**By R. Belmans**

**In the FEM models considered** up to now it was assumed that current densities, permanent magnets and/or given potential distributions were imposed as field exciting sources. Many models of technical relevance can be modelled using the mentioned sources. But electromechanical devices are operated by currents and/or voltages generated by a power supply. Thus, such real devices are fed from electrical circuits. Such external circuits consist of inductances, capacitances, resistances, current and/or voltage sources.

**The two-dimensional** model of an induction motor does not consider the resistor of the cage end-windings. The connection of the rotor bars is not modelled in a simple two-dimensional cross-sectional model. Therefore, additional external circuits with resistive elements have to be defined to model the short-cut rings of the rotor cage.

In the following sections such external circuit equations will be defined and coupled to the two-dimensional finite element equations. The types of stranded and solid conductor will be introduced and implemented in the FEM model. Both types can be handled simultaneously in one model using a mixed formulation.

The same notation as used in the previous sections is assumed and used.

## Time harmonic problem

Regions with eddy currents and applied voltage gradient are called solid conductors.

The partial differential equation for a two-dimensional time-harmonic magnetic problem in solid conductors is:

**To stay in the same notation as in section 5.7 using the relative reluctivity****and the permeability of the free space****we can write:**

**Regions without eddy currents but with applied current density**** represent stranded conductors. They are described by Poisson’s equation:**

**For non-conducting regions Laplace’s equation is valid:**

**Different materials with respect to their conductivity****and permeability****can be considered element by element. The corresponding energy-minimum functionals are for solid conductors:**

**• for stranded conductors:**

**• and for non-conducting regions:**

**Functional within an element** For non-conducting regions and regions without eddy currents (stranded conductors) the 3×3 element matrixremains the same as in the magneto static case (5.123).

**For regions with eddy currents (solid conductors) the functional within an element is:**

**The second term is:**

The terms of the 3×3 eddy current matrix L(" are given by:

**The matrix****is symmetric. In matrix-vcctor notation, the functional within an element becomes:**

**Source vector** For regions without eddy current but with an applied current density(stranded conductors) the source vector remains the same as in the static case (5.130).

The terms of the 3×3 eddy current matrix L(" are given by:

**Windings of electrical machines can consist** of a number of conductors connected in parallel with geometrical dimensions such that at the considered frequency eddy currents can be neglected. For this case, and stranded conductor p, the source vector is written in terms of the current per strand

is the number of turns of the winding or the strands of conductor p andis the area of the stranded conductor in the FEM mesh.

In the case of a solid conductor q, the source vector can be written in terms of its voltage drop

the length of the conductor.

**System of equations** The system of linear equations is assembled in the same way as for the magneto static problem.

Dirichlet boundary conditions are considered in eq.(5.209) and binary conditions are omitted.

**If the currents in the stranded** conductors and the voltage drops over the solid conductors are unknown, extra circuit equations have to be added to the system. These equations can be seen as boundary conditions. The circuit conditions act as a lumped parameters model that is applied to the boundary of the differential problem.

**If****are unknowns, the system in (5.209) becomes:**

**Stranded conductors in eddy current problems** The skin depth of the current into a conducting material is given by;

withthe angular frequency of the current,the permeability and the conductivity of the conductor. In general, it is assumed that the current density is uniformly distributed across the skin depth. However, when a number of conductors are connected in parallel, the induced voltage depends on both the current density and the magnetic vector potential (5.212).

**The entire voltage drop over the stranded conductors is calculated by using the average voltage over the conducting area:**

with

**When a stranded conductor is considered as a region** of filamentary conducting wires, the fill factorfor the conducting material is defined as the ratio of the surface of the conducting material to the entire surface of the conductor. In this way, insulating material can be considered.

**From eq.(5.214)** it can be taken that the total voltage drop consists of a resistive componentand an inductive componentTherefore, in an electric network model, a stranded conductor can be represented by a series connection of an ohinic resistor and a controlled voltage source (Fig. 5.73).

**Fig. 5.73. Network elements representing a stranded conductor.**

**Solid conductors in eddy current problems** In the case of solid conductors, the gradient of the voltage is assumed to be constant over the surface of the conductor. The eddy currents are a function of both the voltage gradient and the magnetic vector potential:

**From eq.(5.215)** it can be taken that the current consists of an admittance currentand an eddy currentcomponent.

**In the circuit analysis**, a solid conductor can be modelled by admittance and a controlled current source connected in parallel (Fig. 5.74).

**Fig. 5.74. Network elements representing a solid conductor.**

## Coupled field-circuit equations

**To solve the time-harmonic magnetic field, the unknown magnetic vector potentials can be calculated by evaluating the:**

• known potentials at the boundary (Dirichlet)

• known current densities in stranded conductors

• known voltage gradients in solid conductors.

**It is recommended,** therefore, that the currents of stranded conductors and the voltage drops in solid conductors are considered as unknowns of the system. The equations (5.215) and (5.214), describing an external electric circuit (Fig. 5.73 and Fig. 5.74), have to be added to the set of field equations to obtain a coupled system.

The coupling of the magnetic field with the-electric circuit equations can be obtained numerically strong or by a weak coupling.

**Using some initial values**, a first computation of the magnetic field is performed. Out of this field solution, the induced voltage drops over the stranded conductors and the eddy currents in solid conductors arc evaluated. By using these results, the electrical quantities of the electric circuit network are calculated, supplying new values for the currents in the stranded conductors and the voltage drops over the solid conductors, to be used for a new calculation of the magnetic field.

