In this brief lecture, I'm going to
explain how to drive the average AC equations of a switching converter. And we're going to do this by simple
extension of the ideas of volt second balance and charge balance
that you already know how to do. We're going to extend those to write
the equations of the average voltages and currents under transient conditions. I'm going to explain just how to do it,
but not derive where it comes from. This is what you need to know to do
the homework, but in the next lecture, which really is perhaps optional we're
going to discuss where it comes from and give a little more idea of
the mathematical foundations. So I'm going to explain with reference
to this Buck-boost converter. So the usual Buck-boost converter
with an input voltage vg(t) and inductor current i(t), and
an output voltage v(t). So as usual, we start in the usual way with a switch
in position one, here is the circuit and we can write the equations for the
inductor voltage and capacitor currents. So the inductor voltage which is ldi/dt in the first position is equal
to the input voltage vg. Likewise the capacitor
current which is cdv/dt is equal to minus the load current
during this first interval. At this point what we
previously did was to replace these quantities on the right hand side of
the equation with their DC components and make the small ripple approximation
to ignore their ripples. So what we want to do today is still
make the small ripple approximation, but don't assume that the quantities
are purely DC, allow them to vary. And so what we will do is replace
the quantities on the right hand side of the equation with their averaged
values where the angle brackets here denote averaging and
I mentioned that on the previous lecture. So say the average value of v(t), where we averaged over
a period TS is equal to 1 over the period times the integral
over one period of v(t) dt. And the integrating period we can, there are different choices we can
make for the integrating period. The one that I described in the last
lecture was that we integrate over a period that is centered at time T and
we integrate a half a period before and after, like this. And what the point of this is that it
removes the switching ripple by averaging over exactly one period and
what is left is the underlying low frequency
component including the DC component. But also this average can
vary during transients. So vg(t) here is replaced
by its average and v(t) is replaced by its average. In fact what this amounts to is instead
of writing the DC component we write the average value. Okay, same thing for
the switch in position two, here is our usual circuit with the usual
equations for the inductor voltage and capacitor current for this interval. We have v(t) and we also have I(t) on
the right-hand sides of these equations. So we will replace I with its average and
v with its average. So we have equations for what the inductor
voltage is during each interval and from those equations as usual we can write
the inductor voltage waveform, which is one value during the first interval and
different value during the second. The solid dark line here
is the actual waveform including the ripples and
the dashed lines here have ignored the ripple making
the small ripple approximation. And for example,
in the second interval, the dark line, our solid line is v(t) while
the dashed line is the average v. So what we do now is we calculate
the average inductor voltage. Previously the average inductor voltage
equals 0 by volt second balance, but now we're going to let the average
inductor voltage equal ldi/dt. Where here we're taking
the derivative of the average inductor current with the ripple removed or
ignored. So this average inductor voltage is
simply d times the value during the first interval plus d prime times the value
during the second interval, here and here. And we equate this then to L times
the derivative of the average I and this is our averaged equation
that describes the inductor during transient as well as DC conditions. And what this shows is how the low
frequency component of the inductor current with the ripple removed
will vary over time, okay? We can do the same thing for the capacitor,
here is the capacitor current waveform. The solid lines include the ripple and the dashed lines employ
the small ripple approximation. So the average capacitor current again,
can be written as d times the value during the first interval here plus d prime times
the value during the second interval. We collect terms and equate this to c
times the derivative of the average v and we get this equation that describes
how the low frequency components of the capacitor voltage
will change with time. Finally, we know that to complete
the model of the converter we may have to write an equation for
the average input current. And in the DC case we found the DC
component of Ig the current coming out of EG and so for the AC model we
will need to do the same thing. So we express Ig during each interval and write the Ig(t) waveform and we apply
the small ripple approximation to it. So when the switch in position one
Ig is equal to the inductor current which we can write by ignoring
the ripple in the inductor current and say Ig is the average I during
the first interval, and it's zero during the second interval
when the transistor is off. So the average Ig then,
here which is the average or low frequency component of
the switched waveform Ig. That average is equal to d times
the average I plus d prime times 0. So here's our equation that we get for
the average inductor current. So to summarize, we can generalize the
ideas of inductor volts second balance and capacitor charge balance to write
the equations of the averaged or low frequency components
of the waveforms without requiring that their DC but
rather they can vary with time. We employ the small ripple approximation, the equations look very similar to
those you've already been writing, but now we allow the waveforms
to have AC variations. Okay, in the next lecture, which is
optional, we're going to discuss some of the mathematical details and
foundations behind averaging. And then in the following lectures,
which aren't optional, we're going to linearize these equations
that we derived in this lecture and then construct small signal AC
equivalence circuit models.