In this optional lecture, I'm going to briefly summarize
the math behind averaging. So in the last lecture, we talked about
the average equation of the inductor in which
L times the derivative of the average inductor current, was taken to be
equal to the average inductor voltage. This actually follows directly from the
direct, the definition of averaging and from the defining
equation of the inductor. And we can see that as follows. Here we'll start with the equation of the
inductor, by definition l d i d t, is the inductor voltage with no averaging
applied. And so, let's then consider, what is the derivitive of the average inductor
current. Where we define the average inductor
current as in the previous lecture. So the derivative of the average inductor
current can be written like this, where inside the brackets is our
definition of how we average. And so let's take the derivative. Now we have the derivative of an integral
here. we're allowed to interchange the order of
differentiation and integration under a certain, certain
set of conditions, that the mathematicians tell us. First the, the function we're integrating,
namely the inductor current, needs to be a continuous function so that we can
take it's derivative. And second, the derivative of the current
is allowed to be discontinuous, but it can only have a finite number of
discontinuities over one switching period. While the derivative of the current is the inductor voltage over L, this is in
general a switched wave form in our converters, so
it has discontinuities. But, if we don't switch too many times per switching period, then we can
certainly, or at least if we switch a finite number of
times per switching period, then we satisfy the
second requirement. And so we're allowed to, reverse the
orders of integration and differentiation. So that's what we get in this bottom
equation then, where we have the integral of the derivative of, of the current.
Okay now, for the for the derivative of I what we will do is take this defining equation of the
inductor and divide by L. So the DIDT is the inductor voltage over
L, and we can substitute that in here for the
derivative. So that's what I've done here. Okay now we can just rearrange terms. actually just take this L and put it over
here to get LDIT. And on the right hand side, what we have, is in fact the average value of the
inductor voltage. So we integrate over one period and divide
by the period, and we get the inte, the average
voltage like this. So what we've shown then is that, the averages obey the inductor equation as
well. now you might say, well that's obvious,
but in fact, it isn't, we have to show it. We don't have any extra terms, and the actual value of L is unchanged by
averaging. you can apply the same arguments to the
capacitors. And so for capacitor with the same arguments we get a similar kind of
equation. Okay. So let's look at the mechanics then of actually doing, computing this average
formally. the first thing I should say is that this
definition of the average is not synchronized with the beginning of the switching period in
general. this is a, this moving average is a
function of time that doesn't hop from one switching period to the next. So what I've drawn is an inductor voltage
wave form VL of T, where the switching period is
starting right here. And then this is drawn at some arbitrary
time T, that doesn't coincide with the switching
period, and is right there. And so our averaging inner volt in,
extends one half switching period, before and after
this time T. Okay?
So this is the interval that we have to average over, to compute the average value
of the inductor voltage at this time, T. To compute this average, if we include the
ripple then we ha, you know the computation gets
more complex. And so what we do is apply the small ripple approximation. And what that small ripple approximation
effectively says, or what we will assume, is that the values of quantity, of continuous
quantities are taken to be constant over this
interval. That the change over this small switching
period is taken to be small. So the inductor voltage during the first interval is the input voltage
VG. If it has ripple we're going to ignore it,
and we're going to replace this VG of T with its
average, average VG. Okay, so the D interval Is, is this one
right here. The D prime interval is the rest of the
switching period. Which, and there's some of it over on the
left side, and there's some more on the right. [COUGH] So, during the D prime interval, we also
assume that the inductor voltage during that, those
two intervals, is the same. so during the second interval, the
inductor voltage is the output voltage V, and we will take it to be it's average, or average V, which
is the same value here, as it is here. And that's the assumption that is made
with a small ripple approximation. Here's the same, same diagram.
With that assumption then, the average inductor voltage is D times this, average
VG for the D interval. Plus D prime times the average V, both
there and there, add up to a total net time of D prime TS. And so we get D prime times the average V
for the second interval. And, and therefore L times the derivative
of the average inductor current, is d times the average VG plus D prime
times the average V. A little more about this business of
averaging. We can view averaging, or this, this way that it, it's been defined here as an
operator. And this is an operator that removes the
ripple, but preserves the underlying low frequency
components of the wave form. And it's an important thing to do because it makes our equations tractable
and simple. So we get a simple set of equations of our
AC equations for the converters. And it's removing the, the switching
frequency components, while preserving the magnitude in phase of the
low frequency components of the wave forms. So really we can think of this averaging operation as being a type of low pass
filtering. Where we're removing the switching
harmonics of the waveform, and preserving the low
frequency components. And we can actually take the Laplace
transform of this averaging operator to see what it does in
the frequency domain. So we'll do that in a second, but first to
just plot it. These are computer generated plots, which
I've shown two lectures ago, but they're real simulations
of a, buck-boost convertor. the top waveform, or top axis are the gate
drive showing the duty cycle changing from a lower value to a higher value.
