IEEE SENSORS JOURNAL, VOL. 13, NO. 12, DECEMBER 2013
4621
Linearization of NTC Thermistor Characteristic
Using Op-Amp Based Inverting Amplifier
Aloke Raj Sarkar, Debangshu Dey, Member, IEEE, and Sugata Munshi
Abstract— A low cost linearizing circuit is developed, placing
the NTC thermistor in a widely used inverting amplifier circuit
using operational amplifier. The performance of the system is
verified experimentally. A linearity of approximately ±1% is
achieved over 30 °C–120 °C. When used for a narrower span,
a much better linearity of ±0.5% is obtained. The gain of the
arrangement can be adjusted over a wide range by simply varying
the feedback resistance. The simplicity of the configuration
promises a greater reliability, and also curtails the deterioration
in the stability of performance, by reducing the cumulation of
drifts in the different circuit components and devices.
Index Terms— NTC thermistor, linearization, temperature
sensors, inverting amplifiers.
I. I NTRODUCTION
S
IGNAL conditioning arrangements for thermistors with
negative temperature coefficient (NTC) have long been
objects of inquest to engineers involved in research and developmental work related to transducers. The NTC thermistors,
by virtue of their high sensitivity, find extensive usage in transducers for measurement and control of temperature and other
physical variables (e.g. flow, humidity) and also as temperature
compensating elements in electronic systems [1]–[4]. Further,
the NTC thermistors have highly nonlinear (approximately
exponential) resistance-temperature characteristics that make
the devising of signal conditioning arrangements, a formidable
task.
In the past, numerous signal conditioning circuits have been
devised for NTC thermistors, aimed at obtaining an output that
has a quasi-linear relation with the temperature being sensed.
These linearizing arrangements involved in the simplest form,
passive components (resistances) either shunting the sensor
or placed in series with it [5]–[7]. They were followed by
logarithmic amplifier based systems [8]–[11] and multivibrator
circuits [12]–[16]. A parallel stream of developmental activities focused on software techniques, i.e. numerical methods
and soft computing techniques for linearization [17]–[26].
With the widespread availability of processor based systems,
particularly with the advent of microcontrollers, the balance
has tilted overwhelmingly in favour of the software based
linearization methods. It is however true that even with the
Manuscript received November 16, 2012; accepted June 1, 2013. Date of
publication June 10, 2013; date of current version October 4, 2013. The
associate editor coordinating the review of this paper and approving it for
publication was Dr. M. Nurul Abedin.
The authors are with the Electrical Engineering Department, Jadavpur
University,
Kolkata, India (e-mail: aloke.sarkar.cemk@gmail.com;
debangshudey80@gmail.com; sugatamunshi@yahoo.com).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSEN.2013.2267332
slashed prices of microprocessors and microcontrollers, they
are still costlier than analog signal processing circuits involving op-amps. Furthermore, there are some software methods
that can not be implemented using microcontrollers. So, their
realization demands the availability of at least existing PC
based systems, since it is absurd to dedicate a PC solely for
the measurement of a single physical variable. Some of the
numerical methods may be programmed into microprocessors
and microcontrollers with lot of difficulty, but the computational burden becomes so heavy for the processor, that there is
an appreciable delay introduced by this measurement system.
As a matter of fact, the most important drawback of software
based linearization methods in general, is the introduction of
measurement delay, which is particularly of concern when the
transducer is used in the feedback path of closed loop control
system. Although, this should not have been of concern with
the fast processors that are easily available nowadays, use of
such processors is often not justified when the sensing a single
physical variable.
