**By R. Belmans**

## Inductances in magnetostatic problems

**Quantities, such as inductances, resitances**, etc. may be determined from a numerical solution by several methods. It is just as in a laboratory, where a quantity may often be measured in many ways eventually leading to different results. Modelling implies the simplification of the complex physical phenomena determining the behaviour of a technical device. A FEM model of a device serves the purpose of predicting certain aspects of its behaviour while neglecting others. The first simplification is introduced by the choice of the formulation of the equations to solve. However, even if the basic formulation is appropriate, different results may be obtained in the post-process if inappropriate definitions of quantities are employed. This effect is illustrated by the calculation of inductances using a magneto-static analysis.

### Linear models

**Consider a simple iron core inductor** drawn in Fig. 5.110. The aim is to compute the inductance of this device to be used in an equivalent electric circuit. It is assumed that the core is loss-free without leakage flux and the winding end-effects are neglected. A two-dimensional linear, electromagnetic analysis of the device is applied.

**Fig. 5.110. Iron core inductor and equivalent electric circuit.**

**Two definitions of the inductance can be found: one based on the flux linked with the winding (5.339) and one based on the energy stored in the inductor:**

with Nthe number of turns, / the terminal current and F the volume of the device.

The flux linkage with the winding in (5.339) can be determined from the vector potential inside the winding area:

with S the surface in which the flux is penetrating. Based on the definition of the magneto static vector potential, the flux can be determined over the vector potential integrated along the length of the winding

**Equation (5.342) can be simplified when considering a two-dimensional Cartesian geometty, with no variation of the field and geometry in z-direction:**

with / the length of the device in z-direction and A the average value of the vector potential at the right and left side of the winding. Due to the symmetry in the given device and the discretisation of the winding cross section, it can be written:

with n the number of elements over the cross section of the winding and the surface of element k. In the general case (no symmetry), the flux linkage can be extracted from the FEM solution via an integration over the winding cross sections by applying:

with J the current density in the fractional winding cross section withthe surface of the winding cross section (including both sides of the winding). This last equation automatically accounts for the number of turns and the orientation of the different sides of one winding, as the sign of both the vector potential and the current density are related. The value for the inductance can be calculated from:

When comparing this equation with the magneto static energy functional, it can be recognised as the term for the linear energy, i.e. the energy supplied from the source:

A sinusoidal variation of the source current will also result in a sinusoidal change of this energy; the system is linear.

In a linear analysisthe calculation of the inductance via the stored magnetic energy in the model, (5.340) gives the same result. The stored magnetic energy per unit volume can be represented as the surface above the material characteristics (Fig. 5.111).

**The amount of the stored energy per unit volume is thus geometrically:**

A sinusoidal variation of the source current yields a sinusoidal variation of the stored energy. Hence, the calculation of the inductances based on flux linkage and stored magnetic energy in the model give equal results.

**Fig. 5.111. Representation of the stored energy in the model at a defined operating point.**

This "linear" definition of an inductivity is equivalent to a measurement by a ballistic flux meter.

### Non-linear models

**For the computation of the inductance based on the flux linkage,** (5.346) can be applied without changes. However, this value differs from the value of the stored energy computed by (5.340). The material is non-linear (Fig. 5.112). A sinusoidal current excitation does not yield a sinusoidal change of the stored magnetic energy due to the saturation effect. Using eq.(5.349) is not appropriate for this sinusoidal operation because it does not consider the effect of higher harmonics. Signals modulated at non-linear characteristic do contain higher harmonics. Ferromagnetic saturation usually generates harmonics of threefold fundamental frequency.

**Fig. 5.112. Stored energy (5.349) in a non-linear model at a defined operating point.**

The calculation of the stored energy is equivalent to an impedance measurement, which includes the measurement of the terminal voltages and currents. If sinusoidal current is applied, there will be odd harmonics in the voltage waveform, which cannot be determined by a magneto static analysis.

### Example: Inductor for a fluorescent lamp

**To illustrate the problem**, consider a ballast inductor from a circuit of a fluorescent lamp (Fig. 5.113). The inductor is about seven times longer than it is wide. It can be treated as a two-dimensional Cartesian problem neglecting end-effects.

**Fig. 5.113. Half symmetry of the inductor.**

**The design goal for the inductor is twofold:** stabilise the current during the heating phase for the electrodes of the lamp and provide a defined over-voltage when the starter opens in order to ignite the fluorescent lamp.

**Fig. 5.114. Circuit of a fluorescent lamp and shape of the inductor.**

**The results of the inductance computation (Fig. 5.115)** using both definitions clearly indicate the difference that occurs for the saturated operation points of the device. This difference is useful in this case as it allows determination of the linear operation range of the device. However, the value of the inductance should be determined based on the flux linkage.

**Fig, 5.115. Two-dimensional flux plot and the computed energy.**