Coupling of field and circuit equations (Electrical Machine) Part 2

By

Cut-set matrix

For a directed graph with b branches andtmp101283_thumb[2]oriented cut-sets, the cut-set matrix is atmp101284_thumb[2]matrixtmp101285_thumb[2]where:

tmp101289_thumb[2]if branch j is in cut-set i, and their directions agree,

tmp101290_thumb[2]if branch j is in cut-set i, and their directions oppose, and

tmp101291_thumb[2]if branchy is not in cut-set i.

For example, there are seven cut-sets in the graph of Fig. 5.77. The cut-set matrix is

tmp101295_thumb[2]

The general form of the KCL in matrix-vector notation is:

tmp101296_thumb[2]

Any sub-matrixtmp101297_thumb[2]consisting of the maximum number of independent rows oftmp101298_thumb[2]is called a basic cut-set matrix. A systematic method for constructing a basic cut-set matrix is through the aid of a tree T. Each tree branch of T, together with some (possibly no) links in the associated co-tree forms a cut-set, called a fundamental cut-set for that tree branch. A sub-matrixtmp101299_thumb[2]constructed with thetmp101300_thumb[2]fundamental cutsets is called a fundamental cut-set matrix.

For example, for the graph of Fig. 5.77 with tree branches (abc), the fundamental cut-set matrix is

tmp101305_thumb[2]

It is obvious that any fundamental cut-set matrix can be partitioned as:

tmp101306_thumb[2]

Relationship between branch variables

When the reduced incidence matrix A, the fundamental loop matrix B and the fundamental cut-set matrix D are built corresponding to the same tree T, the matrices can be partitioned by:

tmp101307_thumb[2]

and the following relationships can be given:

tmp101308_thumb[2]

Circuit analysis

Linear and time invariant networks with lumped parameters can be described by various techniques. The equations are the KVL, the KCL and the branch current-voltage relationships (BCVR). The known quantities are the currents of independent current sources, the voltages of independent voltage sources and the impedances of the branches.

Tableau analysis The unknowns in a tableau analysis are the branch currents and the branch voltages. The system of equations consists of the KCL for each node of the circuit, the KVL for each loop of the circuit and the BCVR for each branch of the circuit.

Example circuit.

Fig. 5.78. Example circuit.

For example, consider the circuit of Fig. 5.78. The system of equations is given by:

tmp101310_thumb[2]

It is obvious that the number of equations is larger than the number of unknowns. There are linear dependent equations in the system. The matrix is sparse, singular and not symmetric.

Modified nodal analysis

The unknowns in a modified nodal analysis (MNA) are the nodal voltages and the currents through the independent voltage sources. For each node per connected circuit component, except the chosen reference node, the KCL is written.

Applying the BCVR to each branch immediately eliminates the currents. The branch voltages are expressed with respect to the nodal voltages. For each independent voltage source, an extra unknown current is added to the set of unknowns. An extra equation, representing the difference between the two nodal voltages of the voltage source to the known voltage drop, is added.

For the circuit of Fig. 5.78, this approach results in:

tmp101311_thumb[2]

The system of equations is sparse and smaller when compared to the tableau analysis. There are no superfluous equations. The unknowns arc the nodal voltages and branch currents.

Compacted modified nodal analysis

In the compacted form of the MNA, the unknown currents through the independent voltage sources are eliminated by substitution of the extra equations concerning the independent voltage sources.

tmp101312_thumb[2]

The system of equations is again reduced, moreover, there are no zero diagonal elements left in the matrix. The matrix is symmetric.

Topological methods

A topological method for circuit analysis is a technique deriving parameters describing the circuit behaviour from the structure of a graph, associated with the network. Some topological methods are the Signal Flow Graph method and the tree-enumeration method.

Signal flow graph

A signal flow graph (SFG) is a weighted directed graph representing a system of linear equations. The nodes represent the variables; the branch weights represent the coefficients in the relations between the variables. A node variable and the sum of the incoming branch weights, multiplied by the node variable from which the incoming branch originates, represent an equation. Consider for example the SFG of Fig. 5.79 and the corresponding system of equations:

tmp101313_thumb[2]

In a SFG, a node with outgoing branches only is called a source node. A node with some incoming branches is called a dependent node. A dependent node with incoming branches only is called a sink node. In Fig. 5.79,tmp101314_thumb[2]is a source node,tmp101315_thumb[2]are dependent nodes andtmp101316_thumb[2]is a sink node.

