Complexity

Research Article

Duality of Complex Systems Built from Higher-Order Elements

Table 1

Dual terms. The symbols F, F₁, and F₂ denote the general constitutive relations (F = implicit, F₁ = for current-controlled elements, and F₂ = for voltage-controlled elements).


System A ↔ System A′

Short circuit: v^(α) = 0	Open circuit: i^(α) = 0
Independent v^(α) source	Independent i^(α) source
Resistor: v = v⁽⁰⁾, i = i⁽⁰⁾	Conductor: v = v⁽⁰⁾, i = i⁽⁰⁾
F(v,i) = 0 or	F(i,v) = 0 or
v = F₁(i) or	i = F₁(v) or
i = F₂(v)	v = F₂(i)
Capacitor: v = v⁽⁰⁾, q = i⁽⁻¹⁾	Inductor: φ = v⁽⁻¹⁾, i = i⁽⁰⁾
F(v,q) = 0 or	F(i,φ) = 0 or
v = F₁(q) or	i = F₁(φ) or
q = F₂(v)	φ = F₂(i)
Memristor: φ = v⁽⁻¹⁾, q = i⁽⁻¹⁾	Memductor: φ = v⁽⁻¹⁾, q = i⁽⁻¹⁾
F(φ,q) = 0 or	F(q,v) = 0 or
φ = F₁(q) or	q = F₁(φ) or
q = F₂(φ)	φ = F₂(q)
…	…
(α,β) HOE: v^(α), i^(β)	(β,α) HOE: v^(β), i^(α)
F(v^(α), i^(β)) = 0 or	F(i^(α), v^(β)) = 0 or
v ^(α) = F₁(i^(β)) or	i^(α) = F₁(v^(β)) or
i^(β) = F₂(v^(α))	v^(β) = F₂(i^(α))