The middle and lower wave forms are the resulting
inductor current and capacitor voltage wave form for the
converter. The wave, the dark wave forms that have
ripple in them are the actual simulated current in
voltage including the ripple. And then the underlying thinner, smooth
curves are what is calculated by the computer
applying this definition of averaging. So what, what I've done for real, actually
here is to take the wave forms with ripple, apply the averaging operator
to them with the computer simulation, and plot the
result. And you can see that the, the average
quantity really does go through right, you know, right through the middle of the, the
ripple, and it is in fact the correct way to, to calculate this moving
average. One other small point I'll make is that if you change the interval of integration, so
instead of say, integrating from T minus TS of, over
2, suppose we just integrate from T to T plus
TS. What you're effectively doing is just
shifting the answer in time. And it would shift the average forward or
backwards in time, depending on the interval of integration
that you choose. So centering the interval of integration
right at time T is the correct one, that adds no phase shift
to the waveform. And gives us, a waveform that actually go,
goes through the middle of the the ripple. If you're, up on your Laplace transforms,
you can, take the Laplace transform of this averaging function.
and actually here's the answer. we have the 1 over TS term out in front
that is this. Integration in the Laplace domain is
dividing by S which gives us this S in the
denominator. And then the 2 exponentials in the numerator are what make the would take care of the limits of integration,
and make the integration period go from TS over 2
in the past, to TS over 2 in the future. Okay. You can let S equal J omega, and turn this into a transfer function, and plot it as a
bode plot. We're going to talk about bode plots next
week. But for now, you, I guess if your not
familiar with that, you'll have to take my word for
it. when we do that, we find that this G halve
of S transfer function with S equals J omega, turns out to be this.
It's a purely real number. There is no, no imaginary part, so there
is zero phase shift in this operator. And it has a magnitude whose plot looks
like this. So here I've plotted the magnitude in
decibels versus frequency on a logarithmic scale, like in
a bode plot. And, what you can see, is that at low
frequency, say, for example at one tenth of the switching frequency, the
magnitude is 0 db, or basically 1. And there's 0 phase shift, so the signal passes through this averaging
operator unchanged. At the switching frequency, the magnitude
of this quantity turns out to be zero. so it's like a notch filter that
completely removes components at the switching frequency, and therefore
it removes the switching ripple. It also removes harmonics of the switching
frequencies. So at twice and three times the switching
frequencies and so on, the magnitude is 0. So switching harmonics are completely
removed. So this is really what we want. And this averaging operator then,
preserves a low frequency components, even at half the switching frequency here
Magnitude is a little less than one. So, there is some change, but it's not
that great. But when we get past half the switching
frequency there's a big change in the wave forms and in particular
the switching harmonics are removed. So, you can debate whether we want to say
that the, operator doesn't change the result at half the
switching frequency or not. But certainly at sufficiently lower
frequencies this averaging operator does not change the components of
the waveform. Okay, we can apply the same arguments to
the capacitor then. What I've drawn here though, is not the capacitor wave format
some arbitrary time t, that I've simply started
at the beginning of one switching period and,
and drawn the wave form over one complete
switching period. and this is what was done in the last
lecture. So, if you like you can draw the wave
forms as I did previously for the inductor at some arbitrary time T
in the middle of the switching period. But the answer that you get for the
convertors and cases that we're considering in this
course is unchanged. You get the same answer if you just talk
about what is the average over one switching period, and
it's simpler to do. And so from now on, that's what we're
going to do. What I, what I would caution you is to say
that for some cases that are more advanced, but that we're not
going to consider in this course. It is important to carefully derive the
average as we did for the inductor wave form. And in particular, I'm thinking about
current mode control. so for those of you who know what current
mode control is, to get an up to date and
accurate model in current mode control, we do have
to carefully draw the wave forms at an arbitrary time T
like this. But that's beyond the scope of this
course. So the process of averaging is indeed well
founded mathematically. and it works very well for the continous conduction mode.