8
1 Device Modeling and Circuit Elements
f R (v, i) = 0, where, differently from the linear resistor in Example 1.1, f R is now
a nonlinear function of voltage v and current i. The nonlinear resistor is said to be
current-controlled (resp., voltage controlled) in the case the CR is given in explicit
form as v = ˆ
v(i) (resp., i = ˆ
i(v)). In the current-controlled case the corresponding
A-pad results to be
F(R) = {( ˆ
v(i 1 (t)), i 1 (t)), ( ˆ
v(i 2 (t)), i 2 (t)), . . . , ( ˆ
v(i n (t)), i n (t)), . . . }.
Since both Examples 1.1 and 1.2 involve the same pair of circuit variables (v, i),
all two-terminal devices R modeled by a circuit element with CR f R (v, i) = 0 are
named resistors.
This notwithstanding, most two-terminal devices cannot be described by a CR
between the variable pair (v(t), i(t)). Important subclasses can be expressed by a
relationship between the variable pairs (q(t), v(t)) or (ϕ(t), i(t)).
Example 1.3 (Nonlinear Capacitor) A subclass of two-terminal devices can be
characterized by an A-pad involving the pair (q(t), v(t)), namely
F(C) = {( ˆ
q(v 1 (t)), v 1 (t)), ( ˆ
q(v 2 (t)), v 2 (t)), . . . , ( ˆ
q(v n (t)), v n (t)), . . . }.
Such F(C) permits to define an ideal circuit element called voltage-controlled
nonlinear capacitor whose CR is given by the algebraic equation f C (q(t), v(t)) =
q(t) − ˆ
q(v(t)) = 0, or q(t) = ˆ
q(v(t)).
Example 1.4 (Nonlinear Inductor) Another important subclass of two-terminal
devices can be characterized by an A-pad involving the electric variable pair
(ϕ(t), i(t)), namely
F(L) = {( ˆ
ϕ(i 1 (t)), i 1 (t)), ( ˆ
ϕ(i 2 (t)), i 2 (t)), . . . , ( ˆ
ϕ(i n (t)), i n (t)), . . . }.
Such F(L) permits to define an ideal circuit element called current-controlled
nonlinear inductor whose CR is given by the algebraic equation f L (ϕ(t), i(t)) =
ϕ(t) − ˆ
ϕ(i(t)) = 0, or ϕ(t) = ˆ
ϕ(i(t)).
The subclass of two-terminal devices characterized in terms of the variable pair
(v, q) (resp., (ϕ, i)) is modeled by a circuit element called capacitor (resp., inductor)
with CR f C (v, q) = 0 (resp., f L (ϕ, i) = 0). For logical consistency, and symmetry
considerations, it is necessary to define a fourth circuit element via the CR
f M (ϕ(t), q(t)) = 0
(1.8)
between the variable pair (ϕ, q). This element was postulated and named the
memristor (acronym for memory-resistor) by Chua in the seminal paper [3]. A
physical device D described by such a circuit element has been fabricated in 2008
as a TiO 2 nanodevice at HP laboratories [4] (see also Chap. 2).
1 Device Modeling and Circuit Elements
f R (v, i) = 0, where, differently from the linear resistor in Example 1.1, f R is now
a nonlinear function of voltage v and current i. The nonlinear resistor is said to be
current-controlled (resp., voltage controlled) in the case the CR is given in explicit
form as v = ˆ
v(i) (resp., i = ˆ
i(v)). In the current-controlled case the corresponding
A-pad results to be
F(R) = {( ˆ
v(i 1 (t)), i 1 (t)), ( ˆ
v(i 2 (t)), i 2 (t)), . . . , ( ˆ
v(i n (t)), i n (t)), . . . }.
Since both Examples 1.1 and 1.2 involve the same pair of circuit variables (v, i),
all two-terminal devices R modeled by a circuit element with CR f R (v, i) = 0 are
named resistors.
This notwithstanding, most two-terminal devices cannot be described by a CR
between the variable pair (v(t), i(t)). Important subclasses can be expressed by a
relationship between the variable pairs (q(t), v(t)) or (ϕ(t), i(t)).
Example 1.3 (Nonlinear Capacitor) A subclass of two-terminal devices can be
characterized by an A-pad involving the pair (q(t), v(t)), namely
F(C) = {( ˆ
q(v 1 (t)), v 1 (t)), ( ˆ
q(v 2 (t)), v 2 (t)), . . . , ( ˆ
q(v n (t)), v n (t)), . . . }.
Such F(C) permits to define an ideal circuit element called voltage-controlled
nonlinear capacitor whose CR is given by the algebraic equation f C (q(t), v(t)) =
q(t) − ˆ
q(v(t)) = 0, or q(t) = ˆ
q(v(t)).
Example 1.4 (Nonlinear Inductor) Another important subclass of two-terminal
devices can be characterized by an A-pad involving the electric variable pair
(ϕ(t), i(t)), namely
F(L) = {( ˆ
ϕ(i 1 (t)), i 1 (t)), ( ˆ
ϕ(i 2 (t)), i 2 (t)), . . . , ( ˆ
ϕ(i n (t)), i n (t)), . . . }.
Such F(L) permits to define an ideal circuit element called current-controlled
nonlinear inductor whose CR is given by the algebraic equation f L (ϕ(t), i(t)) =
ϕ(t) − ˆ
ϕ(i(t)) = 0, or ϕ(t) = ˆ
ϕ(i(t)).
The subclass of two-terminal devices characterized in terms of the variable pair
(v, q) (resp., (ϕ, i)) is modeled by a circuit element called capacitor (resp., inductor)
with CR f C (v, q) = 0 (resp., f L (ϕ, i) = 0). For logical consistency, and symmetry
considerations, it is necessary to define a fourth circuit element via the CR
f M (ϕ(t), q(t)) = 0
(1.8)
between the variable pair (ϕ, q). This element was postulated and named the
memristor (acronym for memory-resistor) by Chua in the seminal paper [3]. A
physical device D described by such a circuit element has been fabricated in 2008
as a TiO 2 nanodevice at HP laboratories [4] (see also Chap. 2).