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Operation and Characterization of NTC Thermistors
Summary
For almost all process in industry and applications for home
appliances, temperature is the variable most frequently
measured. The three most common types of contact
electronic temperature sensors in use today are the
thermocouples, Resistance Temperature Detectors (RTD),
and thermistors.
Thermistors are divided in positive and negative
temperature coefficient thermistors (PTC and NTC),
according to their resistive behave against temperature. This
application note will examine the Negative Temperature
Coefficient Thermal Resistors and its applications in orderto sense the temperature inside the Mod..
After a theoretical background on NTCs, some linearizing
networks, circuit setups and experimental results will be
exposed for temperature acquisition at the HPM elements. It
is important to remark that, the term NTC thermistor, NTC
or just thermistor can be interchangeable along the
manuscript.
The NTC Thermistor
Introduction
The term thermistor is an abbreviation of Thermal Resistors,
these elements are made from different kinds of metal
oxides. Common metals are magnesium, cobalt, nickel,
copper, and iron. The oxides are semiconductors with
resistivity that decreases with temperature, hence the name.
The temperature dependence of resistance is enormous
when compared to other materials. For example, an NTC
thermistors resistance at 100C may be as little as 5.10% of
the thermistors resistance at 25C, while the resistance of a
platinum RTD may double over the same range.
Roughly speaking, NTC thermistors are an order of
magnitude more sensitive than other temperature sensors.
This high temperature sensitivity is one of the mainadvantages of NTC thermistors. Also, high resistance values
are available, which makes lead resistance negligible in
many instances. Thus, there is no need for 4terminal
measurement arrangements. Another advantage is that
fabrication technology is mature and thermistors are
inexpensive, stable, and available in many physical
configurations, and with a wide range of electrical
specifications.
The main disadvantage is that the relationship between
resistance and temperature is nonlinear. However, the
resistancetemperature curve is monotonic and can be very
accurately described with a 3 rd. order polynomial. The
operating temperature is limited to 60C ~ 300C, which is
smaller than that of metal RTDs.
Thermistor Types and Fabrication
Thermistors are available in many configurations including
beads, disks, wafers, SMTs, flakes rods, tape and washers.
Nonbead thermistors are also known as surface electrode
thermistors and their manufacturing process has many
similarities to the construction of ceramic capacitors.
Figure 1. Thermistor types, from left to right: Screwtype,washertype, rodtype (3), disktype, beadtype (4), tape
type, axialtype and SMD [1].
In fact, a disk NTC thermistor may easily be mistaken for a
disk ceramic capacitor. First, powdered metal oxides are
combined with a plastic binder and additives that enhance
stability. The mixture is then formed into sheets that are cut
to component size or formed into pellets and pressed into
disks.
The bodies are then sintered at temperatures in excess of
1,000C that forms the final polycrystalline NTC thermistor
body. The sides are then silvered, leads are attached, and the
thermistors are sealed, varnished, and labeled.
Bead thermistors often resemble small tantalum electrolytic
capacitors. Manufacturing starts with platinum or copper
alloy wires and slurry of the metal oxide and suitable binder.
Drops of the slurry are dabbed onto the wires. The surface
tension pulls the drops into small elliptical beads. The string
of beads is then allowed to dry and then sintered at high
temperature. During sintering, the beads shrink and form an
excellent electrical connection with the wires. Next, the
wires are cut to form the individual thermistors. The next

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figure shows several possible cutting options. Finally, the
thermistors are coated and most often hermetically sealed
with glass.
Glass bead are small and range from 0.25 ~ 1.5 mm in
diameter. The small size means fast thermal response (low
dissipation constant) and bead thermistors have high
stability, but they are more costly to manufacture than
surface electrode thermistors.
Uses of Thermistors
NTC thermistors have two broad areas of applications: The
first is where the thermistors are used to sense temperature
in appliances such as coffee makers, refrigerators and
freezers, dehumidifiers, room air conditioners,
meteorological instrumentation, deep ocean temperature
probes, dialysis equipment, neonatal warmers, battery
charger temperature monitoring, intravenous catheters,
control of liquid crystal displays (LCDs are temperature
sensitive and the brightness/contrast depends on the ambient
temperature; a feedback loop to sense the ambient
temperature and adjust the LCD brightness/contrast
appropriately).
A subclassification is temperature compensation of
electronic components (e. g. fan speed control). Normally,
cooling fans in electronic equipment and switch mode
power supplies (SMPS) are powered by brushless DC
motors that run at constant speed.
Inrush current limiting is the second application area. Unless
appropriate precautions are taken, then many electronic
circuits are prone to high inrush currents. An example is that
of a power supply where the smoothing capacitors are
initially discharged. When power is turned on, the
capacitors present very low impedance, and the initial
current is limited by the capacitors stray resistance, andlarge currents can flow, possibly damaging the diodes. Once
powered, the currents are within the design specifications.
One solution is to specify components that can handle the
peak inrush currents, but this is costly and often impractical.
NTC thermistors often provide a simple and effective
solution.
An NTC thermistor is placed in series with a main current
path of the electronic device that needs protection. Initially,
the NTC thermistor has a high resistance and limits the
current that can flow. However, the dissipated power
(I2RTHERM, where I is the current through the NTC and
RTHERM is the rated NTCs resistance) heats the thermistor
and lowers its resistance. This decreases its resistance andincreases the current, which increases the dissipated power,
which leads to more heating, and so on [1].
Eventually the NTC reaches a thermal equilibrium where an
increase in temperature does not lead to a significant
decrease in resistance. The final resistance is a fraction of
the initial resistance and is small from the circuit s point ofview. NTC thermistors are very useful components and not
really too hard to work with. The main challenge is probably
to understand the datasheets and what all those numbers
mean.
Figure 2. Thermistor ResistanceTemperaturecharacteristic.
NTC Thermistor Physical Features
Electrical Characteristics
The voltagecurrent characteristic of an NTC thermistor(rated for [email protected]), is shown in figure 3, and its behave
is typical of mostly a wide variety of thermistors.
It is possible to observe that at a very small current the
I2RTHERM losses in the thermistor are very small and the
thermistor is essentially linear. At higher currents I2RTHERM
losses cause selfheating and this reduces the resistance, but
the thermistor still has a positive resistance (increase in I
results in an increase in V).
As the current increases, the selfheating causes the
resistance to decrease even more. Eventually, a point is
reached when an increase in current (and dissipated power)
heats the thermistor so much that the resulting decrease in
resistance causes the voltage across the thermistor to drop.
This is the part of the slope where the graph has a negative
slope, and is the negative resistance region of the thermistor.

