S4w's Wind Generator Notes

Inductors

To begin this series of tutorials we must deeply understand how inductors work since they are the most important part in a DC-DC converter together with the switches and proper layout.

What are inductors?

Inductors are nothing more than energy storage devices just like capacitors. The difference between them is simply the way they store the energy. Inductance is measured in Henries. Inductors store energy in the form of a magnetic field and the amount of energy (in joules) stored in an inductor is given by:

Eq. 1

If we plot a graph of the energy stored by various inductors of different values we can see that there is more energy stored in them when there is more current regardless of the other inductors being of higher values. This is because the energy stored in the inductors grows exponentially with respect to the current through them but only grows linearly with respect to the inductance value. The energy that can be stored in an inductor unfortunately is not infinite, it has a limit and it's reached when the inductor saturates. But we will get into that later.

Properties of an inductor:

Something that we must always keep in mind are the properties of inductors. Here I list them separately:

Property #1:

Inductors behave as a short circuit in the presence of DC current when they saturate. Reaching this point takes some time.

Saturation occurs when the magnetic material, in our case probably ferrite, can no longer fit any more flux lines though it, thus, a further increase in current will not result in a greater amount of energy stored as a magnetic field. At this point the inductor is pretty much a short circuit because its effective inductance have dropped.

As we probably know, switch mode converters are operated by applying a series of square pulses with a predetermined width (PWM). With this in mind lets see what is the transient response of an inductor. Transient response is how the inductor behaves from the moment current is applied to it at t=0 to the moment that it saturates or vise versa.

The current in an inductor at any time is given by:

Eq. 2

= Maximum current through the inductor.
= Initial current through inductor when circuit was closed.
Where Tc is the inductor's time constant and its given by:
Tc = L/R Eq. 3

Without getting into messy mathematics you will have to take my word for it. Notice how the inductor current grows linearly with respect to time. At t=0 there is no current flowing through the inductor. Not because it is not connected to the electric circuit but because inductors tend to resist changes in current. This brings us to our second property.

Property #2:

On the presence of very high frequencies inductors act as an open circuit. Since the inductor was at 0 Amps before closing the circuit it will try to remain at 0 Amps for as long as it can. This is the same property that makes filters work. When there is a sudden change in current the inductor will react to keep things as they where thus preventing that change to be seen on the other side.

Combining Eq. 1, Eq. 2 and Eq. 3 we get Eq. 4 and can now plot a more realistic graph of how inductors store energy during each cycle of the PWM signal.

Eq. 4

This graph does not take into consideration the effects of saturation. I will explain more about that subject when we get into actually designing an inductor for the buck converter.

The most common variables that you will find are B and H which are often called “magnetic field”. In reality there is very little difference between them. H is the magnetic field in free space, B is the magnetic field inside a magnetic material such as ferrite. But they are exactly the same thing. The relationship between them is given by a property of the magnetic material in which the field is going through. This property is called permeability and its denoted by the Greek letter μ.

Eq. 5

The permeability of a magnetic material can be thought of as the gain in an operational amplifier, because it concentrates the magnetic field making it stronger than it would be if the magnetic material wouldn't be there.

The value of a given magnetic core is also very common. This number tells us what inductance value in millihenries is achievable for the magnetic core in question per 1000 turns.

The Buck Converter

The purpose of a buck converter is mainly to efficiently convert from one voltage level to another. The same power that goes in is what it should come out ideally. In the real world all converter types have inefficiencies and can be calculated the same way for all of them.

Eq. 6

The buck topology looks like in the picture below. There are other methods of building a buck converter (synchronous rectification) that will also be explained later. For now we can concentrate on the very basics of its operation.

The switch at the positive rail that connects and disconnects the output from the input can be either a MOSFET, a BJT or an IGBT. The amount of time that this switch is closed determines the output voltage. For example, if your switching frequency has a period of 1uS and the pulse width is 50% ( the switch will be turned on 50% of the time, 0.5uS) then the output voltage will be 50% of the input voltage. If the input is 12V then the average output voltage will be 6V.

During the on state (switch closed) the supply will be providing the voltage and current to the load normally through the inductor. Then, during the off state the polarity will reverse through the inductor as a result of the collapsing magnetic field and the current will now be supplied to the load by the inductor through the diode. During the on state the diode is simply sitting there reverse biased, it is only when the switch is closed that the diode is forward biased.

Now that we know how buck converters and inductors basically work we can go ahead and start setting design goals for our converter.

Design Goals

= 15V
= 5V
= 10A
F = 20kHz
= 1A

Eq. 7

Eq. 8

Selecting The Right Core

Using the above formula (Eq. 8) we can calculate the value of the inductor that we need for this task, and that value is 82.5uH. The value of should be about 20% MAX of . The lower the value of the better efficiency your converter will have. This is because magnetic materials have something called remanence that opposes to changes of the magnetic flux that passes through them. So, the less you try to change it the better efficiency the converter will have. Now we will use a Magnetics Inc. catalog to find the right core for this converter.

The chart helps us locate what geometry factor we need to consider for our application depending on the amount of energy that the inductor will have to store per cycle, actually twice that amount.

With the data that we have from our design goals we can easily get that number by using the equation provided by Magnetics Inc. They want the value of the inductance in milihenries so:

Now we look for that value in the selection chart and match it with a core geometry. In this case is core geometry 77868. Using the catalog we extract the information about that core:

μ= 26
= 30
Path Length= 20.0cm

With this data in hand we can now find out how many turns of wire are we going to need in order to get the desired inductance. It is calculated as follows:

Eq. 9

By using this formula we get that we will be needing 53 turns of wire. Next, we need to calculate N/HI with the following formula:

Eq. 10

And we get: H/NI=0.063

With this data we now need to calculate the magnetic field that is being applied to the core at maximum current to see if the core will actually be able to handle that much flux. If it cant handle it then we would have to select a core that is a size bigger and repeat the same operation until we find a suitable core. The Magnetics Inc. catalog just gives us a starting point to select the right core.

Finally, to calculate the magnetic field being applied to the core we will use the following formula:

Eq. 11

H=33.3Oe

Now, looking at the DC BIAS graph also provided by Magnetics Inc. we can determine if the core is good or not. We must never allow or inductance value to go under 80% of the initial value that we calculated as we would be risking going into saturation.

From the chart and our calculated value for the magnetic field we can see that the curve for 26 permeability core goes down by only 5% of the initial value and this IS GREAT!

If you were to find that the permeability goes down 80% or more, the process would have to be repeated again until you find a suitable core.

Next Tutorial will be about synchronous Buck converters.