3.1 Introduction to Kirchhoff Laws
101
provide (n − 1) node equations and b − (n − 1) loop equations that are independent
of each other. Thus, overall Kirchhoff laws allow to write b independent linear
algebraic equations in terms of the 2b unknowns given by the branch variables,
i.e., the branch currents i 1 , i 2 , . . . , i b and the branch voltages v 1 , v 2 , . . . , v b .
By joining the b equations derived from the constitutive relations (CRs) associated with the b two-terminal circuit elements in the network and the independent b
equations obtained applying Kirchhoff laws we obtain a system of 2b equations in
2b unknowns. It is apparent that such a system is linear/nonlinear algebraic/dynamic
according to the mathematical properties of the CRs. The circuit analysis methods
derived from this approach are briefly discussed in the next Sect. 3.2. Before
presenting circuit analysis methods it is convenient to introduce a compact matrix
formulation of KCLs and KVLs via basic notions of graph theory.
3.1.1 Basics of Graph Theory
The interconnection properties of a network are fully described by its graph,
according to the next definition.
Definition 3.3 (Graph of a Network) A graph G is specified by the set of n nodes
{ 1 , 2 , . . . , n } together with a set of b branches {β 1 , β 2 , . . . , β b }. If each branch
is given an orientation, then G is a directed graph (or digraph).
Hereinafter the associated reference direction is considered for each branch,
i.e., the branch current is assumed to go from the terminal “+” to the terminal
“−” associated with the branch voltage. 2 In addition, each branch in G is oriented
according to the branch current. The direction of a branch is indicated by an arrow
in the same direction of the positive branch current. When we are not interested
with the reference directions of branch voltages and currents, all the arrows in G
may be removed. Such simpler graph is called undirected graph associated with the
network N .
Example 3.1 (Directed Graph G) Figure 3.1 shows a network N and its associated
directed graph G with n = 4 nodes and b = 6 branches.
We assume that G is a (directed or undirected) connected graph, that is, there
exists a path between any two nodes of the graph. The path between two nodes i
and j is defined as a set of p branches {β 1 , β 2 , . . . , β p } such that: (a) consecutive
branches β i and β i+1 always have a common end point; (b) no node is the endpoint
of more than two branches in the set; (c) i is the endpoint of exactly one branch in
the set, and so is j .
2 Sometimes, for drawing convenience, in the subsequent chapters the reference direction for the
branch voltage is also denoted by an arrow, where the arrow-head corresponds to the “+” terminal
and the arrow-end to the “−” terminal.
101
provide (n − 1) node equations and b − (n − 1) loop equations that are independent
of each other. Thus, overall Kirchhoff laws allow to write b independent linear
algebraic equations in terms of the 2b unknowns given by the branch variables,
i.e., the branch currents i 1 , i 2 , . . . , i b and the branch voltages v 1 , v 2 , . . . , v b .
By joining the b equations derived from the constitutive relations (CRs) associated with the b two-terminal circuit elements in the network and the independent b
equations obtained applying Kirchhoff laws we obtain a system of 2b equations in
2b unknowns. It is apparent that such a system is linear/nonlinear algebraic/dynamic
according to the mathematical properties of the CRs. The circuit analysis methods
derived from this approach are briefly discussed in the next Sect. 3.2. Before
presenting circuit analysis methods it is convenient to introduce a compact matrix
formulation of KCLs and KVLs via basic notions of graph theory.
3.1.1 Basics of Graph Theory
The interconnection properties of a network are fully described by its graph,
according to the next definition.
Definition 3.3 (Graph of a Network) A graph G is specified by the set of n nodes
{ 1 , 2 , . . . , n } together with a set of b branches {β 1 , β 2 , . . . , β b }. If each branch
is given an orientation, then G is a directed graph (or digraph).
Hereinafter the associated reference direction is considered for each branch,
i.e., the branch current is assumed to go from the terminal “+” to the terminal
“−” associated with the branch voltage. 2 In addition, each branch in G is oriented
according to the branch current. The direction of a branch is indicated by an arrow
in the same direction of the positive branch current. When we are not interested
with the reference directions of branch voltages and currents, all the arrows in G
may be removed. Such simpler graph is called undirected graph associated with the
network N .
Example 3.1 (Directed Graph G) Figure 3.1 shows a network N and its associated
directed graph G with n = 4 nodes and b = 6 branches.
We assume that G is a (directed or undirected) connected graph, that is, there
exists a path between any two nodes of the graph. The path between two nodes i
and j is defined as a set of p branches {β 1 , β 2 , . . . , β p } such that: (a) consecutive
branches β i and β i+1 always have a common end point; (b) no node is the endpoint
of more than two branches in the set; (c) i is the endpoint of exactly one branch in
the set, and so is j .
2 Sometimes, for drawing convenience, in the subsequent chapters the reference direction for the
branch voltage is also denoted by an arrow, where the arrow-head corresponds to the “+” terminal
and the arrow-end to the “−” terminal.