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Figure 3. Voltage vs. Current of a thermistor.
Commonly the four constants that determine the thermistor
characteristics are [1]:
Rated Resistance/Rated Temperature (or Tolerance):
There are two dominant factors that that determine the
resistance tolerance.
The first is the manufacturing tolerance (TF) in the NTCs
nominal resistance. The second factor is the tolerance in the
Material Constant . Tolerance is normally referred to the
nominal resistance (RNOM, R25 orR0) at the specification
temperature (TNOM or T0), typically at 25C. The
approximate relationship between resistance and
temperature is given as follows:
1 2
1 1
1 2
T T
R R e
where,
R1: Resistance () at absolute temperature T1 (K).
R2: Resistance () at absolute temperature T2 (K).
: Material Constant (K).
NTC thermistor resistance R at any Temperature T is
determined from previous equation, observe figure 2 and
figure 4.
Figure 4. ResistanceTemperature tolerance.
Material Constant (or Sensitivity Index or B): This
constant expresses a change rate in resistance between two
temperatures, which is derived from the equation:
1 2 1 2 2
1 2 1
1 2
1 2
1 2
ln lnln
1 1
log log2.3026
1 1
R R TT R
T T R
T T
R R
T T
where,
R1: Resistance () at absolute temperature T
1(K).
R2: Resistance () at absolute temperature T2 (K).
: Material Constant (K).
The term constant is misleading since is a function of
temperature. Alternatively, different (T1, R1) pairs in the
equation above give different values for . Some
manufactures provide a table of as a function of
temperature, while others may provide it at two points in the
rated operating range.
In general, the material constant value ranges are25C~85C=
2,000K ~ 6,000K. The higher the value, the higher thechange rate in resistance per 1C.
Thermal Dissipation Constant : Is the expression of a
degree of radiation from surface and lead wires of a
thermistor element when an electric current is applied to
heat it up. It can be determined by the following equation as
the ratio between power consumption applied to a
thermistor and a degree of temperature increased by the
power:

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2
A A
P I R
T T T T
where,
: Thermal dissipation constant (mW/C).
P: Power consumption in the thermistor (mW).
T: Temperature of heat equilibrium after rising (C).
TA: Ambient temperature (C).
I: Current flowing in the thermistor at temperature T(mA).
R: Resistance of a thermistor at temperature T(k).
In order to measure the temperature accurately and to
control precisely, it is important to look closely at the value
of and minimize the electric current so that the
measurement error caused by selfheating is eliminated.
Generally the thermal dissipation constant shows a value
when a discrete element is placed in still air. That value may
change for an assembled thermal sensor.
Thermal Time Constant : This constant indicates how
fast the resistance value of a thermistor follows the change
of the surrounding temperature or electric current injected.
This constant is expressed by the time to reach the 63.21%
(or 11/e), of a difference between initial and final achieving
temperatures of a thermistor element. An example of the
thermal time constant is shown in figure 5.
Figure 5. Thermal Time Constant on a thermistor.
Other constant that can be found in vendors datasheets and
could be useful for certain designs are as follows:
Temperature Coefficient: The relative change in resistance
R1T1 at a temperature T1 is as:
1
2
1 1 1
1 dR
R dT T
where,
R1: Resistance () at absolute temperature T1 (K).
This coefficient is measured in percent per C or percent
per K and is valid only over small temperature ranges.
Interchangeability/Curve Matching: This is expressed as
a temperature tolerance over a temperature range. However,
it is possible to manufacture NTC thermistors with
temperature tolerances as small as 0.005C over a 0.100C
range. Interchangeability gauges how close the resistance
temperature curves of two thermistors match. High
interchangeability helps keep costs down since equipment
does not need to be calibrated or adjusted for individual
thermistors. Interchangeability is also a major advantagewhere NTC thermistors are used as cheap, disposable
temperature probes (e. g. medical applications).
ResistanceTemperature Operation
There are three basic electrical configurations that account
for virtually all the applications in which NTC thermistors
may be used:
1. CurrentTime characteristics.
2. VoltageCurrent characteristics.
3. ResistanceTemperature characteristics.
This application note will be focused on the third point
which is more relevant in the temperature sensing at the
HPM power modules.
For most applications based on RTcharacteristic, the self
heating effect is undesirable and it is necessary to work as
close to zeropower as possible.
Zeropower is a term that is often encountered in NTC
thermistor literature. When current flows through the NTC
thermistor it heats itself, which changes the resistance.
When this is small enough to neglect it is called the zero
power condition.
By definition in MILPRF23648, the power is considered
negligible when any further decrease in power will result inno more than 0.1% of change in resistance (i. e. 1/10 of the
specified measurement tolerance) [2].
Graphically this is the region of the currentvoltage graph
where that has a constant positive slope. Mathematically the
concept of zeropower or noselfheating implementation
will be defined in the following section.

