Inductance plays an important role in the characterization of magnetic devices, electrical machines, sensors and actuators. The concept of a non-variant (time-independent), linear inductance of wire-like coils is discussed in every electrical engineering book. However, its extension to variant, nonlinear, distributed coil cases is far from obvious.
Time-variance is essential when the geometry of the device is changing: actuators and electrical machines, for example. In this case, the inductance depends on a stroke (in a 1-D motion case) which, in turn, depends on time.
Many magnetic devices apply iron for the conductance of magnetic flux. Most iron has a nonlinear B-H curve. Because of this nonlinear feature, two kinds of inductance must be considered: differential and secant. The secant inductance is the ratio of the total flux over current. The differential inductance is the ratio of flux change over a current excitation change.
The flux of a single wire coil can be defined as the surface integral of the flux density. However, when the size of the wire is not negligible, it is not clear which contour spans the surface. The field within the coil must be taken into account. Even larger difficulties occur when the current is not constant: for example solid rotor or squirrel-caged induction machines.
The voltage induced in a variant coil can be decomposed into two major components: transformer voltage and motion-induced voltage.
The transformer voltage is induced in coils by the rate change of exciting currents. It is present even if the geometry of the system is constant, meaning the coils don't move or expand. To obtain the transformer voltage, the knowledge of flux change (i.e., that of differential flux) is necessary when the exciting currents are perturbed.
The motion-induced voltage (sometimes called back-EMF) is related to the geometry change of the system. It is present even if the currents are kept constant. To obtain the motion-induced voltage, the knowledge of absolute flux in the coils as a function of stroke is necessary.
Obtaining the proper differential and absolute flux values requires consistent calculations of magnetic absolute and incremental energies and co-energies. For current-technology elements, the linear perturbation procedure can be used to calculate the differential inductance and the absolute flux using the incremental (IENE) and the co-energy (COEN) element records, respectively.
Consider a magnetic excitation system consisting of n coils each fed by a current, Ii. The flux linkage ψi of the coils is defined as the surface integral of the flux density over the area multiplied by the number of turns, Ni, of the of the pertinent coil. The relationship between the flux linkage and currents can be described by the secant inductance matrix, [Ls]:
 | (5–164) |
where:
| {ψ} = vector of coil flux linkages |
| t = time |
| {I} = vector of coil currents. |
| {ψo} = vector of flux linkages for zero coil currents (effect of permanent magnets) |
Main diagonal element terms of [Ls] are called self inductance, whereas off diagonal terms are the mutual inductance coefficients. [Ls] is symmetric which can be proved by the principle of energy conservation.
In general, the inductance coefficients depend on time, t, and on the currents. The time dependent case is called time variant which is characteristic when the coils move. The inductance computation used by the program is restricted to time invariant cases. Note that time variant problems may be reduced to a series of invariant analyses with fixed coil positions. The inductance coefficient depends on the currents when nonlinear magnetic material is present in the domain.
The voltage vector, {U}, of the coils can be expressed as:
 | (5–165) |
In the time invariant nonlinear case
 | (5–166) |
The expression in the bracket is called the differential inductance matrix, [Ld]. The circuit behavior of a coil system is governed by [Ld]: the induced voltage is directly proportional to the differential inductance matrix and the time derivative of the coil currents. In general, [Ld] depends on the currents, therefore it should be evaluated for each operating point.
After a magnetic field analysis, the secant inductance matrix coefficients, Lsij, of a coupled coil system could be calculated at postprocessing by computing flux linkage as the surface integral of the flux density, {B}. The differential inductance coefficients could be obtained by perturbing the operating currents with some current increments and calculating numerical derivatives. However, this method is cumbersome and neither accurate nor efficient. A much more convenient and efficient method is offered by the energy perturbation method developed by Demerdash and Arkadan([226]), Demerdash and Nehl([227]) and Nehl et al.([228]). The energy perturbation method is based on the following formula:
 | (5–167) |
where W is the magnetic energy, Ii and Ij are the currents of coils i and j. The first step of this procedure is to obtain an operating point solution for nominal current loads by a nonlinear analysis. In the second step, linear analyses are carried out with properly perturbed current loads and a tangent reluctivity tensor, νt, evaluated at the operating point. For a self coefficient, two, and for a mutual coefficient, four, incremental analyses are required. In the third step the magnetic energies are obtained from the incremental solutions and the coefficients are calculated according to Equation 5–167.
