Back in 2006, I wrote about inductor testing. I'd like to go into more detail about magnetic components, now that I have some excellent experience on the subject. This page will start roughly where I left off, and explain things from there.

The circuit which I use to test inductors, to this day, is still:

This isn't exactly the circuit, because I need the function generator and oscilloscope on the same ground. So the current sense resistor is actually in series with the MOSFET source (which is actually ground). This creates a negative-going current waveform, and only displays the charging part, but that's okay. Incidentially, you don't want to run it at so high a duty cycle -- that sucks a lot of current, making things heat up and sucking down the power supply. I typically use a 1/20 duty cycle.

One shortcoming of this type of tester is that it is unidirectional. The significance of this is, it charges the inductor to some level, then lets it discharge to its natural level. For gapped cores, this isn't a problem, but with ungapped cores, the point on the B-H curve it returns to is B_{r}, the remenance. Typical values for ferrites are B_{sat} = 0.4T and B_{r} = 0.1T, so you only get 3/4ths of the B-H curve by testing in this manner. That's not too bad, but it is something to keep in mind when taking quantitative measurements. Moral of the story: use a bipolar tester if you are interested in the full B-H curve.

The significance of this can be seen in cores with high remenance. These cores can give *really weird* results. Example: often seen in ATX power supplies is a small plastic-cased toroid, usually with 5-6 turns of heavy copper looped around it. (Sometimes this "choke" is wrapped with heatshrink, and sometimes it may be a special ferrite instead -- don't be fooled, it's no ordinary ferrite!) Attempting to test one of these cores with this tester will show low initial inductivity, with very rapid saturation. But the poor waveform isn't the worst. You can test this inductor using the classic LC-resonant-with-signal-generator approach and find the small-signal inductivity. You will discover it's quite high, maybe 1μH/t^{2}. How can that be, when its large-signal response is so awful? Because: the small-signal test uses AC. This type of core has high remenance, meaning you can only recharge it maybe 0.1 or 0.05T in a given direction before it saturates: that means a short on-time before it saturates again. But if you reverse the direction each time, you'll get plenty of flux (maybe 0.4T worth for ferrites, or 1-1.5T for metglas and stripwound types).

What good is a high remenance? Well, a permanent magnet has high remenance, but it also has high coercivity (it takes a lot of amps to demagnetize one). These things don't, they're quite easy to demagnetize, so they aren't useful as permanent magnets. Well, if you connect a diode in series with this inductor (which is basically what I've done in the tester, using the MOSFET) and supplying an AC signal, a small amount of DC will be rectified, which biases the core, causing more DC to be rectified and so on; pretty soon, the inductor has ratcheted itself pretty far up the B-H curve, to the point of saturation. Inbetween cycles, current drops to zero, the diode stops conducting and voltage "flies back" (it's still an inductor, so it still needs zero DC voltage across it, so there has to be a reaction at some point). Well, if you sink some current into this pulse, you can unwind more of the flux, causing the inductor to be more inductive. As a result, less DC gets through. By adding a small current to this device, you can control a much larger current: you have an amplifier! (The gain depends on how much residual flux is left in the core, which is quantified by the "squareness factor".)

Now, why do you see these mag amps in power supplies? Because they need 3.3V, but are only wound for 5 and 12V. What they do is this: if you half-wave rectify the 5V winding and filter it, you'll get half, or 2.5V, which is too little. If you full wave rectify, of course, you get 5V, which is too much. So this mag amp inductor goes in series with one leg, so when it's "out of circuit" (biased to full inductance), the output is essentially half-wave rectified, so it will make 2.5V. If the inductor is left alone, it will magnetize and pass full AC, full-wave rectifying 5V. By adding a little error amplifier to control the current, the output is regulated at 3.3V, independent of what the other outputs are doing -- often, the 12V rail sags from a somewhat heavier load than the 5V rail, but 3.3V is always very close to 3.3V.

Now that I've said some more about the kind of behavior to expect from the inductor tester, including an application for some more unusual materials, I'd better explain just what it is that's going on at the physical level. Now, when I think of magnets, I don't think of analogies, I just think of magnets, I'm comfortable with magnets and fields and their quantities. So what I shall do is introduce these quantities, what they mean in accessible terms (volts, amps, time and geometry), and how they relate. There will be some math, but since it's all ratios (well, I might sneak in an integral, so what), it is very easy to follow, just as easy as electronic circuitry.