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Heat Transfer Characteristics
When a thermistor is connected in an electrical circuit,
power is dissipated as heat and the body temperature of the
thermistor will rise above the ambient temperature of its
environment. The rate at which energy is supplied must be
equal the rate at which energy is lost or dissipated plus the
rate at which energy is absorbed (the energy storage
capacity of the device), and is as:
sup loss absorved dE dE dE
dt dt dt
Then, the thermal energy that is supplied to the thermistor in
an electrical circuit is equal to the power dissipated in the
thermistor:
sup 2dE
P I Rdt
Hence, thermal energy lost from the thermistor to its
surroundings and which is proportional to the temperature
rise of the thermistor is:
loss AdE
T T Tdt
And finally, the thermal energy absorbed by the thermistor
to produce specific amount of rise in temperature is
expressed as follows:
absorvedth
dE dT dT sm C
dt dt dt
where,
s: Specific heat (J/grsK).
m: Thermistors mass (grs.).
Cth: Heat Capacity (J/K).
It is important to realize that while the heat capacity Cth of a
thermistor is a property of the thermistor material, the
dissipation factor is not constant. For, example it depend on
the environment the thermistor is. In water a thermistor has
higher dissipation factor than the thermistor in still air, since
the water conducts heat better.
Since the thermal time constant depends on the dissipation
factor, is follow that it too is not a true constant, but depends
on the environment the thermistor is placed in. Thus,
manufacturers normally give the dissipation factor in bothair and water.
Therefore, in function of the thermistor electrical behave at
any instant in time after power has been applied to the
circuit; it is possible to write the thermistor heat transfer
equation as:
th AdT
P C T Tdt
Utilizing previous definitions and equations, it is possible
now to define mathematically the concept of zeropower
measurement. If the power at the general thermal transfer
equation is set as P0, then the following equitation is
obtained:
0 th A
A
th
dTC T T
dt
T TdT
dt C
The ratio Cth/th is equivalent to the thermal time constant of
the thermistor . For example, consider a thermistor
operated in the zeropower (no selfheating) condition at an
initial temperature T0. Now if the thermistor is placed in an
environment with ambient temperature TA. Then it is
possible to solve the zeropower equation above for the
thermistor body temperature as a function of time:
0
0
th
th
t
C
A A
t
A A
T T T T e
T T T e
The larger , the longer it takes for the thermistor to reach
thermal equilibrium when it is subjected to a sudden change
in temperature, and the longer it takes for the accompanying
resistance change to reach its final value (i. e. 63.21% of its
final value).
Thus far, the thermal properties of the NTC have been based
upon a simple device structure with a single time constant.
When any thermistor device is encapsulated into sensor
housing, the simple exponential response functions nolonger exist. The mass of the housing and the thermal
conductivity of the material used in the sensor will normally
increase the dissipation constant of the thermistor and will
invariably increase the thermal response time.
The thermal properties are somewhat difficult to predict by
mathematical modeling and manufacturing variances will
introduce enough uncertainty so the testing of the finished
sensor is usually required to obtain data on the response
time and dissipation constant .
ResistanceTemperature
LinearizationThe HartSteinhart Thermistor Equation
There are two models presently to explain the electrical
mechanism for the NTC thermistors. One explanation
involves the so called hopping model and the other
explanation is based upon the energy band model. Both
conduction models have difficulty when it comes to a
complete explanation of the RT characteristics of metal
oxide thermistors [3]. Fortunately, there are a number of
equations that can be used to define the resistance

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temperature of the devices. The more recent literature on
thermistors account for the nonlinearity of RT
characteristics by using the standard curve fitting technique.
The most known technique is the HartSteinhart curve
fitting equation [4].
The HartSteinhart equation is named after two
oceanographers during their investigation on deep sea [5].
The equation was published in 1968 and is derived from the
mathematical curvefitting techniques and examination of
resistance versus temperature characteristics of thermistor
devices.
In particular, using the plot of the natural logarithm of the
resistance value versus the inverse of temperature for a
thermistor component, an equation of the following form is
developed:
2
0 1 2
1ln ln ln
n
nA A R A R A R
T
where,
T: Temperature (K).
R: Resistance ().
A0...An: Polynomial coefficients.
The order of the polynomial to be used to model the
relationship between RT depends on the accuracy of the
model that is required and on the nonlinearity of the
relationship for a particular thermistor.
It is generally accepted that the use of a third order
polynomial gives a very good correlation with measure data,
and that the squared term is not significant. The equation
then is reduced to a simpler form, and is given as:
31
ln lnA B R C RT
where,
T: Temperature (K).
R: Resistance ().
A, B, C: Constant factors for the thermistor that is being
modeled.
Although characteristic curves are useful for derivinginterpolation equations, it is more common for
manufacturers to provide nominal thermistor resistance
values at a standard reference temperature (usually specified
as 25C), as well as resistanceratio versus temperature
characteristics.
Thermistor Calibration and Testing
Some applications have accuracy requirements which are
tighter than the conventional limits on interchangeable
devices. For these applications the thermistors must be
calibrated. To use one of the interpolation equations over a
specified range, the thermistor must be calibrated at two or
more temperatures.
The accuracy of the calculated RT characteristic over the
temperature range depends upon the proper selection of
equation and reference temperatures as well as upon the
calibration uncertainties.
Obviously, not all thermistors or assemblies can becalibrated at all temperatures over the range. There will be
limitations which are imposed by the type of thermistor and
its nominal resistance as well as by the materials used in the
construction of the assembly.
When a current source and digital voltmeter are used for
calibration, suitable averaging and integration techniques
are used to eliminate noise spikes. Thermal electromotive
forces are eliminated by either subtracting the zero current
readings or averaging forward and reverse polarity readings.
There are several calibration plans and the types of
thermistor to which they apply. As an informative remark
the plan utilized for glass enclosed beads, currently usedinside the HPMs, is disclosure; the application of this
procedure is beyond the focus of this application note.
The method for all glass enclosed beads and probes as well
as epoxy encapsulated discs or chips and sensor assemblies
using these devices is as follows: A precision constant
temperature bath is set using two or more thermistor
temperature standards [6], [7]. Resistance measurements are
performed using a precision Wheatstone bridge or a stable
precision current source and digital voltmeter in conjunction
with a data acquisition system verified against standard
resistors and an ohmic standard precision resistance decade.
Testing Equipment UncertaintyThe first step in setting up a thermistor test system is to
determine the level of uncertainty allowable for the
application. Determining the level of uncertainty is an
important part of the process used for setting up a thermistor
testing system.
The National Institute of Standards and Technology (NIST)
[8], and the International Organization for Standardization
[9] have formed an international consensus to adopt the
guidelines recommended by the International Committee for
Weights and Measures (CIPM) to provide a uniform
approach to expressing uncertainty in measurement. In these
guidelines, terms such as accuracy, repeatability, and
reproducibility have definitions that may differ from those
used by some equipment manufacturers. For example, at the
NIST guidelines, accuracy is defined as a qualitative
concept and should not be used quantitatively. The current
approach is to report a measurement result accompanied by
a quantitative statement of its uncertainty [8].
Because the cost of equipment increases as the level of
uncertainty decreases, it is important not to over specify the
equipment. Generally speaking, test system uncertainty
should be 4 to 10 times better than that of the device to be

