24
1 Device Modeling and Circuit Elements
expressed as an algebraic relationship involving only v (α) and i (β) , so that we need
to classify this element as a dynamic two-terminal element.
Example 1.12 Consider a two-terminal element given by a nonlinear capacitor
described by the equation i = C(v)dv/dt, i.e., an algebraic (α , β ) = (0, −1)element, in parallel to a nonlinear voltage-controlled resistor i = ˆ
i(v), i.e., an
(α , β ) = (0, 0)-element. The parallel obeys to i = C(v)dv/dt + ˆ
i(v). Also
in this case it is not possible to find (α, β) such that the CR of this two-terminal
element is expressed by an algebraic relationship between v (α) and i (β) . Hence,
such an element is dynamic.
It can be seen that any two-terminal element obtained by interconnecting
nonlinear elements with different (α, β) from the circuit-element-array results in
a dynamic element. Since, by definition, any lumped element that is not algebraic
is dynamic, the class of dynamic elements is therefore much larger than that of
algebraic elements.
Most realistic circuit models of devices are made of the interconnection of
algebraic elements with different (α, β). Consequently, most realistic circuit models
are dynamic elements. These elements are usually described by a CR given by a state
equation and an output equation of the following form
˙
x = f(x, η)
(1.17)
ξ = g(x, η)
where η, ξ ∈ R, x = (x 1 , x 2 , . . . , x n ) T ∈ R n is a vector of internal state variables,
f : R n+1 → R n and g : R n+1 → R.
We can now define four basic classes of dynamic two-terminal elements that are
analogous to the four basic algebraic two-terminal elements in Definition 1.1.
Definition 1.5 A two-terminal element is called an R-, L-, C-, or M-dynamic
element if and only if it can be described by a CR having the form (1.17), where
{ξ, η} denotes {(v, i)}, {(ϕ, i)}, {(v, q)}, {(ϕ, q)}, respectively.
As an example, four distinct relevant circuit families of this kind are singled out
as follows.
1. Current-Controlled R-dynamic two-terminal element
˙
x = f(x, i)
(1.18)
v = g(x, i)
2. Current-Controlled L-dynamic two-terminal element
˙
x = f(x, i)
(1.19)
ϕ = g(x, i)
1 Device Modeling and Circuit Elements
expressed as an algebraic relationship involving only v (α) and i (β) , so that we need
to classify this element as a dynamic two-terminal element.
Example 1.12 Consider a two-terminal element given by a nonlinear capacitor
described by the equation i = C(v)dv/dt, i.e., an algebraic (α , β ) = (0, −1)element, in parallel to a nonlinear voltage-controlled resistor i = ˆ
i(v), i.e., an
(α , β ) = (0, 0)-element. The parallel obeys to i = C(v)dv/dt + ˆ
i(v). Also
in this case it is not possible to find (α, β) such that the CR of this two-terminal
element is expressed by an algebraic relationship between v (α) and i (β) . Hence,
such an element is dynamic.
It can be seen that any two-terminal element obtained by interconnecting
nonlinear elements with different (α, β) from the circuit-element-array results in
a dynamic element. Since, by definition, any lumped element that is not algebraic
is dynamic, the class of dynamic elements is therefore much larger than that of
algebraic elements.
Most realistic circuit models of devices are made of the interconnection of
algebraic elements with different (α, β). Consequently, most realistic circuit models
are dynamic elements. These elements are usually described by a CR given by a state
equation and an output equation of the following form
˙
x = f(x, η)
(1.17)
ξ = g(x, η)
where η, ξ ∈ R, x = (x 1 , x 2 , . . . , x n ) T ∈ R n is a vector of internal state variables,
f : R n+1 → R n and g : R n+1 → R.
We can now define four basic classes of dynamic two-terminal elements that are
analogous to the four basic algebraic two-terminal elements in Definition 1.1.
Definition 1.5 A two-terminal element is called an R-, L-, C-, or M-dynamic
element if and only if it can be described by a CR having the form (1.17), where
{ξ, η} denotes {(v, i)}, {(ϕ, i)}, {(v, q)}, {(ϕ, q)}, respectively.
As an example, four distinct relevant circuit families of this kind are singled out
as follows.
1. Current-Controlled R-dynamic two-terminal element
˙
x = f(x, i)
(1.18)
v = g(x, i)
2. Current-Controlled L-dynamic two-terminal element
˙
x = f(x, i)
(1.19)
ϕ = g(x, i)