# Designing high-current chokes is easy

-June 24, 2004

Somewhat unusually, this Design Idea deals with a formula rather than a circuit. You might think that all the basic formulas of magnetic phenomena were discovered more than a century ago. In fact, they probably were, but, at the time, some were of little practical interest and were essentially disregarded and never included in books or formula tables. I developed the formula describe here because I had to design many inductive components subjected to high peak currents, such as dc filter chokes, ac reactors for resonant converters, and flyback transformers. In such cases, you have to consider two main aspects: One is the current-carrying capacity of the wire, and the other is the peak induction that the core material supports. The first point is well-known and relatively easy to deal with, but the magnetic induction is much more problematic to determine.

The traditional methods of selecting a suitable core size and air gap are generally based on tables or graphical information. Examples of such methods include Hannah curves and energy-storage-capacity graphs. I found these methods cumbersome, inflexible, and almost impossible to automate; hence, I looked for a better approach. I wanted a formula as compact and elegant as the one that is at the base of a symmetrical converter's design: N=V/(4BFA), where N is the number of turns required to achieve the target induction, Visthe voltage applied to the winding in volts, Bisthe peak magnetic induction in the core material in tesla, F is the frequency of operation in hertz, and A is the effective core area in square meters.

This formula is attractive because you need only essential parameters; you need not mess around with the permeability or the length of the magnetic path, for example. By combining and algebraically manipulating the fundamental equations of the magnetic formula, I arrived at a similarly simple equality applicable to inductors: N=(LI)/(BA), where L is the inductance in henries, and I is the instantaneous peak current in amperes. Here again, you need no more parameters than the bare minimum. Using this formula, a typical design procedure is:

1. Select a core size that seems likely to suit your application (The selection information that the manufacturer provides can be useful.)

2. Use the formula and the core's data sheet to compute the number of turns required for the worst-case situation—in other words, the maximum peak current and magnetic induction below the saturation limit for the whole temperature range.

3. Check that the resulting winding does not exceed the capacity of the coil former; if it does, select the next-higher size.

4. Compute the air gap required to achieve the target inductance using the manufacturer's data or the following formula (approximate):

where µ0 is the permeability of a vacuum (4π×10–7), and k is a factor that depends on the implementation of the air gap. For a single air gap, as in a potentiometer core in which the center pillar is machined, k=2. If, instead, you use spacers such as in a U-core, the air gap is split in two, and the factor k=1. If you need high accuracy for the inductance value, you should build a sample to optimize the gap. Also, for small or large gaps, the formula loses its accuracy because it assumes that the magnetic material has a negligible reluctance compared with the air gap. If the gap is small or if the core material has a low permeability, the assumption about negligible reluctance is no longer true. At the other extreme, the first-order term of the formula does not sufficiently compensate for the apparent increase in the core area that fringe fields cause. Thus, discrepancies can exist between the calculated and the measured values.

The relationship N=(LI)/(BA) can also be useful in a different manner. You may want to reverse-engineer off-the-shelf components to check that they do not risk saturation at the intended peak current. (In converter circuits, the peak current can be much higher than the rms current.) To do this reverse-engineering, you can use the form B=(LI)/(NA). For most general-purpose ferrites, a peak induction of 0.2 to 0.25 tesla is acceptable, whereas materials for power applications can tolerate more than 0.4 tesla. Metal-powder cores accept inductions as high as 1 tesla. If you want to know what maximum current is acceptable for a component, then the following form is convenient: I=(BNA)/L. At first sight, this formula looks counterintuitive or even erroneous, because it seems to imply that you can increase the current for a given induction if you also increase the number of turns. How can this situation be?

Increasing the current or the number of turns results in an increase in ampere-turns that the core sees, which should also increase the induction. The key to understanding this apparent paradox is to take into account what the formula implies: If L has to remain constant with more turns, the air gap must be wider to reduce the apparent permeability (µ) of the core, resulting in a greater current capacity, although the air gap appears nowhere in the formula. The paradox may explain why hardly anyone ever mentions this family of formulas. If you try to superficially make sense of the implications of the formulas, you have to conclude that there must be a mistake somewhere.

You can also apply the results to open-circuit magnetic components, such as cylindrical coils wound on a rod of magnetic material. In this case, the air gap becomes almost as large as the core, yielding two implications: Because the surrounding vacuum or air contributes as much as the core itself to the inductance, you can double the core area the formula uses with respect to the physical value, and, even when saturation does occur, the effect is much less brutal than in a closed magnetic circuit. Second, the simplified inductance formula is no longer valid.

To conclude, the user-friendly versions of the formulas, expressed in more convenient units are: N=0.01(LI)/(BA), with L in microhenries, I in amperes, B in tesla, and A in square centimeters.

And the user-friendly expression for the air gap is

where Δ is in millimeters, A is in square centimeters, and L is in microhenries.

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