The important magnetic quantities (using SI units) are as follows. Note that I've extended the units by specifying turns, to distinguish between physical quantities and make circuit values absolutely unambiguous.

• B - Magnetic flux density, in tesla per turn (T/t). Tesla are a unit which you may not really know much about. Sure, you may know that 1T is considered a pretty strong field, at least in air. 1.2T is pretty common in iron based cores, although it's somehow less impressive there (you can't put your hand inside the field, I guess?). The Earth's magnetic field is something like 0.1mT, which is fairly weak, but noticable, and useful. That's all well and good, but what does a tesla actually mean in electronics? Well, a tesla is defined as the magnetic flux density, or Wb/m^{2}. That means, whatever this field is going through (a piece of space, a core, a coil), that object has an area, and the magnetic flux density times the area equals the flux through it. If that flux is pretty well trapped in a core, then the amount of flux is proportional to the effective area of that core, A_{e}, so that Φ = BA_{e}. Because it's a total amount, every turn of wire around it will see the same flux, so it is a per-turn unit.

• Φ, Φ/N - Magnetic flux, in webers (Wb) or webers per turn (Wb/t). Flux is the amount of magnetic field going through an area. If that area is enclosed by multiple turns, the flux is multiplied by the number of turns. Flux and webers don't really mean anything yet, so let's look at the unit more closely: 1 Wb = 1V·s. Well where the heck do we get voltseconds from... ohhh, why that would be the on-time of the inductor tester, wouldn't it? It applies a fairly constant voltage for some time, then the core starts saturating, and it better turn off soon after. But isn't the current saturating it? Well, both, really. The reason is, they are equivalent, but more on that when we get down to permeability. The most important thing about flux is its use in Faraday's Law of Induction: E = -*d*Φ*/dt*. That's all that EMF is -- the change of flux. Alternately, for an applied EMF, Φ changes at a known rate -- this is essentially why the current through an inductor rises at a constant rate. ∫E*dt* = Φ, so if we apply some arbitrary waveform (square, sine, or something uglier -- doesn't matter, we can measure it!), the area underneath it *equals* the amount of flux through the winding. In this article, Φ shall be used as the circuit parameter, the amount of area under the waveform on the oscilloscope, and Φ/N used for the amount of flux in a core.

• E - Electromotive force, in volts (V). Electromotive force is the "voltage" applied to a coil (actually, it's the voltage in space, period, whether there's a coil there or not, but for our purposes, we will work with wires). Where there is a closed path for that EMF to be carried along, a flux will be created. Now, be very careful with the difference between EMF and circuit volts -- when testing an inductor at very high current, you might discover that the current charges, and keeps on charging, then it starts flattening out and approaches a static value (that is, dI/dt goes to zero). This is not the magnetics' fault, in fact it is occuring because EMF dropped to zero. But how can that be, if you still measure a voltage across the coil? Ah, but are you measuring it across the primary? Which is now carrying a big stinkin' current? Indeed, you are now measuring the resistive voltage drop -- the actual EMF is zero, which can be measured on a secondary winding (Faraday's law, it will only pick up the EMF, no resistive voltage drops). The major assumption in my inductor tester is that you're exciting it with pulses significantly shorter than the L/R time constant, so EMF is approximately constant and current rises linearly with time (until saturation).

• NI - Current, in ampere-turns (A·t). Also known as: Magnetomotive force (MMF). Current is the total amount of amperage flowing around the area of flux we're interested in (i.e., around the core, if there is one). Amps are a circuit property, so we can measure the magnetization of a particular inductor (fixed geometry) in amps. Under static conditions, current can saturate the core (with no EMF), but as Faraday says, it takes flux to get there -- equivalent to charging a capacitor to a fixed voltage, but it took current to get there.

• H - Magnetic field strength, in ampere-turns per meter (A·t/m). Magnetic field strength is essentially the field way of saying how much magnetization you've applied. So a circuit engineer might be more interested in how many amperes a choke handles before saturation, but the designer of that choke needs to know the magnetization field, because he gets to design its geometry. Amps per length is kind of a weird measurement; the length is the length of a field line, which might be circular for a toroid coil, straight for the infinite solenoid, curved in weird ways for realistic inductors, or trapped inside a core, in which case it's the effective length of that core, l_{e}.

