Exponential
Model NTC Thermistors
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Exponential
Model of NTC
Thermistors:
0/50 Beta Value
(ß) or
Sensitivity Index:
As
outlined in the
previous section,
a simple approximation
for the relationship
between Resistance
and Temperature
for an NTC thermistor
assumes an exponential
relationship between
them. This approximation
is based on simple
curve fitting to
experimental data
and also on an
intuitive feel
for electrical
behaviour of semiconductor
devices.
The exponential approximation
is a mathematical model
that applies an equation
that can be expressed in
the form:
RT
=  exp(ß/T)
............ (equation
# 4)
Where:
RT is the
Resistance in
ohms at temperature
T
T is the absolute Temperature
in Kelvin
A is a linear factor
"exp" is the exponential
function
ß is the exponential factor
known as "beta" value
or sensitivity index of the thermistor
material.
The ß value is
a very important
parameter in the
description and
specification of
thermistor materials
and thermistor
components. When
the natural log
of both sides of
the equation is
taken, the relationship
becomes:
ln(RT)
= C + ( ß/T)
.................
(equation # 5)
Where C is
a constant factor,
(C = ln(A)) from
the equation above.
If ln(RT) is plotted versus
1/T, (as in graph #4) then
the slope of the resulting
curve will be equal to beta, ß.
This equation provides
a reasonable approximation
to measured data, but as
mentioned in the previous
section, the thermistor
materials are not ideal
materials. For the exponential
model to apply over a large
temperature range (greater
than 50 °C) the beta
value has to vary, therefore
the beta value is not constant
over extensive ranges. In
fact, the beta value is
also temperature dependent
and it decreases with temperature.
Although
this simple exponential
model for the relationship
between Resistance
and Temperature
of athermistor
is limited over
large temperature
spans, concepts
derived from it
are of importance
in the thermistor
industry and in
the specification
of NTC thermistors.
Some of these concepts
are developed in
the following sections
with the intention
of explaining some
of the basic calculations
and specifications
used in the industry.
Practical
application of
the beta value:
It is common practice to
specify thermistor materials
in terms of beta value
over a particular temperature
span.
For a temperature T1 and
thermistor resistance R1
at this temperature T1:
R1 = A exp (ß/
T1).
For
a temperature T2
and thermistor
resistance R2 at
this temperature
T2:
R2 = A
exp ((ß/
T2).
Taking
the ratio: R1
/ R2 = exp(ß(1/
T1 - 1/ T2))
The
expression for ß then
becomes:
Where:
ß has units of temperature
(Kelvin)
"ln" represents the natural
logarithm (log base e)
inverse of the exponential
function.
In this form,
published beta values
can be used to calculate
resistance or temperature
values when other
items in the equation
are known.
The beta value can then
be regarded a quantitative
value of thermistor materials
that is assigned as a material
constant and that indicates
the relationship of material
resistivity to temperature.
Application
of the beta
value is demonstrated
in a numerical
example at
the end of
this section.
The general information
on sensitivity of material
resistivity to temperature
that can be interpreted
from the beta value is
indicated in Graph #
5. This shows the resistivities
of BetaTHERM’s Standard
Thermistor Materials versus
their 0/50 °C Beta
Values.
The
beta value is derived
from a mathematical
approximation.
For this mathematical
approximation to
apply over a large
temperature range,
beta has to vary
with temperature.
This variation
of beta value with
temperature is
indicated in Graph
# 6. The variation
is greater at the
low temperature
end of the curve.
This variation
should be borne
in mind when using
beta values for
calculations over
temperature ranges.
Because the beta value
is an indication of the
relationship between the
resistivity of thermistor
material and temperature,
it can also be used to
calculate alpha (a) value
(temperature coefficient)
for a thermistor made from
the same material. Recalling
the definition of the alpha
value as the percentage
change in resistance per °C,
given by equation #
3
and,
expressing R as
a function of T
using the exponential
model, it can be
shown that a good
approximation for
the temperature
coefficient or
alpha value, at
a temperature T
Kelvin, in terms
of beta is :
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The
beta value is a
single expression
that can be regarded
as a material constant.
It depends on basic material properties,
and beta values
derived from measurements
provide an indication
of general thermistor
material quality.
Although deviations in
beta value from nominal
values affect the tolerance
of thermistors and are
indicative of material
quality, such deviations
are not widely published
by thermistor manufacturers.
Typically, manufacturers
will list the nominal Beta
Value only or will list
the nominal Beta Value
with a tolerance expressed
in K.
BetaTHERM provide information
on beta value deviations
for the range of thermistor
materials that it produces.
This information is contained
in Table #2. This
lists the beta values and
deviations for high precision
BetaCURVE thermistors and
includes data for low precision
tolerance BetaCHIP thermistors
also.
Graphs of Material resistivity
versus Beta value, and
Beta Value versus temperature
are included also to illustrate
how Beta is regarded as
a material characteristic
that is related to resistivity.
The relationship between
Beta value and temperature
shows the temperature regions
for which Beta value is
approximately constant
for different materials.
Sample
calculations are
included in this
section also to
illustrate some
typical engineering
uses of beta values.
Beta
Values
and tolerances
for BetaCURVE
and
BetaCHIP series Thermistors. |
| |
0/50
Beta Value
(K) |
| Material
Curve # |
BetaCURVE |
BetaCHIP |
| 1 |
3108
+/-0.40% |
3108
+/-1.7% |
| 2 |
3263
+/-0.38% |
3263
+/-0.9% |
| 3 |
3892
+/-0.32% |
3892
+/-1.0% |
| 4 |
3575
+/-0.30% |
3575
+/-1.0% |
| 5 |
3811
+/-0.33% |
3811
+/-0.9% |
| 6 |
4143
+/-0.30% |
4143
+/-1.3% |
| 7 |
3422
+/-0.36% |
3422
+/-1.0% |
| 9 |
4582
+/-0.63% |
4582
+/-1.9% |
Table
# 2
Material Resistivity vs.