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tested. A 4:1 ratio is adequate for most applications; for
more stringent requirements, a 10:1 ratio may be necessary
and will probably result in a more costly system [10], [11]
(e. g. using the 4:1 ratio, a thermistor with a tolerance of
0.2C should be tested on a system with an overall
uncertainty of (0.2C)/4 or 0.05C; if a 10:1 ratio were
required, the overall system uncertainty would need to be
0.02C).
To calculate the uncertainty of the overall test system, the
uncertainties of the individual components are combined
using a statistical approach [8][11]. Each component is
represented as an estimated standard deviation, or the
standard uncertainty. The two statistical methods most
commonly used by NIST are the combined standard
uncertainty and the expanded uncertainty [8].
The combined standard uncertainty uC is obtained by
combining the individual standard uncertainties using the
usual method for combining standard deviations. This
method is called the law of propagation of uncertainty (i. e.
RMS).
The expanded uncertainty Uis obtained by multiplying thecombined standard uncertainty by a coverage factor k,
which typically has a value between 2 and 3 (i.e., U= kuC).
For a normal distribution and k = 2 or 3, the expanded
uncertainty defines an interval having a level of confidence
of 95.45% or 99.73%, respectively. The NIST policy is to
use the expanded uncertainty method with the coverage
factork= 2 for all measurements other than those to which
the combined uncertainty method traditionally has been
applied. The expanded uncertainty of a system thus can be
determined once the uncertainties of the bath, the
temperature standard, and the resistance measuring
instrument are known.
Module NTC Thermistor
A Word on Mod.s Thermal Characteristics
In power electronics, semiconductor devices are operated as
switches, taking on various static and dynamic states in
cycles. In any of these states, one power dissipation or
energy dissipation component is generated, heating the
semiconductor, and adding to the to the total power losses
of the switch.
At the Mod., the commutation components are enclosed on
a single unit in order to, between another reasons, minimize
stray elements which contribute to electric loss. However,
the proximity of these switching elements has the potentialof increase the thermal dissipation on the overall modules
realstate.
Therefore, suitable power semiconductor rating and above
all, cooling measures must be taken to ensure that the
maximum junction temperature specified by the
manufacturer is complied with at any standard moment of
converter operation. To facilitate the tight temperature
monitoring during Mod.s electrical operation, a NTC
thermistor is incorporated to the module system, figure 6.
Figure 6. Inner Mod. realstate architecture, the NTC
thermistor is located in the upperleft corner.
As can be observed, the NTC thermistor is located on the
DCB (Direct Copper Bonding), which afterwards will be
potted with silicone gel in order to enhance electrical
isolation of the components during normal operation, to
protect the electronic components from mechanical stresses
and pollutants from the environment [12].
It is important to remark that silicone gel will help to avoid,
in certain amount, convection and radiative thermal
exchange towards the NTC thermistor. Hence, it is fair to
assume that the NTC thermistor will acquire the temperature
through conduction effect from the DBC where the powercomponents will displace the thermal components during
their operation.
Thermal analysis of Mod.s is beyond the scope of this
Article, for more information about thermal properties on
power modules, consult the references
ResistanceTemperature ImplementationCircuits
Applications that are based upon the RT characteristics
include temperature measurement, control, and
compensation. Also included are those applications forwhich the temperature of the thermistor is related to some
other physical phenomena. Unlike the application based
upon the currenttime or voltagecurrent characteristics,
these applications require that the thermistor be operated in
zeropower condition.
In the previous treatment of theRTcharacteristic, data was
presented on the derivation of interpolation equations that
can be used for NTC thermistors. The various equations
discussed, when used under the proper set of conditions, can

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adequately and accurately define the zeropower RT
characteristic of the NTC thermistors.
There are a variety of instrumentation/telemetry circuits in
which a thermistor may be used for temperature
measurements. In most cases, a major criterion is that the
circuit provides an output that is linear with temperature.
When the use of a constantcurrent source is desired, the
circuit used should be a twoterminal network that exhibits alinear resistancetemperature characteristic. The output of
this network is a linear voltagetemperature function. Under
these conditions, a digital voltmeter connected across the
network can display temperature directly when the proper
combination of current and resistance level are selected.
If the use of a constant voltage source is more desirable, the
circuit used should be a twoterminal network that exhibits a
linear conductancetemperature characteristic. Conversely,
the output of this network is a linear currenttemperature
function. Consequently, the design of thermistor networks
for most instrumentation/telemetry applications is focused
on creating linearRT or linear conductancetemperature
circuits.
Vin
t
RsetVO(T)
T0
VO(T)
Vin
Rtherm
Linear
Approximation
Figure 7. Voltage divider configuration.
Voltage Divider: The simplest thermistor network used in
many applications is the voltage divider circuit shown in
figure 7. In this circuit, the output voltage is taken across the
fixed resistor. This has the advantages of providing an
increasing output voltage for increasing temperatures and
allows the loading effect of any external measurement
circuitry to be included into the computations for the resistor
R and thus the loading will not affect the output voltage as
temperature varies.
The output voltage as a function of temperature can be
expressed as follows:
setO inset therm
RV T V
R R
where,
Vin: Circuit polarization (V).Rtherm: NTC zeropower resistance at temperature T().
Rset: Voltage divider/linearizing resistance ().
VO(T): Resultant output voltage (V).
From the plot of the output voltage, we can observe that a
range of temperatures exists where the circuit is reasonably
linear with good sensitivity at certain range. Therefore, the
objective will be to solve for a fixed resistor value Rset that
provides optimum linearity for a given resistance
temperature characteristic and a given temperature range.
A very useful approach to the solution of a linear voltage
divider circuit is to normalize the output voltage with
respect to the input voltage. The result will be a standard
output function (per unit volt) that can be used in many
design problems. In this case, the normalization is obtained
utilizing previous equation; the normalized output is as
follows:
11
O
in therm
set
V T
V R
R
In most thermistor literature, the thermistor referencetemperature T0 is 25C (298.15K) and the thermistors are
cataloged by their nominal resistance value at 25C (defined
as Rtherm0, the zeropower resistance at a standard reference
temperature).
Thus, the thermistor resistance is normalized with respect to
its resistance at the specified temperature as:
0
0
thermtherm therm therm therm
therm
Rr r R R
R
In the actual solution of many applications problems, it is
desirable forT0 andRtherm0 to be specified at the midpoint of
the intended operating temperature range.
The ratio Sof the zeropower resistance of the thermistor at
the desired reference temperature to the fixed value resistor
in the voltage divider circuit is as:
0therm
set
RS
R
From where the transfer function is derived as follows:

