There's a fundamental difference between an inductor used for AC without any DC current, and an inductor that can pass DC current and still function as an inductor. Most AC inductors would saturate with more than a few milliamps of DC current. A DC Inductor can operate with high DC currents and still function as an inductor. The first image, below shows a magnetic core B-H Loop. B is the flux density with units of Gauss or Tesla and H is the Coercive Force with units of Oersted, which is ampere-turn/core area. H is the represented by the width of the B-H loop and B is represented by the height.
In order to allow a magnetic core to pass DC without saturating, we must change the shape of the B-H loop by essentially rotating it to the right along the center of its B axis, which somewhat flattens it out as can be seen in the second image. This is done by gapping the core so that the magnetic circuit has a reluctance that's the sum of the high-permeability magnetic core material plus the gap, which although not made of air, has the same magnetic properties as air.
The magnetic circuit has similarities to an electric circuit where the applied ampere-turns (H) is equivalent to applied voltage in an electrical circuit. The resulting current is equivalent to the magnetic flux. The circuit resistance is called reluctance in a magnetic circuit and just as in an electric circuit obeying ohms law, the reluctances in a magnetic circuit add. Reluctance has units of 1/Henry. The air gap can be thought of as a high reluctance compared to the low reluctance of the magnetic core. When combined in a magnetic circuit, as can be seen in the blue B-H loop, for a given flux density level, the H is considerably higher, meaning the ampere-turns are higher, meaning for a given number of turns, the amps (current) is higher in the core without saturating. This is why, by using a gap in the magnetic core, we can apply DC current yet not saturate the core.
There are two ways to have a gap in a magnetic circuit. One way is a discrete gap and the other is to have a distributed gap. In a discrete gap, a relatively small part of the core is cut out or machined out to leave a gap. In a distributed gap, the core is crushed into small, sand-like pieces that get mixed in with a binder and held together with epoxy. The amount of binder put in the mix determines the amount of effective gap. A distributed gap core has the advantage that the core loses will be less due to minimal flux fringing at the gap, which can cause severe heating in discrete-gapped cores. In fact, in a discrete-gapped core, many times there will be multiple gaps cut into the core so as to have several smaller gaps instead of one larger gap. Having smaller gaps minimizes the flux fringing which minimizes core heating to fringing loss in the core. And if you take it to the extreme or to the limit, a distributed gap core is nothing more than a discrete-gapped core with an almost infinite number of tiny gaps.
Designing a distributed-gap inductor is relatively easy as the number of turns can be calculated from the desired inductance. The core size and material is first chosen, then the core's Al (A sub L) value in mH/1000T or equivalently, nH/T^2 is read from the core's datasheet and the inductance computed as follows: L = K*n^2, or by rearranging the equation, n=SQRT(L/K), where K= the Al value and n is the number of turns, L is the inductance in the units of the K given. If K is given in units of mH/1000T, then just use nH/T^2, which is 1*10^-9Henry/Turn^2 for the units of K. The two are equivalent and some core manufacturers use one over the other. As an example, let's say the desired inductance is 20 uH and the core has an Al value of 370 nH/T^2. Then the number of turns is computed as: n=SQRT(20E-6/370E-9) = 7.35 turns, so you can choose to use 7T or 8T.
