1.3 Higher-Order Circuit Elements
23
elements [9, 10], behave as elements from the periodic table. Such resistors with
negative resistances show a quadratic dependence of the small-signal conductance
or resistance on the frequency and they correspond to the (1, −1) and (−1, 1)elements of the periodic table. The article [11] introduced the FDPC (Frequency
Dependent Positive Conductance) whose conductance depends on the fourth power
of frequency and thus corresponds the (2, −2)-element of the periodic table. It is
also worth remarking that the inerter, proposed in 2002 in [12], is the mechanical
analog of the (1, 0) circuit element of the periodic table.
The following properties hold [1].
Property 1.1 (Element Independence Property) The (infinitely many) two-terminal
elements in the Circuit-Element-Array are all independent of each other in the sense
that no element defined by a nonlinear v (α) − i (β) curve can be synthesized by a
combination of any other elements in the array.
Property 1.2 (Element Closure Property) Arbitrary interconnection of elements of
the same type (i.e., same α and β) is equivalent to another element of the same type.
1.3.4 Algebraic and Dynamic Elements
From a circuit-theoretic foundation point of view, it is desirable to classify the
universe of all lumped circuit elements into two mutually exclusive categories.
The first category is that of the algebraic elements and coincides with the (α, β)elements, i.e., the circuit elements in the periodic table. According to Definition 1.2,
a two-terminal element is algebraic if and only if its CR can be expressed by
an algebraic relationship involving at most two dynamically independent electric
variables v (α) and i (β) , where α and β are integers. The second category is that of
the dynamic elements, i.e., all elements that are not algebraic.
Example 1.10 Consider a nonlinear capacitor described by the equation i =
C(v)dv/dt. Such a CR involves three different variables i, v, and v (1) = dv/dt.
However, by recasting the equation in the form q = ˆ
q(v), where we let C(v) =
ˆ
q (v), we conclude that the capacitor needs to be classified as an algebraic (α, β) =
(0, −1)-element.
Example 1.11 (ac Dynamic Hysteresis Model of an Inductor) A realistic model of
a hysteretic iron-core inductor operating under periodic excitations is given by a
two-terminal element with CR
v = g(i − f (v
(−1) ))
where g(·) and f (·) are continuous monotone-increasing functions obtained from
the hysteresis loops [1]. Here the CR involves three terminal variables, i.e.,
{v, i, v (−1) }. In this case, it is not possible to find (α, β) such that the CR relation is
23
elements [9, 10], behave as elements from the periodic table. Such resistors with
negative resistances show a quadratic dependence of the small-signal conductance
or resistance on the frequency and they correspond to the (1, −1) and (−1, 1)elements of the periodic table. The article [11] introduced the FDPC (Frequency
Dependent Positive Conductance) whose conductance depends on the fourth power
of frequency and thus corresponds the (2, −2)-element of the periodic table. It is
also worth remarking that the inerter, proposed in 2002 in [12], is the mechanical
analog of the (1, 0) circuit element of the periodic table.
The following properties hold [1].
Property 1.1 (Element Independence Property) The (infinitely many) two-terminal
elements in the Circuit-Element-Array are all independent of each other in the sense
that no element defined by a nonlinear v (α) − i (β) curve can be synthesized by a
combination of any other elements in the array.
Property 1.2 (Element Closure Property) Arbitrary interconnection of elements of
the same type (i.e., same α and β) is equivalent to another element of the same type.
1.3.4 Algebraic and Dynamic Elements
From a circuit-theoretic foundation point of view, it is desirable to classify the
universe of all lumped circuit elements into two mutually exclusive categories.
The first category is that of the algebraic elements and coincides with the (α, β)elements, i.e., the circuit elements in the periodic table. According to Definition 1.2,
a two-terminal element is algebraic if and only if its CR can be expressed by
an algebraic relationship involving at most two dynamically independent electric
variables v (α) and i (β) , where α and β are integers. The second category is that of
the dynamic elements, i.e., all elements that are not algebraic.
Example 1.10 Consider a nonlinear capacitor described by the equation i =
C(v)dv/dt. Such a CR involves three different variables i, v, and v (1) = dv/dt.
However, by recasting the equation in the form q = ˆ
q(v), where we let C(v) =
ˆ
q (v), we conclude that the capacitor needs to be classified as an algebraic (α, β) =
(0, −1)-element.
Example 1.11 (ac Dynamic Hysteresis Model of an Inductor) A realistic model of
a hysteretic iron-core inductor operating under periodic excitations is given by a
two-terminal element with CR
v = g(i − f (v
(−1) ))
where g(·) and f (·) are continuous monotone-increasing functions obtained from
the hysteresis loops [1]. Here the CR involves three terminal variables, i.e.,
{v, i, v (−1) }. In this case, it is not possible to find (α, β) such that the CR relation is