Chapter 3
RLC Networks Equations and Analysis
Methods
Let us consider a circuit 1 N made of an arbitrary interconnection of (two-terminal)
circuit elements. In general, analysis and design of a circuit require to solve a
network problem where numerous mathematical (circuit) variables are established
and then a set of equations is generated describing the behavior of the composite
network. Fundamental principles of Circuit Theory permit to show that circuit
equations depend on (a) the structure of the network and (b) the properties of
the devices which it contains. If the network contains nonlinear elements, then the
equations are usually more difficult to solve.
A graph is an abstraction of a physically implemented network, i.e., it represents
the structure or skeleton of the network. Graph Theory is therefore expected to be
valuable in the network analysis. In particular, the general results of graph theory
provide the theoretical basis to produce circuit equations in the simplest form.
Since the choice of the method should be based on mathematical convenience, a
method yielding equations in the simplest (normal) form is indeed highly desirable.
In the following, the laws of Kirchhoff with some fundamental concepts of graph
theory are summarized in order to show how many electrical variables and how
many circuit equations are needed to fully describe a network N. Several different
methods for deriving the set of circuit equations are available in the literature.
It is well known that methods based on classical dynamics (e.g., Lagrange and
Hamiltonian principles) yield equations similar to those derived by Kirchhoff laws.
The chapter starts reviewing Kirchhoff current and voltage laws and then, after
discussing the basics of graph theory, arrives at introducing the main forms of
tableau equations to describe dynamic circuits, i.e., circuits containing inductors
and capacitors in addition to resistive elements. Tableau equations include in general
linear algebraic equations corresponding to Kirchhoff laws and dynamic equations
(given in integral or differential forms) representing the constitutive relations of
1 In many instances the term network is also used in literature. Hereinafter, “circuit” and “network”
are used interchangeably.
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_3
99
RLC Networks Equations and Analysis
Methods
Let us consider a circuit 1 N made of an arbitrary interconnection of (two-terminal)
circuit elements. In general, analysis and design of a circuit require to solve a
network problem where numerous mathematical (circuit) variables are established
and then a set of equations is generated describing the behavior of the composite
network. Fundamental principles of Circuit Theory permit to show that circuit
equations depend on (a) the structure of the network and (b) the properties of
the devices which it contains. If the network contains nonlinear elements, then the
equations are usually more difficult to solve.
A graph is an abstraction of a physically implemented network, i.e., it represents
the structure or skeleton of the network. Graph Theory is therefore expected to be
valuable in the network analysis. In particular, the general results of graph theory
provide the theoretical basis to produce circuit equations in the simplest form.
Since the choice of the method should be based on mathematical convenience, a
method yielding equations in the simplest (normal) form is indeed highly desirable.
In the following, the laws of Kirchhoff with some fundamental concepts of graph
theory are summarized in order to show how many electrical variables and how
many circuit equations are needed to fully describe a network N. Several different
methods for deriving the set of circuit equations are available in the literature.
It is well known that methods based on classical dynamics (e.g., Lagrange and
Hamiltonian principles) yield equations similar to those derived by Kirchhoff laws.
The chapter starts reviewing Kirchhoff current and voltage laws and then, after
discussing the basics of graph theory, arrives at introducing the main forms of
tableau equations to describe dynamic circuits, i.e., circuits containing inductors
and capacitors in addition to resistive elements. Tableau equations include in general
linear algebraic equations corresponding to Kirchhoff laws and dynamic equations
(given in integral or differential forms) representing the constitutive relations of
1 In many instances the term network is also used in literature. Hereinafter, “circuit” and “network”
are used interchangeably.
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_3
99