Excitation Coils: Field-Circuit Coupling

In electromagnetic simulations, you can connect excitation coil regions to an electric circuit using excitation coil circuit elements.

The electromagnetic field solution and the circuit solution are implicitly coupled: you can define circuit element properties using data from the electromagnetic analysis (for example, you can calculate the resistance of the circuit element using a report for the coil region). Similarly, you can use circuit data to define electromagnetic properties for the region.

An implicitly coupled field-circuit simulation requires the solution of a system of discrete equations for the field and the additional circuit state variables.

For electric machines, as with batteries, the control system can contain devices with very high switching frequencies. The machine’s electric inductance smooths out those switching events and therefore allows decoupling the field simulation from the circuit. The circuit responds to voltages and currents applied at machine terminals. For low frequency modelling, the field simulation only responds to currents for closed loops. However, for the purpose of electric machine simulations, the current-carrying elements are taken to be closed loops.

Along with the switching events on the control/circuit side, similar events occur on the field simulation side. These events are most pronounced for machines with very little air gap between rotor and stator, as in the case of switched reluctance machines. These events are happening when, due to a change in rotor position, the flux path suddenly changes.

When extracting parameters for a suitable coupling, the linearized residual equation needs to be solved for a number of different right-hand sides—a situation where direct sparse matrix solvers can become comparable to iterative solvers in terms of overall computational expense.

Circuit Side

Assume that there are $N$ coils in the field simulation domain. The currents $I_{r}$ (with $r = 1, \dots, N$ ) that flow in each of these coils form a vector of currents $I = (I_{1}, \dots, I_{N})$ . Similarly, the voltage differences $V_{r}$ at the terminals of each coil can be described with a vector $V = (V_{1}, \dots, V_{N})$ . In the circuit, the response to induced currents and motor rotation can be expressed by the following equation:

Figure 1. EQUATION_DISPLAY

\begin{matrix} V & = R I + \partial_{t} Ψ (I (t), θ (t)) \\ = R I + L^{\partial I} I^{'} (t) + \partial_{θ} Ψ θ^{'} (t) \end{matrix}

(4379)

where $R$ is the DC resistance. $R$ can be calculated using special reports for the regions selected for the excitation coil model. The first line is a direct approach for calculating the time derivative of the flux linkage $Ψ$ , which in fact is available using volume integral reports of the specific electromotive force. However, using this approach to capture motion dependence can be numerically unstable for certain machine types.

Extraction of parameters in the second line of the equation should address these issues. These parameters are in the differential inductance matrix $L^{\partial I}$ , which is generally a full matrix, with some entries being close to 0 according to the vicinity of the respective pair of coils. The last term, $\partial_{θ} Ψ θ^{'} (t)$ , refers to the motion-induced voltages. See chapter 3 of [850] for an extraction approach.

Since the inductance matrix is a full matrix, it needs to be represented as a multi-connection component other than the current two-pole circuit element.

Field Side

A transient simulation, based on a magnetostatics approximation, in which all current-carrying regions are stranded conductors, solves the following model, subject to proper boundary conditions, described below:

Figure 2. EQUATION_DISPLAY

\nabla\times (v (A) \nabla\times A) = J

(4380)

where:

$v (A)$ is the reciprocal of the materials permeability.
$J$ is the current density distribution, which is determined by the shape of the conductors and electric circuit states.

Eqn. (4380), when discretized, leads to a nonlinear equation for $a$ with currents or other sources represented in the right-hand side $b$ .

Figure 3. EQUATION_DISPLAY

M (a) a = b

(4381)

The last two summands in Eqn. (4379) result in solving a number of linearized systems of the above equation.

Inductance Matrix

The inductance matrix

L^{\partial I}

describes the change of the systems flux linkage with respect to a current change. For its derivation, the rotor position is regarded as fixed.

Basic data exchange between the circuit and the field is done on the field side through the Finite Element Excitation Coil model and on the circuit side through the Lumped Parameter model.

The right-hand side of Eqn. (4381) is a linear combination of vectors where each represents an individual excitation coil $b = \sum_{i = 1}^{N} w_{i} I_{i}$ .

The solution to Eqn. (4381) applies an iterative process to drive the residual $r (a) = M (a) a - b$ down to zero using

Figure 4. EQUATION_DISPLAY

r (a + δ a) - r (a) = J R (a) δ a

(4382)

$r (a + δ a) \approx 0$ , therefore $- r (a) = J R (a) δ a$ .

This can also be used to gauge the deviation from $a^{*}$ as a reaction to small signal changes in the right-hand side $b^{*} + δ b$ as a first order expansion given from:

Figure 5. EQUATION_DISPLAY

M (a^{*} + δ a) = M (a^{*}) a^{*} + J (a^{*}) δ a = b + δ b

(4383)

The first-order reaction to changes in the right-hand side can be computed by solving $J R (a^{*}) δ a = δ b$ .

Coming back to the flux linkage of the system, which is a linear map from the solution for a given excitation coil, the variation in the right-hand side is set to represent a small change in applied currents in the excitation coil $δ b_{i} = w_{i} δ I_{i}$ :

Figure 6. EQUATION_DISPLAY

δ ψ_{r} = \underset{j}{Σ} w_{r} δ a_{j} = \underset{j}{Σ} w_{r} J^{- 1} (a^{*}) δ b_{j} = \underset{j}{Σ} [w_{r} J^{- 1} (a^{*}) w_{j}] δ I_{j}

(4384)

Note that $w_{r} J^{- 1} (a^{*}) w_{j} = L_{r j}^{\partial} (a^{*})$ .

Hence, getting the induction matrix components requires solving the Jacobian for different right-hand sides and multiplying the result by the weighting vector $w_{r}$ for the flux linkage of coil $r$ .

For an electric machine, the coils belonging to one phase can be collected in the flux linkage report. This usually means solving for three phases and obtaining a 3x3 inductance matrix.

For asynchronous machines, the solution must also consider the induced currents in the rotor cage.

Electromotive Force

Simulation of the electromotive force requires getting the change in the flux linkage of the coil due to rotation. The change in torque with coil current must be calculated. The torque $T (a)$ is a simple cylinder surface integral of the following field function:

Figure 7. EQUATION_DISPLAY

E_{r} = \frac{T (a^{*}) - T (a^{*} + λ δ a |_{δ θ = 0})}{λ δ I_{r}}, λ = κ \frac{‖ a^{*} ‖}{‖ δ a ‖}

(4385)

$κ$ is a small parameter with a suggested range of 0.01 to 0.05. So the solutions $δ a_{i}$ for the coils are post-processed twice.

Summary and Potential User Experience

An induction matrix is computed in Simcenter STAR-CCM+ by using the Jacobian matrix for the nonlinear solution with additional right-hand sides. You can then specify the connectivity of the coils to form a phase and in this way reduce the number of pole pairs in the induction matrix circuit element:

This can be done by defining the induction matrix element first and then have an operation which adds a connected set of coils. Each of these sets ends up defining a pole pair. As an example, for the above circuit, one would join coils 1 and 2 to form phase $u$ , and so forth.

A further element is needed for cases with motion: the motion-induced voltages per phase leg have to be added to each contact pair.