This course introduces more advanced concepts of switched-mode converter circuits. Realization of the power semiconductors in inverters or in converters having bidirectional power flow is explained. Power diodes, power MOSFETs, and IGBTs are explained, along with the origins of their switching times. Equivalent circuit models are refined to include the effects of switching loss. The discontinuous conduction mode is described and analyzed. A number of well-known converter circuit topologies are explored, including those with transformer isolation. The homework assignments include a boost converter and an H-bridge inverter used in a grid-interfaced solar inverter system, as well as transformer-isolated forward and flyback converters. After completing this course, you will: ● Understand how to implement the power semiconductor devices in a switching converter ● Understand the origins of the discontinuous conduction mode and be able to solve converters operating in DCM ● Understand the basic dc-dc converter and dc-ac inverter circuits ● Understand how to implement transformer isolation in a dc-dc converter, including the popular forward and flyback converter topologies. Completion of the first course Introduction to Power Electronics is the assumed prerequisite for this course.
Sect. 5.1 Origin of DCM and Mode Boundary
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Del curso dictado por University of Colorado Boulder
Converter Circuits
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Ch 5: Discontinuous Conduction Mode
The discontinuous conduction mode (DCM) arising from unidirectional switch realization. Analysis of mode boundaries and output voltage.
Conoce a los instructores
Dr. Robert EricksonProfessor
Electrical, Computer, and Energy Engineering
Having discuessed how to realize switches using semi conducter
devices and to implement things such as single quadrant or two quadrant
switches in a switching converter We're now in a position
to discuss what, what is known as the discontinuous conduction mode.
And chapter five covers the first the
origin of the discontinuous conduction mode and
then how to solve it, and we'll do some examples of
solving Bach and boost converters that
operate in this discontinuous conduction mode.
The converters that we've discussed so far in this course operate in what is called
the continuous conduction mode by contrast to the
mode of operation we're going to discuss, now.
the discontinuous conduction mode
occurs because there is a ripple in the
current or voltage waveform that is applied to one
or more of the switches inside the converter,
and this ripple is larger than the DC component.
and as a result, the, the current or
the voltage attempts to reverse through the switch.
And now, we've realized the switch to be a unidirectional
switch, and we try then to reverse the polarity.
the switch doesn't behave in a way that it was intended,
and so it will switch on or off at a time that
does not coincide with the driver signal switching, and so we get additional
intervals or sub-intervals within the switching period as a result.
And this completely changes the characteristics
of the converter.
The most traditional, discontinuous mode of operation occurs
because the inductor current ripple is greater than the
DC component of current, and this causes the
current through the switches to try to reverse direction.
the diode won't allow it.
The diode will turn off, instead.
And, we get a third period during the switching period, and this
changes the characteristics of the converter substantially, so
when the discontinuous mode happens, what we're going to
find is that the output voltage now becomes load-dependent.
In continuous mode, say for a buck converter, the
output voltage is the duty cycle times the input voltage.
Any dependence on load is small and only comes
from loss at the effects of loss in the converters.
in the discontinuous mode though, we have a first
order dependence of the output voltage on the load current.
so the properties of the converter change, drastically.
We then all of a sudden, we have a high output impedance.
We, we're going to find that the, or it turns out that the dynamics of the
converter that we're going to discuss in several weeks, those dynamics change
very substantially.
And in fact, they're simpler in discontinuous mode.
we also find that when you remove the
load, the discontinuous mode causes the output voltage
to not be controllable, and it can do bad things, so we have to account for that.
Some converters are designed on purpose to always work in discontinuous mode.
One of the good things about it is that the current goes to zero before the end
of the switching period, the diode turns off, and
then there's no reverse recovery when the MOSFET is
next turn on. So there can be less switching loss.
But on the other hand, there are higher p currents, and so
we can get more conduction loss, and there's a trade off then.
even if we don't intentionally design a converter to
work in this mode many converters will operate in
discontinuous mode at low output power, and so we
have to be able to analyze and know what's going
to happen then at those operating points.
So what we're going to do in this lecture is discuss the origins of
the mode and find the mode boundaries,
the conditions under which we operate and continue
as our discontinuous mode, and then in
the next several lectures we will analyze converters
to find their output voltages and other
things when they operate in the discontinuous mode.
So, here's an example.
We have a, a basic buck converter, and we've realized the
switches with conventional single-quadrant switches of a transistor and a diode.
here is what the inductor current wave form looks like in the continuous mode.
And it's the wave part we've been drawing all along this in this
class, so it has a DC component that we're labeling capital I plus
it has some switching ripple that has a peak magnitude of delta I.
Kay?
We previously analyzed this circuit, and we know
how to calculate capital I and delta I now.
here are the expressions.
So the DC component, capital I, is the load current, V over R.
And delta I, the, the, peak to average ripple is the slope times the time,
which we can write, in terms of VG, like this.
and the ripple depends on the duty
cycle, VG, the inductance and the switching period.
Okay, the interesting thing about this is that the DC component depends on the load
resistance, or on the, the current that the
load decides to draw, but the ripple doesn't.
