The previous lectures treat the proximity loss when the current waveform
is sinusoidal. In switching converters, often
our winding currents are non-sinusoidal and may be switched or have switching
ripple and so on. In such cases, we can
use Fourier analysis to work out the effects of
harmonics in the waveform. Maxwell's equations are linear, and so superposition applies and it's valid to apply
Fourier analysis, where we calculate
the harmonic content of our waveform and then apply the proximity analysis with Dowell's equation to each
harmonic of the waveform. So for example, let's
suppose we have, say a transformer winding that has a pulsating
current such as this. This is a waveform
we might have in the primary or secondary winding, say of a forward converter. With a pulsating
waveform like this, we can use Fourier analysis to find the Fourier series of
the current waveform and we can write it in some
function like this with a DC component and
with a Fourier series. Here, j is not the square
root of minus one, j is the harmonic number. With Fourier analysis,
we can actually work out that for this
pulsating waveform, the Fourier
coefficients, which are the RMS values of the current, are given by this expression. Their magnitude depends on the
peak value of the current, I_pk, and on the duty cycle. The DC component is D times I_pk, which is the average
value as usual. To calculate the copper loss in the winding that has
this pulsating current, we calculate the loss
at each frequency, so the DC copper loss is the dc current squared
times the dc resistance. For each harmonic, the Ac copper loss is given by a function
that looks like this, and this is in fact
Dowell's equation, assuming there's no interleaving, and assuming we have capital
M layers that are winding. One little thing to note here, the value of F changes
for each harmonic. F recall is the
conductor thickness divided by the skin depth, and the skin depth
is a function of frequency that goes like one over the square
root of frequency. So for harmonic j, F j will be h over Delta j, where Delta j is the skin
depth for harmonic j. Since this goes again, like square root of frequency, we can write that F j is the
square root of j times F1, where F1 is the value of F for
the fundamental component. So as we go to a higher and higher frequency
or higher harmonic number, our value of F gets
larger and larger, but it's a weak function of j, the square root function. So we can write Dowell's
equation for each harmonic, and then we need to sum
the Fourier series. So plug these into the sum
for the Fourier series to calculate I squared
R and add them up. Summing the series sounds like a pretty complicated thing, what I usually do is
write a computer program, maybe a MATLAB program
to sum them up, and I'll go up to some
high harmonic number and then just stop, terminate at some point. It's interesting to define
a harmonic loss factor, which tells us how the harmonics increase the proximity
and total copper loss. Here, I define a
harmonic loss factor, is the total copper loss
of the fundamental and the harmonics divided by the copper loss of just
the fundamental component. So with that definition then, we can write that the
total copper loss is the copper loss from the DC component plus
this copper loss from the fundamental and all of the
harmonics where I 1-squared R_dc is the RMS
fundamental component times the dc resistance. We multiply that by F_R
at the fundamental, to get the proximity loss included in the
fundamental component, and then we multiply by F_H, which tells us the effect of all the harmonics on
the total copper loss. The value of F_H, of course, depends on the Fourier series
of the original current. Just to see what happens for the waveform of this
forward converter, here's what F_H looks
like at D of 0.5. So F_H again, is the factor by which the harmonics increase
the total copper loss. At low frequency, these curves tend to a number
that's bigger than one, and that's because
the RMS value of the fundamental is smaller than the RMS value of the full site, the full pulsating waveform. So the total harmonic
distortion of the waveform will
increase the DC loss. At high frequency, we go to a larger number
that also is fixed, and this is the high
frequency limit with very large conductor. The interesting thing here, is that at intermediate
frequencies, F_H becomes large, what's happening here is that at these intermediate
frequencies are intermediate values
of F conductor size. The effect of the
harmonics is large, the fundamental component makes a relatively low copper loss, whereas the harmonics make a large increase in
total copper loss. So the harmonics have
the biggest effect for values of F1 a little
less than one. From our previous lectures, F1 a little less than
one was where we wanted to design the winding. So that choice is fine when
we have a sinusoidal current, but when we have
significant harmonics, it might not be the best choice. Here's the same plot
for duty cycle of 0.3. The curves look smaller, but actually the
scale is increased. The last scale maxed out at 10, this one maxes out at 100, and actually with
lower duty cycle, the effect of the
harmonics is getting worse, and idea 0.1, it's even worse yet with
curves approaching 100. So the harmonics can have
a significant effect depending on how you've
designed your winding, and the moral of that story is that when we have
significant harmonics, we need to include them and
sum the Fourier series. So harmonics can significantly
affect the answer and they can affect that conclusion on the
optimum wire size, and they can affect
in both directions. If you have a significant
dc component, the optimum solution
might be to use a thicker conductor to reduce the copper loss of
the dc component. On the other hand, if you have significant high-frequency
components, such as in harmonics, it may be that the optimum
is a smaller conductor. Here, we get into writing some computer programs
to evaluate this, in our quest to find the optimum.