• μ - Permeability, in henries per meter per turn squared (H/(m·t^{2})). This is like the bulk version of inductivity. In analogy to resistance and capacitance, you might expect the units are henry-meter per square meter, which is indeed the case. However, the dimensions are not of some prism with electrodes on the end, as is the case for capacitors and resistors (which are both electric devices). In magnetics, the length measurement is always a closed loop, so you go around the core, and the area is the cross-section of that core. Instead of electrodes, you have a coil around the core. But there's more to permeability than just that. Magnetic flux density has units of Wb/m^{2} and magnetization has units of A/m; the ratio is Wb·m/(A·m^{2}) = H/m, so permeability is directly the conversion ratio between magnetization and magnetic flux density. There are two kinds of permeability: one you always need, μ_{0}, the permeability of free space, and one you need for cores, μ_{r}, relative permeability (dimensionless). μ = μ_{r}μ_{0} is the actual value inside a core. μ_{0} = 4π × 10^{-7} is fairly easy to remember (or in decimals, 1.257μH/m).

• L - Inductance, in henries (H); A_{L} - Inductivity in H/t^{2}. Okay, so what is a henry? H is defined as Wb/A, so it's the amount of flux through a winding that's obtained by appling some current to it. Which in turn means H = V·s/A, which is very much the familiar circuit definition of an ideal inductor: E = L *dI/dt*. The henry is the circuit value; inductivity is a property of the core (i.e., fixed geometry), so it depends on turns. Now I can explain here why flux is written "per turn" above, and why amps go with turns: consider that some amount of flux is changing in a core. This is caused by the change in current through N turns, or NI. This causes a magnetization of H = NI/l_{e}, which causes a flux density of B = μH and a flux of Φ = BA_{e}. A change in flux induces an EMF *per turn* in the winding, so the total flux the wire sees is NΦ, or putting it all together, E = *dI/dt* N^{2}μ_{r}μ_{0}A_{e}/l_{e}. This is the definition of inductance once again, but this time derived from the physical quanities; the inductance is L = N^{2}μ_{r}μ_{0}A_{e}/l_{e}, and if you want to know inductivity, divide by N^{2} to get A_{L} = μ_{r}μ_{0}A_{e}/l_{e} alone (which, if you've been following my choice of units, you should discover this has units of H/t^{2}, just as it should).

What if the core is not continuous? Air gapped cores are very common. What to do? The trick is, the magnetomotive force is applied to all paths in series, as if MMF is a voltage source which divides among blocks of magnetic volume, linked by equal current (flux). So instead of H = NI/l, it's NI = H_{c}l_{c} + H_{g}l_{g}. We aren't too interested in the H field, however, since flux is constant, and areas are equal (or close enough), the flux densities are equal, so by invoking B = μH, NI = B_{c}l_{c}/(μ_{r}μ_{0}) + B_{g}l_{g}/μ_{0}, so B = μ_{0}NI / (l_{c}/μ_{r} + l_{g}). This equation is basically converting the core into a free-space equivalent, then adding it with the air gap. The equivalent happens to be μ_{r} times shorter -- makes sense, it's more permeable, so you don't drop much magnetization across it.

That means the inductance formula derived earlier can be modified for gap by replacing l_{c}/μ_{r} with (l_{c}/μ_{r} + l_{g}), or A_{L} = μ_{0}A_{e} / (l_{c}/μ_{r} + l_{g}). Notice if l_{g} = 0, this reduces to the ungapped formula.

Increasing the air gap seems pretty useless if it just decreases inductance. However, the air gap stores energy, so even though inductance drops, the amount of current required for the same flux density goes up proportionally (effectively, permeability is decreased). The result is an increase in energy storage.