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1 1( )
11
O
in therm therm
set
V TG T
V Sr R
R
The transfer function G(T) is dependent upon the circuit
constant S and the resistanceratio versus temperature
characteristic rtherm. If we allow the circuit constant to
assume a series of constant values and solve for the transferfunction, we shall generate a family ofScurves.
Figure 8 illustrates a family of such curves. These curves
were generated using the resistanceratio temperature
characteristic given for the NTC Thermistor.
Figure 8. Transfer function G(T)curvesfor a NTC thermistor.
It is obvious from the design curves that a value for the
circuit constant Sexists such that optimum linearity can be
achieved for the divider network over a specified
temperature range. The design curves can be used to provide
a graphical solution or a first approximation for many
applications. For the best solution to a design problem an
analytical approach is required. There are two analytical
methods employed to solve for the optimum linearity
conditions of the divider network: the Inflection Point
Method and the Equal Slope Method.
In the inflection point method, the inflection point is the
position where the slope of the curve is a maximum;
therefore, it is desired to have the change point of the
standard function occur at the midpoint of the operating
temperature range. The sensitivity of the divider network
would therefore be at a maximum at this point.
This method is recommended for the solution of
temperature control applications. However, does not provide
good linearity over wide temperature ranges. Its use should
be restricted to temperature spans that are narrow enough
for to be considered constant and thus the intrinsic
equations can be used. At the inflection point, the slope of
the standard curve (first derivative with respect to
temperature) is at a maximum and the curvature (second
derivative with respect to temperature) is zero. The
reference temperature will be selected as the midpoint
temperature of the desired operating range.
At the equal slope method it is desired to set the slopes of
the standard function equal to each other at the endpoints of
the temperature range (Tmin and Tmax). This method can
provide good linearity over wider temperature ranges. When
using this method for solution, the polynomial equations for
theRTcharacteristic are used.
Both discussed above have been based on a single
thermistor voltage divider. When the thermistor is
connected to more complex circuits which contain only
resistances and voltage sources, the problem can be reduced
back to the simple voltage divider by considering the
Thevenin equivalent circuit as seen at the thermistor
terminals.
Voltage Divider Variants: Figure 9 shows two simple
modifications to the basic voltage divider which can be
converted to/from a Thevenin equivalent circuit as requiredfor any given application. The voltage divider of figure 9a
is used where it is desired to reduce the output signal while
figure 9b is used where it is desired to reduce the source
voltage and translate the output signal by adding a bias
voltage. Of the two circuits, figure 9b is commonly used,
especially in bridge circuits. It permits the use of
conventional source voltages and reduces the voltage placed
across the thermistor to an acceptable level of selfheating.
The bias voltage can be compensated in the bridge design.
Bridge Architectures: Bridge circuits are actually two
voltage divider circuits. In most applications, the bridge
consists of a linear thermistor voltage divider and a fixed
resistor voltage divider.
For differential temperature applications, the bridge consists
of matching thermistor linear voltage dividers. Figure 10a
illustrates a basic Wheatstone bridge circuit with one
linearized thermistor voltage divider and Figure 10b
illustrates the Wheatstone bridge circuit used for differential
temperature applications.
Both of the circuits in figure 10 represent cases where the
load resistance is infinite and thus does not affect the output
voltage of the voltage divider or dividers.

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Vin
t
RsetVO(T)
RthermR1
R2
Vin
t
RsetVO(T)
RthermR1
a)
b )
Figure 9. Different Voltage divider configurations.
VO1(T)
Vin
t
Rset
UB
RthermR1
R2vB VO(T)
Vin
t
Rset1
UB
Rtherm1t
VO2(T)Rset2
Rtherm2
a)
b)
Figure 10. Wheatstone bridgeconfigurations (infinite load).
When the Wheatstone bridge circuit is more complex and
the load resistance cannot be considered infinite, the
Thevenin theorem is used to reduce the circuit to its
equivalent form. Figure 11a shows the basic Wheatstone
bridge circuit for a finite load resistance, while figure 11b
shows the Thevenin equivalent circuit.
Ohmmeter Circuit: Another circuit which is commonly
employed in temperature measurement applications is the
basic ohmmeter circuit which is shown in figure 12. This
circuit is also a basic voltage divider of sorts. It is generally
used for low cost temperature measurement applications;
thus, the trimming potentiometer may not always be in the
circuit. In this architecture the objective is to produce a
linear current.
This current can be expressed as a constant times the
standards function G(T). The value of the constant is the
source voltage divided by the circuit resistance as seen by
the thermistor.
Vin
t
Rset
RthermR1
R2
VTHEV
t
RTHEV
Rtherm
RL
iL
a)
b )
Figure 11. Wheatstone bridge configuration (finite load).
Note that the circuit consisting of a thermistor in series with
a fixed resistance is a linear conductance versus temperature
network. The voltage divider circuits, the Wheatstone bridge
circuits and the Ohmmeter circuit discussed so far have all
been examples of linear conductance versus temperature
networks. They may all be solved by the use of the standard
function Scurves, the inflection point method or the equal
slope method as preferred.
Linear Resistance Networks: Many applications based
upon the RT characteristic require the use of a linearized
resistance network. The linear conductancetemperature
networks are driven by a constant voltage source, whereas,
the linearRTnetworks will be driven by a constant current
source. Note that one is the dual of the other.