Ripple depends on all kinds of things, but the one thing
it doesn't depend on is the low resistance or the load current.
So what happens if we say increase the value of'
R or our all of a sudden decides it doesn't
need much current?
Well, that will make the dc component capital
I go down, but it won't change the ripple.
what's important here is actually the effect of that on a
diode, so when the diode is on it, conducts the inductor current,
and the diode current wave form looks like this, so when the
diode is on, it has the same capital I plus the ripple.
And the minimum diode current happens right there at the end of the switching
period. And that minimum value is the DC component
of current minus the ripple. Kay?
So, the diode current has to stay positive for the diode to work.
And, when we account for the ripple, the, the, minimum diode current is
actually less than capital I. So, let's consider increasing the value
of R to the point where the ripple equals capital I.
Actually, we reduce capital I, so it's equal to the ripple.
And the inductor current looks like this, and the diode current
follows the inductor current for the second half of the period.
So you can see that the diode's current starts out
at I plus delta i, and it ends I minus delta
i which for this particular load current is equal to zero.
Okay, what happens if we increase the load resistance even more?
Well, here is that case.
DC component of lo, load current and inductor
current, capital I, is now less than delta i.
So even though the average current and the load current are positive, they're less
than delta i, and so the diode current and inductor current go to zero before
the end of the switching period. Okay, once that happens, the
diode will turn off, and it will not allow the inductor current to continue in
the same direction and go negative. So we get a third interval now
this, from the point the diode turns off until the end of the
switching period where, the diode and the MOSFET are off and the inductor
current just sits at zero current.
So now, our switching period is divided into 3 intervals.
There's the D1 interval, the first
interval, which we're also going to call DTS.
It's the, D is the tran-, the MOSFET duty cycle,
and D1 is the duty cycle of the first interval.
Then they're the same here.
we have a D2 interval now, when the diode conducts.
And we have a D3
interval when nobody conducts.
So, this is the discontinuous mode of operation, and
this extra interval changes all the equations of the converter.
So first of all, we need to be able to write the equations of the mode boundary.
And we now, we have a we have a way to do that, that the mode boundary
happens when delta i is equal to the DC component capital I.
Kay?
and in fact, the conditions then are that
we're in DCM, or discontinuous mode, when, when capital
I is less than delta I.
Now, at the boundary, the continuous mode equations
are still valid, so we can plug the continuous
mode equations that we had for capital I and
Delta I into these equations to find the boundary.
So here's, here's the expression for capital I in terms of VG.
Here's the expressions for the delta I.
And so we can plug them into this.
And we can solve, so what, the three VG's cancel out.
we can move the two l and t s over to the left hand side.
And cancel the d's it looks like. And we'll get this expression.
So this is the expression or the condition for operation in the discontinuous mode.
Kay, the quantity on the left-hand side is
a function of element values in the switching period.
it's traditional to call this quantity K, and so in the power
electronics business, we generally call capital K is two L over RTS.
And this two L over RTS factor keeps coming up
in many of our equations, over and over from here on.
the right hand side is the critical value of K at the
boundary between modes, so we call that K crit.
so the right hand side is a function of duty cycle.
It will vary with duty cycle.
but, when K is equal to K crit, then we're at the mode
boundary, and if K is less than K crit, then we operate in
the discontinuous conduction mode. For the buck
converter, we, we find here that K crit is equal to D prime.
For different converters, they'll have, K crit
will be a different function of duty cycle.
But in general, we can write this expression for the mode boundary, in
some equation of the form K less than K crit of D for discontinuous mode.
So here's a plot of that. K crit for the buck
converter is one minus D or D
prime. So here's a plot of K crit versus D.
So K crit is one at D of zero.
K crit is zero at D of one, and it looks like that.
Okay, so we compare that to K. So suppose K is less than one.
But the, the maximum value of K crit is one,
so K is less than one, says some value like here.
Then we'll be in discontinuous mode over this range where K is less than K
crit, and then at higher duty cycles over this range we'll be in continuous mode.
On the right side here is another example where K is bigger than one, and in
that case, K is always greater than K
crit, so we're always in continuous mode then.
For the different converters Buck, Boost and Buck-Boost it with their K
crit expressions labeled so, K crit for the Buck is D prime.
We're going to show in an upcoming lecture, that
K crit for the Boost is D times D prime
squared, and on the homework, you're going to work out for
the buck boost, that K crit is D prime squared.
for the buck,
the maximum value of K crit for any D between zero and one is one.
So if K is greater than one, then we're always in continuous mode.
But if K is less than one, then we have to worry about discontinuous mode.
For the boost, this, this maximum value turns out to be four twenty-sevenths.
We'll discuss that.
it's an interesting number that comes out of nature here.
the boost is less likely to
run in discontinuous mode, but if K
is smaller than four twenty-sevenths, then it can.
for the buck boost, the maximum value of K crit is one.
So again, the K less than one, we may run into discontinuous mode.
Okay, so that's the mode boundary in the next lecture.
We're going to solve for the output voltage.
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