If a material saturates at some flux density B_{max}, then it will always have the same maximum flux, Φ_{max} = NB_{max}A_{e}. (In practice, it may be slightly higher, because some flux slips around the gap through fringing fields.) However, because the gap increases the magnetic path length, the current required to reach B_{max} is I = B_{max}(l_{c}/μ_{r} + l_{g}) / (μ_{0}N). The energy is IΦ_{max} = B_{max}^{2}A_{e}(l_{c}/μ_{r} + l_{g})/μ_{0}, which increases with l_{g} and particularly with the square of B_{max}. Notice it does not depend on turns -- energy is constant, you essentially use the number of turns to match that energy to your circuit's specific V/I requirements.

If permeability is low enough, you can reach high magnetization levels without an explicit air gap. Both of these qualities combine in powdered iron cores, which have low permeability (30 to 120) and high saturation (1.2-1.5T). (Nitpickers might note that powdered iron cores have a lot of internal airgap, which is true. It's not specified how much, so we use the effective permeability instead, and are careless enough to call it relative permeability.)

One more thing that's important: all magnetic materials that aren't air are significantly nonlinear. This is typically specified by the B-H curve, which describes flux density vs. magnetization, and is given such parameters as B_{r}, the remenance (which I went into a little at the top), and H_{c}, the coercive force.

In general, the B-H curve traces out a hysteresis loop, whose area is energy density (for B in tesla and H in amps per meter, T·A/m = V·A·s/m^{3} = J/m^{3}). I won't go into the characteristics of permanent magnets here, just make light of this nonlinearity as a loss component. Clearly, if you traverse the complete B-H curve in one cycle, you will have burned exactly that energy density, or if it's traversed repeatedly at some frequency, you'll have 2f times, which is now in power density.

For our purposes, the most important aspect of the B-H curve is how it flattens out for large magnetization levels. Simply put, the physical processes going on inside the material can only help you so much; eventually, applying more magnetization produces diminishing returns in flux density. This is called saturation. A few examples: ferrites saturate at 0.3-0.5T, mild steel around 0.6T, and various powdered, amorphous and rolled steel alloys cover the 1-1.8T range, with 0.8T being typical for MPP types, 1.2T for transformer iron (laminated silicon steel) and 1.5T for "high flux" types. There are absolutely no materials available with a higher saturation flux density than about 2T -- if you want a stronger field, you have to generate it the hard way, with brute force amperage. (There may be exotic materials with higher magnetization levels, but they require cryogenic cooling or special construction. Most high intensity magnets today are superconducting, which use brute force amperage, not permeability.)

What happens when a material saturates? Flux density stops increasing, rather sharply. With *d*Φ*/dt* ≈ 0, EMF goes to zero. So if you're applying a constant voltage to a coil in saturation, current is going to explode until I = V/R. Effectively, what happens is μ_{r} goes down to 1, so your core starts looking like air. It's not actually air, because it's magnetized and it does have a field, but the field isn't doing your transformer or inductor any good unless it's changing, and it won't change any more in this direction (incremental permeability is 1). As a result, inductance drops, current rises quickly, and after a few L/R time constants, it levels off at the resistive amount.

There are special cases where saturation can be used to one's advantage, however, in practical transformers and inductors, saturation is to be avoided. Since hysteresis loss tracks flux density quite closely, a high flux density will cause a large power density, making it heat up. It may be advantageous to keep flux density much lower than saturation, especially for large transformers at high frequencies.

First of all, what does a transformer do? It must transform a voltage or current (or impedance) by a fixed ratio, with little else added -- no series stuff, no shunt stuff. The transformation should work for frequencies as low as possible (an ideal transformer works down to DC). You're making an inductor, so it will inevitably have shunt inductance, but you want it as large as possible so it becomes negligible. You want as much flux as possible, so you can accommodate as low a frequency as possible. We can calculate flux and inductance, so we can solve this pretty easily. Transformer design, as far as general magnetism is concerned, is the easiest process.

Let me illustrate the design of a transformer with an example. Say a transformer is required with these specs:

- Primary: 80V peak (square wave), 20A capacity
- Bandwidth: 20-100kHz (plus square wave harmonics)
- Peak magnetizing current less than 10% of full load current
- Turns ratio 10:1

We start by calculating the required flux: an 80V square wave at a minimum frequency of 20kHz is a rectangle of 80V by 25μs (the half wave period), or Φ_{max} = 2000μWb. But this is only the change; at equilibrium, the flux will cross zero halfway through, so the actual peak flux is only 1000μWb. Tip: we can ignore this and go with the higher value, using it for a factor-of-two overhead. This also helps during startup, when the flux starts at zero and a full half-cycle is applied to the transformer. Since we know the total flux, we can find the flux per turn, Φ_{max}/N_{p}. At this frequency, hysteresis loss won't be a big deal, so we can choose a peak flux density near saturation, about 0.4T, so the primary winding has to be N_{p} = Φ_{max}/(B_{max}A_{e}).

