Summary
This page summarizes the use of circuits electromagnetic FEA in general, and EMDtool in particular. Please note that the concept ‘circuit’ here refers to anything with a current density - whether imposed or induced, uniform or eddy-esque - not just intentionally-wound windings. Thus, circuits include also solid conductive shafts and permanent magnets, not just three-phase windings and rotor cages.
Circuits in electromagnetics
At this point, you have hopefully read the basics of electromagnetic FEA. As you hopefully remember, a pure magnetic problem is of the form
\[\mathbf{S} \left(\mathbf{a} \right) \mathbf{a} = \mathbf{f} ,\]from which the only unknown to be solved is the vector of vector potentials \(\mathbf{a}\).
In the simplest case of a ‘circuit’, we only have a current density known in advance, in which case only the load vector \(\mathbf{f}\) is modified.
However, in the general case, each circuit will introduce some extra unknowns to the problem. Often, they can include some net currents flowing in the conductors, and/or some electric potential or voltage terms depending on the exact formulation used and the time of the circuit.
In any case, let us denote these extra unknown(s) by the vector \(\mathbf{c}\), where the letter c conviently stands for both Circuit and Constraint (which is the mathematical role of most circuits in FEA).
Thus, our vector of unknowns gets larger and now consists of two distinct parts - the vector potentials and the circuit variables.
\(\mathbf{a} \rightarrow \begin{bmatrix} \mathbf{a} \\ \mathbf{c} \end{bmatrix} := \mathbf{x}\).
Furthermore, circuits typically introduce some first-order time-dependency into the problem, changing the governing equation into
\[\mathbf{S}_\text{tot} \mathbf{x} + \mathbf{M}_\text{tot} \frac{\text{d}\mathbf{x}}{\text{d}t} = \mathbf{f}_\text{tot}.\]Here, the subscript \(\text{tot}\) is used to emphasize the fact that the matrices and vectors are no longer the pure-magnetic FE matrices, but instead consist of the magnetic and circuit parts both. The matrix \(\mathbf{M}_\text{tot}\) is the so-called mass matrix, its name owning to the first applications of FEA being in linear mechanics.
Splitting the system of equations
Now, let us apply a similar splitting to the matrices, as we did into our vector of unknowns \(\mathbf{x}\).
Indeed, we get for the matrices
\[\mathbf{S}_\text{tot} = \begin{bmatrix} \mathbf{S} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{S}_\text{aa} & \mathbf{S}_\text{ac} \\ \mathbf{S}_\text{ca} & \mathbf{S}_\text{cc} \end{bmatrix}\]and
\[\mathbf{M}_\text{tot} = \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{M}_\text{aa} & \mathbf{M}_\text{ac} \\ \mathbf{M}_\text{ca} & \mathbf{M}_\text{cc} \end{bmatrix}\]On the right-hand side, the first term denotes the pure-magnetic contribution, while the latter is the contribution from the circuits. For clarity, the matrices have been split into blocks, reflecting the division between the pure-FE and circuit variables.
Similarly, the load vector is split into
\(\mathbf{f}_\text{tot} = \begin{bmatrix} \mathbf{f} \\ \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{f}_{a,c} \\ \mathbf{f}_\text{c} \end{bmatrix}\).
Again here, the first right-hand-side term refers to the pure-FE contribution, while the latter vector is the circuit contribution. Note that the circuits can contribute both to the first blow-row of the vector (typically via an imposed current density) and to the second row (typically via voltage equations of some kind).
Multiple circuits
Often, there are multiple circuits in a FE model. These circuits are also (typically, and so far always in EMDtool) independent of each other - only interacting via the vector potential \(\mathbf{a}\), not the circuit variables \(\mathbf{c}\). This independence is - luckily - reflected in the structure of the circuit matrices.
Indeed, let us denote the different circuit by the running subscript \({}_1, {}_2, \ldots\). Also, let us divide the vector of unknowns into the vector potential and the contribution of each circuit separately:
\[\mathbf{x} = \begin{bmatrix} \mathbf{a} \\ \mathbf{c}_1 \\ \mathbf{c}_1 \\ \vdots \end{bmatrix}\]Then, we can apply a similar splitting to the circuit arrays:
\[\mathbf{S}_\text{aa} = \mathbf{S}_{\text{aa},1} + \mathbf{S}_{\text{aa},2} + \ldots\] \[\mathbf{S}_\text{ac} = \begin{bmatrix} \mathbf{S}_{\text{ac}, 1} & \mathbf{S}_{\text{ac}, 2} & \ldots\end{bmatrix}\] \[\mathbf{S}_\text{ca} = \begin{bmatrix} \mathbf{S}_{\text{ca}, 1} \\ \mathbf{S}_{\text{ca}, 2} \\ \vdots\end{bmatrix}\] \[\mathbf{S}_\text{cc} = \begin{bmatrix} \mathbf{S}_{\text{cc}, 1} & & \\ & \mathbf{S}_{\text{cc}, 2} & \\ & & \ddots \end{bmatrix}\] \[\begin{bmatrix} \mathbf{f}_{a,c} \\ \mathbf{f}_\text{c} \end{bmatrix} = \begin{bmatrix} \mathbf{f}_{a,c,1} \\ \mathbf{f}_\text{c,1} \end{bmatrix} + \begin{bmatrix} \mathbf{f}_{a,c,2} \\ \mathbf{f}_\text{c,2} \end{bmatrix} + \ldots\]So, what this means that the contribution of each circuit can be assembled independently, with no regard to the number and type of other circuits in the problem. The individual contributions are then combined (by the CircuitSet helper class) via either summation or matrix block operations into the final circuit matrices, after which the actual solver can focus on the general form of the problem
Useful miscellaneous information
Next, several useful bits of miscellaneous circuit-related information is presented.
