In the last lecture, we talked about the
proximity effect and how currents in the windings are associated with flux in the
air next to the winding, and we're calling this
flux now the leakage flux. Multiple layers and multiple
turns cause this flux to build up and induce
additional proximity losses. It's possible to
do things such as interleaving of windings
to reduce this effect. To understand that, we draw what are called MMF diagrams or magnetomotive force diagrams that describe this flux in the air
surrounding the windings. That's the topic of this lecture, and here we're going to call
that flux leakage flux, and generally in a transformer
that is indeed what it is. Here I've drawn a simple example, it's a core with
high permeability and we have some turns of round wire on it
in a transformer, here I've got eight primary turns and eight secondary turns. Again, we can think of
these turns as going around the core like this. I guess this bottom
turn will come back on the next layer of turns
and will go like that, and then there's the final
turn for the primary, then perhaps our secondary
is on top of that. It actually doesn't matter which leg of the core it goes around, it's topologically equivalent,
though we can just think of the secondary
winding may be as being on top of the primary. So our turns are
going around like this and then we get
to the next layer, and our turns continue to connect and then
we come out there. We have really turns going
around the post of the core and this is a drawing really of a cross-section of what
the wires look like. This would be a
transformer example that's eight turns
to eight turns. With this simple geometry, each piece of wire or each turn carries
the same current i, coming in the primary and
going out the secondary. What is the flux then induced by these currents
in this geometry? In the transformer, we had an equivalent circuit
that's the T-model, so I have this ideal transformer
that's eight to eight, we have a magnetizing inductance then we have leakage inductances. The magnetizing inductance
models flux that is common to the two windings or coupling the two windings and we think
of that as the mutual flux, which really is going
entirely in the core. While the leakage fluxes in
the windings are fluxes that link some of the turns
of wire but not others. Generally we think of leakage
fluxes as flowing at least partly in the air and not just being
confined to the core, where they link both the
primary and secondary windings. This flux next to
the windings that is associated with the
proximity effect is in fact leakage flux. We have, for example, current in this first layer of primary turns and there is flux that travels in the air
adjacent to the first layer. We can model that flux
as being proportional to the current in
the adjacent turns. This flux is driven by the current in the
windings and in fact, with Ampere's Law, we can calculate the
size of this flux. Recall Ampere's law says, if we consider the
path a flux line takes in its closed
loop that the total MMF around that path that drives the magnetic
field and drives flux is proportional to the
current enclosed by the path. If we say have some path
with flux lines going this way through the air in the
middle of the winding, then the MMF that drives magnetic field in that
path is equal to the total current enclosed or in the air in the winding window
inside of the path. That this MMF is equal to the H field times the
distance of the path. Let's suppose that the core
has large permeability, much greater than the
permeability of air, and what that means is that the MMF in the core
is very small, it's very low reluctance and
nearly all of the MMF is in the air driving the flux across basically the air gap or in
the air through the winding. If we take this path right
here for this flux line, basically all of the
MMF is in the air and if we say that our
window has some height, LW, then the MMF is this H right here times LW. That's equal to the
current inside the path, and as drawn here, we have four turns each
carrying current i. So this is 4 times i. Now, why did I draw the flux is going vertically through the air? Why isn't there any going horizontally or in diagonally
or in some other direction? Well, let's consider a
path going this way. As you can see, the net current
inside that path is zero. There are two primary turns and two secondary turns with currents flowing in the
opposite direction. The net current cancels
out and is zero. Because of the geometry
of the winding, we don't expect to have horizontal components
of magnetic field, we just have vertical components. Now, of course, there may be some little curves around
the round wire and so on, which we're not
going to model here. We can talk about the MMF, which is from one side of the
core window to the other. That is what drives the
flux through the winding. We can plot that as a function
of the distance here, which we're going to call x. This MMF is a function of x and we can plot how
it goes up and down, and this is called
the MMF diagram. We can see that for this path, the MMF is 4i. Really our MMF goes
from zero at this end, and when we get here,
we've got four. We draw it like that. When we get to here, our path encloses total of eight turns each with current i. So we go up to 8i. At this point, our path
includes plus 8i and then minus 4i for the center layer of
the secondary winding. So we go back down to a net 4i. Then finally at the
far end of the window, we're back down to zero. Here's a plot of that, and what I've done here is simply sketch the conductors and
what currents they carry, and directly under that, I've drawn this MMF diagram
like on the previous slide. Basically you just
count the amount of current to the left of the point. At any point, say right here, you count the amount
of current to the left and that
adds up to the MMF. This diagram also
is assuming that the current is uniformly distributed across the conductor. If you choose a point x that's in the middle
of the winding, this is assuming that
