6.2 Second-Order Memristor Oscillators
243
Y 0 >
a/3b
then the unique EP P is asymptotically stable (Fig. 6.17). Moreover, by arguing
like in the previous point, there are two impasse points at P 1 and P 2 = −P 1 .
Now, point P 1 is a backward impasse point (any backward trajectory starting at
( ˆ
X, ˆ
Y ) with ˆ
Y ∈ (Y 0 ,
√
a/3b) ∪ (
√
a/3b, −
√
a/3b) and ˆ
X = h( ˆ
Y ) is stuck at P 1
after a finite backward time), whereas P 2 is a forward impasse point (any forward
trajectory starting at ( ˆ
X, ˆ
Y ) with ˆ
Y ∈ (
√
a/3b, −
√
a/3b) ∪ (−
√
a/3b, −∞) and
ˆ
X = h( ˆ
Y ) is stuck at P 2 after a finite forward time);
3. the case Y 0 < −
√
a/3b can be analyzed, mutatis mutandis, by arguing as in the
previous point. Now, P 1 (resp., P 2 ) is a forward (resp., backward) impasse point,
while P is the only asymptotically stable EP.
Such pathological situation due to the coexistence of equilibria and impasse
points imply that the M–C circuit model in Fig. 6.14 is defective. The circuit
needs to be remodeled by including parasitic inductances and/or capacitances
at appropriate locations in order to characterize relaxation oscillations and jump
phenomena that are observed in experiments with the considered circuit. This issue
will be discussed in the next section.
6.2.2 M–L–C Circuit with Relaxation Oscillations
Any physical (memristor) circuit is of course characterized by a well-defined
dynamic behavior for any t ≥ t 0 . Impasse points as those observed in the M–
C circuit of the previous section represent nonphysical phenomena due to a poor
circuit modeling (Chap. 4). It is known from the approach discussed in Chap. 4 that
the impasse points in the M–C circuit of Fig. 6.14 can be hopefully broken by
introducing a small inductance L (e.g., representing the inductance of connecting
wires) connected in series with the memristor. As a result, the series M–L–C
circuit in the class LM, depicted in Fig. 6.18, is obtained. Suppose the memristor
M is still defined by (6.22) and let v C (t 0 ) = v C 0 , i L (t 0 ) = i L 0 , q M (t 0 ) = q M 0
be the initial conditions at t 0 for the state variables in the (v, i)-domain. Then,
q C (t 0 ) = q C 0 = Cv C 0 and ϕ L (t 0 ) = ϕ L 0 = Li L 0 . The corresponding circuit in
the (ϕ, q)-domain is reported in Fig. 6.19.
6.2.2.1 Formulation of the Circuit Equations
Analysis by inspection of the circuit in Fig. 6.19 permits to write the following KϕL,
KqLs, and CRs of circuit elements
ϕ C (t; t 0 ) = ϕ L (t; t 0 ) + ϕ M (t; t 0 )
(6.28a)
243
Y 0 >
a/3b
then the unique EP P is asymptotically stable (Fig. 6.17). Moreover, by arguing
like in the previous point, there are two impasse points at P 1 and P 2 = −P 1 .
Now, point P 1 is a backward impasse point (any backward trajectory starting at
( ˆ
X, ˆ
Y ) with ˆ
Y ∈ (Y 0 ,
√
a/3b) ∪ (
√
a/3b, −
√
a/3b) and ˆ
X = h( ˆ
Y ) is stuck at P 1
after a finite backward time), whereas P 2 is a forward impasse point (any forward
trajectory starting at ( ˆ
X, ˆ
Y ) with ˆ
Y ∈ (
√
a/3b, −
√
a/3b) ∪ (−
√
a/3b, −∞) and
ˆ
X = h( ˆ
Y ) is stuck at P 2 after a finite forward time);
3. the case Y 0 < −
√
a/3b can be analyzed, mutatis mutandis, by arguing as in the
previous point. Now, P 1 (resp., P 2 ) is a forward (resp., backward) impasse point,
while P is the only asymptotically stable EP.
Such pathological situation due to the coexistence of equilibria and impasse
points imply that the M–C circuit model in Fig. 6.14 is defective. The circuit
needs to be remodeled by including parasitic inductances and/or capacitances
at appropriate locations in order to characterize relaxation oscillations and jump
phenomena that are observed in experiments with the considered circuit. This issue
will be discussed in the next section.
6.2.2 M–L–C Circuit with Relaxation Oscillations
Any physical (memristor) circuit is of course characterized by a well-defined
dynamic behavior for any t ≥ t 0 . Impasse points as those observed in the M–
C circuit of the previous section represent nonphysical phenomena due to a poor
circuit modeling (Chap. 4). It is known from the approach discussed in Chap. 4 that
the impasse points in the M–C circuit of Fig. 6.14 can be hopefully broken by
introducing a small inductance L (e.g., representing the inductance of connecting
wires) connected in series with the memristor. As a result, the series M–L–C
circuit in the class LM, depicted in Fig. 6.18, is obtained. Suppose the memristor
M is still defined by (6.22) and let v C (t 0 ) = v C 0 , i L (t 0 ) = i L 0 , q M (t 0 ) = q M 0
be the initial conditions at t 0 for the state variables in the (v, i)-domain. Then,
q C (t 0 ) = q C 0 = Cv C 0 and ϕ L (t 0 ) = ϕ L 0 = Li L 0 . The corresponding circuit in
the (ϕ, q)-domain is reported in Fig. 6.19.
6.2.2.1 Formulation of the Circuit Equations
Analysis by inspection of the circuit in Fig. 6.19 permits to write the following KϕL,
KqLs, and CRs of circuit elements
ϕ C (t; t 0 ) = ϕ L (t; t 0 ) + ϕ M (t; t 0 )
(6.28a)