Some basic concepts:

**Magnetic field strength** is defined as

H = I * N / length = Ampere-turns / meter
Thus you can produce any desired field strength by increasing current or number of turns, and by decreasing the length over which the current is applied.

**Magnetic flux density** is defined as

B = Phi / A = Flux / Area = Vs / m^2 = Teslas
Page 122 of the Epcos catalog provides these formulas along with a description of hysteresis and the mechanism that describes the behavior of magnetic materials. The flux density (B) is related to field strength (H) by

B = u0 * ur(H) * H
The B-H curve shows that, at low levels of H, the flux density follows a curve close to that of a coil in air or vacuum, but soon the curve rises more sharply, corresponding to a higher permeability, which is caused by magnetic domains in the material aligning with the field. Eventually, all of these domains align, and the flux density flattens out, so that it does not increase with field strength, except perhaps a slight amount according to the permeability of vacuum u0. This is called saturation.

The B-H curve also shows the

**hysteresis** of the magnetic material, which is caused by remanent magnetism or alignment of the magnetic domains, which do not reverse until a sufficiently strong opposite field is applied. This hysteresis varies considerably with the material, being less for "soft" materials such as iron, and greater for "hard" materials such as ferrite. The area of the hysteresis curve is proportional to the losses of the material.

An inductor is a component which stores energy in the form of current and the electromagnetic field it produces, which is analogous to a capacitor which stores energy in the form of voltage and an electrostatic field. The energy stored in the inductor is:

E = 0.5 * I^2 * L
**Inductance** (L) is related to the slope of the B-H curve, so a steep curve showing a fast rise of flux density with field strength is analogous to a fast rise of voltage with current, and this shows high permeability as well as high inductance. The flux density B may be thought of as voltage and the field strength H as current, so their product is power. It is not quite as simple as that, and the Epcos document goes into real and imaginary components of permeability (complex permeability), which may be thought of as resistive and reactive, or dissipated energy versus stored energy, and thus efficiency.

An air gap may be introduced in the magnetic circuit to lower the permeability but also allow a greater amount of stored energy. This is because the lower permeability allows a much higher current and magnetic field strength. The magnetic material will still saturate at the same flux density, which corresponds to voltage, but the current contributes energy according to a square function. Thus the lower inductance from the reduced permeability with an air gap allows higher current and greater stored energy.

A fixed air gap inductor exhibits a rather sharp saturation curve at the point where most of the magnetic material saturates at the same flux density, depending somewhat on its physical shape, whereas materials such as powdered iron have a distributed air gap, and variations in the size cause saturation to occur less sharply.

The amount of power that can be handled by an inductor or transformer depends on the amount of energy that can be stored in its inductance and how fast the field can be reversed to transfer it. This is determined a little differently for a transformer as compared to an inductor.

A transformer is based on volt-seconds and is limited by the saturation of the magnetic material. According to

http://www.smps.us/magnetics.html this limit does not depend on the core's magnetic properties or air gaps, but on the frequency and wave shape (dV/dt). This will determine the maximum power based on heating of the core. Additionally, there will be resistive losses in the windings, and the combination will determine the maximum power for a given design.

An inductor is based more on current and saturation, at which point the effective inductance is reduced and the maximum energy storage is attained. There will typically be a period of time where the inductor will be storing energy applied to it, and another period of time where it will be released into a load. Unlike a transformer, there will be a net DC current flow, and the effective inductance depends on this value.

To determine the power that is to be handled by an inductive component, one may use the amount of energy that can be stored and the maximum frequency that can be applied. For instance, a 100 uH inductor rated at 10 amps can store 0.5*100*10^2 = 5000 uJ or 0.005 Watt-seconds. If you can drive this inductor at 100 kHz, you can transfer 0.005*100 = 0.5 kW or 500 watts. It may actually be 1/2 that amount because part of the time will be storing energy and the remainder will be releasing it.

I'll go into details on selecting a core material, shape, and size, and windings, in a subsequent post.

Note that this is based on my own understanding of magnetics, and I may not have it quite right. Please correct me or clarify as needed.