[BLANK_AUDIO] So far in the course we have talked about converters containing only
inductors, capacitors and switches. We've ignored one of the other primary
elements used in switching converters, namely the
magnetic transformer. So, for the next several lectures, we're going to discuss converters with
transformers what the basic circuits are, and how to model them and extend what we already know to
understand the operation of transformer isolated
converters. There are a number of reasons to incorporate a transformer into a
converter. One of the major ones is when safety
requirements require that we have transformer isolation from the power line
now one way to do this is to put a 60 cycle
transformer at the input of our system. But 60 cycle or 50 cycle transformers
elsewhere in the world these transformers are very
heavy and, and large and now a days we generally
try to avoid them whenever we can. the size of the transformer scales more or
less inversely with frequency. So if we can build a transformer that operates at the converter switching
frequency, say 100 kilohertz instead of 60 or 50 hertz the size of the
transformer and its weight can be reduced by a very
large margin. Here's an example of a what a iPhone
charger that plugs into the a. 120 volt 60 cycle wall and produces a five
volt output to drive the USB connected devices and power them.
Inside this charger unit is, is a fly back converter
which is a transformer isolated version of the
buck boost converter. And it contains a high frequency transformer that operates at the switching
frequency. [NOISE] Another reason to use a transformer is
when we have a large step up or step down of
voltage. Suppose we wanted, we had 100 volts and we
wanted to make one volt. We could build a buck converter and
operate it at a duty cycle of 1%, but such a small duty cycle may have, may mean that the
conduction time of the transistor is equal to the switching
times of the transistors. And we find then that the efficiency is
very low. And most of the time, we're switching
instead of conducting. So instead, we could build a converter
that includes a transformer, that has a turns
ratio, and use the turns ratio to do most of the
work of stepping down the voltage. And that converter then operates at a much higher duty cycle, where the efficiency is
better. Yet another reason is we sometimes build multiple output
converters that supply multiple voltages. And these are simply additional secondary
windings on the transformer inside the converter, and the voltages approximately
scale by the turns ratios of the windings. Here's a diagram of a transformer that is a physical magnetic transformer, it has
multiple windings, each with some number of turns, so winding
one has n1 turns, winding two has n2 turns, and so
on. And we've labeled the voltages and
currents of each winding. here the convention used with polarity
marks on the windings is we're defining, for now, all the currents as flowing into
the respective polarity mark or dot. And we're defining the voltages as plus
with respect to the dots. To understand how a converter works that
contains a transformer like this. We need to first model the transformer. And to write a set of equations that
describe the behave, its behavior. What we're going to do is to employ this equivalent circuit model for
the transformer. And so. The physical transformer on the left is
replaced with this equivalent circuit model on the
right. Where everything inside my dashed lines
here, is internal to the model and is modelling things that are going on inside the
transformer. So we replace our physical transformer
with this equivalent circuit model. And this equivalent circuit model has
elements that we can solve using standard circuit
analysis. So the equivalent circuit model we will
use is most of the time, is this simple first order
model that contains an ideal transformer.
Inside these dotted lines. and then in parallel with one of the
windings of the ideal transformer is an inductor that is called the magnetizing
inductance of the transformer. 'Kay? So there isn't actually a physical extra
inductor put on our transformer. This is built into the transformer. And everything inside this dashed line is a equivalent circuit that is modelling
correctly, the terminal equations between the voltages
and currents of the windings. Okay, the definig equations of the ideal
transformer, we'd seen before the voltage of each winding divided by its
turns is equal for all of the windings. So V1 over N1 equals V2 over N2 and so on,
and this makes the voltages scale according to
the turns ratios of the ideal transformer. The equations of the currents going into
the windings of the ideal transformer. is this one. And this is, again we've seen before also,
the sum of the amp turns flowing into the dots of the ideal transformer is zero.
