuum gaps, not the atoms themselves. (Not if they
align easily, anyway. Many atoms don't 'help' the field any, don't
line up, and so the magnetic field doesn't get a free ride from them.
But iron works really good this way.)
All these iron atoms, giving the magnetic field a free ride over to
the next bit of vacuum, in effect short out or bypass that much of the
distance. So that distance doesn't count in the magnetic path length.
Really, the actual magnetic path length is just the vacuum parts that
the magnetic field must punch through and consume energy bridging. So
if we could just somehow only add up the vacuum parts and ignore the
iron atom parts of the loop length (l_e), we'd be able to figure out
just how much of the iron core loop length is actually just vacuum
(from the magnetic field point of view.) If we knew that, we'd just
use equation (2). That value for l_e would be much, much shorter than
what our tape measure would say about it, obviously.
However, we have tape measures and iron is easy to hold and see and
measuring the bits of vacuum isn't easy to measure. So the other
alternative, since the measurable l_e is so much longer due to all
those iron atoms in there, is to come up with something in the
numerator of the equation to compensate. In other words, add another
factor. This is mu_r seen in equation (1). It is there to compensate
l_e in the denominator for the error due to the fact that we measure
the total length using a tape measure instead of using some magical
means of adding up all those bits of vacuum hops that the magnetic
field actually must make and ignoring all those iron atoms. Since the
loop length of the iron core measured with common measuring means is
so much longer than that of the vacuum (which is what we really want
to know) hops the magnetic field will make, we'd calculate far too
little an inductance figure if we didn't add in this new factor.
In effect, __all__ of the energy gets stored in vacuum. Always has
been that way, always will be. For atoms that don't align at all, no
bridging takes place and the vacuum space they occupy gets filled with
it's part of the magetic energy hopping past them. For atoms that do
align easily, they act as dead-shorts so that the magnetic field
expends very little energy bridging across the tiny bit of space they
occupy along the path.
Jon