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VTHEV
t
Rset
Rtherm
RM
Rp
Figure 12. Ohmmeter configuration circuit.
RP
Rtherm
t
Rtherm
t
RP
RS
Rtherm
t
RP
RS
R1
RP
R1
Rtherm
t
a)
b )
c )
d )
Figure 13. Linearization networks.
Figure 13 illustrates the basic linearRT networks used in
most compensation applications. The simplest network is
obviously that shown in figure 13a. If we normalize the
network resistance with respect to the shunt resistor, we
observe that the standard function G(T) can be used for the
design of linear resistance networks. In order to increase the
overall network resistance to a higher value, a series resistor
can be inserted as illustrated by figure 13b. This can also be
done to increase the voltage drop across the network when a
constant current is applied to the terminals (as will be
mentioned afterwards).
Obviously, the linearRTcharacteristic is translated by the
series resistor and the slope remains unchanged. Figure 13c
shows the circuit of figure 13b with the addition of a resistor
in series with the thermistor. This circuit is used to permit
the use of a standard value for the thermistor. The standard
value thermistor must be slightly lower than the desired
value for optimum linearity and both thermistors must have
the same resistance ratiotemperature characteristic. Figure
30d shows the basic circuit of figure 13a with the addition
of a resistor in series with the thermistor, for the purpose of
utilizing a standard value of thermistor.
Going back to the network on figure 13a, it is possible to
obtain a better linearization when the fixed resistor and thenominal temperature value of the NTC thermistor are
related by the following formula:
0 2P therm
TR R
T
where,
Rthermo0: NTC nominal resistance value at 25C ().
: Material Constant (K).
T: Centered temperature 298.15K (25C).
Figure 14. Linearization curves for NTC thermistor.
The best linearization is obtained by laying the turning point
in the middle of the operating temperature range. Figure 14
shows the curve for a NTC thermistor with Rthermo0= 10k
(@ 25C) and a material constant (25C~50C) = 3450K;
hence, the calculated normalizing parallel resistor is equal to

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RP = 7.79k (i. e. 7.7k). This value gives a good linear
span for a range between 0C ~ 65C (other values can be
traced for different span needs). The rate of rise of the
linearized characteristic is given by:
0
0
2 2
1
therm
therm
P
RdR
dT TR
R
It is important to remark that the sensitivity of the measured
temperature decreases with linearization.
In the case where the curve is necessary to be shifted for
higher impedance value (with same linear slope), then it is
necessary to adopt the normalizing circuit of figure 13b. The
resultant family of curves for previous given values are
shown in figure 15.
Figure 15. Impedance of normalized curvesshifted due to series resistance.
It is obvious that the values selected should be a tradeoff in
order to have all the measurements at zeropower mode.
Then, if the circuit is biased in the previous example with
Vin = 1V, the voltage measured at the normalized network
VNTC, and the power displaced on it will be as plotted in
figure 16.
It is possible to observe that at the peak of the power curve
(i. e. Pdiss 0.25mW) and knowing that the dissipationconstant for this element is equal to 1.4mW/C, then the
selfheating will be accounted as 0.178C which in that
point (around 92C) correspond to a temperature error of
approximately 0.2%, enough accurate for any kind of
applications.
Figure 16. Signal voltage and power dissipationcurves of the linearized NTC thermistor.
Vin
t
Rset
RthermR1
R2
a)
+ VO(T)
Rfeed
Vin
R2
R1
b)
+
VO(T)
Rfeed
Rset
tRtherm
DZ
T0
VO(T)
c)
Figure 17. OpAmp application with NTC thermistor.
Operational Amplifier Circuits: As observed in previous
section, generally to obtain a smooth measurement from a
thermistor it is necessary to utilize some sort of linearizing

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aid network; however, the necessity of measurement at zero
power resistance on those architectures make the whole
setup susceptible of noises due to the small scale of
voltage/current utilized.
One solution is employ high performance precision
instrumentation amplifiers with railtorail I/O. Which have
the advantage of very low DC errors, longterm stability and
very low 1/f noise. Two examples of a Wheatstone bridge
with an OpAmp are shown in figure 17.
Figure 17a shows a temperature Wheatstone bridge with an
OpAmp acting as differential amplifier, this kind of
circuitry can have very high sensitivity (zener diode can be
omitted if bias voltage is set for zeropower resistance). By
the other hand, on figure 17b, the OpAmp acts as a
Schmitttrigger which generates the transfer characteristic
given in figure 17c.
Another variant from figure 17a can be seen in figure 18.
This circuit is a temperature dependent reference voltage
that can be implemented using thermistor/resistive parallel
combination illustrated in figure 13a as feedback element in
an operational amplifier circuit.
Vin
Rset
R1
R2
+ VO(T)
DZ
tRthermRP
VX
Figure 18. Amplifier gain changed by NTC thermistor.
In this circuit, a zener diode reference is used to drive the
inverting input of an OpAmp. The gain of the amplifier
portion of the circuit is:
0 1therm P
X
set
R RV T V
R
IfR1= 8.06k,R2= 1k,Rset= 549 andRP= 10k with a
zener voltage of 2.5V are used (for a NTC thermistor of
[email protected]), it will generate the 0.276V at the input to the
operational amplifierVX.When the temperature of the NTC thermistor is equal to 0C
Rtherm is approximately 32,650.8. The value of the parallel
combination of this resistor and RP is equal to 7655.38.
This gives a operational amplifier gain of 14.94 V/V or an
output voltage V0(T) of 4.093V.
When the temperature of the NTC thermistor is 50C, the
resistance of the thermistor is approximately 3601.
Following the same calculations above, the operational
amplifier gain becomes 5.8226V/V, giving a 1.595V at the
output of the amplifier. This could be use in a logic circuitry
utilizing any 12bit DAC.
Another OpAmp based topology is shown in figure 19a;
due toRP andRset the voltage at point Uvaries linearly with
the NTC thermistor temperature. The voltage at point V is
equal to that of point U when the NTC thermistor is 0C.
Both voltages are fed to the comparator circuitry and
sampled according to the clock pulses figure 19b. The
output pulse train can be utilized in any digital circuit.
Vin
t
Rset
RthermR1
R2
a)
VO2(T)
t0
b)
R3
RP
Sawtooth Gen.
Clock Gen.
VO1(T)
Vpulse(T)
U
V
VO2(T)
VO1(T)
Vpulse(T)
0C Ref.
Figure 19. Bridge sensing with 0C offset.
It is obvious that for certain applications, the partcount is
not desirable; therefore, exist other solutions that have a
high component integration allowing a very good
temperature monitor precision at relative low cost.
One of those solutions is an integrated circuit optimized for
use in 10k NTC thermistor. This IC provides the necessary
NTC thermistor excitation and generates an output voltage
proportional to the difference in resistances applied to theinputs. It uses only one precision resistor plus the NTC
thermistor reducing the partcount issue. It maintain
excellent accuracy for temperature control applications,
figure 20. Several other topologies based on precision
instrumentation amplifiers with railtorail I/O [17] can be
observed in Appendix A. In more advance architectures
towards digital acquisition of temperature there is one
topology that can be extended to any kind of logical control,
figure 21.