Thus, the primary incentives for analog circuits and digital
hardware based linearizers being clear, it will not be improper
to point out yet another reason in favour of the linearizing
circuits. The linearizer circuit may be used to accomplish the
first stage of linearization, to improve the linearity to a certain
extent. The resulting signal, after digitization, is processed
by a software that implements an algorithm to achieve the
second stage of linearization [27]. Due to this two-tier structure of the linearizing arrangement, the second–stage can be
relatively simple, e.g., a read-only-memory (ROM) based lookup table method in conjunction with linear interpolation. These
arrangements may be effective since temperature measurement
systems using NTC thermistors have a pronounced nonlinearity of the transfer characteristic.
Once we decide to opt for linearizing circuits, it is desired
that the circuit configuration should be as simple as possible,
provided of course that the linearization achieved is not
compromised to a significant extent. A simple circuit is not
only cheap and compact, it also has two other important merits.
With the decrease in the number of circuit components and
devices used, the probability of failure also decreases, i.e., the
reliability of the system improves considerably. Furthermore,
the drift of the circuit characteristic is also lessened, as the
drifts of fewer number of items are cumulated.
It is therefore not surprising that even in recent times, quite
a few works on linearization circuits for NTC thermistors
come trickling in. Two linearization circuits have been investigated in recent times that place the thermistor as one of the
timing resistors for a 555 timer based free-running (astable)
1530-437X © 2013 IEEE
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4622
IEEE SENSORS JOURNAL, VOL. 13, NO. 12, DECEMBER 2013
multivibrator circuit. The circuits ultimately have both analog
and frequency outputs [28], [29]. The study made by Abdulwahab et al. [28] only identifies qualitatively the temperature
zones over a span of 0 °C to 120 °C, for which the transfer
characteristic is most linear. On the other hand, in the work
reported by Nenova et al. [29], a more systematic analysis
has been carried out to obtain an optimal value of the other
timing resistor of the multivibrator circuit. The experiment
performed on a thermistor yields a nonlinearity not exceeding
±1.7% over 0 °C to 120 °C and ±1% over and 0 °C to
100 °C. Rosa et al. [30] report a signal conditioning circuit,
comprising a one-bit sigma–delta modulator in which some
of the block functions are performed by an NTC thermistor.
The performance of the arrangement has been studied only
for a restricted range of 40 °C to 65 °C. Mohan et al.
proposed a digital hardware based linearizer employing a dual
slope analog to digital converter [31]. Although the system is
complicated, its performance is quite good.
Motivated by the necessity to develop a linearizing circuit as
simple as possible but competent enough to achieve a linearity
comparable with those obtained with relatively complicated
circuits, the present paper reports a low-cost linearizing circuit
for NTC thermistors, employing an op-amp based inverting
amplifier. The thermistor is placed in series with the linearizing
resistance in the feed forward path. Unlike the usual practice,
the value of the linearizing resistance has been selected
by defining a normalized deviation from linearity, and by
numerically minimizing the sum-square value of this deviation considering several points across the intended working
temperature. Experimental results show that the performance
of the proposed scheme exhibits acceptable linearity which
is comparable with the results obtained from other circuits
reported in literatures [29]. However, the proposed circuit has
a much simpler configuration.
II. P ROPOSED L INEARIZING C IRCUIT
Although several expressions are in use to represent the
transfer curves of NTC thermistors, the most popular one is
given by
R = RT0 e
β
1
1
T − T0
(a)
(b)
Fig. 1. Proposed linearizing scheme for NTC thermistor. (a) Circuit diagram.
(b) Photograph of experimental setup.
increases, RT will decrease and consequently V o will increase.
It is desired that Vo (T ) should have a linear relation with the
temperature T (K) to which the thermistor is exposed, over
the temperature range of interest extending from TL (K) to
TU (K).
For NTC thermistor based transducers, the widely accepted
method of obtaining the linearizing resistance value is as
follows:
V0 (T ) can be expanded into a Taylor’s series about a
reference temperature Tr , as
(1)
Where, RT = Thermistor resistance at a temperature T (K).
RT0 = Thermistor resistance at a reference temperature of
T0 (K).
