In the previous lectures, I have discussed the
proximity effect for what really is the limiting case
of very thick conductors. We could also call this
the high frequency limit. This is the case where the conductor thickness is much greater than the skin depth. We can view this then as a limiting case where the
current distribution is these decaying exponentials
inside the conductor, and essentially no current flows in the center of the conductor. We saw when we had
multiple layers, these conductors made
circulating currents in which the magnetomotive
force would build up. So we can have current going one direction at one
side of the conductor, the opposite direction down the other side so that there are circulating any currents
inside the conductors, and that this causes
the MMFs to build up as we get more layers, and we get significant
proximity effect. Of course, we have
the opposite extreme as well of a very thin conductor, which we could also
consider as the DC limit, where the conductor thickness is much less than a skin depth. In this case, the current in
the conductor is uniform, simply all flows
in one direction, and we have essentially a
constant current density. In this case, we don't
have proximity losses, and the resistance of the wire tends
towards the DC limit. Now there are problems with
both of these limiting cases. Of course, in the
high frequency limit or for a very thick conductors, we get significant
proximity effect that can significantly
increase the copper loss. In the DC limit or
for a thin conductor, the problem there is that when we make our conductor thin, we have high DC resistance. How do we build a conductor
that has low resistance, large thickness, and can carry a high current at the same time? Well, naturally, it
turns out that the optimum is in between
these limits, that between the low-frequency
and high-frequency limits, there is an optimum. In other words, there's
an optimum thickness that minimizes the total loss. To find that optimum, we need to solve
Maxwell's equations for the real current distributions
in the conductor, when the conductor has a moderate thickness that is in the vicinity of a skin depth. In this lecture, I'll
describe that solution. There's a classic solution in the literature called
dowels equation, which is a
one-dimensional solution of Maxwell's equations. It's assuming the flux, takes these straight
lines alongside a simple layer that has this simple rectangular geometry as we've discussed in
the previous lecture. Through this solution
of Maxwell's equations, we can work out the equation of the actual copper loss as a function of any
conductor thickness. So what we will do
is assume that we've drawn the MMF diagram
of the winding. We know what the H field is or the MMF on the two
sides of a layer, we're going to assume that the waveforms are sinusoidal
so that we can do a phasor or AC sinusoidal
analysis for the linear system, and we can solve
Maxwell's equations. I'm not going to go through
the whole solution, but I am going to summarize the results and talk about
how to use the results, and we will use these
results then to design good transformers and
inductor windings. Here is the solution. The equations on this page
are called dowels equation. The first equation here is
the copper loss in one layer. It's a function of the DC
resistance of that layer. Phi, we talked about
in the last lecture, it's the ratio of the conductor thickness to the skin depth. We have this transcendental
equation with G_1 of Phi and G_2 of Phi that are given by these hyperbolic sines and
cosines as shown here. It's also a function of the magneto motive forces on
the two sides of the layer. It's not obvious looking at the transcendental functions
of how they behave. This is something that probably
is not design-oriented. However, it is something
that is enclosed form. You can type these equations into a spreadsheet like Excel, or into a calculator, or into MATLAB and
easily evaluate them. When we draw the MMF
diagram of a winding, the change in the MMF from
one side to the other of a layer is determined by the number of the
amps in the layer. The change in the MMF from
applying Ampere's law will be equal to determinants or the amp
turns of that layer. So we can relate the MMFs to
the current in the layer. Further, we're going
to define a quantity, lowercase m, and define
lowercase m, with this equation, where the MMF on one side of
the layer is proportional to the m times of the
layer with constant of proportionality
equal to m. In fact, when we drew the MMF diagram
in previous lectures, we get something that
would look like this, for example, and the
MMF would build up. We can also almost think of m as being the layer
number in such a case. Here, if this is nI, 2nI, 3nI, the lowercase m is one, two or three, and it's the layer number in this
particular winding. Combining these two equations, the change in MMF from one side of the
winding to the other, we can write as m minus 1 over m by simply solving
these equations. We can eliminate the MMF
[inaudible] equation in terms of the amp turns or current in the layer and the quantity
m for that layer. If you take this
and substitute this into those equations
on the previous slide, here's what we get, and this is a very common way to express [inaudible] equation. Here, P is the copper
loss of the layer. This Q prime has the transcendental functions
from the previous slide. It's written then in
terms of the layer number m. Here's a plot of that. This is the vertical axis here, it's what we sometimes call F_R. It's the ratio of the
actual copper loss to the copper loss that would be caused by the DC
resistance of the winding. This is the factor by which the proximity effect
increases the copper loss. This is on a log
scale, as you can see. One is what we
would like to have, so we have a resistance
equal to the DC resistance, but here we're
getting 10 and 100, we are getting big numbers
from the proximity effect. Phi here was the
conductor thickness. It's the ratio of the
conductor thickness, h, to the skin depth. This is really the
thickness of the conductor. Then lowercase, m,
is the layer number. What we can see is that for
thin conductors with low Phi, we go to the DC resistance. For very thick conductors, then the proximity effect
becomes important, and especially when
we're in a conductor that is in a high
number of layers, a high MMF environment, then we can get orders of magnitude increase
in the copper loss. This plot suggests that
what we should do is use very thin conductors
than be down here. That's not quite the right
answer because if you use thin conductors then
the DC resistance is high. We're starting out
with thin wire that has high DC resistance
to begin with. What we want to think
about here is not the factor by which
the loss increases, but just the absolute number, how big is the loss? It turns out that there is an optimum conductor
thickness that minimizes the total copper loss. This is a similar plot, except that the
denominator is a constant. What's been done here
is the numerator, P, is the total copper
loss in the layer, but the denominator is
the DC loss that we would get in a conductor having a
thickness of one skin depth. It's I squared times RDC, where RDC is for a conductor
having Phi equal to 1, or one skin depth. At that, denominator simply
normalizes everything. What we can see
here is that as we go to very thin conductors, the DC resistance becomes large. As we go to very
thick conductors, we head towards the
high frequency limit, which is what we calculated in earlier lectures that were
for very thick conductors. If you look at the plots, for a given value of
m, there's a minimum. For example, with m equals 3, there's a minimum right
there.For layer 3, the optimum conductor
thickness is this value, which is 1.7 skin depths, and that will minimize
the total loss. The minimum values depend
on the layer number, but there are more or less in the vicinity of Phi equals 1. The plot on this slide and on the previous slide
are the solutions of [inaudible] equation
for the general case of arbitrary conductor thickness. This is the equation
that we use in practice to optimize a transformer
or inductor winding. In upcoming lectures, we'll
do some examples with this.