AV formulation
Most commonly, the so-called AV mathematical formulation is used for handling circuits in finite-element analysis. It’s background is in the potential formulation of Maxwell’s equations, where the electric field is written as
\(\mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}\),
where \(\mathbf{V}\) is called the electric scalar potential.
From the electric field, the current density is then known to be \(\mathbf{J} = \sigma \mathbf{E}\), where \(\sigma\) is the electrical conductivity. As the typically-analysed frequency ranges have no charge concentration to speak of, the governing equation for the newly-introduced thus-far unknown scalar potential is the current conservation condition
\(\nabla \cdot \mathbf{J} = \nabla \cdot \left( -\nabla V - \frac{\partial \mathbf{A}}{\partial t} \right) = 0\).
AVI-formulation
The AV formulation is sufficient for handling massive conductive bodies in 3D analysis. It does work in 2D, too, but does not alone allow relating any current variables to the current density distribution. In order to do this, some extra steps are needed.
Before moving on to currents, one detail: In 2D electromagnetics, the electric field typically has a component in the z-direction only. Therefore, it is sometimes expressed with the voltage difference over the ends (in z-direction) of a conducting body
\(\nabla V = - \frac{u}{l}\).
Here, \(u\) is the voltage difference, while \(l\) is the length of the problem in the z-direction (out-of-plane).
Incidentally, this modified AV formulation allows to naturally introduce currents into the problem. In 2D analysis, the net (axial) current in a conducting body is
\(I = \int\limits_{A_\text{conductor}} \mathbf{J} \mathrm{d}S = \sigma \int\limits_{A_\text{conductor}} { \frac{u}{l} - \frac{\partial \mathbf{A}}{\partial t} } \mathrm{d}S\).
Now, by introducing the cross-sectional area of the conductor \(A\), the first term of the integral is simplified into
\(\sigma \int\limits_{A_\text{conductor}} \frac{u}{l} \mathrm{d}S = \frac{\sigma A}{l} u = \frac{1}{R} u\),
where $R$ is the DC resistance of the conducting body, again in the z-direction.
Substituting this back into the net current equation, multiplying it by \(R\) and re-ordering and simplifying a bit yields an important equation
\(u = RI + \frac{l}{A} \int\limits_{A_\text{conductor}} \frac{\partial \mathbf{A}}{\partial t} \mathrm{d}S\).
Although somewhat tedious to derive, this equation is very easy to relate to traditional circuit analysis. Indeed, it simply states that the net axial voltage over the conductor is the sum of the DC-resistance-related voltage drop plus the average induced voltage.
Nowm we still have to define a governing equation for the current \(I\). A few common cases are briefly described.
Zero net current
Often, the net current \(I\) is assumed to be zero in each individual conductor. This is often the case when modelling unwanted eddy currents in individual conductors, such as permanent magnets.
In this case, the vector of circuit variables \(\mathbf{b}\) only consists of the voltages \(u\). In EMDtool, the BlockCircuit is a perfect choice for this case.
Zero net current in a conductor spanning the entire cross-section
Now, one special case is worth mentioning. Often, only the minimum symmetric sector of an electric machine is modelled - often a pole-pitch or two. However, the problem might also contain solid conductors that span the entire physical geometry and thus extend beyond the symmetry sector. Solid shafts, rotor cores, and stator housings are all common examples. For the typical end-user, the SheetCircuit class is suitable for this case.
Also in this case, a zero-net-current assumption is commonly assumed. However, how the condition is enforced depends on the periodicity of the problem. Indeed, the condition is automatically satisfied for anti-periodic (periodicity coefficient equal to -1) problems, in which there are no circuit variables \(\mathbf{c}\) associated. With periodic problems (periodicity coefficient equal to 1), \(\mathbf{c}\) consists of a single voltage \(u\).
Full circuit coupling
In many other cases, the per-conductor net current is not known. Thus, it must be solved together along with the other variables, and constrained by their own governing equation.