But this gets a bit more complicated since, as the DC current increases, the inductance decreases. The core manufacturers have a graph for this. So, lets say for example, we want 20 uH inductance at a 10A DC current. First, the Permeability vs DC Bias Chart is in units of permeability for the y-axis and Oersted for the x-axis. For permeability, it's easy. The top line is a value of 1, or 100% of the inductance value with no DC bias current. And when it reads 0.8, that's 80% of the inductance value at zero DC current. For the x-axis, we need the core area as Oersted is ampere*turns/core area and depends on the size of the core. The core area for distributed gap cores can be found in the datasheet for the core used. I'm going to use a Micrometals (formerly Arnold Magnetics) data catalog. Continuing our example, I'll initially chose an MPP core with 60u permeability with a core size of OD: 5.218, ID: 3.094, and Height: 1.00" with Ac=6.710 cm^2. The inductor design may need some iteration. The catalog says the inductance factor, K=156 mH/1000T, but let's use the alternate units of K=156 nH/T^2. For L=20uH, n=SQRt(20E-6/156E-9) = 11.3 turns. Let's round up to 12 turns. We'll round up vs rounding down as the inductance always goes down with DC current bias. To check the true inductance value at 10ADC current, well use the chart and find the u=60 curve. First, we need to compute the magnetic field strength in Oersted as follows:
H = 10A * 12 turns/6.71 cm^2 = 17.88 Oersted, or 17.88 ampere-turn/cm^2. Now, back to the Permeability vs DC Bias Chart in the Micrometals catalog for the MPP material at the chosen core size and the u=60 curve. finding 17.88 on the x-axis and moving up to intercept the 60u curve, we read the Percent Permeability value of about 98%. Let's step back a moment and compute the inductance with 12 turns, which is L=12^2*156 nH/T^2 = 22.46 uH. Now, we multiply this value by 0.98 to account for the drop in inductance vs DC current and we get L= 22.46 * 0.98 = 22.0 uH.
In many cases, there will be a far more substantial drop in inductance down to 50% or even less depending on current. These computations must be done to determine your actual inductance at maximum DC current. Now, I'll discuss the types of distributed gap core materials. There are only about three commercial ones, MolyPermalloy (MPP), HiFlux or Sendust (depending on brand), and Powdered iron. The cost of MPP is highest cost and highest performance and Powdered Iron is the lowest cost and lowest performance. The core losses in a switching converter will be higher using powdered iron versus MPP. The computations shown above are the same for all three types of materials. The manufacturer's catalog will have all the necessary curves and data. And fortunately, the competitors tend to have the same size cores listed in their catalogs. There is one other core material I should mention here and that's Metglas, which is an amorphous magnetic material. It has better performance in terms of core loss than even MPP, but is quite expensive. Most all distributed gap cores are toroidal in shape.
One of the characteristics of a distributed-gap inductor is that you don't reach a point with increasing DC current where the core saturates. Instead, the inductance value keeps declining with increasing DC current. It is not this way with a discrete-gap DC inductor. Instead, the inductance tends to be stable over a wide range of DC current until the core saturation point is reached and then your switching converter will most likely blow up. It happens all of a sudden versus gradually as with the distributed gap core inductor. However, the distributed-gap DC inductance cannot be easily adjust whereas it can be adjusted with a discrete gap core by changing the gap size and this is an advantage in some applications.
Now, for the design of a discrete-gap DC inductor. Often, these are used when high inductance values are needed as the core sizes get much larger (in terms of core volume) than the distributed gap cores get. These are often made of Silicon steel and are in the shape of the letters E & I, but the discrete gap can be done using pot cores, toroids, E_E cores, E-I cores, U-I Cores, and more. To insert the gap in an E-I lamination core, we only need gap the center leg, which can be done by putting a gap in the center leg made of a combination of G-10 (or FR-4) fiberglass-epoxy (available up to at least 1" thick) and Nomex 410 Aramid fiber paper (available up to 30-mils thick). Gapping the center-leg will also cause the outer legs to have an air gap, but this is normally not a problem. However, it must be taken into consideration when computing the gap size as now the gap is effectively twice the gap length of the center leg gap. Note that the fringing flux at the gap boundaries will decrease the effective gap length as the fringing flux will flow around the gap, thus effectively bypassing part of the gap. To minimize the complexity, we'll ignore fringing flux here, so just be aware of it's effect of decreasing the effective gap length.