β = A material constant for the thermistor (K)
Equation (1) approximates the resistance curve over a
narrow span of temperature. However, because of its maneuverability, it finds extensive use in the designing of signal
conditioning circuits for NTC thermistors.
The present paper puts forth a low cost and simple analog
signal conditioning circuit for NTC thermistors depicted in
Fig. 1.
The output voltage signal of the circuit is
Vi R f
(2)
Vo (T ) = −
r + RT
It can be easily seen that since the excitation voltage
Vi is negative, Vo (T ) is always positive. Furthermore, if T
V0 (T ) = V0 (Tr ) + τ V0 (Tr ) +
τ 2 τ3
V0 (Tr ) + V0 (Tr ) + .......
2!
3!
(3)
where
τ = T ∼ Tr
and
V0 (Tr ) =
d V0 (T ) d 2 V0 (T ) ,
V
(T
)
=
r
d T T =Tr 0
d T 2 T =Tr
and so on.
Truncating the series after the third term, we get
τ 2 V (Tr )
(4)
2! 0
Over and above this truncation operation, if we further introduce the condition that V0 (Tr ) = 0, then an approximately
linear relation between V0 and T is obtained. Usually, the
V0 (T ) ≈ V0 (Tr ) + τ V0 (Tr ) +
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SARKAR et al.: LINEARIZATION OF NTC THERMISTOR CHARACTERISTIC USING OP-AMP BASED INVERTING AMPLIFIER
midpoint TM of the temperature range of interest is considered
as the reference temperature Tr .
d 2 V0 (T ) ∴
= V0 (TM ) = 0
(5)
d T 2 T =TM
U
is the midpoint of the temperature range
where, TM = TL +T
2
of interest. In other words, by proper choice of the resistance
r , the output voltage versus temperature curve is forced to
have an inflection at T = TM .
From equation (2) and (5), we get
f (Tm ) = 0
(6)
1
r + Rr
(7)
4623
TABLE I
T HERMISTORS U SED IN E XPERIMENT
Thermistors
Thermistor I
(100 Ω)
Thermistor II
(1 kΩ)
Thermistor III
(5 kΩ)
RT0 in
(kΩ)
β – Values
(K)
Linearizing
Resistance
(r) in (Ω)
Excitation
voltage
(Vi)
in (V)
Feedback
Resistance (Rf)
in (kΩ)
0.11
2847.4
17.02
1.0
0.14
1.34
3056.4
193.55
1.0
1.53
4.86
3963.3
504.85
1
3.50
where,
f (T ) =
Equation (6) yields
r=
2RT2m
RTm
− RTm
(8)
Considering eqn. (1) representing the thermistor transfer
relation, the following expressions are obtained.
β
(9)
R TM = R TM − 2
TM
and
RTM = RTM
β
TM3
Thus, finally we get,
r = r o = R TM
2+
β
TM
β − 2TM
β + 2TM
(10)
(11)
However, in this paper the required value of r has been
obtained in a different manner. For this purpose, the normalized deviation from linearity at a temperature T (K) is
defined as
T − TL
Vo (T ) − Vo (TL )
(12)
−
D=
Vo (TU ) − Vo (TL ) TU − TL
The sum-square value of D is obtained numerically by measuring the thermistor resistance at say N number of temperatures,
and is given by
N
D 2 (r, Ti )
S=
(13)
i=1
The optimum value of r is that for which S is minimum.
To start with, an approximate value r1 of r is computed by
introducing the condition for perfect linearity as
Vo (TM ) − Vo (TL ) = Vo (TU ) − Vo (TL )
(14)
Then, the optimum value of r is obtained by numerically
minimizing S. This is done by varying r over a range
spanning from a suitable value less than r1 to one greater
than r1 , but necessarily including the value ro obtained
from eqn. (11).