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Figure 20. NTC thermistor signal amplifier.
10k
Rtherm
t
a)
RREF
100
CONTROLLER
C
GP1
GP2
GP3
b)
10 Mtherm REF C
TR k R
T
t0
VC
TM TC
Figure 21. NTC thermistor Calibrator/Sensor.
In this topology, on the first step the sensing circuit is
implemented by setting GP1 and GP2 of the controller as
inputs. Additionally, GP0 is set low to discharge the
capacitor, C.
Once C is discharged, the configuration of GP0 is changed
to an input and GP1 is set to a high output. A timer counts
the amount of time before GP0 changes to 1, giving the time
TM in figure 21b. At this point, GP1 and GP2 are again set
as inputs and GP0 as an output low.
Once the integrating capacitorC, has time to discharge, GP2
is set to a high output and GP0 as an input. A timer counts
the amount of time before GP0 changes to 1, giving the time
TC. The difference on timing between TM and TC will
determine the actual resistance (and temperature), in the
thermistor and after consulting a lookup table stored at the
controller.
The values ofRREFand C are calculated according to the
number of bits of resolution required. RREF should be
approximately one half the highest resistance value to be
measured, hence:
10 ln 1
res
biastherm
ref
tC
VR k
V
where,
Rtherm: NTC nominal resistance value 25C ().
tres: Time to acquire the required resolution bits (sec).
Vbias: Threshold voltage of controller being used (V).
Vref: reference voltage (V).
ResistanceTemperature Experiments
General Information on Mounting Requirements: The
mounting instructions outlined below are taken from several
Application Notes [13][16]. These recommendations are
based on the knowledge acquired during laboratory and
field examinations.
The power modules are intended to be mounted on a PCBcircuit board from the pin side and to a heatsink from the
backside. The contact area of the module and heatsink must
be free of any particles or damages.
Before the module is installed onto the heatsink, it is
necessary to apply a thin film of thermal compound of
approximately 100 ~ 200m. As a ruler of thumb, a small
rim of thermal compound around the edge of the module
should be visible after the module is attached.
To fasten the module to the heatsink in a simple and reliable
way, the power module has a pair of screw flanges, figure
22; which should be bolted with M4 type screws. One screw
should be slotted in a flange but not tightened until the
opposite screw is in its place. After both crews are inserted,
then the screws are tightened one after the other with a
recommended mounting torque of 2.0 ~ 2.3Nm.

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Figure 22. Mod.and mounting draft.
If the power module is correctly installed onto the heatsink,then an optimal thermal resistance between module and
heatsink is ensured. For more detailed information on
mounting configurations, package characteristics and power
modules requirements, please visit the knowledge base web
page at:
Materials and Methods: The circuits to be tested are the
most significant in the industry environment: the Voltage
Divider configuration (figure 7), the Wheatstone Bridge
(figure 10) and the Linearization Network(figure 13b). All
these topologies will utilize the NTC Thermistor which its
RTvalues can be observed in Appendix B and the physical
characteristics are as follows:
Operating Temperature Range: 50C ~ +200C.
Thermal Dissipation Constant : ~ 1.4mW/C
Thermal Time Constant : ~ 10sec.
Material Constant : 3450K 2% (25C ~ 50C).
The measurements will be carried utilizing the Agilent
34411A DMM, with 4wire Kelvin terminals at 90min
warmup and integration of 100PLC (PowerLine Cycles),
accuracy specification error equal to 0.06C at TCAL 5C
and 0.003C at TCAL +10C [18], which fits by far the 4:1
and even the 10:1 uncertainty level ratio (i. e. NTC
tolerance at 25C = 1.25C, uncertainty level 10:1 =
0.125C).
The modules are similar as figure 22, and will be divided in
two categories, Sample A: complete standard module
(power components and NTC immersed in silicone gel),
Sample B: same as Sample A but without silicone gel.
Sampleswill be subjected to temperature range from 30C
to 150C; what is more, in Sample A the measure of the lag
on the thermal constant caused by the silicone gel will be
obtained.
The measurements will be taken at 10C at 10min intervals
in a thermal isolated enclosure to suppress micro thermal
currents and other stray effects; also, in order to achieve
zeropower measurements the NTC will be polarized at the
measurement instant (i.e. [email protected]).
Figure 23 shows the experimental and the manufacturers
values of both samples without any normalizing network.
Observe the variation found against the original data.
This variation goes in hand with the already mentioned
thermal dissipation constant alteration in the Heat Transfer
Characteristics section.
It is important to keep in mind this alteration during
temperature measurements because the value tolerance
given by the manufacturers data is shifted; for example, a
variation of 5% will occur between 0C and 100C and not
at 3% for 0C as specified on the datasheet.
Figure 23. Experimental resistive values andtemperature variations on the addition of silicone gel.
Now, for the Voltage Divider network, note thatRSET is not
subjected to the temperature variation due to it is considered
that this element is elsewhere in the PCB circuit.