The inductance computation method is based on Gyimesi and Ostergaard([230]) who used Smythe's procedure([150]).
The incremental energy Wij is defined by
 | (5–168) |
where {ΔH} and {ΔB} denote the increase of magnetic field and flux density due to current increments, ΔIi and ΔIj. The coefficients can be obtained from
 | (5–169) |
This allows an efficient method that has the following advantages:
For any coefficient, self or mutual, only one incremental analysis is required.
There is no need to evaluate the absolute magnetic energy. Instead, an “incremental energy” is calculated according to a simple expression.
The calculation of incremental analysis is more efficient because the factorized stiffness matrix can be applied (no inversion is needed). Only incremental load vectors should be evaluated.
For elements PLANE233, SOLID236 and SOLID237, the linear perturbation procedure can be used to derive the differential inductance from the element incremental energy (output as IENE). The incremental energy is calculated using Equation 5–168, where ΔH and ΔB are the linear perturbation analysis magnetic field and flux densities corresponding to the perturbation current loads ΔIi and ΔIj.
The absolute flux linkages of a time-variant multi-coil system can be written in general:
 | (5–170) |
where:
| {X} = vector of strokes |
The induced voltages in the coils are the time derivative of the flux linkages, according to Equation 5–165. After differentiation:
 | (5–171) |
 | (5–172) |
where:
| {V} = vector of stroke velocities |
The first term is called transformer voltage (it is related to the change of the exciting current). The proportional term between the transformer voltage and current rate is the differential inductance matrix according to Equation 5–166.
The second term is the motion-induced voltage or back-EMF (it is related to the change of strokes). The time derivative of the stroke is the velocity, hence the motion-induced voltage is proportional to the velocity.
Whereas the differential inductance can be obtained from the
differential flux due to current perturbation as described in Differential Inductance Definition, Review of Inductance Computation Methods,
and Inductance Computation Method Used. The computation of the motion
induced voltage requires the knowledge of absolute flux. In order
to apply Equation 5–172, the absolute flux
should be mapped out as a function of strokes for a given current
excitation ad the derivative
provides the matrix link between back-EMF and
velocity.
The absolute flux is related to the system co-energy by:
 | (5–173) |
According to Equation 5–173, the absolute flux can be obtained with an energy perturbation method by changing the excitation current for a given stroke position and taking the derivative of the system co-energy.
The increment of co-energy can be obtained by:
 | (5–174) |
where:
 = change of co-energy due to change
of current Ii |
| ΔHi = change of magnetic field due to change of current Ii |
The incremental co-energy in Equation 5–174 is output as COEN in a linear perturbation analysis using elements PLANE233, SOLID236 and SOLID237.
The differential inductance matrix and the absolute flux linkages of coils can be calculated using linear perturbation analysis for the current-technology electromagnetic elements.
The differential inductance computation is based on the energy perturbation procedure using Equation 5–168 and Equation 5–169.
The absolute flux computation is based on the co-energy perturbation procedure using Equation 5–173 and Equation 5–174.
The output can be applied to compute the voltages induced in the coils using Equation 5–172.
The absolute magnetic energy is defined by:
 | (5–175) |
and the absolute magnetic co-energy is defined by:
 | (5–176) |
See Figure 5.2: Energy and Co-energy for Non-Permanent Magnets and Figure 5.3: Energy and Co-energy for Permanent Magnets for the graphical representation of these energy definitions. Equations and provide the incremental magnetic energy and incremental magnetic co-energy definitions used for inductance and absolute flux computations.
For current-technology electromagnetic elements, the absolute magnetic energy and co-energy is output as MENE and COEN element records, respectively.
Equation 5–168 and Equation 5–174 provide the incremental magnetic energy and incremental magnetic co-energy definitions used for inductance and absolute flux computations.
In addition to the magnetic energy and co-energy, elements PLANE233, SOLID236 and SOLID237 calculate the apparent energy defined as:
 | (5–177) |