In this lecture, we will develop the basic analytical techniques that allow us to

solve for the voltages and currents of any arbitrary switching converter.

Specifically, we'll employ the small ripple approximation to

simplify the equations and we'll derive the principles of inductor volt,

second balance, and capacitor charge balance that

give us the DC equations of the converter circuit.

So, you recall from the previous lectures we discussed the buck converter,

having a switch and having a low-pass filter that removes the switching harmonics,

but allows the DC components or the DC component of the waveform through to the output.

Now, in a practical converter,

we can't build a perfect low-pass filter,

the low-pass filter will have attenuation,

but it won't completely remove the switching harmonics of the waveform, and therefore,

the output voltage of a practical converter,

might look like this,

where there's a DC component, capital V, that,

as we previously found,

is equal to the duty cycle times the input voltage, Vg.

Then in addition to that,

we have some small ripple that is at the switching frequency and it's harmonics,

that where small amount of

the switching ripple gets through the filter and appears in the output.

As sketched here, this ripple

is actually probably a lot larger than it would be in practice.

So, in a practical circuit,

we'll have some kind of specification on how large this switching ripple can be,

and generally, it's a very small number.

So, in an output voltage of say,

a volt or two for a computer power supply,

this ripple might be 10 millivolts or some very small number.

So therefore, we can write the equation of

the output voltage V of T as being equal to capital V,

which here, I'm using the capital letters to denote the DC components.

So, capital V is the desired DC component equal to DVg,

then plus some AC variation that's called V ripple of T here,

that is the undesired switching harmonics

that make it through the filter to the output voltage.

So in a well-designed convertor,

this LC filter will have lots of attenuation and

the ripple will be very small compared to the DC component.

So, we call this the small ripple approximation,

where the ripple is small compared to the desired DC component,

and under certain circumstances,

we will neglect the ripple and simply

approximate the output voltage by it's desired DC component.

This has the effect of decoupling the differential equations of the circuit,

and making them very simple to solve.

Okay. So, this is called

the small ripple approximation to ignore this component and simply

approximate V of T by capital V. Okay.

Brief discussion here, as we're going to see in a minute,

the small ripple approximation formally can be

applied only to continuous wave forms that have small ripple.

So, we don't apply the small ripple approximation to switched waveforms in the circuits,

such as the switch output voltage,

Vs of t. Instead,

we apply the small ripple approximation to continuous waveforms and

specifically the inductor currents and capacitor voltages of the circuit.