Now in addition to the ideal transformer, we're adding one more element.
this L sub N is called the magnetizing inductance, and its current, I sub N, is
called the magnetizing current. The magnetizing inductance is actually
real. And the way to see that is to do the
thought experiment of, suppose we connect something up to our
transformer, like some AC voltage. Connect it up to our primary winding. And suppose we disconnect all the other
windings. So there, winding two and winding three
here are simply open circuited. What do we expect this to, to do do? Well this is a real, physical magnetic
device. It has some number of turns of wire on a
iron core or ferrite core. and if you have simply some turns of wire
on one winding that is on a core, we would expect that to
behave like an inductor. And the magnetizing inductance then
actually models that inductance. So the magnetizing inductance referred to
the first winding, where it's drawn here, is the
inductance that you would get if you simply put in
one turns of wire on the core. Magnetizing inductance does behave as a
real inductor. It obeys the voltage is L di dt, or in
this case V1 is Lm diM dt, and it has to obey volts second balance. The average value of the voltage applied
to this inductor must be zero. So we can apply volt second balance to
find the steady state behavior. Of the, of circuits containing physical
magnetic transformers. Magnetizing inductnace also explains why
you can't put DC through a transformer. So we can't, for example, apply a DC voltage, Vg, to our
transformer. Because at DC, the magnetizing inductance
being an inductor has zero impedance, its impedance goes to a short circuit at
DC and then shorts out our DC voltage. And if we apply a DC voltage volt seconds
won't balance on, on LM. The magnetizing inductance is a physical
inductor in that sense. It models magnetization of the transformer
core. The transformer core has a BH
characteristic. If you apply too many volt seconds to the
transformer core it will saturate and the transformer will
turn into a short circuit. Which generally is bad in most of our
converters. so if the applied volt seconds on the winding is
too large, what happens when you apply a voltage is
we integrate. So by the by Faraday's law, the flux in
the core is given, or proportional to the integral
of the applied voltage. As we apply volage, we integrate up the BH
loop. When we get into saturation, then the
current in the windings and the magnetizing
inductance becomes very large. And magnetizing inductance essentially
turns into a short circuit. Which is called saturation of the
transformer, and it's generally something we want to
avoid. So then, we apply volt-second balance to transformers when we analyze converters
that contain them. So the procedure here is that we replace
the physical transformer with this equivalent circuit model that
contains an ideal transformer plus a magnetizing inductance in parallel
with one of the windings. And then we apply volt second balance to
this magnetizing inductance just like we do to any other inductor in the
converter and as we've been doing a in the past and
understanding how the volt seconds balance then is part of the
analysis of, of the converter. Okay we sometimes use the terminology transformer reset and transformer reset is
the mechanism by which the converter circuit
makes the volt seconds balance on the
transformer. In many converter circuits it's not easy
to add a transformer. We have to add additional circuitry to make the transformer reset and get the
volt second balance to happen. So we're going to discuss that with
respect to a transformer isolated version of the buck converter
known as the forward converter. To understand how the transformer gets
reset and how we can get the volt second to balance. One last point this equivalent circuit
model is a simple first order model that is adequate for
most things we're going to do in the analysis of converters
but if desired, we can further refine this model
to model other things. So for example, if we want to model the
resistance of the windings, like we've done in the, in
previous exercises for modeling losses in inductors, we can model resistance of
the windings of the transformer by putting resistors in series
with each of these windings. And in our equivalent circuit model, then,
these resistors appear, basically, at the
terminals of the model. So in these places. Sometimes we model what's called core loss
in the transformer, and we can do that by putting a resistor in parallel with a magnetizing inductance, traditional way
to do that. And sometimes we model what are called
leakage inductances that are effectively series inductances in
series with these windings. Okay, we're not going to model those in
this course but they do exist, there are higher order effect in the transformer but
there were times where we need to, to include
them. So in the upcoming lectures we're going to
use this transformer model to understand how some of the basic transformer isolated converter
circuits work. We're going to look at transformer
isolated versions of a buck converter. And the buck-boost converter. And we'll apply this model and understand the transformer reset,
generalizing our ideas of volt-second balance and charge balance
to include volt-second balance on the
magnetizing inductance. I'll also briefly list some of the other
common transformer-isolated converters.