 

Sipal flow graph.

Fig. 5.79. Sipal flow graph.

Formulation of the signal flow graph

It is assumed that the voltage sources do not contain loops, and that the current sources do not contain cut-sets. It is possible to select a tree T such that all voltage sourcestmp101324_thumb[2]are tree branches, and all current sourcestmp101325_thumb[2]are links. The impedance tree branches are characterised by the impedance matrixtmp101326_thumb[2] and the immittance co-tree branches are characterised by the admittance matrixtmp101327_thumb[2]The following steps can construct a SFG:

1. Apply KVL to express each element oftmp101332_thumb[2]in terms of elements oftmp101333_thumb[2]

2. Apply KCL to express each element oftmp101334_thumb[2]in terms of elements oftmp101335_thumb[2]andtmp101336_thumb[2]

3. For impedance tree branches, each voltage is expressed in terms of the current through the branch:tmp101337_thumb[2]

4. For immittance co-tree branches, each current is expressed in terms of the voltage across the branch:tmp101338_thumb[2]

A SFG formed in this way displays the KCL, KVL and BCRV relations in their most primitive way. As a consequence, it is called a primitive signal-flow graph. The primitive SFG of the circuit from Fig. 5.79 is represented in Fig. 5.80.

The number of nodes in the SFG can very easily be largely reduced by the use of a compacted signal flow graph, which is obtained from the primitive signal flow graph by eliminating all sink nodes and the variablestmp101339_thumb[2]The compacted SFG of the circuit of Fig. 5.78 is represented in Fig. 5.81.

Primitive signal flow graph.

Fig. 5.80. Primitive signal flow graph.

 Compacted signal flow graph.

Fig. 5.81. Compacted signal flow graph.

Circuit analysis for the coupled field-external circuit system

The solid conductors are considered as an admittance branch and a controlled current source connected in parallel:

tmp101350_thumb[2]

A is the vector of the magnetic vector potentials andtmp101351_thumb[2]is the matrix with the coupling terms of (5.215). The stranded conductors are considered an impedance branch and a dependent voltage source connected in series:

tmp101353_thumb[2]

wheretmp101354_thumb[2]is the matrix with the coupling terms obtained by the discrete integration of (5.214).

Modified and compacted modified nodal analysis

The tableau analysis is not suited for numerical calculation because the dependent equations result in a singular system matrix.

In an MNA, for each connected sub-circuit a reference node is chosen. On all other nodes an unknown voltage is defined. For each independent voltage source and for each stranded conductor, an unknown current is defined. For each node, the KCL is written. The currents are immediately written in terms of the nodal voltages by means of the BCVR. In the case of a stranded conductor branch or an independent voltage source branch, the current is unknown. For each of these branches, an extra KVL equation describes the nodal voltages of the endpoints in terms of the unknown currents and the magnetic vector potentials.

Electric circuit with stranded and solid conductors. This method has been applied to the circuit in Fig. 5.82.

Fig. 5.82. Electric circuit with stranded and solid conductors. This method has been applied to the circuit in Fig. 5.82.

tmp101357_thumb[2]

The factortmp101358_thumb[2]can symmetrise the coupled magnetic-electric system of equations for a 2D time-harmonic problem.

The presents of zero elements in the diagonal of the matrix makes the choice of an appropriate method for solving the system of equations more difficult.

As with the CMNA, the extra KVL equations can be substituted in both the KCL equations and the field equations, tending to a smaller and symmetric matrix with a fully occupied diagonal. However, the implementation of the dense circuit equations into the sparse system of field equations makes the solution procedure difficult.

The major problem of the nodal approach in the circuit analysis is that the description of the circuit is based on the voltage only. A more suitable description would be based on a hybrid use of both unknown voltages and currents. Here, the difficulty is the systematic determination of the circuit components described by Currents, voltages and the interface parts in the circuit.

Signal flow graph for coupled magnetic-electric problems

The circuit in Fig. 5.83 is considered.

Branches are assembled in the tree with preference:

• voltage sources, solid conductors, impedances and stranded conductors.

