In the previous several lectures about the proximity effect, I've been using the
terms turns of wire and foil or layers almost
interchangeably. This is something
that's actually done in the literature and I want to explain it now in
this brief lecture. In the last lecture, I used this example
of a transformer with primary and secondary
windings in which the flux lines are
vertical through the window or air gap
running between the turns. We didn't have flux running horizontally because of the
symmetry of the winding. The winding actually made magnetomotive force that
drove the flux vertically. So then we could think about
a layer of wire such as this first layer of four turns inducing flux that goes in this uniform
vertical direction. Indeed, later in that lecture, I approximated the round wire
turns as uniform layers. In the next lecture, I'm going to discuss the one dimensional solution of Maxwell's equations
for this case. To do this, we need to approximate
the turns of wire with a simple geometry and what we will use as
a uniform layer that spans the entire width or height of the window in the core. So recall if this is
our core here and here, we'll talk about a window, width and have
layers that stretch all the way from the top
to the bottom that have a nice simple geometry that we can then
solve analytically. The literature contains
an approximation that is used in many
of the papers in which a layer of round turns
of wire are massaged and stretched into a uniform layer that looks like say copper foil. These are based on
heuristic arguments, we can view them as
an approximation but the approximation
seems to work pretty well and it's adopted in many papers
by many authors. Here's a summary of what's done. Let's say we have a
layer of turns of round wire where the wire
has diameter d. What we want to do is basically
smoosh them or push them into an equivalent
nice rectangular, regular conductor of wire. We would like our
equations to maintain the same wire resistance to correctly model
the copper loss. If we start with
wire of diameter d, the first thing to do is
to square the circle into a square having a
length h on a side, and we want to choose
h so that the area of this square wire is the same as the area of the
original round wire. Let's see. The area of the
round wire would be Pi r squared or Pi times
d over 2 squared, and that should be equal to
the area of the square wire, which is h squared. So we can solve for h. Take the square root of
both sides and we'll get h equals square root of Pi over 4 times d. So here's how we define h. The next
thing we do is we push these square conductors together to get rid of the gaps between the squares and what
we'll get next and see then, is a single conductor
having thickness h, but it's shorter because we've eliminated these spaces
between the conductors. The final thing we want to do is stretch this conductor
so that it fills the entire width of the
window in the core, which is this length L_w. This distance here is the distance h times
the number of turns. Let's call that n. So we have to stretch this length hn to L_w, the width of the window. Yet, when we stretch, this area now becomes
bigger than this area. The problem with that is
that we would predict that the resistance of the conductor is lower than it actually is. So I have to compensate for that. Here's what's done in
the literature for that. We define what's called porosity. It says if the
stretched conductor is full of little holes like a sponge that effectively reduce its area back down
to the original area, the porosity then is defined as the ratio of the
cross-sectional areas. It's the ratio of the original area divided
by this new area. If you plug in our
expression for h, you get this for the porosity. What we do then is we change the effective resistivity
of the copper to reduce it by the porosity so that the resistance of the
conductor comes out correct. With an actual winding and
with real turns of round wire, you can plug into these
expressions and for a round wire, a typical value of
this porosity is 0.8, so that 80% of the area is actually filled with copper conductor in
the original winding. If we have a foil winding that is the entire
height of the window, then we could get
a porosity of one. Now, reduced porosity means that effectively the
resistivity is lower. The resistivity affects
the skin depth, and so we get an effective
skin depth as well. We can plug into the
expression for skin depth and find that there is an
effective skin depth in Delta prime, that is the original skin depth divided by the square
root of the porosity. So the porosity affects
the skin depth. In the upcoming lectures, we're going to define
this quantity Phi, which is the ratio of the conductor thickness
to the skin depth. Conductors with
large values of Phi are greatly affected by the skin effect and
the proximity effect, whereas conductors where
Phi is much less than one will have very little impact
from the skin effect. If we have a layer
of round conductors, we can talk about
an effective Phi, which is h divided by the
effective skin depth. So this effective Phi turns
out to be this expression. So we have a simple approximation in which a layer of round wire is approximated by a foil
winding of uniform geometry. In the next lecture, we will use this uniform
geometry to write the equations or the solution of Maxwell's equations
in the general case.