Since magnetizing current is specified, we can calculate the required inductance. The applied voltage is 80V, the maximum peak current is 2A, and the time is 12.5μs (because, like the flux, it crosses zero at 1/4th and 3/4th of a cycle), so L = VΔt/ΔI gives 500μH.

A commonly quoted current density for transformer use is J = 2000A/in^{2}, but high frequency transformers are smaller and require fewer turns than line transformers, so we can get by with a higher value, like 5000A/in^{2}. This implies using 0.004in^{2} cross section wire for the primary, or 71 mil diameter. This is close to 13AWG, but solid 13AWG isn't a good idea because of skin effect, current crowding and eddy currents. I won't go into the details of these phenomena, but they have the effect of reducing the wire cross section -- maybe it isn't a great idea to use a high current density in the first place! But there is a solution: by using many thin, individually insulated wires in parallel, we can minimize these effects, thereby maximizing efficiency with a minimum of copper. The downside is that 'litz' wire is expensive and hard to find (especially in this size). An excellent calculator for litz wire suggests that only 20 strands of 26AWG is acceptable for this transformer.

If litz is unavailable, it can be made by hand with an awful lot of enameled wire, or solid or stranded wire 2-4 sizes larger (10AWG or so in this example) can be used. The irony: because litz is made up of lots of wires, it takes up a lot more space than its copper content alone. So you end up taking up almost as much winding area either way!

All that wire takes up space. The area enclosed by the core path is called the winding window, for obvious reasons. We need a window large enough to accommodate all the wire we're putting through it. Assuming a typical fill factor of 0.5, and remembering that *twice as much wire* goes through the core, because all the amp-turns the primary is carrying is also carried by the secondary, the area required is A_{w} = 4N_{p}I_{p}/J.

The area product is used as a figure of merit for transformer cores. Since we already know N_{p}, we can substitute and find A_{e}A_{w} = 4I_{p}Φ_{max}/(JB_{max}). For this example, the required area product is 51,613 mm^{4}.

Now we know how much flux is required, what size wire will handle the current, what the minimum inductance is, and some idea about the size of the core required. The only thing left to do is go shopping!

The best source for core data comes from the manufacturer. I am not affiliated with these companies in any way. I don't claim to make an exhaustive list, only a representative sample.

- Fair-Rite Products Corporation Ferrite toroids and shapes in many materials (numbered mixes)
- Magnetics, Incorporated Powder toroids, ferrite toroids and shapes, and stripwound cores; letter or named mixes
- Micrometals Powdered iron toroids and shapes
- Ferroxcube powder toroids, ferrite toroids and shapes; coded mixes

With a similar disclaimer, a few suppliers (and who they primarily stock) are:

- Adams Magnetic Products; Ferroxcube and Magnetics Inc.
- Amidon Corporation; Magnetics Inc., Fair-Rite, Micrometals
- CWS ByteMark; Magnetics Inc., Fair-Rite, Micrometals

The basic selection process is, for those suppliers which list short data on their inventory, go down the list and find some likely candidates. Go to the manufacturer (if you can determine who made it; since the manufacturer often is not stated, you have to memorize mix codes) and find the data on that core. Or is you can't find a likely product in any supplier's catalog, check the manufacturers first and then see if similar products are available.

For this example, a medium to high permeability ferrite toroid is suitable, which includes Fair-Rite types 75, 76 and 77, Magnetics' R, P, J and W, and most Ferroxcube 3Cxx and 3Ex types. Going through the list, a 1" toroid has 15.5mm i.d. and A_{e} = 30.8 mm^{2}, so its A_{w} is 189 mm^{2} and area product is only 5812 mm^{4}, much too little. In other words, there's no way in hell you can squeeze on enough turns to meet the flux requirement, while also meeting the current density requirement without resorting to superconductors! At the bottom of the list is a whopping 4" toroid, with area product 1.44M mm^{4}. This would work well, but would be bulky, expensive and hard to find. In the middle of the list, a 1.7" toroid has A_{e}A_{w} = 74,599 mm^{4}, very close. This is pretty small, and in practice I would pick a comfortably larger size in the 2-3" range.

