1.1 Axiomatic Approach to Device Modeling
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be retrieved via F(D). In other words, F(D) acts as an analog memory bank, thus
a lookup table containing the results of all possible experiments on D.
Since it is impossible to measure and then store the results of an infinite number
of experiments, it is clearly impractical to use F(D) to characterize D, i.e., the above
Gedanken Experiment is only a thought experiment. However, a large number of
real-world two-terminal devices D can be described via mathematical equations,
i.e., a mathematical model of D, obtained either by a physical approach or a
black-box approach, consisting of one or more (algebraic, differential, or integral)
equations that provide an accurate approximation to the corresponding measured
admissible signal pairs (v(t), i(t)). This leads to the conceptual definition of an
“ideal” circuit element whose associated AVIS pairs are generated (rather than
measured) via some prescribed mathematical model referred to as Constitutive
Relation (or CR for short). 2
Example 1.1 (Linear Resistor) A particular class of physical devices R are
described as a black-box with a mathematical equation (Ohm’s law)
v(t) = Ri(t)
or in implicit form
f R (v, i) = v − Ri = 0
(1.1)
where R in Ohm (Ω) is a parameter named resistance. In this case the mathematical
model (1.1) permits to generate the A-pad
F(R) = {(Ri 1 (t), i 1 (t)), (Ri 2 (t), i 2 (t)), . . . , (Ri n (t), i n (t)), . . . }.
In circuit theory, (1.1) is the CR of an (ideal) circuit element referred to as linear
resistor. Along this line of reasoning the CR (1.1) can be thought of as a compact
mathematical model for generating the infinite set F(R) of a physical resistor.
1.1.2 Remarks on the Concept of a Device Model
A model of a device is not an equivalent circuit (i.e., an arbitrary interconnection
of circuit elements) of D because no real device can be exactly mimicked by a
circuit or mathematical model. In fact, depending on the application (e.g., amplitude
and frequency of operation), a given device may have many distinct models. As
reported in [1]: “There is no “best model” for all occasions. The best model in a
given situation is the simplest model capable of yielding realistic solutions.”
2 From now on the term device will be used only when referring to a physical device, while any
“ideal” device will be referred to as a circuit element, or simply an element.
5
be retrieved via F(D). In other words, F(D) acts as an analog memory bank, thus
a lookup table containing the results of all possible experiments on D.
Since it is impossible to measure and then store the results of an infinite number
of experiments, it is clearly impractical to use F(D) to characterize D, i.e., the above
Gedanken Experiment is only a thought experiment. However, a large number of
real-world two-terminal devices D can be described via mathematical equations,
i.e., a mathematical model of D, obtained either by a physical approach or a
black-box approach, consisting of one or more (algebraic, differential, or integral)
equations that provide an accurate approximation to the corresponding measured
admissible signal pairs (v(t), i(t)). This leads to the conceptual definition of an
“ideal” circuit element whose associated AVIS pairs are generated (rather than
measured) via some prescribed mathematical model referred to as Constitutive
Relation (or CR for short). 2
Example 1.1 (Linear Resistor) A particular class of physical devices R are
described as a black-box with a mathematical equation (Ohm’s law)
v(t) = Ri(t)
or in implicit form
f R (v, i) = v − Ri = 0
(1.1)
where R in Ohm (Ω) is a parameter named resistance. In this case the mathematical
model (1.1) permits to generate the A-pad
F(R) = {(Ri 1 (t), i 1 (t)), (Ri 2 (t), i 2 (t)), . . . , (Ri n (t), i n (t)), . . . }.
In circuit theory, (1.1) is the CR of an (ideal) circuit element referred to as linear
resistor. Along this line of reasoning the CR (1.1) can be thought of as a compact
mathematical model for generating the infinite set F(R) of a physical resistor.
1.1.2 Remarks on the Concept of a Device Model
A model of a device is not an equivalent circuit (i.e., an arbitrary interconnection
of circuit elements) of D because no real device can be exactly mimicked by a
circuit or mathematical model. In fact, depending on the application (e.g., amplitude
and frequency of operation), a given device may have many distinct models. As
reported in [1]: “There is no “best model” for all occasions. The best model in a
given situation is the simplest model capable of yielding realistic solutions.”
2 From now on the term device will be used only when referring to a physical device, while any
“ideal” device will be referred to as a circuit element, or simply an element.