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5 Flux-Charge Analysis Method of Memristor Circuits
5.7 Analogy Between a Nonlinear RLC Circuit and a
Memristor Circuit
Consider the following analogies between electric quantities in the (v, i)-domain
and (ϕ, q)-domain, respectively:
• v(t) (voltage) is analogous to ϕ(t) (flux or voltage momentum)
• i(t) (current) is analogous to q(t) (charge or current momentum)
The discussion in the previous sections demonstrates that there hold the following analogies between two-terminal elements in the (v, i)-domain and (ϕ, q)domain.
• A capacitor in the (ϕ, q)-domain is analogous to a capacitor in the (v, i)-domain.
It is worth recalling that the model of a capacitor in the (ϕ, q)-domain has a
charge source impressing a charge proportional to the initial condition for the
state variable in the (v, i)-domain.
• An inductor in the (ϕ, q)-domain is analogous to an inductor in the (v, i)-domain.
The model of an inductor in the (ϕ, q)-domain has a flux source impressing a flux
proportional to the initial condition for the state variable in the (v, i)-domain.
• A flux-source (resp., a charge source) in the (ϕ, q)-domain is analogous to a
voltage-source (resp., a current source) in the (v, i)-domain.
• A linear resistor in the (ϕ, q)-domain is analogous to a linear resistor in the (v, i)domain.
Consider a flux-controlled memristor. The CR in the (ϕ, q)-domain is given by
an algebraic relationship between the electric quantities at its terminals ϕ M (t; t 0 )
and q M (t; t 0 ), i.e.,
q M (t; t 0 ) = f (ϕ M (t; t 0 ) + ϕ M 0 ) − f (ϕ M 0 )
.
= ˜
f (ϕ M (t; t 0 ); ϕ M 0 ).
This shows that
• for any fixed ϕ M 0 , a flux-controlled memristor is analogous in the (ϕ, q)-domain
to a voltage-controlled nonlinear resistor in the (v, i)-domain, i.e., an element
defined by an algebraic relationship between voltage and current at its terminals.
It is important to stress that, different from a resistor, a flux-controlled memristor
holds memory of the past history of its voltage through the source ϕ M 0 =
t 0
−∞ v M (τ )dτ associated with its initial condition (see Fig. 5.12). To further
highlight this difference, we depict in Fig. 5.19 the nonlinear characteristic of the
memristor q M (t; t 0 ) = f (ϕ M (t; t 0 )) = ˜
f (ϕ M (t; t 0 ); 0) with ϕ M 0 = 0, and
the characteristic q M (t; t 0 ) = ˜
f (ϕ M (t; t 0 ); ϕ M 0 ) of the memristor charged at an
initial flux ϕ M 0 = 0. Note that the latter is obtained by translating to the point
(ϕ M 0 , f (ϕ M 0 )) the former characteristic. It is clear that the initial condition has a
relevant influence on the shape of the nonlinear characteristic and behavior of the
memristor within a circuit.
5 Flux-Charge Analysis Method of Memristor Circuits
5.7 Analogy Between a Nonlinear RLC Circuit and a
Memristor Circuit
Consider the following analogies between electric quantities in the (v, i)-domain
and (ϕ, q)-domain, respectively:
• v(t) (voltage) is analogous to ϕ(t) (flux or voltage momentum)
• i(t) (current) is analogous to q(t) (charge or current momentum)
The discussion in the previous sections demonstrates that there hold the following analogies between two-terminal elements in the (v, i)-domain and (ϕ, q)domain.
• A capacitor in the (ϕ, q)-domain is analogous to a capacitor in the (v, i)-domain.
It is worth recalling that the model of a capacitor in the (ϕ, q)-domain has a
charge source impressing a charge proportional to the initial condition for the
state variable in the (v, i)-domain.
• An inductor in the (ϕ, q)-domain is analogous to an inductor in the (v, i)-domain.
The model of an inductor in the (ϕ, q)-domain has a flux source impressing a flux
proportional to the initial condition for the state variable in the (v, i)-domain.
• A flux-source (resp., a charge source) in the (ϕ, q)-domain is analogous to a
voltage-source (resp., a current source) in the (v, i)-domain.
• A linear resistor in the (ϕ, q)-domain is analogous to a linear resistor in the (v, i)domain.
Consider a flux-controlled memristor. The CR in the (ϕ, q)-domain is given by
an algebraic relationship between the electric quantities at its terminals ϕ M (t; t 0 )
and q M (t; t 0 ), i.e.,
q M (t; t 0 ) = f (ϕ M (t; t 0 ) + ϕ M 0 ) − f (ϕ M 0 )
.
= ˜
f (ϕ M (t; t 0 ); ϕ M 0 ).
This shows that
• for any fixed ϕ M 0 , a flux-controlled memristor is analogous in the (ϕ, q)-domain
to a voltage-controlled nonlinear resistor in the (v, i)-domain, i.e., an element
defined by an algebraic relationship between voltage and current at its terminals.
It is important to stress that, different from a resistor, a flux-controlled memristor
holds memory of the past history of its voltage through the source ϕ M 0 =
t 0
−∞ v M (τ )dτ associated with its initial condition (see Fig. 5.12). To further
highlight this difference, we depict in Fig. 5.19 the nonlinear characteristic of the
memristor q M (t; t 0 ) = f (ϕ M (t; t 0 )) = ˜
f (ϕ M (t; t 0 ); 0) with ϕ M 0 = 0, and
the characteristic q M (t; t 0 ) = ˜
f (ϕ M (t; t 0 ); ϕ M 0 ) of the memristor charged at an
initial flux ϕ M 0 = 0. Note that the latter is obtained by translating to the point
(ϕ M 0 , f (ϕ M 0 )) the former characteristic. It is clear that the initial condition has a
relevant influence on the shape of the nonlinear characteristic and behavior of the
memristor within a circuit.