In the design of a discrete-gap DC inductor, both the DC current and the AC current must be taken into consideration to avoid core saturation as 1/2 the AC flux will add to the DC flux to get to the peak flux that must remain below saturation for the core material being used. I'll use an example of Silicon Steel with an M6 grade that has a saturation flux density of 15KGauss. This is the equation for computing the peak flux density in the core:
Bpk = 0.4*PI*N*(Idc + Iac/2)/(lg + MPL/um) Gauss, where: Bpk = Peak Flux Density in the care, PI = 3.14159, N = turns, Idc = DC current, Iac = AC Current, lg = length of the gap (cm), MPL = magnetic path length (cm), and um = core material permeability. This must be computed after the number of turns is determined and the gap length as those values are needed. I'll leave this to the reader with the reminder that the computed value must be below the saturation flux density of the material used with a suitable safety factor (10% to 20% typical safety factor).
There is more than one approach to designing this type of inductor. One way is to chose an appropriate core size using the area product approach (see Transformer and Inductor Design Handbook by Col. W.T. McLyman), then compute the maximum number of turns of the needed wire size that will fit in the core's window, then compute the gap last. Another approach is to choose an oversized core and pick a reasonable number of turns that's substantially less than can fit on the core, and then compute the needed gap. In the case of a DC inductor, we'll need to check that we're well below core saturation after the design is computed and if it's at or over saturation, then iterate the design by using more turns or a larger core or both. I'll give an example without getting into how the core size was chosen. An initial design can be done and the needed wire area from the number of turns and the wire size can be used to calculate whether or not the needed wire will fit in the available winding window. If not, iterate the design by using a larger core. If there's a lot of room left over, the core size can be reduced and the design iterated.
Here's an example. DC Current: 10 ADC, AC Current: 5 AMPS AC, Needed inductance: 50 mH. We'll start with an M6 grade Silicon Steel Lamination of size EI-150 with a saturation flux density of 15 KGauss. The core area is 13.06 cm^2 based on a 0.9 stacking factor (butt joint). Note that the inductor must be designed using a butt joint if using laminations or else the gap cannot be inserted if the core is actually laminated (interleaved). A butt joint is where all the Es are butted-up together for one side and all the Is are butted-up for the other side. The outer pieces at the top and bottom most position can be arranged so that the Is can be retained easily.
For 10amps DC, the wire size can be determined by current density and we'll use 500 CircularMils/amp. 500 CM/A * 10A = 5,000 Circular Mils wire cross sectional area. Using a magnet wire chart, this calls for AWG#: 13, which has 5,184 C.M. with an OD of 0.0754" nominal using heavy build wire. The EI-150 core has a window area of 1.6875 in^2. The window length is 2.25" and the windows height is .75". Therefore, if layer wound, one layer can hold 2.25"/0.0754" = 29 Turns/layer and 0.75/0.0754 = 9 layers without considering interlayer insulation. Therefore, the maximum turns it can hold at 85% window fill is: 29T/layer * 9 layers * 0.85 = 222 turns. Note that you can never count on filling 100% of the available window space and 85% for layer winding might not be conservative enough. But this is an example, so we'll continue.
The inductance equation is: L=0.4*PI*N^2*Ac*10^-8/lg, rearranging to solve for lg, we get lg = 0.4*PI*N^2*Ac*10^-8/L, for our example, lg = 0.4PI*222^2*13.06*10^-8/50E-3 = 0.161 cm = 0.064" or about a 64 mil gap for a 50 mH inductor. This is quite a large gap, so we can reduce the number of turns a lot and recalculate,. Let's cut the turns by about 2/3rds for 222*1/3 = 74 turns. We already know this will fit in the available window. We'll recompute the gap:
lg = 0.4*PI*74^2*13.06*10^-8/50E-3 = 0.018 cm => 7 mils. This is a standard Nomex 410 thickness so we can use the Nomex sheet cut to the center area dimensions of 1.5" x 1.5" for the gap. This is not taking into consideration that we're gapping both the center leg and the outer legs, which would require a gap of about 1/2 the size or about 3.5 mils thick.
This is not an optimal design in terms of cost and size, but it would work and the inductance can be adjusted up or down by changing the gap size. The bigger the gap, the lower the inductance and vise-versa. I'll end it here and I'll be glad to answer any questions I see in the comment section.