114
3 RLC Networks Equations and Analysis Methods
(e.g., a sinusoidal source) we need to consider the class of non-autonomous SEs in
normal form
˙
x = f(x, t),
t ≥ t 0
with the initial condition
x(t 0 ) ∈ R
n
where now the vector field f : R n+1 → R n has an explicit dependence on time t.
There are some important reasons for writing the SEs of a dynamic circuit.
Since it is usually impossible to solve the dynamic equations of a nonlinear circuit
explicitly, the interest is in the study of the qualitative properties of its solutions
(existence, uniqueness and boundedness of solutions, equilibrium points, stability,
oscillations, and complex dynamics). It is well known that there are effective tools
and a well-developed mathematical theory for studying the qualitative properties of
SEs in the normal form (3.43). It is remarkable for example that it is usually possible
to provide simple and easily checkable conditions on the vector field to guarantee
that, given any initial state, there exists a unique solution of the IVP (3.43) and (3.44)
on some time interval. More physically, this means that the knowledge of the initial
condition, the structure of the circuit, and the applied sources guarantees a unique
evolution of the state for t > t 0 . Another important reason is that any properly
modeled circuits have a well-defined SE. In fact, as we will discuss in more detail in
Chap. 4, circuits that do not admit an SE representation 3 may be bad modeled from
a physical viewpoint due to the presence of singular points, named impasse points,
where the solutions cannot be prolonged forward or backward in time. Finally, it is
noted that most numerical methods for solving nonlinear differential equations are
formulated in terms of the standard form (3.43).
Several techniques are available to write the SEs of a dynamic circuit. For simple
low-dimensional circuits it is often possible to proceed by inspection [1]. Otherwise,
one can first write the DAEs describing the tableau, the node, or the loop equations
and then try to cast them by substitution into an SE (cf. Example 3.4). Next, we
discuss a systematic and effective technique where use is made of the decomposition
of the network into a dynamic part and a resistive part. The hybrid representation of
a multiport resistive network obtained via this decomposition is then used to write
the SEs. For simplicity, in the remainder of this chapter we suppose that all elements,
except possibly the voltage and current sources, are time-invariant. The extension
to the time-varying case is straightforward. For didactic reasons, we start with the
discussion of linear RLC networks and then address the case of nonlinear RLC
networks.
3 We have already encountered one such circuit in Example 3.4.
3 RLC Networks Equations and Analysis Methods
(e.g., a sinusoidal source) we need to consider the class of non-autonomous SEs in
normal form
˙
x = f(x, t),
t ≥ t 0
with the initial condition
x(t 0 ) ∈ R
n
where now the vector field f : R n+1 → R n has an explicit dependence on time t.
There are some important reasons for writing the SEs of a dynamic circuit.
Since it is usually impossible to solve the dynamic equations of a nonlinear circuit
explicitly, the interest is in the study of the qualitative properties of its solutions
(existence, uniqueness and boundedness of solutions, equilibrium points, stability,
oscillations, and complex dynamics). It is well known that there are effective tools
and a well-developed mathematical theory for studying the qualitative properties of
SEs in the normal form (3.43). It is remarkable for example that it is usually possible
to provide simple and easily checkable conditions on the vector field to guarantee
that, given any initial state, there exists a unique solution of the IVP (3.43) and (3.44)
on some time interval. More physically, this means that the knowledge of the initial
condition, the structure of the circuit, and the applied sources guarantees a unique
evolution of the state for t > t 0 . Another important reason is that any properly
modeled circuits have a well-defined SE. In fact, as we will discuss in more detail in
Chap. 4, circuits that do not admit an SE representation 3 may be bad modeled from
a physical viewpoint due to the presence of singular points, named impasse points,
where the solutions cannot be prolonged forward or backward in time. Finally, it is
noted that most numerical methods for solving nonlinear differential equations are
formulated in terms of the standard form (3.43).
Several techniques are available to write the SEs of a dynamic circuit. For simple
low-dimensional circuits it is often possible to proceed by inspection [1]. Otherwise,
one can first write the DAEs describing the tableau, the node, or the loop equations
and then try to cast them by substitution into an SE (cf. Example 3.4). Next, we
discuss a systematic and effective technique where use is made of the decomposition
of the network into a dynamic part and a resistive part. The hybrid representation of
a multiport resistive network obtained via this decomposition is then used to write
the SEs. For simplicity, in the remainder of this chapter we suppose that all elements,
except possibly the voltage and current sources, are time-invariant. The extension
to the time-varying case is straightforward. For didactic reasons, we start with the
discussion of linear RLC networks and then address the case of nonlinear RLC
networks.
3 We have already encountered one such circuit in Example 3.4.