the current with uniform distribution will
increase linearly with x. We'll come back to
that in a minute and account for the scan effect. But for now we'll just
draw it this way. Places where there is high MMF between the winding is
places where we expect to have more eddy
currents in the wires and where we'll have
more proximity effect. Here's another example. This is a two
winding transformer, and here I've just drawn layers. This layer I've just
drawn is a rectangle, perhaps it actually is comprised of a bunch of turns of wire, or maybe it is just one single rectangular conductor
is drawn here. But what's important here is what is the total
current in this layer, and that's what
determines the MMF. If we have total current
i in the first layer, then the MMF diagram
will go up to i, to the right of the first layer. It'll go up to 2i here and
3i here, just like this. Then with the secondary
layers carrying minus i will go back down and we'll have an
MMF diagram like that. Now with proximity effect, here's what really happens. We discussed this last
time in the last lecture. In the first layer, all of the current in
the first layer actually crowds on the
right-hand surface of that first layer within more or less one skin depth
of its right side surface. In fact, if we move
across this wire, there is no current
enclosed in our path until we get to within one skin depth approximately
of the right-hand side. After that, we have this decaying
exponential function for our current density
according to the skin effect. I'll draw our little decaying
exponential function. What actually happens
is the MMF goes up to get to be i at the
surface of the first conductor, our MMF that drives flux
across the air gap between the first and second
conductors is driven by this current i, and it induces an equal
and opposite current in the adjacent surface
of the second layer. If we move past that to look at, say what happens here in the middle of the
second conductor, the minus i cancels the plus i and we go back down to zero. In the middle of the
second conductor, there is no MMF, and in fact, the second conductor
has currents on its surfaces that are
rejecting the current, expelling it so it doesn't flow in the middle
of the conductor. What actually happens then, with this skin effect, is that there is no MMF in
the middle of any conductor, and we have surface
currents that achieve that. Then we see 2i on the right side of that
second conductor, as we discussed in
the last lecture, and our MMF goes up to 2i to
it's right in the air gap, that gets canceled out by the surface currents on
the third conductor, and we have a decaying
exponential down to zero, and so forth. You could draw these
straight lines, but in fact we have these decaying
exponential functions and we have zero MMFs in the
center of any conductor. Something we often do to
reduce the proximity effect and with great success is
to interleave windings. Here's an example of
this same transformer, with three primary layers
and three secondary layers, and what we do here is we interleave the primary
and secondary, so we alternate, primary, secondary, primary,
secondary like this. If you draw the MMF
diagram in this case, this first secondary, its current is equal and opposite to the current in the first primary, and we simply have the currents
on the adjacent surfaces, here and here within the
skin depth of the surface, producing equal and
opposite currents. There is no further build-up
of circulating currents in these windings and
the MMF diagram just goes up and down like this. In fact, in these surfaces
between these layers, there's no flux and
there is no MMF, and we have relatively low
leakage flux here and here, it only goes up to i and
never builds up to 3i. You can see that there's much less eddy currents in the windings and there'll be a much lower proximity losses. Interleaving is a good thing. Now there are
practical reasons why sometimes we can't interleave, but this gives us a tool
then to work out what the effect of that
is and perhaps do partial interleaving
or something similar. We can discuss that using these MMF diagrams and
analyze the effects. Here's a partially interleaved
transformer where we have four layers of secondary and three
layers of primary, and there's a turns ratio. So really this is a
transformer that is 3:4, suppose we put two
layers of secondary, then in the middle we have
our three layers of primary, then we have the last two
layers of the secondary. We can draw our MMF diagrams
in this space here, we'll go to minus 3/4i MMF. In the next space here, the total current to
the left is what, minus 1.5i, right there. With the primary, we got plus 1i, so this MMF is what? Minus 1.5 and then plus 1, so here we're at minus 1.5i. In this space, we'll
add another 1i, so we'll be at plus 1.5i, and here we add another one, so we're at plus 1.5i, then we go back down
by three-quarters, so here we're at 3/4i and
then we're back down to zero. When we have diagrams like this, we will work out then
what are the currents on each surface as a function of
these MMFs in the air gaps, and then work out the
proximity effect. Why would you want
to do something like this and
partially interleave? Well, in a lot of
transformers we're required to have insulation
between the layers. The simple enamel insulation on magnet wire is good for about 70 volts and
then it breaks down, if we have higher voltages
and often where there are safety-related things
like insulating users from the power line, we'll have requirements of
5,000 volts or similar things, and so we may put tape
between the primary and the secondary to achieve
this insulation, and it's hard to
fully interleave or it's expensive labor to do that, so perhaps in this example, we only need tape here
and here and this is something perhaps that's more practical and
economical to build. We've seen how to draw
MMF diagrams both in simple cases and in more
complicated interleave cases. We will use these MMF diagrams
to analyze and calculate the proximity effects in complicated or arbitrary
winding geometries.