Exponential
Model NTC Thermistors
Please
click on
the section
below to
view your
area of
interest:
|
|
|
|
|
|
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 :
 |
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.
|
