So there are t basic cutsets in a graph In this example the basic cutsets are 1

# So there are t basic cutsets in a graph in this

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So, there are t basic cutsets in a graph. In this example, the basic cutsets are { 1 , 3, 6 } { 2 , 3, 5 } { 4 , 5, 6 } The importance of basic cutsets is the formulation of independent KCL equations: tree branches
Prof. C.K. Tse: Graph Theory & Systematic Analysis 12 Basic loops A basic loop is a loop containing only one co-tree branch. So, there are l basic loops in a graph. In this example, the basic loops are { 1, 2, 3 } { 2, 4, 5 } { 1, 4, 6 } The importance of basic loops is the formulation of independent KVL equations: co-tree branches
Prof. C.K. Tse: Graph Theory & Systematic Analysis 13 Independent KCL/KVL equations A different choice of tree gives a different set of basic cutsets and basic loops. The set of independent KCL and KVL equations found is not unique. But any set of independent KCL and KVL equations gives essentially the same information about the circuit. So, it doesn’t matter which tree is chosen. Once a tree is chosen, a set of independent KCL and KVL equations is found. Any other KCL or KVL equation is derivable from the independent set. That means, we DON’T NEED to find more than t KCL or b–t KVL equations, since anything more than the basic set is redundant and a waste of effort!
Prof. C.K. Tse: Graph Theory & Systematic Analysis 14 Matrix representations There are three fundamental matrices representing the graph of a given circuit: They are very useful in computer-aided systematic analysis. 1. Node-incidence matrix (A-matrix) 2. Basic cutset matrix (Q-matrix) 3. Basic loop matrix (B-matrix)
Prof. C.K. Tse: Graph Theory & Systematic Analysis 15 Node-incidence matrix (A-matrix) The A-matrix describes the way a circuit is connected. It is very important in computer simulation. The columns in a A-matrix correspond to the branches; and the rows correspond to the nodes.
Prof. C.K. Tse: Graph Theory & Systematic Analysis 16 Basic cutset matrix (Q-matrix) The Q-matrix describes the way the basic cutset is chosen. Each column corresponds to a branch ( b columns). Each row corresponds to a basic cutset ( t rows). Construction For each row: Put a “+1” in the entry corresponding to the cutset tree branch. Put a “0” in the entry corresponding to other tree branches. Put a “+1” or “–1” in the entry corresponding to each cutset co-tree branch; “+” if it is consistent with the tree branch direction and “–” otherwise. Q = [ 1 | Q 1 ]
Prof. C.K. Tse: Graph Theory & Systematic Analysis 17 Basic loop matrix (B-matrix) The B-matrix describes the way the basic loop is chosen. Each column corresponds to a branch ( b columns). Each row corresponds to a basic loop ( b t rows). Construction For each row: Put a “+1” in the entry corresponding to the loop co-tree branch. Put a “0” in the entry corresponding to other co-tree branches. Put a “+1” or “–1” in the entry corresponding to each loop tree branch; “+” if it is consistent with the co-tree branch direction and “–” otherwise.

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• Summer '16
• Martin Chow
• Graph Theory, Mesh Analysis, branch, Voltage source, Voltage drop, Systematic Analysis

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