3.1 Introduction to Kirchhoff Laws
105
where I n−1 is the (n−1)×(n−1) identity matrix and Q l is a submatrix of n−1 rows
and l columns with entries 0, 1, and −1. Clearly, the vector of the branch currents i
can be decomposed in the corresponding i t = (i 1 , i 2 , . . . , i n−1 ) T twig currents and
i l = (i n , i n+1 , . . . , i b ) T link currents. As a consequence, the n − 1 KCL equations
can be written as follows:
i t = −Q l i l .
(3.7)
The corresponding decomposition of the branch voltage vector v =
(v 1 , v 2 , . . . , v b ) T into the twig voltage vector v t = (v 1 , v 2 , . . . , v n−1 ) T and link
voltage vector v l = (v n , v n+1 , . . . , v b ) T permits to write the KVL equations in the
form
v = Q
T v t
(3.8)
from which it is easily derived that
v l = Q
T
l v t .
(3.9)
Equations (3.4) and (3.8) provide the matrix form of Kirchhoff equations based
on the fundamental cut-set matrix Q.
3.1.3 The Incidence Matrix A
Any cut-set separates the set of nodes { 1 , 2 , . . . , n } of G into two subsets.
Writing the KCL at each node in such subsets, and adding the results, we obtain
the cut-set equations. In general, Kirchhoff equations (3.4) and (3.8) represent a
generalization of KCL and KVL based on the nodes and the incidence matrix. It can
be shown that for many digraphs, a tree T can be picked in such a way that Q is
identical to the reduced incident matrix for a particular datum node.
Suppose that for the connected digraph G, we write the n KCL equations for
each node. Then we choose a datum node (e.g., node n ) and we throw away
the corresponding KCL equation. The remaining n − 1 KCL equations at nodes
1 , 2 , . . . , n-1 are linearly independent and they can be written in matrix form by
means of the reduced incident matrix A which is of dimension (n − 1) × b. The
corresponding n − 1 KCLs read
Ai = 0
(3.10)
where the elements of A are specified as follows:
105
where I n−1 is the (n−1)×(n−1) identity matrix and Q l is a submatrix of n−1 rows
and l columns with entries 0, 1, and −1. Clearly, the vector of the branch currents i
can be decomposed in the corresponding i t = (i 1 , i 2 , . . . , i n−1 ) T twig currents and
i l = (i n , i n+1 , . . . , i b ) T link currents. As a consequence, the n − 1 KCL equations
can be written as follows:
i t = −Q l i l .
(3.7)
The corresponding decomposition of the branch voltage vector v =
(v 1 , v 2 , . . . , v b ) T into the twig voltage vector v t = (v 1 , v 2 , . . . , v n−1 ) T and link
voltage vector v l = (v n , v n+1 , . . . , v b ) T permits to write the KVL equations in the
form
v = Q
T v t
(3.8)
from which it is easily derived that
v l = Q
T
l v t .
(3.9)
Equations (3.4) and (3.8) provide the matrix form of Kirchhoff equations based
on the fundamental cut-set matrix Q.
3.1.3 The Incidence Matrix A
Any cut-set separates the set of nodes { 1 , 2 , . . . , n } of G into two subsets.
Writing the KCL at each node in such subsets, and adding the results, we obtain
the cut-set equations. In general, Kirchhoff equations (3.4) and (3.8) represent a
generalization of KCL and KVL based on the nodes and the incidence matrix. It can
be shown that for many digraphs, a tree T can be picked in such a way that Q is
identical to the reduced incident matrix for a particular datum node.
Suppose that for the connected digraph G, we write the n KCL equations for
each node. Then we choose a datum node (e.g., node n ) and we throw away
the corresponding KCL equation. The remaining n − 1 KCL equations at nodes
1 , 2 , . . . , n-1 are linearly independent and they can be written in matrix form by
means of the reduced incident matrix A which is of dimension (n − 1) × b. The
corresponding n − 1 KCLs read
Ai = 0
(3.10)
where the elements of A are specified as follows: