104
3 RLC Networks Equations and Analysis Methods
Theorem 3.1 (Fundamental Theorem of Graphs) Given a connected digraph G
with n nodes and b branches and a tree T of G:
1. There is a unique path (dismiss the branch orientation) along the tree between
any pairs of nodes.
2. There are n − 1 twigs and l = b − (n − 1) links.
3. Every twig of T together with some links defines a unique cut-set, named the
fundamental cut-set associated with the twig.
4. Every link of T and the unique path on the tree between its two nodes constitute
a unique loop, named the fundamental loop associated with the link.
The proof can be found in Section 3 of Chapter 12 in [1]. These theoretical
results allow us to introduce the uniquely defined reduced cut-set and loop matrices
associated with a tree, thereby paving the way for expressing Kirchhoff laws and
developing cut-set analysis and loop analysis in matrix-vector form.
3.1.2 The Fundamental Cut-Set Matrix Q Associated with a
Tree
Let us consider a connected digraph G with n nodes and b branches and single
out a tree T . Theorem 3.1 guarantees that there exist n − 1 twigs and each twig
identifies a unique cut-set. Hence, n−1 KCL equations can be derived from the n−1
fundamental cut-sets. Certainly, these n − 1 KCL equations are linearly independent
because each equation has one and only one twig current. In matrix form the KCL
equations based on the fundamental cut-sets of a tree can be written as:
Qi = 0
(3.4)
where Q is an (n − 1) × b matrix called the fundamental cut-set matrix associated
with a tree T and i = (i 1 , . . . , i b ) T is the b×1 column vector (the symbol T denotes
the transpose operator) of branch currents. The jk-th entry of Q is defined as:
q jk =
⎧
⎨
⎩
+1 if branch k belongs to cut-set j and has the same direction
−1 if branch k belongs to cut-set j and has the opposite direction
0 if branch k does not belong to cut-set j .
(3.5)
Here, the direction of each cut-set j is defined as the direction of its associated
twig. By labeling the b branches of the graph G in such a way that first there
are the twigs, i.e., 1, 2, . . . , n − 1, and then the links, i.e., n, n + 1, . . . , b, the
fundamental cut-set matrix associated with the corresponding tree T can be written
in a convenient form as follows:
Q = [I n−1 , Q l ]
(3.6)
3 RLC Networks Equations and Analysis Methods
Theorem 3.1 (Fundamental Theorem of Graphs) Given a connected digraph G
with n nodes and b branches and a tree T of G:
1. There is a unique path (dismiss the branch orientation) along the tree between
any pairs of nodes.
2. There are n − 1 twigs and l = b − (n − 1) links.
3. Every twig of T together with some links defines a unique cut-set, named the
fundamental cut-set associated with the twig.
4. Every link of T and the unique path on the tree between its two nodes constitute
a unique loop, named the fundamental loop associated with the link.
The proof can be found in Section 3 of Chapter 12 in [1]. These theoretical
results allow us to introduce the uniquely defined reduced cut-set and loop matrices
associated with a tree, thereby paving the way for expressing Kirchhoff laws and
developing cut-set analysis and loop analysis in matrix-vector form.
3.1.2 The Fundamental Cut-Set Matrix Q Associated with a
Tree
Let us consider a connected digraph G with n nodes and b branches and single
out a tree T . Theorem 3.1 guarantees that there exist n − 1 twigs and each twig
identifies a unique cut-set. Hence, n−1 KCL equations can be derived from the n−1
fundamental cut-sets. Certainly, these n − 1 KCL equations are linearly independent
because each equation has one and only one twig current. In matrix form the KCL
equations based on the fundamental cut-sets of a tree can be written as:
Qi = 0
(3.4)
where Q is an (n − 1) × b matrix called the fundamental cut-set matrix associated
with a tree T and i = (i 1 , . . . , i b ) T is the b×1 column vector (the symbol T denotes
the transpose operator) of branch currents. The jk-th entry of Q is defined as:
q jk =
⎧
⎨
⎩
+1 if branch k belongs to cut-set j and has the same direction
−1 if branch k belongs to cut-set j and has the opposite direction
0 if branch k does not belong to cut-set j .
(3.5)
Here, the direction of each cut-set j is defined as the direction of its associated
twig. By labeling the b branches of the graph G in such a way that first there
are the twigs, i.e., 1, 2, . . . , n − 1, and then the links, i.e., n, n + 1, . . . , b, the
fundamental cut-set matrix associated with the corresponding tree T can be written
in a convenient form as follows:
Q = [I n−1 , Q l ]
(3.6)