A very good example would be a stator winding wound from rectangular conductors, whether of the hairpin or the more traditional form-wound kind. As before, each individual conductor in each slot would have a voltage variable \(u\) associated with it, together forming the vector \(\mathbf{u}\).
However, the number of independent current variables would normally be significantly smaller. After all, we know that most of the conductors would be in series with each other, carrying the same net current. Thus, for clarity’s sake, let us get rid of the per-conductor current variable \(I\) and introduce the vector of all such variables in the entire winding:
\(I \rightarrow \mathbf{i}_\text{conductor}\).
Furthermore, let \(\mathbf{i}\) be the vector of independent phase currents. Finally, let us introduce a loop matrix \(\mathbf{L}\) defining the dependency between conductor and independent currents:
\(\mathbf{L}_{ij} := \begin{cases} +1 & \text{current $j$ traverses conductor $i$ to the positive direction} \\ -1 & \text{current $j$ traverses conductor $i$ to the negative direction} \\ 0 & \text{otherwise} \end{cases}\).
The matrix \(\mathbf{L}\) is the so-called loop matrix, and can either be formed manually / hard-coded for common cases such as cage or distributed windings, or in general cage formed automatically with the fundamental loop method.
Now, the governing equations for the phase currents \(\mathbf{i}\) are still needed. These are obtained from the Kirchoff’s voltage law, stating that the sum of all voltage terms - whether sources or impedance-related voltage drops - must sum up to zero. This can be concisely written as
\(\mathbf{L}^\intercal\mathbf{u} + \left(\mathbf{R}_\text{EW} + \frac{\partial}{\partial t}\mathbf{L}_\text{EW}\right)\mathbf{i}=\mathbf{U}_\text{source}\).
Here, the \(\mathbf{L}^\intercal\mathbf{u}\) term stands for the sum of the conductor voltages from the finite-element problem. The \(\mathbf{R}_\text{EW}\) and \(\mathbf{L}_\text{EW}\) matrices are the ‘end-winding’ resistance and inductance matrices. They don’t have to be actually associated with a physical end-winding - hence the quotation marks - braking resistors and line chokes would also be considered inside these matrices. Rather, the subscript \(\text{EW}\) can be understood as a general term for circuit components with no corresponding equivalent in the FE problem. Finally, \(\mathbf{U}_\text{source}\) is the vector of source (a priori known) voltages for each current loop.
Circuits in EMDtool
EMDtool handles circuits in a way that reflects the theory presented so far. All circuits are subclasses of the abstract CircuitBase class, and thus must implement some abstract methods and properties imposed by it. The details are best left to the Matlab environment; however some important high-level details are discussed here.
.get_matrices method
Like we saw earlier, each circuit in a FE problem is governed by its own stiffness- and mass/damping-like matrices (acting on the vector of unknowns \(\mathbf{x}\) and its time-derivative \(\partial / \partial t \mathbf{x}\) respectively), which in turn can be split into 2x2 block matrices
\[\mathbf{S}_\text{circuit} = \begin{bmatrix} \mathbf{S}_\text{aa} & \mathbf{S}_\text{ac} \\ \mathbf{S}_\text{ca} & \mathbf{S}_\text{cc} \end{bmatrix}\] \[\mathbf{M}_\text{circuit} = \begin{bmatrix} \mathbf{M}_\text{aa} & \mathbf{M}_\text{ac} \\ \mathbf{M}_\text{ca} & \mathbf{M}_\text{cc} \end{bmatrix}\]reflecting the relationship between the pure-FE vector of unknowns \(\mathbf{a}\) and the vector of circuit unknowns \(\mathbf{c}\).
Unsurprisingly, it is the task of the .get_matrices method of each circuit object to return the corresponding matrix blocks:
[Saa, Maa, Sac, Mac, Sca, Mca, Scc, Mcc] = circuit.get_matrices()
This method is normally called by a CircuitSet object, combining the individual circuit matrices into the final mass and stiffness matrices.
.set_load method
This method adds the contribution of the circuit into the load vector \(\mathbf{f}_\text{tot}\). For now at least, this method works by incrementing a global vector given as an input argument and returning the incremented vector - rather than returning the vector block component. This behaviour may change in the future.
.init method
This method initializes the circuit for a given MagneticsProblem. Normally, several matrices are precomputed and stored in circuit.matrices.
.init_for_simulation method
This method is called before solving a particular problem - static, harmonic, or time-stepping analysis. Normally, only very light-weight operations are performed here, such as setting the frequency for harmonic (frequency-domain) analysis.
.losses method
Returns the total average circuit losses for a given solution, plus whatever data computed by the particular circuit class.
.get_ndof method
Returns number of circuit variables, i.e. the length of the \(\mathbf{c}\) vector.
.set_dof_bias method
This method sets the circuit aware of its relative position among all the circuits in the problem. This is mainly useful if the circuit needs to fetch its circuit variables \(\mathbf{c}\) from the global solution vector \(\mathbf{x}\).