What did I actually do? (No, this wasn't just a hypothetical example!) I found an extraordinary deal at All Electronics for 3.38" toroids, $3.50 each, so I bought a bunch. Without any specs, it was a risk, but even if they ended up unsuitable, I'd only be out 30 bucks, not a big deal in the long run. A week later, a box arrived, carrying a shrinkwrapped card of Magnetics 0W48613TC cores -- huzzah, data after all! (Sure would've been helpful if whoever wrote the inventory listing had noted that...) These cores are W type, which is high permeability (~10k), which isn't ideal for power applications and tends to be lossy over 100kHz, but it'll still work quite well. The area product is 457,233 mm^{4}, plenty oversized, but that means I can get away with fewer turns and lots more winding space, which will come in handy. If I hadn't found this great deal (at the time of writing, this toroid is no longer in All Electronics' inventory!), I would've settled for a 2.9" type J or 77 from Amidon, which costs $15-18 each! Indeed, surplus stores often have good deals on magnetics, even if they may not have exactly what you need, or much data on them.

Now that a core has been selected, the windings can be designed. Since A_{e} = 189mm^{2}, N_{p} = 26t. That's not very convienient, because the ratio is 10:1 and N_{s} has to be whole (no fractional turns on a toroid!). 20 and 30 are the next closest multiples. The secondary has to be pretty heavy (200A suggests water-cooled copper tubing!), so a minimum of turns is desirable. 20t cuts into the steady state safety factor, but for other reasons, this will still be acceptable. (If a smaller 2-3" core were used, 30:3 would be required.)

We can now verify the inductivity and magnetizing current limit for this transformer. From the datasheet, l_{e} = 215mm and μ_{r} = 10,000, so the inductivity is 11μH/t^{2} (exactly the datasheet value), and the inductance 4.42mH, 8.8 times the required value. Evidently, we could use permeability as low as 1,100 and still meet the magnetizing requirement.

Here is the completed transformer inside my prototype 1kW induction heater supply. It operates over 20-100kHz, supplying rectified line voltage (i.e., ±80V square wave) to the primary, inducing voltage in two turns of water-cooled 1/4" copper tubing secondary. The core is secured to a fiberglass angle bracket with wire ties. A fan cools the capacitor and primary winding, which is only 12AWG hookup wire, which will definitely get hot. It is 200°C rated mil-spec wire, so practically as long as it doesn't start glowing, it will be okay.

- Find the maximum flux required by the circuit. For a square wave of full duty cycle, Φ
_{max}= V_{pk}/2F. For a sine wave, Φ_{max}= V_{rms}/2.22F. - Select a core material of medium to high permeability, typically R, P, J or W, 75 or 77, 3C90, 3E3, etc. For high frequency use, C, L, 43, 3F3 ferrites, and high permeability MPP cores are suitable. Powdered irons are not suitable. Choose a peak flux density typical for the material (usually 0.3-0.4T for ferrite, 0.8T for MPP or 1.2T for laminated iron, or less at high frequency due to hysteresis loss).
- Estimate how much copper is required to handle the current. The required cross-sectional area (including a fill factor of 0.5) is A
_{w}= 4N_{p}I_{p}/J. A typical current density is J = 7.75A/mm^{2}. You can guesstimate a number of primary turns to get started, or you can calculate the area product, A_{e}A_{w}= 4I_{p}Φ_{max}/(JB_{max}). Many toroids are in a certain proportion, so the approximate minimum diameter can be found from (50A_{e}A_{w})^{0.25}. - Select cores from the list and evaluate them. Find the required primary turns, N
_{p}= Φ_{max}/(B_{max}A_{e}), and decide if the core is suitable. - Once a core has been selected, find the number of secondary turns. Adjust primary turns if necessary. Check that the winding will fit with the desired amount of free space in the winding window.
- Go shopping. Find a supplier who stocks this core, or find another suitable core which they stock. This may require cross-checking between manufacturer listings and inventory before the final core is found.

