Associated Reference Convention A current direction is chosen entering each positively referenced terminal v1 i1 D i1 i2 D v1 v2 v2 ij i2 D i1 vj i j 1 v1 v j 1 Device Graph DIGRAPH Directed Graph 1 1 2 2 1 j j 1 Associated Reference Convention v1 i1 i2 D v2 2 port Device Device Graph v1 v2 vn i1 i2 D in n port Device 1 1 2 2 n e1 D6 v6 1 i1 v1 D1 D2 v3 e2 D3 i4 2 v4 v2 i3 3 e3 i5 D5 D4 1 v5 4 2 1 3 3 KVL around closed node sequence 2 6 i2 4 5 4 1 3 2 1 2 3 4 2 1 3 4 2 v2 v3 v1 0 v3 v5 v4 0 1 v2 v5 v4 v1 0 These 3 KVL equations are not linearly independent because the 3rd equation can be obtained by adding the first 2 equations v2 v3 v1 v3 v5 v4 v2 v1 v5 v4 0 Circuits containing n terminal devices can have many distinct digraphs due to different arbitrary choices of the datum terminal for each n terminal device Although the KCL and KVL equations associated with 2 different digraphs of a given circuit are different they contain the same information because each set of equations can be derived from the other A Circuit with 3 different digraphs 1 Choose 3 as datum for D 1 2 i2 i3 D i1 i4 v4 v2 v3 v1 1 1 4 3 i1 v1 as datum for D 2 v2 i2 D i3 v3 1 2 i4 v4 1 v1 i1 1 D v2 2 1 4 3 as datum for D v3 2 3 3 3 Choose 3 2 3 2 Choose 2 1 i3 i2 3 1 2 i4 v4 3 2 2 1 4 3 1 1 v1 i1 i3 i2 v2 2 2 D i5 i4 4 4 i6 v6 2 port v4 device v5 v3 5 5 3 3 1 4 2 2 3 1 6 5 4 3 5 disconnected digraph KCL at 2 i3 i4 0 KCL at 4 i5 i6 0 KVL around 2 3 2 KVL around 4 5 4 v4 v3 0 v6 v5 0 1 1 v1 i1 i3 i2 v2 D 2 2 v3 i5 i4 4 4 i6 v6 2 port v4 device v5 5 5 3 3 1 4 2 2 3 1 5 6 4 i7 0 3 5 Adding a wire connecting one node from each separate component does not change KVL or KCL equations 7 is a cut set i7 0 1 1 v1 i1 i3 i2 v2 D 2 v3 2 i5 i4 4 4 i6 v6 2 port v4 device v5 5 5 3 3 1 4 2 2 3 1 4 3 6 5 5 Adding a wire connecting one node from each separate component does not change KVL or KCL equations 1 1 i1 v1 i3 i2 v2 2 D v3 2 i5 i4 4 4 i6 v6 2 port v4 device v5 5 5 3 3 1 4 2 2 3 1 4 6 5 3 5 HINGED DIGRAPH 1 2 2 3 3 and 5 are now the same node 1 4 6 Since nodes they can be combined into 3 one node and the redrawn 5 digraph is called a 4 hinged graph 1 e2 1 e1 KCL Equations 2 e3 3 3 2 6 1 2 4 5 3 i1 i2 i6 0 i1 i3 i4 0 i2 i3 i5 0 4 Ai 0 node no 1 1 1 2 1 3 0 Branch no 2 3 4 5 6 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 A i1 i 2 i3 0 i4 0 i5 0 i6 i 0 A is called the reduced Incidence Matrix of the diagraph G relative to datum node 4 1 e2 1 Choose 4 as datum node for digraph G e1 2 e3 3 3 2 6 KCL Equations 4 5 1 2 3 i1 i2 i6 0 i1 i3 i4 0 i2 i3 i5 0 4 Independent KCL Equations Ai 0 Independent KVL Equations node no 1 1 1 2 1 3 0 Branch no 2 3 4 5 6 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 A i1 i 2 i3 0 0 i 4 i5 0 i6 i v1 e1 e2 v1 1 1 0 v 1 0 1 v2 e1 e3 2 e v3 0 1 1 1 v3 e2 e3 e2 v4 e2 v4 0 1 0 e v5 0 0 1 3 v5 e3 e v6 e1 v6 1 0 0 v A T 0 1 e 2 1 choose 6 as datum and let e 1 e 2 e 4 be new node to datum voltages e 1 2 3 KCL Equations 3 2 4 3 5 e 4 1 2 4 4 Independent KCL Equations i 0 A Independent KVL Equations i1 i2 i6 0 i1 i3 i4 0 i4 i5 i6 0 i1 Branch no node no 1 2 3 4 5 6 i2 1 1 1 0 0 0 1 i 3 0 0 2 1 0 1 1 0 0 i 4 4 0 0 0 1 1 1 i 0 5 i6 A 0 i v1 e 1 e2 v1 1 1 0 v 1 0 0 1 e v 2 2 e v3 0 1 0 1 v3 e 2 e 2 v4 e 2 e 4 v4 0 1 1 e v5 0 0 1 4 v5 e 4 v6 e 1 e 4 v6 1 0 1 e v T A Reduced Incidence Matrix A Let G be a connected digraph with n nodes and b branches Pick any node as the datum node and label the remaining nodes arbitrarily from 1 to n 1 Label the branches arbitrarily from 1 to b The reduced incidence matrix A of G is an n 1 x b matrix where each row j corresponds to node j and each column k corresponds to branch k and where the jkth element ajk of 1 ajk 1 0 A is constructed as follow if branch k leaves node j if branch k enters node j if branch k in not connected to node j How to write An Independent System of KCL and KVL Equations Let N be any connected circuit and let the digraph G associated with N contain n nodes and b branches Choose an arbitrary datum node and define the associated node to datum voltage e the branch voltage vector v and the branch current vector i Then we have the vector following system of independent KCL and KVL equations n 1 Independent KCL Equations Ai 0 b Independent KVL Equations T v A e
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