Doug Craigen, PhD

Integrated Engineering Software

The concept of inductance of a coil has proven very useful in a variety of applications. A simple measurement of inductance permits a variety of predictions including:

- The time response of the current flow through the coil when attached to a given circuit
- The amount of magnetic flux passing through the coil
- The energy of the magnetic system

However, in reality there may be many inconsistencies found when comparing different measurements and/or design simulations of inductance. There are two main reasons for this:

- There may be eddy currents in nearby materials making the “inductance” time or frequency dependent.
- The equivalence of the many applications of inductance does not apply when the current is strong enough that ferromagnetic material affecting the coil is behaving in a non-linear fashion.

In both these cases there is no single measurement of “*the inductance*” of the coil that will work universally. What it meant by a measurement or simulation of “*inductance*” needs to be defined before the number obtained will be of practical use. Problem #2 will be the focus of this article. The coil used to illustrate the problem is shown to the right. The amount of current in the coil is high enough that a significant amount of the steel core of this inductor is in a nonlinear portion of the magnetization curve for any real steel material.

For background reading, the classic paper by L.T. Rader and E.C. Litscher “**Some Aspects of Inductance When Iron is Present**,” (Trans. AIEE, vol. 63, pp. 133, March 1944) is highly recommended.

The goal of this article is to compare three definitions of inductance as applied to the coil shown above:

- Inductance relates magnetic energy to current in the coil: E = 1/2 LI
^{2}, or L = 2E/I^{2} - Inductance relates flux linkage to current in the coil: Ф = LI , or L = Ф/I
- Inductance relates the change in current in a coil to the change in flux linkage: L = (dФ)/dI

### The Inductor Model

The inductor model shown above was drawn and simulated in Integrated Engineering Software’s 2D magnetic simulation program MAGNETO. It consists of a cylindrical 1010 Steel core, 100 mm long with a radius of 4 mm. Surrounding the core is a 1000 turn coil with an outer radius of 6 mm. Various currents are applied to the coil to study the relationship between different inductance definitions for a real steel material. A parametric run was setup as shown below to directly compare different scenarios. This parametric provides flux linkage in the coil and total magnetic energy for three different total Amp-turn current values and both linear and nonlinear versions of the material.

The comparison of 5000 and 5001 Amp-turns enables a differential definition of inductance to be used.

### Results

The model was run with both Boundary Element (BEM) and Finite Element (FEM) solvers within MAGNETO. These are independent solvers using different calculation methods. The extent of agreement between them is a measure of the accuracy of the results. The Parametric Results were pasted into Excel for computation of “inductance”.

For the results shown below each flux linkage was divided by the current per turn (total current/1000) to obtain an inductance number. Also the difference between the flux linkage at 5000 and 5001 total amp turns divided by the difference in current per turn gives a “differential inductance” result. Each energy result was multiplied by two and divided by current per turn squared to obtain energy based inductance numbers for the same current and material.

### Linear Calculations

To confirm the equivalence of all formulations the first set of results are calculated with “Linear” 1010 Steel. That is, instead of a magnetization curve the initial low field permeability of 1126 is applied to the material.

All results are quite consistent. Any variations between FEM & BEM as well as between different calculations of Inductance from the data are in the 5th decimal place. The FEM-BEM agreement confirms the accuracy of the simulation and the results show that all definitions of inductance produce the same value for linear material.

### Nonlinear Calculations

The nonlinear results below still show good agreement between the BEM & FEM solvers for any given row (see red note below). However, the different definitions of inductance lead to results that vary by a factor of 10. Note that the low current case predicts the same inductance as the linear material for both flux and energy definitions.

### * FEM & Differential Inductance:

The results above are shown for a basic (quick) analysis. With this analysis the FEM & BEM results are quite discrepant for the differential inductance calculation. Although the correct result is not known a priori, the FEM result is clearly incorrect – the differential inductance near the knee of the magnetization curve must be smaller than the inductance at the start of the curve. This is discussed later, including how to properly do this case with FEM.

### Explanation

The magnetization curve below shows the typical shape for ferromagnetic materials. At the selected point where the B field is nearly 2 Tesla, the “permeability” has two possible meanings varying by a factor of 10, depending on whether the permeability is considered to be B/H or dB/dH at the point. Since inductance will be proportional to the permeability, the inductance would be expected to show values varying by the same factor if this is the operating range for the coil core. That is confirmed in the results shown above.

Which is the correct inductance depends on the actual problem being solved. If the device is biased by a large field, but there are effects being studied related to a varying current which only slight perturbs the field, then the differential definition is the most appropriate.

If the applied field is oscillating between large and small (or zero) values then possibly the B/H definition of permeability and the Ф/I /definition of inductance will be the most appropriate. This case will require much closer consideration to decide what is useful.

If the inductance is being related to energy measurements, a third distinct case comes up:

- If L ≡Ф/I the contribution of the point is only determined by H, B at that point
- If L ≡dФ/dI the contribution of the point is determined by H, B in the vicinity of the point, giving the tangent behavior
- If L ≡2E/I
^{2}the energy is calculated from the area under the curve. Hence all (H, B) values up to the operating point affect the result.

### * How to use FEM for Differential Inductance:

This is an especially difficult calculation for the FEM method because it inherently has more noise in the results than BEM. There is a very small relative change in current chosen (total amp turns are only increased by 0.02%) so the flux linkages vary in the 4th or 5th decimal place for this analysis. Noise in the result obscures the real effect. Slower, more precise analysis (not shown here) where a lot more elements were used in FEM resulted in a value closer to the BEM prediction. Two easier methods to reconcile the results and confirm what is correct are shown below. In addition to the first case run, the total amp-turns are varied from 4995 to 5005 (for a 10 A difference), then 4950 to 5050 (for a 100 A difference) and from 4500 to 5500 (for a 1000 A difference). All of these are progressively cruder approximations to the slope of the magnetization curve, but make the noise in the result less significant. The original study was solved with a self-adaptive mesh, because this generally produces a better result than could be obtained by manually specifying the mesh. This is especially important given the range of material properties and current, which means the various scenarios have quite different ideal meshes. However, since the particular case of differential inductance is comparing very similar physical setups the mesh required will be very similar. So, to reduce noise due to different meshes the model was solved once with 5000 A, then the solver was set to Manual mesh so that the same mesh would be used for all analyses.

The results show that the effect of the selected dI is very small over the range 1-100 A. By dI=1000 A the L value is starting to show significant change. The adaptive FEM has too much noise due to mesh variations to be reliable until dI is in the range of 100-1000 A for this model. The superiority of the second FEM solution shows how a trained user can sometimes find ways to configure the solver to produce more reliable results than the default settings.

Although there is a widespread misconception that FEM is always superior to BEM for nonlinear calculations, this is a clear case where good nonlinear results were obtained easily with BEM, but required special setup and a much longer analysis using FEM.

### Conclusion

If the “inductance” of a device operating in a nonlinear range is to be used, it is important that any measurement or simulation to obtain the property be made appropriately to the intended use, or the predictions could be quite incorrect.