To solve the entire coupled field-circuit problem, an iterative procedure can be applied. Both partial problems are solved in successive steps. This approach is a numerically weak coupling of the two systems.

**To obtain a numerically strong coupling,** it is possible to assemble all unknowns in one vector and to combine all equations describing the system in one matrix. All equations describing the system are solved simultaneously. This approach is called a numerically strong coupling of the systems. If all stranded conductors are voltage driven and if all solid conductors are current driven, the coupled matrix is given by:

Multiplying (5.215) with a factor

and (5.214) withresults in a symmetrical matrix.

**Mixed stranded and solid conductors** Mixed stranded and solid conductors in a connected network cause problems describing the circuit. The matrices obtained by a separate analysis of stranded and solid conductors can not be arranged together.

The circuit theory indicates a problem while enumerating tree and co-tree branches. In Fig. 5.75a one of the three solid conductors has to be considered as a link. In Fig. 5.75b only one of the stranded conductors can be considered as a tree branch.

**Replacing the magnetic branches,** as indicated in Fig. 5.73 and Fig. 5.74, fails. In this case cut-sets containing a stranded conductor branch include solid conductor branches. Loops holding a solid conductor branch include stranded conductor branches.

To solve this problem, the connected network has to be represented in another form.

**Fig. 5.75. a) Star-connected stranded conductors and b) solid conductors connected in parallel.**

## Network topology

**With a network topology, lumped parameter networks obeying three basic laws:**

• Kirchhoff voltage law (KVL),

• Kirchhoff current law (KCL) and

**• Branch current-voltage relations (BCVR) can be studied. A complete description of the network delivers information on:**

• the connection of branches,

• the reference directions for branch currents and voltages and

• the branch characteristics.

**Items 1 and 2 can be represented by a directed graph (Fig. 5.76).** By defining loops, cut-sets and a tree, the network description can be performed systematically. Loops are the sub-graphs to which KVL is applied. Cut-sets are the sub-graphs to which a generalised KCL is applied. The concept of a tree is a tool for a systematic formulation of independent KCL and KVL equations.

**Fig. 5.76. Directed graph.**

### Definitions Path

**A set of branches is called a path between two nodes p and q, if the branches can be labelled in such a way that:**

**• consecutive branches always have a common endpoint,**

**• no node is the endpoint of more than two branches in the set, and**

**• p is the endpoint of exactly one branch in the set, and so is q.**

Thus, A path is just a route between two nodes. In Fig. 5.76 branches (dhib) form a path between nodes 1 and 2.

### Connected graph

An undirected graph is a connected graph if there exists a path between any two nodes of the graph. A network is connected if the associated graph is connected. The graph in Fig. 5.76 is connected.

### Loop

**A sub-graph of a graph is called a loop if**

• the sub-graph is connected, and

• every node of the sub-graph has exactly two branches incident at it.

For example, in Fig. 5.76 the branches (abed) form a loop.

### Tree, co-tree, branches and links

A sub-graph of a connected graph is called a tree if

**• the sub-graph is connected,**

**• the sub-graph contains all nodes of the graph, and**

**• the sub-graph contains no loops.**

**For example, in Fig. 5.76,** the branches (aedgi) form a tree. The branches belonging to a tree are called tree branches, and those that do not belong to a tree are called links. All the links of a given tree T form what is called a co-tree with respect to the tree T. It can be shown that for a connected graph with n nodes, any tree T has exactly n-l tree branches. Furthermore, each set of n-l branches without loops constitutes a tree.

### Cut-set

A set of branches of a connected graph is said to be a cut-set if

**• the removal of the set of branches** (but not their endpoints) results in a graph that is not connected and

**• after the removal of the set of branches**, the restoration of any branch from the set will result in a connected graph again.

**For example in Fig. 5.76**, the branches (aed) or (dgjb) form a cut-set.

### Incidence matrix

**An incidence matrix of a directed graph with n nodes and b branches is a n x b matrix****where:**

•if branch j is incident to node i, and the arrow is pointing away from node i,

•if branch j is incident to node /, and the arrow is pointing towards node i, and

•if branch j is not incident to node i.

**Fig. 5.77. Directed graph.**

**For example, for the directed graph from Fig. 5.77:**

**One of the rows of****is linearly dependent on the other rows.** The matrix A obtained fromby omitting a row is called a reduced incidence matrix.is called the complete incidence matrix.

The KCL for the network can be written in matrix-vector notation as

The maximum set of independent KCL equations, obtained from the nodes of a connected network, can be expressed as

A can be partitioned aswhere the columns ofcorrespond to the tree branches of a chosen tree T, and the columns of correspond to the links.

### Loop matrix

For a directed graph with b branches andoriented loops, the loop matrix is amatrixwhere:

•if branch j is in loop i, and their directions agree,

•if branch j is in loop i, and their directions oppose, and

•if branch j is not in loop i.

For example in Fig. 5.77 there are seven loops.

**The loop matrix is:**

The KVL for all loops can be expressed in matrix-vector notation as

Any sub-matrixconsisting of the maximum number of independent rows ofis called a basic loop matrix. Thus, the independent KVL equations may be expressed as

**A systematic method of constructing a basic** loop matrix is through the aid of a tree T. Each link of the associated co-tree, together with the unique path through T, forms a loop, called the fundamental loop for that link. A sub-matrix ofconstructed by the use oflinks is called a fundamental loop matrix

**For example,** in Fig. 5.77 a tree 7 consists of the branches (abc). The corresponding fundamental loop matrix is

**It is obvious that any fundamental loop matrix can be partitioned as:**