Coupling of field and circuit equations (Electrical Machine) Part 1

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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:

tmp101132_thumb

To stay in the same notation as in section 5.7 using the relative reluctivitytmp101133_thumband the permeability of the free spacetmp101134_thumbwe can write:

tmp101139_thumb

Regions without eddy currents but with applied current densitytmp101140_thumb represent stranded conductors. They are described by Poisson’s equation:

tmp101142_thumb

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

tmp101144_thumb

Different materials with respect to their conductivitytmp101145_thumband permeabilitytmp101146_thumbcan be considered element by element. The corresponding energy-minimum functionals are for solid conductors:

tmp101149_thumb

• for stranded conductors:

tmp101152_thumb

• and for non-conducting regions:

tmp101153_thumb

Functional within an element For non-conducting regions and regions without eddy currents (stranded conductors) the 3×3 element matrixtmp101154_thumbremains the same as in the magneto static case (5.123).

tmp101156_thumb

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

tmp101158_thumb

The second term is:

tmp101159_thumb

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

tmp101161_thumb

The matrixtmp101163_thumbis symmetric. In matrix-vcctor notation, the functional within an element becomes:

 

tmp101165_thumb

Source vector For regions without eddy current but with an applied current densitytmp101167_thumb(stranded conductors) the source vectortmp101168_thumb remains the same as in the static case (5.130).

tmp101171_thumb

The terms of the 3×3 eddy current matrix L(" are given by: tmp101174_thumb

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 strandtmp101175_thumbtmp101177_thumb

tmp101179_thumbis the number of turns of the winding or the strands of conductor p andtmp101180_thumbis 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 droptmp101181_thumb

tmp101188_thumb

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.

tmp101190_thumb

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.

Iftmp101191_thumbare unknowns, the system in (5.209) becomes:

tmp101193_thumb

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

tmp101194_thumb

withtmp101195_thumbthe angular frequency of the current,tmp101196_thumbthe permeability andtmp101197_thumb 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:

tmp101201_thumb

with

tmp101202_thumb

When a stranded conductor is considered as a region of filamentary conducting wires, the fill factortmp101203_thumbfor 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.

tmp101206_thumb

From eq.(5.214) it can be taken that the total voltage drop consists of a resistive componenttmp101207_thumband an inductive componenttmp101209_thumbTherefore, 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).

Network elements representing a stranded conductor.

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:

tmp101213_thumb

From eq.(5.215) it can be taken that the current consists of an admittance currenttmp101214_thumband an eddy currenttmp101215_thumbcomponent.

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

Network elements representing a solid conductor.

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:

tmp101219_thumb

Multiplying (5.215) with a factor

tmp101220_thumb

and (5.214) withtmp101221_thumbresults 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.

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

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.

Directed graph.

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 matrixtmp101224_thumbwhere:

tmp101225_thumbif branch j is incident to node i, and the arrow is pointing away from node i,

tmp101226_thumbif branch j is incident to node /, and the arrow is pointing towards node i, and

tmp101227_thumbif branch j is not incident to node i.

Directed graph.

Fig. 5.77. Directed graph.

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

tmp101233_thumb

One of the rows oftmp101234_thumbis linearly dependent on the other rows. The matrix A obtained fromtmp101235_thumbby omitting a row is called a reduced incidence matrix.tmp101236_thumbis called the complete incidence matrix.

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

tmp101240_thumb

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

tmp101244_thumb

A can be partitioned astmp101245_thumbwhere the columns oftmp101246_thumbcorrespond to the tree branches of a chosen tree T, and the columns of tmp101247_thumbcorrespond to the links.

Loop matrix

For a directed graph with b branches andtmp101248_thumboriented loops, the loop matrix is atmp101249_thumbmatrixtmp101250_thumbwhere:

tmp101257_thumbif branch j is in loop i, and their directions agree,

tmp101258_thumbif branch j is in loop i, and their directions oppose, and

tmp101259_thumbif branch j is not in loop i.

For example in Fig. 5.77 there are seven loops.

The loop matrix is:

tmp101263_thumb

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

tmp101264_thumb

Any sub-matrixtmp101265_thumbconsisting of the maximum number of independent rows oftmp101266_thumbis called a basic loop matrix. Thus, thetmp101267_thumb independent KVL equations may be expressed as

tmp101271_thumb

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 oftmp101275_thumbconstructed by the use oftmp101276_thumblinks is called a fundamental loop matrixtmp101277_thumb

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

tmp101281_thumb

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

tmp101282_thumb