Mathematical
Modelling Thermistors
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Mathematical
Modelling of
Thermistors:
One
of the core physical
assumptions of
the band gap theory
of solid state
physics is that
charge carrier
concentration has
an exponential
dependence on absolute
temperature. The
charge carriers
can be either negatively
charged (n-type)
electrons, or positively
charged (p-type)
holes. The p-type
conduction mechanism
is the trapping
of electrons by
fixed positively
charged sites.
The electrons move
between these sites
so that the net
effect is that
of mobile positively
charged carriers
moving in an electric
field, in the opposite
direction to negatively
charged electrons.
It can be demonstrated
theoretically from the
chemical composition of
the components and experimentally
from Hall effect measurements
that the metal oxide thermistor
materials are p-type semiconductors.
More details on the determination
of carrier types can be
found in reference text
books on ceramic materials.
The expression for the
density of carriers available
in "ideal" semiconductor
material can be derived
directly from application
of Quantum Mechanics to
Solid State theory. This
involves applying the Fermi-Dirac
distribution to the calculation
of energy states for charge
carriers in the material.
The process leads to an
equation that describes
the intrinsic carrier density
in terms of some material
constants and physical
constants. The results
derived in this way are
expressed in an equation
of the form:
Where :
ni is the intrinsic
carrier density in appropriate
units
T is
the absolute
temperature
in Kelvin.
k is
Boltzmann’s
constant
(1.38066
x 10-23
) J/K
Further analysis of "ideal" semiconductor
materials using principles
of solid state physics
relates the electrical
resistivity of the material
to the carrier density.
The resulting expression
is in the form of :
Where r is the material
resistivity, in appropriate
units, such as ohm-cm.
The equation can be reduced
to a simpler format by
writing it as:
This equation relates the
resistivity of semiconductor
material to the exponential
of the reciprocal of absolute
temperature, directly from
fundamental principles
of solid state physics.
As stated previously, an
equation of this form is
relevant for an "ideal" material
with a regular crystal
structure. Although the
metal oxide thermistor
materials are not "ideal",
intuition based on the
foregoing discussion would
suggest that there should
be an exponential aspect
to the relationship between material
resistivity and the reciprocal
of absolute temperature.
Inspection of a graph of
the natural log of measured
Resistivity values versus
the reciprocal of absolute
temperature for thermistor
materials, indicate that
such a model is appropriate.
Graph # 4
The
Resistance (R)
of a piece of material
of resistivity  (ohm-cm)
is proportional
to this resistivity
value.
R = x
(t/A)
where:
R is the resistance
in ohms,
t is thickness
of the material (length
of current path),
A is the cross-sectional
area.
It follows that the expression
for resistance as a function
of temperature can be stated
as:
Where RT denotes Resistance
in ohms at temperature
T Kelvin.
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