11.4 Constitutive Relations of Two-Terminal Elements in LME
399
– n ρ meminductors ML ρ
2. LME ⊂ N e is obtained interconnecting N R to the following nonlinear
dynamical two-terminal elements D :
– n γ μ flux-controlled memristors M ϕ (each having in parallel a chargecontrolled nonlinear capacitor C q or a σ -controlled memcapacitor MC σ )
– n λμ charge-controlled memristors M q (each having in series a flux-controlled
nonlinear inductor L ϕ or a ρ-controlled meminductor ML ρ )
– n γ charge-controlled nonlinear capacitors C q
– n λ flux-controlled nonlinear inductors L ϕ
– n σ memcapacitors MC σ
– n ρ meminductors ML ρ .
The subclasses LME and LME include several mem-circuits 3 studied in the
literature. Henceforth, we focus on developing a systematic approach for the SEs
in the (ϕ, q)-domain of LME. 4 To this end, the CRs of any nonlinear dynamical
two-terminal element D ∈ {M ϕ , M q , C, L, MC σ , ML ρ } in LME are derived in the
next section and used to write the SEs in Sect. 11.5. We then discuss in Sect. 11.7
via a number of specific examples how we can write the SEs for circuits in the class
LME .
11.4 Constitutive Relations of Two-Terminal Elements in
LME
In this section, the CRs are written both in the (ϕ, q)-domain and in the (v, i)domain, with the aim of identifying the state variable(s) associated with D and
the element order in both domains. The “element order” of D in a given (v α , i β )domain (denoted by O
(α,β)
D ) is defined as the number of differential/integral
operators appearing in its CR. By comparing state variables and element order, it
is possible to ascertain the reduction of O
(α,β)
D
in the CRs passing from the (v, i)domain to the (ϕ, q)-domain. In the following, the CRs of M ϕ and M q are briefly
summarized, whereas the CRs of MC σ and ML ρ and their possible equivalent
circuit representations in the (ϕ, q)-domain are presented in detail. For the CRs and
equivalent circuit in the (ϕ, q)-domain of linear resistors R, capacitors C, inductors
L, and independent sources, we refer the reader to Chap. 5.
3 Hereinafter, nonlinear dynamical circuits with mem-elements in LME are named mem-circuits.
4 Note that LM ⊂ LME ⊂ N e .
399
– n ρ meminductors ML ρ
2. LME ⊂ N e is obtained interconnecting N R to the following nonlinear
dynamical two-terminal elements D :
– n γ μ flux-controlled memristors M ϕ (each having in parallel a chargecontrolled nonlinear capacitor C q or a σ -controlled memcapacitor MC σ )
– n λμ charge-controlled memristors M q (each having in series a flux-controlled
nonlinear inductor L ϕ or a ρ-controlled meminductor ML ρ )
– n γ charge-controlled nonlinear capacitors C q
– n λ flux-controlled nonlinear inductors L ϕ
– n σ memcapacitors MC σ
– n ρ meminductors ML ρ .
The subclasses LME and LME include several mem-circuits 3 studied in the
literature. Henceforth, we focus on developing a systematic approach for the SEs
in the (ϕ, q)-domain of LME. 4 To this end, the CRs of any nonlinear dynamical
two-terminal element D ∈ {M ϕ , M q , C, L, MC σ , ML ρ } in LME are derived in the
next section and used to write the SEs in Sect. 11.5. We then discuss in Sect. 11.7
via a number of specific examples how we can write the SEs for circuits in the class
LME .
11.4 Constitutive Relations of Two-Terminal Elements in
LME
In this section, the CRs are written both in the (ϕ, q)-domain and in the (v, i)domain, with the aim of identifying the state variable(s) associated with D and
the element order in both domains. The “element order” of D in a given (v α , i β )domain (denoted by O
(α,β)
D ) is defined as the number of differential/integral
operators appearing in its CR. By comparing state variables and element order, it
is possible to ascertain the reduction of O
(α,β)
D
in the CRs passing from the (v, i)domain to the (ϕ, q)-domain. In the following, the CRs of M ϕ and M q are briefly
summarized, whereas the CRs of MC σ and ML ρ and their possible equivalent
circuit representations in the (ϕ, q)-domain are presented in detail. For the CRs and
equivalent circuit in the (ϕ, q)-domain of linear resistors R, capacitors C, inductors
L, and independent sources, we refer the reader to Chap. 5.
3 Hereinafter, nonlinear dynamical circuits with mem-elements in LME are named mem-circuits.
4 Note that LM ⊂ LME ⊂ N e .