Beta Value for BetaTHERM
Thermistor Materials (Curves
)
Graph
# 5
Beta
Values vs. temperature
for Betatherm
Materials
Graph
# 6
Examples
of calculations
using Beta values:
(The 0/50 Beta value means
a Beta value calculated
from resistance data at
0°C and at 50°C)
Example: Calculate
the 0/50 Beta value for
BetaTHERM’s 30K5A1
thermistor, given the
following information:
R1 is
resistance
value measured
at T1,
0°C
= 94980
ohms
R2 is
resistance
value measured
at T2,
50°C=
10969 ohms
The relevant formula for
calculating the beta value
is:
Where:
ß has
units of temperature
(Kelvin)
ln represents the natural
logarithm (log base e)
inverse of the exponential
function.
Temperatures are expressed
in Kelvin.
This can then be written
as:
 |
Since
the Beta value
of a thermistor
material is derived
from an exponential
model, it can also
be used to calculate
resistance values
of a thermistor
at a particular
temperature when
the resistance
at another temperature
is known. This
is demonstrated
in the following
example:
Example:
Calculate
the resistance
of a BetaTHERM
10K3 thermistor
at 33°C
using the
following
information:
The resistance of the thermistor
at 25 °C is 10000 ohms,
the 0-50 Beta value is
3892 K
Using the equation:
R1 / R2 = exp(ß (1/
T1 - 1/ T2))
and writing it in the form:
R2 = R1 / (exp(ß (1/
T1 - 1/ T2)))
where
T1 = 25°C = 298.15
K, R1 = 10000 ohms.
T2 = 33°C = 306.15
K, ß= 3892 K
Using these values, the
value of R2 is calculated
as:
R2 = 10000 / (exp(0.3411094421367))
Resistance at 33°C
calculated using Beta
value:
R2 = 7109.81ohms.
The value of resistance
of a 10K3 device that is
published in the R /T tables
is 7097.2 ohms.
The sensitivity of the
resistance of the device
to temperature in this
region is 304 ohms per °C.
The difference between
the published resistance
value and the resistance
value that is calculated
using the beta value is
equivalent to .04 °C .
Using the Beta 0-50 value
in the range 0-50 °C,
the maximum errors are
at the ends of the range,
and are of the order of 0.22 °C.
When the tolerance of
the Beta value is taken
into account, then the
errors at the ends of the
range for material 3
BetaChip products are
of the order of 0.45°C,
while the errors at the
ends of the range for material
3 BetaCurve products
are of the order of 0.29 °C.
| The
main purposes
of these
examples
were : |
 |
To
show some typical
uses of Beta
values in calculations. |
|
To
demonstrate
the difference
between BetaCHIP
and BetaCURVE |
| |
product.
In particular
to show that
the tighter
tolerance BetaCURVE
product has
lower errors
than the BetaCHIP
product in
converting
from Resistance
to temperature
using the exponential
model. |
|
To
establish the
relevance of
Beta as a material
characteristic. |
|
To
show some of
the limitations
of the exponential
model |
| |
so
that further
modelling of
the R / T characteristics
of thermistors
can be considered. |
Further
Modelling of NTC
Thermistor R/T
Characteristics:
The concepts introduced
so far to relate the resistance
of a thermistor to the
temperature have been primarily
based on the characterization
and specification of materials and
on the use of material
parameters rather than
on component parameters.
While
the temperature
coefficient or
alpha value can
be used to calculate
the temperatures
corresponding to
various resistance
values of a thermistor,
the method is rather
limited. A look-up
table of Resistance
versus Temperature
values for the
thermistor is required
and details of
alpha values at
various points
are needed also.
It is very useful
and relevant in
certain situations.
The
use of Beta
Values or sensitivity
index, and
the associated
exponential model
are useful for
material specification,
and for the comparison
of the sensitivity
of bulk materials.
The method is somewhat
limited for general
use in relating
the resistance
of a thermistor
to temperature
over extensive
ranges mainly because
of the temperature
dependence of the
Beta Value itself.
In
general applications
NTC thermistors
are used to measure
temperature, and
this is accomplished
by measuring the
resistance of the
thermistor and
then using that
resistance value
to make an estimate
of temperature.
The various means
of relating resistance
of a thermistor
to the temperature
that have been
discussed so far
are not ideally
suited to this,
as outlined above. The
requirement is
for a single equation
that can be used
easily to relate
resistance and
temperature of
thermistors. The
requirement is
all the more important
to optimize the
use of programmable
calculators, computer
spreadsheets and
microcontrollers.
Because
the conduction
mechanism in metal
oxide semiconductors
is a complex one,
it is difficult
to explain accurately
by applying mathematics
to a physical model. The
method used for
accurate mathematical
modelling of the
Resistance versus
Temperature characteristic
of a thermistor
is to obtain accurate
measurements of
Resistance and
Temperature of
components and
to apply curve
fitting techniques
to model the relationship
between them.
The next section of this
catalog describes the mathematical
model, which is in general
use throughout the industry,
to give a single equation
that relates the Resistance
and Temperature of an NTC
thermistor component. The
equation is called the Steinhart-Hart
Equation, and it is
used by all thermistor
manufacturers.
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