The preferred order for links is:

• current sources, stranded conductors, admittances and solid conductors (Fig. 5.83).

The fundamental cut-set matrix and the fundamental loop matrix are partitioned in components. They are associated with the stranded conductors being linkstmp101360_thumb[2]tree branchestmp101361_thumb[2]solid conductors being tree branchestmp101362_thumb[2]linkstmp101363_thumb[2]independent sources (/ and v) and the immittance tree branches (7) and links (L).

Applying the Kirchhoff current law (KCL) for each fundamental cut-set and the Kirchhoff voltage law (KVL) for each fundamental loop arranges a SFG. The unknowns of the system are the link currents and the tree branch voltages. The SFG of a stranded and a solid conductor is shown in Fig. 5.84. The Signal Flow sub-graphs of the circuit of Fig. 5.83 are represented in Fig. 5.85. Table 5.7 shows the equivalencies between the SFG and the matrix calculus.

Table 5.7. Equivalence between circuit theory, SFG and matrix calculus.

circuit analysis

signal flow graph

matrix notation

Kirchhoff current law

current nodes

tmp10-1368

Kirchhoff voltage law

voltage nodes

tmp10-1369

branch relations

vertical connections

tmp10-1370

cut-set transformation

tmp10-1371 tmp10-1372

loop transformation

tmp10-1373 tmp10-1374

compacting

tmp10-1375 tmp10-1376

Electric circuit with a) stranded and b) solid conductors.

Fig. 5.83. Electric circuit with a) stranded and b) solid conductors.

SFG of a a) stranded and b) solid conductor.

Fig. 5.84. SFG of a a) stranded and b) solid conductor.

 Non-coupled signal flow graph.

Fig. 5.85. Non-coupled signal flow graph.

Joining the graphs does not change the graph nodes so long as the dependent nodes of the first graph correspond to source nodes of the other graph and vice versa. Therefore, branch current-voltage relations (BCVR) are added either as impedances or admittances (Fig. 5.85). Two dependent nodes are joined together to one zero node by changing the sign of all incoming branch weights of one of the sub-graphs. The former unknown of that node disappears. This happens for the stranded conductor links and the solid conductor tree branches (Fig. 5.86).

Coupled signal flow graph.

Fig. 5.86. Coupled signal flow graph.

Combining magnetically coupled branches causes three difficulties:

• Stranded conductor tree branches are described by a dependent current (Fig. 5.85a).

• Solid conductor links are described by a dependent voltage (Fig. 5.85b).

• The coupling terms are not symmetric.

Three operations can solve the above mentioned problems: 1. Partial cut-set transformation

The preferences while choosing tree branches result in a fundamental cut-set associated with a stranded conductor tree branch which contains only current sources and stranded conductors. A partial cut-set transformation

tmp101381_thumb[2]

contracts the graph in directiontmp101382_thumb[2](Fig. 5.86). The current of the stranded conductor tree branch is expressed as a combination of independent currents and other stranded conductor currents (Table 5.7).

Partial loop transformation

A fundamental loop associated with a solid conductor link only exists of voltage sources and solid conductors. A partial loop transformation

tmp101385_thumb[2]

contracts the graph in direction [3 (Fig. 5.86). The voltage of the solid conductor link is expressed as a combination of independent voltages and other solid conductor voltages. 3. Symmetrising the system

A contraction of the BCVR (direction y in Fig. 5.86,Table 5.7) leads to a compact signal flow graph (CSFG) (Fig. 5.87).

Compact signal flow graph.

Fig. 5.87. Compact signal flow graph.

From the CSFG, the coupling matrices are extracted in a simple way. The unknown graph nodes become system unknowns. The dependent graph nodes represent matrix equations. The coupling terms are kept symmetric. Compared to tableau analysis, MNA and CMNA, a reduction of additional circuit equations is obtained. Multiplying the circuit loop equations with % and the circuit cut-set equations withtmp101388_thumb[2]tmp101389_thumb[2]

where

tmp101390_thumb[2]

with

tmp10-1391

 

 

 

tmp10-1392

and

tmp101393_thumb[2]

In the case of a quasi-static problem, K is complex symmetric. S is symmetric becausetmp101394_thumb[2]and thattmp101395_thumb[2]andtmp101396_thumb[2]are diagonal matrices.