When a voltage is applied to a finite inductance for some period of time it gives rise to a current though some number of turns…..the applied volt-sec’s are equivalent to some amp-Henries….
It is the current though some number of turns, known as amp-turns, or equivalently as a magneto-motive force (MMF), that induces some amount of flux through some reluctance…
This is the electrical equivalent to some applied voltage, giving rise to some current though some resistance/impedance….
The equation for this is:
Now reluctance is a measure of how difficult, or easy, it is for a magnetic field to permeate through some material. It is also the reciprocal of the AL value for some material. AL is the inductance per squared turn around a material….and the formula for inductance is as follows:
Ok….so let’s add this to the equation for MMF
Let’s define something called flux density, B, as the flux per unit area….
Now let’s multiply each side by the area and turns, then re-arrange….
This yields the equation between amp-Henries and flux density, turns and area…
Now as first stated amp-Henries is equivalent to volt-sec’s….so let’s show this….
Probably the easiest way is to use a well-known inductor equation…
There are some interesting things to note here….one is the equation implies for some applied constant voltage with time there will always be a linear increase in current due to this, because inductance does not change yes?
In fact the inductance does change, lowering with increased current…this is due to the B-H curve of the core material…..see below a basic B-H curve….
As can be seen, there is a region over which the permeability (the slope of the B-H curve) is fairly linear. Inductance is a function of permeability and since the magnetic area, path length and turns cannot change, what can change inductance is the permeability of the core material.
If that permeability is within the two regions above, the two red lines in one direction and the two green lines in the other direction, the inductance will be fairly constant and unchanging….but apply a bit too much current such that you move outside of these lines and you can see the slope fairly quickly heads toward a zero value.
This means your inductance will drop. Now, why is this? A core can only handle so much flux density…at some point it gets “full” and doesn’t want anymore. It is at this point that increases in current result in virtually no increase in flux density and in order for the equation to be satisfied, the inductance must drop with increases in current while the flux density remains fairly well fixed at some MAX value.
We call this “core saturation” and your inductor, or transformer, at this point becomes fairly useless….
Now the most important thing to realize from the below equation is that the right hand side is how you have designed your transformer or inductor to respond to the inputs on the left hand side….
If you know how long you will apply a voltage for each period of the on/off time (duty cycle) then you must choose a core with an appropriate area and turns with a flux density within the “linear region” above…
If you know what inductance you want for some current you need to choose an appropriate core area and turns, ensuring the flux density lies within the “linear regions” shown above.
Basically, you need to ensure the following:
It best not to design so that this inequality is barely met…it is better to give yourself some room and make sure the right side of the equation is more than barely larger than the left side of the equation.
This is just the basics needed to ensure you design with enough core area and turns, using a core whose MAX flux density is well enough above what you will induce, to get you started.
1. The next consideration would be core power loss….is it too much? The area you see in-between the two lines of the B-H curve represent the power loss of the core; this is the hysteresis loss.
2. Next consideration would be wire size for the given ampacity….is your wire large enough to handle the current?
3. Does your wire fit in the bobbin for the core, or do you need to go larger in core size and reduce turns?
4. Does it need to be Litz wire to avoid unnecessary skin effect issues?
5. What is the power loss due to the copper?
6. How does it compare with the core loss?
7. Are the core and copper losses roughly equal? This yields the most efficient designs….
8. Have you designed in appropriate isolation – creepage and clearance….will your design pass some required HIPOT testing?
9. Etc, etc, etc….
The design process is often iterative. First try may end up too big leaving empty too much area in the bobbin window, or too small to fit all the turns of wire and tape…whatever the result, move to the next size down or up and try again until you get something you are happy with.
Whether you are designing a flyback transformer, push-pull transformer or a PFC inductor knowing these equations and following these rules will be necessary.