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The value of polarization voltage is set as Vin = 1V. After
several testing iterations the best linear fitting gives RSET as
3.6k. Hence, the graphic of the experimental output
voltages VO(T) for the standard commercial sample (Sample
A) is shown at figure 24.
Utilizing the same RSET and bias voltage values obtained in
previous network analysis and for 0C, 25C, 80C and
100C setting points; the curves obtained at the Wheatstone
Bridge (figure 10a), in function of the voltage divider are
as:
2
1 2
0
0
B in
setin
therm set
B B
Rv V
R R
RV T V
R R
U v V T
Figure 24. Voltage Divider network response.
Figure 25. Wheatstone bridge setting pointsat several temperatures.
The results are shown in figure 25 in function of UB and
adjusted R1 and R2 in a manner to obtain the indicated vB
values. These curves are useful as guideline for triggering
any set point in the cooling system (observe the curves
crossing UB at the required temperature settings).
Finally, in the Linearization Network (figure 13b),
maintaining the value ofRS as 3.6k and varying the RP
value, the family of curves obtained is shown in figure 26.
As can be seen, the value ofRP = 5.1k is the values which
approaches to a more linear response from all those family
curves.
Figure 26. Linearization Network tuning toobtain the best linear fitting.

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Appendix A
Figure A1. Bridge to Digital Controller.
Figure A2. Generating Output Offset Voltage.
Figure A3. Highside current Shunt mode.
Figure A4. Lowside current Shunt mode.
Figure A5. Lowside V current Shunt mode.
Figure A6. Highside current Shunt mode.

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Appendix B
Temp (C) RNTC(k) Temp (C) RNTC(k)
30 121.9 37 6.408
20 72.37 38 6.182
10 44.41 39 5.966
0 28.08 40 5.7591 26.86 41 5.56
2 25.7 42 5.369
3 24.6 43 5.185
4 23.55 44 5.009
5 22.56 45 4.839
6 21.61 46 4.676
7 20.7 47 4.52
8 19.84 48 4.369
9 19.02 49 4.224
10 18.24 50 4.085
11 17.49 51 3.951
12 16.78 52 3.822
13 16.11 53 3.698
14 15.46 54 3.579
15 14.84 55 3.464
16 14.25 56 3.354
17 13.69 57 3.247
18 13.15 58 3.144
19 12.64 59 3.04520 12.15 60 2.95
21 11.68 61 2.858
22 11.23 62 2.77
23 10.8 63 2.685
24 10.39 64 2.602
25 10 65 2.523
26 9.625 66 2.446
27 9.266 67 2.372
28 8.922 68 2.301
29 8.592 69 2.232
30 8.277 70 2.166
31 7.975 71 2.102
32 7.685 72 2.04
33 7.408 73 1.98
34 7.142 74 1.923
35 6.887 75 1.867
36 6.642 76 1.813

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Temp (C) RNTC(k) Temp (C) RNTC(k)
77 1.761 117 0.6127
78 1.711 118 0.5982
79 1.662 119 0.5841
80 1.615 120 0.5705
81 1.57 121 0.557282 1.526 122 0.5443
83 1.483 123 0.5317
84 1.442 124 0.5195
85 1.402 125 0.5076
86 1.364 126 0.496
87 1.326 127 0.4848
88 1.29 128 0.4738
89 1.255 129 0.4632
90 1.221 130 0.4528
91 1.189 131 0.4428
92 1.157 132 0.4329
93 1.126 133 0.4234
94 1.096 134 0.4141
95 1.068 135 0.405
96 1.04 136 0.3962
97 1.013 137 0.3876
98 0.9863 138 0.3792
99 0.9609 139 0.3711
100 0.9362 140 0.3632101 0.9123 141 0.3554
102 0.8891 142 0.3479
103 0.8665 143 0.3405
104 0.8447 144 0.3334
105 0.8235 145 0.3264
106 0.8029 146 0.3196
107 0.783 147 0.313
108 0.7636 148 0.3065
109 0.7448 149 0.3002
110 0.7266 150 0.2941111 0.7088
112 0.6916
113 0.6749
114 0.6587Fitting equation : 35.036 16.212 7566.263 13303.198 46.733
T T
thermR e e
115 0.6429
116 0.6276

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References
[1] D. Hill and H. Tuller, Ceramic Sensors: Theory and Practice, Ceramic Materials for Electronics, R. Buchanan, ed., MarcelDekker, Inc., New York, 1991.
[2] MILPRF23648F, Performance Specification: Resistors, Thermal (Thermistor) Insulated, General Specification For; Jan. 2009.[3] P. V. E. McClintock et. al.,Matter at Low Temperatures,Blackie, ISBN 0216915945, 1984.[4] M. Sapoff et al. The Exactness of Fit of ResistanceTemperature Data of Thermistors with ThirdDegree Polynomials,
Temperature, Its Measurement and Control in Science and Industry, Vol. 5, James F. Schooley, ed., American Institute of Physics,New York, NY, p. 875, 1982.
[5] J.S. Steinhart and S.R. Hart, Calibration Curves for Thermistors, Deep Sea Research, 15(497), 1968.[6] IPTS68,Bureau International des Poids et Mesures, 1968.[7] ITS90,Bureau International des Poids et Mesures, ISBN 9282221083, Dec. 1990.[8] B.N. Taylor and C. E. Kuyatt. Sept. 1994. "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement
Results," NIST Technical Note 1297, U.S. Government Printing Office, Washington, DC.[9] "ISO, Guide to the expression of Uncertainty in Measurement," ISO Technical Advisory Group 4 (TAG 4), Working Group 3(WG 3),
Oct. 1993.
[10] G.N. Gray and H.C. Chandon. 1972. "Development of a Comparison Temperature Calibration Capability," Temperature, ItsMeasurement and Control in Science and Industry, Vol 4, Instrument Society of America:1369.
[11] B. Pitcock. 1995. "Elements of a Standards Lab That Supports a Manufacturing Facility," Bench Briefs, Pub. No. 59646003E,2nd/3rd/4th Quarters, HewlettPackard Co., Mountain View, CA.
[12] W. W. Sheng and R. P. Colino, Power Electronic Modules: Design and Manufacture, ed., CRC Press, Boca Raton Florida, 2005.