142
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
to a small, medium, and large value of are depicted in Fig. 4.10. It is worth to
stress that, in each case, all trajectories tend to a unique periodic solution (a stable
limit cycle). This can be rigorously shown using Poincaré–Bendixson theorem [5].
For small (say < 0.2), the limit cycle is approximately a smooth ellipse, and
the waveforms of v C (t) and i L (t) are approximately sinusoidal (see Fig. 4.10 for
= 0.1). For medium values of (say 0.3 < < < 4), the limit cycle becomes
distorted as shown in the same figure for = 3. In this case, the waveforms v C (t)
and i L (t) are no longer sinusoidal and do not admit of even an approximate closed
form expression. For large (say > 20), the limit cycle is seen to cycling closely
to the curve v C = −i L +
1
3 i 3
L , except at the corners, where it becomes nearly
vertical (cf. Fig. 4.10 for = 30). The corresponding waveforms for v C (t) and i L (t)
consist of nearly instantaneous transition from the lower branch to the upper branch,
and vice versa, a peculiar feature also named jump phenomenon. The oscillation
waveforms of v C (t) and i L (t) are far from being sinusoidal. Such oscillators are
named relaxation oscillators.
4.2.2 How to Break Impasse Points
The impasse points in the circuit in Fig. 4.3 of Example 4.2 can be removed when
taking into account suitable parasitic elements. Indeed, suppose to insert in the
circuit in Fig. 4.3, in series with C (recall that we have chosen a normalized value
C = 1), a small inductance L p modeling for instance the parasitic inductance of
connecting wires, thus obtaining the circuit in Fig. 4.11. Note that this is in the form
of a negative resistance oscillator.
The conditions for the existence of a global SE are now satisfied and the SEs
have been already obtained as
dv C
dt
= −i L p
(4.6)
di L p
dt
=
v C − ˆ
v(i L p )
L p
.
Since ˆ
v(·) is piecewise-linear, the vector field defining the SEs (4.6) satisfies a
Lipschitz condition, hence local existence and uniqueness of the solution for any
initial condition (v C (0), i L p (0)) T are guaranteed [5, Theorem 3.1]. Furthermore, the
norm of the same vector field increases at most linearly with the norm of the state
vector (v C , i L p ) T , hence any solution is defined in the whole interval t ∈ [0, +∞)
[5, Theorem 3.2]. This rules out in particular the presence of impasse points.
Clearly, any value of the inductance L p would rule out the presence of impasse
points. Namely, impasse can be broken also by deliberating inserting an inductance
in series as in Fig. 4.11 without the need to account for parasitic inductances.
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
to a small, medium, and large value of are depicted in Fig. 4.10. It is worth to
stress that, in each case, all trajectories tend to a unique periodic solution (a stable
limit cycle). This can be rigorously shown using Poincaré–Bendixson theorem [5].
For small (say < 0.2), the limit cycle is approximately a smooth ellipse, and
the waveforms of v C (t) and i L (t) are approximately sinusoidal (see Fig. 4.10 for
= 0.1). For medium values of (say 0.3 < < < 4), the limit cycle becomes
distorted as shown in the same figure for = 3. In this case, the waveforms v C (t)
and i L (t) are no longer sinusoidal and do not admit of even an approximate closed
form expression. For large (say > 20), the limit cycle is seen to cycling closely
to the curve v C = −i L +
1
3 i 3
L , except at the corners, where it becomes nearly
vertical (cf. Fig. 4.10 for = 30). The corresponding waveforms for v C (t) and i L (t)
consist of nearly instantaneous transition from the lower branch to the upper branch,
and vice versa, a peculiar feature also named jump phenomenon. The oscillation
waveforms of v C (t) and i L (t) are far from being sinusoidal. Such oscillators are
named relaxation oscillators.
4.2.2 How to Break Impasse Points
The impasse points in the circuit in Fig. 4.3 of Example 4.2 can be removed when
taking into account suitable parasitic elements. Indeed, suppose to insert in the
circuit in Fig. 4.3, in series with C (recall that we have chosen a normalized value
C = 1), a small inductance L p modeling for instance the parasitic inductance of
connecting wires, thus obtaining the circuit in Fig. 4.11. Note that this is in the form
of a negative resistance oscillator.
The conditions for the existence of a global SE are now satisfied and the SEs
have been already obtained as
dv C
dt
= −i L p
(4.6)
di L p
dt
=
v C − ˆ
v(i L p )
L p
.
Since ˆ
v(·) is piecewise-linear, the vector field defining the SEs (4.6) satisfies a
Lipschitz condition, hence local existence and uniqueness of the solution for any
initial condition (v C (0), i L p (0)) T are guaranteed [5, Theorem 3.1]. Furthermore, the
norm of the same vector field increases at most linearly with the norm of the state
vector (v C , i L p ) T , hence any solution is defined in the whole interval t ∈ [0, +∞)
[5, Theorem 3.2]. This rules out in particular the presence of impasse points.
Clearly, any value of the inductance L p would rule out the presence of impasse
points. Namely, impasse can be broken also by deliberating inserting an inductance
in series as in Fig. 4.11 without the need to account for parasitic inductances.