Although inductors often have secondary windings (making them transformers), for our purposes, energy storage is the primary goal; the windings are simply there to deliver it as required by the circuit. Whereas inductance is an undesirable trait in a transformer causing increased current draw, it is the most important quality of an inductor.

As has been seen in the section on saturation, the presence of a core actually *reduces* the energy storage of space, because it saturates easily. So why would we want cores at all? The reason is, circuit impedances aren't quite low enough. If average circuit resistances (including copper) were about 100 times lower, and voltage drops (like diodes) were about 10 times lower, it would be very efficient to use air cored coils for most power switching applications -- they wouldn't be swamped by resistivity, and the amount of magnetic field generated would be very reasonable for the amount of EMF obtained. The alternative would be if μ_{0} were about 100 times higher, so that the inductivity were proportionally higher. Alas, neither is the case; we are stuck with the magnetic force being relatively 'weak'. This is the reason why cores are important: they concentrate the magnetic field, so that the energy can be focused into a small volume of space (the air gap), allowing an excellent match between the phenomenon of inductance and the circuits which use it.

Whereas transformers are best made with high-permeability materials like ungapped (or very small gap) ferrite, inductors are more free-form in this choice. There are two options in general use today: either a gapped high-permeability ferrite, or an ungapped low-permeability ferrite or powder core. Powdered irons are available with μ_{r} from 1 to 175, and MPP types reach as high as 800.

As with transformers, flux is still the central factor. Although a transformer's flux can be determined by the applied voltage, because the current returns to zero twice every cycle, the same is not generally true here. Instead, we use the two defining factors, current and inductance. This isn't acceptable for transformer design, because we assume transformers have infinite inductance.

If we want the core to just reach saturation at the peak applied current, or more specifically, any given value of B_{max}, then the applied flux is simply Φ_{max} = LI_{pk}. This flux flows through the core and air gap (if present)

The two most important parameters defining an inductor are the inductance and peak current. This is true regardless of waveform (AC or DC, returning to zero or not), so it's a good starting point. We can use an energy argument to find the total amount of airgap required. The energy density of a magnetic field is the Maxwell stress, σ_{m} = e_{m} = B^{2}/(2μ_{0}), which is in units of Pa ≡ N/m^{2}, but this can also be expressed as J/m^{3}, an energy density. (Indeed, pressure *is* energy density, and you can use pressure differences or energy differences to analyze a physics problem and reach identical conclusions.) At a typical B_{max} of 0.4T, ferrite can support 64 kJ/m^{3} in its air gap, and powdered iron, 573 kJ/m^{3}. Energy storage is one advantage of powdered iron over ferrite; the downside is, it's far lossier, so you can't get away with as large a change in flux density. This is why gapped ferrites are preferred for high-ΔB applications, like flyback converters, and powdered irons for low-ΔB applications, like filter chokes.

The energy in an inductor is E = LI^{2}/2, so the required air gap volume is v_{g} = μ_{0}LI^{2}/B_{max}^{2}. For an ungapped powdered iron core, the air gap is distributed, so the volume is μ_{r} times larger, or v_{c} = μ_{r}μ_{0}LI^{2}/B_{max}^{2}. The required winding window is at least A_{w} = 2NI/J (same as for a transformer, but since we don't have a secondary, it's just the area of one winding, not double). Making a few material and geometric assumptions, we can simplify things a little more.

This section is incomplete. I've found inductor design to be much more iterative than designing transformers. I'm still not sure why this is. One reason may be the diversity of geometries available (rod, bobbin, toroid, gapped shapes, etc.), and the dependance of geometry on the other relations. I welcome any comments you may have on the subject.

Unfortunately, I don't really have any solid references that I'm comfortable with. That's part of the reason I wrote this article: to cement my knowledge of the subject, present it in terms that are important to me (and hopefully to others), and provide a concise reference embracing the subject. For instance, transformer winding guides only tell you what you need to know, without exploring magnetism to any depth. I've read several textbooks, ranging from motors to electromagnetism, and none are geared specifically towards what the electrical engineer needs to know about his or her circuit elements.

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