100
3 RLC Networks Equations and Analysis Methods
capacitors and inductors. The chapter then addresses the relevant issue of how to
write a state equation (SE) representation for a given dynamic network, pointing
out that networks that do not admit an SE representation might be ill-defined
from a mathematical and a physical viewpoint. Conditions ensuring the existence
of the SEs are provided. Since they are couched in topological form, they can
often be easily checked by inspection. The discussion and application examples
refer to networks containing two-terminal elements as (possibly) nonlinear resistors,
inductors, and capacitors, in addition to independent voltage or current sources.
Such networks are referred to in the following as RLC networks. All the results on
graphs and on SEs may be extended also to networks containing multiterminal or
multiport resistors (e.g., ideal operational amplifiers or resistive controlled sources),
capacitors, and inductors.
3.1 Introduction to Kirchhoff Laws
The laws governing the interactions among the circuit elements in a network N are
the two Kirchhoff laws, which are briefly introduced by means of the following
simple topological concepts:
• a node is the connecting point of at least two terminals of distinct circuit elements
• a branch is associated with each two-terminal element (a.k.a. one-port) connected
between two nodes
• a loop is any closed path starting from any node, passing over different branches
and nodes and ending at the same node, where just only two branches are incident
with each node.
It follows that a branch is represented by a line with two endpoints and a node
includes at least two endpoints.
The two Kirchhoff laws can be stated as follows:
Definition 3.1 (Kirchhoff Current Law (KCL)) The algebraic sum of all currents
entering (leaving) any node of any network N is zero at all times t.
Definition 3.2 (Kirchhoff Voltage Law (KVL)) The algebraic sum of all voltages
around any loop of a network N is zero at all times t.
Although Kirchhoff laws can be derived from Maxwell’s equations in a more
general circuit model based on the electromagnetic field theory, KCL and KVL are
given as postulates or experimental laws in the spirit of this book. The equations
obtained by applying Kirchhoff laws are independent of the nature of the single
elements, but they depend only on the way in which elements are interconnected.
Their extension to a circuit also including elements with more than two terminals is
straightforward if one is referring to the circuit graph.
For the sake of clarity, let us assume that a network consists of b branches and n
nodes. By using basic concept of graph theory, it is easy to show that KCL and KVL
3 RLC Networks Equations and Analysis Methods
capacitors and inductors. The chapter then addresses the relevant issue of how to
write a state equation (SE) representation for a given dynamic network, pointing
out that networks that do not admit an SE representation might be ill-defined
from a mathematical and a physical viewpoint. Conditions ensuring the existence
of the SEs are provided. Since they are couched in topological form, they can
often be easily checked by inspection. The discussion and application examples
refer to networks containing two-terminal elements as (possibly) nonlinear resistors,
inductors, and capacitors, in addition to independent voltage or current sources.
Such networks are referred to in the following as RLC networks. All the results on
graphs and on SEs may be extended also to networks containing multiterminal or
multiport resistors (e.g., ideal operational amplifiers or resistive controlled sources),
capacitors, and inductors.
3.1 Introduction to Kirchhoff Laws
The laws governing the interactions among the circuit elements in a network N are
the two Kirchhoff laws, which are briefly introduced by means of the following
simple topological concepts:
• a node is the connecting point of at least two terminals of distinct circuit elements
• a branch is associated with each two-terminal element (a.k.a. one-port) connected
between two nodes
• a loop is any closed path starting from any node, passing over different branches
and nodes and ending at the same node, where just only two branches are incident
with each node.
It follows that a branch is represented by a line with two endpoints and a node
includes at least two endpoints.
The two Kirchhoff laws can be stated as follows:
Definition 3.1 (Kirchhoff Current Law (KCL)) The algebraic sum of all currents
entering (leaving) any node of any network N is zero at all times t.
Definition 3.2 (Kirchhoff Voltage Law (KVL)) The algebraic sum of all voltages
around any loop of a network N is zero at all times t.
Although Kirchhoff laws can be derived from Maxwell’s equations in a more
general circuit model based on the electromagnetic field theory, KCL and KVL are
given as postulates or experimental laws in the spirit of this book. The equations
obtained by applying Kirchhoff laws are independent of the nature of the single
elements, but they depend only on the way in which elements are interconnected.
Their extension to a circuit also including elements with more than two terminals is
straightforward if one is referring to the circuit graph.
For the sake of clarity, let us assume that a network consists of b branches and n
nodes. By using basic concept of graph theory, it is easy to show that KCL and KVL