Once a tree is chosen, a set of independent KCL and KVL equations is found. So, it doesn t matter which tree is chosen. But any set of independent KCL and KVL equations gives essentially the same information about the circuit. The set of independent KCL and KVL equations found is not unique. In this example, the basic cutsets are co-tree branches The importance of basic loops is the formulation of independent KVL equations: Systematic Analysis 12ġ3 Independent KCL/KVL equations A different choice of tree gives a different set of basic cutsets and basic loops. So, there are t basic cutsets in a graph. co-tree Systematic Analysis 9ġ0 Basic relations Let n = number of nodes b = number of branches t = number of tree branches l = number of co-tree branches We have, for all planar graphs, t = n 1 l = b t = b n + 1 Systematic Analysis 10ġ1 Basic cutsets A basic cutset is a cutset containing only one tree branch. After a tree is chosen, the remaining branches form a co-tree. Thus, a tree is a maximal set of branches that contains no loop. Moreover, including one more branch to this set will create a loop. Systematic Analysis 6ħ KCL The following are all KCL equations for the circuit below: I a + I b + I d = 0 I c + I d + I b = 0 I c + I d + I e = 0 Systematic Analysis 7Ĩ Problem: Find I y Usual way: Find I z Then find I x Then find I w Then we get I y I w Alternative way: Using KCL for an appropriate cutset, the problem is as simple as I y = 0! Systematic Analysis 8ĩ Tree and co-tree A tree is a set of branches of a graph which contains no loop.
We may say that the sum of currents going from one sub-graph to the other is zero. KCL more generally stated in terms of cutset with appropriately chosen directions Usually the cutset separates the graph into two subgraphs. Examples: branches f, b, d, c SPECIAL CASE Branches emerging from a node form a cutset always a cutset Systematic Analysis 5Ħ Kirchhoff s laws again KVL same as before. For example, branches a, c, d branches a, b, e, c Systematic Analysis 4ĥ Cutset A cutset is a set of branches of a graph, which upon removal will cause the graph to separate into two disconnected sub-graphs. 1 Electronic Circuits 1 Graph theory and systematic analysis Contents: Graph theory Tree and cotree Basic cutsets and loops Independent Kirchhoff s law equations Systematic analysis of resistive circuits Cutset-voltage method Loop-current method Systematic Analysis 1Ģ Graph and digraph Consists of branches and nodes Describes the interconnection of the elements Graph Digraph arrows indicate directions of currents and voltages polarities Systematic Analysis 2ģ Sign convention Stick to the following sign convention Current direction same as arrow direction Voltage polarity arrow goes from + to through the element + V I Systematic Analysis 3Ĥ Loop A loop is a set of branches of a graph forming a closed path.