0:12

Now let's work a second example of a buck converter

containing an inductor with some DC resistance.

So here's our converter.

We will model the inductor in the same way we did previously for the boost converter.

Where we have a lumped element model of an ideal inductor in series

with an effective resistor that models the winding resistance, okay?

We can go through the same procedure to work out volt second balance on

the ideal part of the inductor model and charge balance on the capacitor.

And if we do this, I won't go through the details but if we do it,

here are the equations that we get.

So we get this equation from volt second balance on the inductor.

It's like in the ideal case except that it has an additional term from

the voltage drop across the winding resistance.

And we get this term from capacitor charge balance that is in fact the same as

in the ideal case.

1:23

That was DVg minus the inductor current times the inductor

current resistance minus the output voltage.

And for a charge balance, we got the average capacitor current which was 0,

was equal to the inductor current minus the load current.

So from the first equation, we have a source,

a dependent source that depends on Vg, that we can write like this.

2:04

Here is where our ideal inductor model goes.

That's effectively a short circuit.

And then the current I flowing through this voltage.

We have right now a term that is the voltage drop

across the winding resistance.

2:21

And then we have the output voltage right here.

In the buck converter,

we know the inductor is always connected to the capacitor node.

And so, in fact, we expect our inductor model to connect directly there.

So this would be the capacitor node.

3:00

So here is the equivalent circuit that we get from the two equations.

This model is okay as far as it goes, but

there is a big question of where is the DC transformer?

In a buck converter, we would expect to have a DC transformer with

a turns ratio of 1 to D equal to the conversion ratio.

And here we have a voltage source that is DVg that looks like maybe it's part

of a transformer.

But where is the rest of the transformer?

And for that matter, where is the input voltage source Vg?

The problem here is that we're missing an equation that describes

how the converter connects to the input terminals, or up to Vg.

We need another equation to describe that connection.

4:01

Is that in our original converter, both inductor and

capacitor are on the same side of the switch.

So we don't have a volt second balance or charge balance equation

that describes the connection to the input port of the converter.

And so we need to manufacture another equation that describes the input port.

4:26

Okay, so we have an input port here.

We know there's a voltage source Vg, but

we need to write an equation of what the converter does.

What current does it draw out of Vg?

And so the way to get that is to write an auxiliary equation or

a third equation that really tells us the DC component of ig.

Which is the current drawn out of Vg, or drawn out by the converter,

out of the node at the input port or input terminals of the converter.

In general, ig is a dependent quantity.

It switches when the switch changes between positions 1 and 2.

It's a discontinuous waveform.

The small ripple approximation does not apply to it.

But we need to sketch what ig is and

express it in terms of things that have small ripple, like the inductor current.

And then find its DC component to give us another equation that we can use to

complete our model.

6:31

So ig looks like this, okay?

What we need to do is calculate the DC component of ig, or

the average value of this waveform, okay?

So we follow il during the first interval.

And the first thing we'll do is make the small ripple approximation on the inductor

current.

Again, the inductor current has smaller ripple and is a continuous waveform,

so we can replace IL(t) by it's DC component, capital IL in usual manner.

And then find the average value of this waveform,

which is just D times the capital IL plus D-prime times 0.

So this is an equation, it's an auxiliary equation for

a quantity that we need to know.

It's a dependent quantity we need to know in order to complete our model.

We can construct an equivalent circuit that goes with this and

then append it to the rest of the model.

So this equation is really the equation

at the node of the input terminals of the converter.

And what it says is that the current Ig which is

the the DC component of current drawn out of Vg.

So I'm going to draw our Vg source and

label the current coming out of Vg as capital IG for our DC model.

And this is equal to DIL, so it is a dependent source that is

dependent on the inductor current IL, elsewhere in the model.

And we can draw a dependent source then,

current source, and label its value DIL.

[COUGH] And when you append this to the rest of the model,

now we can get a complete equivalent circuit.

8:34

So here is the input port equivalent circuit that we just derived.

Here is the equivalent circuit that we got from volts second balance and

charge balance.

And you can see that there are now two dependent sources that can again be

combined into a DC transformer.

And they have turns ratio of 1 to D.

We follow the voltages Vg gets multiplied by to D to produce the secondary voltage.

And the dots follow the plus terminals and are always on the top.

And so here's now an equivalent circuit for the buck converter.

That includes the DC transformer that we expect.

So in general, when we have switching at one of the ports of the converter,

we may need to write an equation for an auxiliary variable.

In this case, the node equation at the input terminals of the converter.

That gives us a complete set of equations to describe the equivalent circuit.

So here we found the waveform of Ig(t),

we expressed it in terms of things that had small ripple such as the inductor

current, and use the small ripple approximation on the inductor current.

And we were able to express the DC component of Ig then as a function

of the DC IL to get an equation that could give us the complete transformer.

In general, we may have a similar situation at the output

if there are switched waveforms on the output side of the converter, and

we may have to do a similar thing there, okay?

So in converters such as the buck or other converters,

even say a buck-boost is another one that have switching on the input port,

we will in general need an equation for the average Ig.