It is worth mentioning that the excitation voltage Vi and
the value of the feedback resistance R f influence none of the
Fig. 2. Resistance versus temperature characteristic of thermistors under test.
values ro and r1 , and not also the final value of r . It is also
imperative that the sensitivity of the proposed circuit depends
on Vi and on R f . However, while Vi cannot be increased much
in order due to avoid self-heating of the thermistor, one may
play with the value of R f to have a full scale output over a
wide range.
III. E XPERIMENTAL R ESULTS
The performance of the proposed analog signal conditioning arrangement has been tested with three separate NTC
thermistors having different nominal resistances (RT 0 ) and
β values.
The temperature range covered, extends from 30 °C
(i.e. TL =303 K) to 120 °C (i.e. TU = 393 K). Thus the
midpoint of the range is 75 °C (i.e. TM = 348K). For each
sensor, the β-constant has been obtained experimentally by
measuring the resistance at 71 known temperature at intervals
of 1 °C, and then fitting the values to eqn. (1) using the
method of least squares. A platinum resistance temperature
sensor (Pt-100) has been considered as the standard. The
nominal resistances, the β values, the values of the linearizing
resistance r , the excitation voltages used, and the feedback
resistance values are given in Table I.
Fig. 2 shows the resistance vs. temperature characteristics of the three thermistors used during experimentation.
Fig. 3 depicts the experimentally obtained output voltage (V0 )
versus temperature (T ) characteristic of the signal conditioning
circuit, for the thermistors under study.
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4624
IEEE SENSORS JOURNAL, VOL. 13, NO. 12, DECEMBER 2013
(a)
Fig. 4.
Percentage error versus temperature (°C).
calculated for each temperature as,
E (T ) =
(b)
(c)
Fig. 3. Output voltage (V0 ) versus temperature (T) characteristic of the signal
conditioning circuit. (a) Thermistor-I. (b) Thermistor-II. (c) Thermistor-III.
The curves have been generated by taking measurements
at intervals of 1 °C. The actual temperature values have been
obtained using the Pt-100 standard. In each case, the ideal
characteristic is the best-fit least-square straight line for the
experimental data.
The deviation from linearity, which in this case is known as
the independent non-linearity error (in percentage), has been
Actual Vo (T ) − Ideal Vo (T )
× 100
Full Scale Vo
(15)
and the deviations from linearity have been plotted against
temperature, in Fig. 4.
It can be seen that for thermistors I and II the nonlinearity
error lies within approximately ±1% for the temperature range
of 30 °C to 120 °C. Moreover, for thermistor I the error is
approximately within ±0.5% in the range 35 °C to 115 °C.
For thermistor III, the nonlinearity error lies approximately
within ±1% for the temperature range of 38 °C to 115 °C
and within ±0.5% between 50 °C to 110 °C. The performance
of the proposed circuit is compared with those of the other
linearizing circuits reported in recent times, and a comparative study is presented in Table-II. The circuit developed by
Rosa et al. [30], exhibits an excellent linearity over a restricted
range of 40 °C to 65 °C. Since it involves a sigma-delta
modulator, the hardware complexity is also an important
disadvantage, as it implies diminished reliability with the
possibility of an escalated drift. Very good linearization can
also be attained by the dual-slope digital converter based
arrangement put forward by Mohan et al. [31]. However, of
all the circuits considered, this scheme has the most intricate
configuration. Therefore, it suffers from the disadvantages
already mentioned, to a greater extent. Furthermore, in this
work, the measurements over 0 °C to 120 °C have been taken
at intervals of 5 °C. Therefore, it is not clear whether or
not measurements at lesser intervals (e.g., 1 °C) would result
in a degraded linearity compared to what has been reported
(i.e. ±0.2%). The astable multivibrator circuit investigated by
Nenova et al. [29] can achieve a decent linearity of the transfer
characteristic, and the circuit complexity is much less than
those of [30], [31]. In the present paper a linearizer has been
examined, that has a substantially simple circuit configuration
compared to those reported in recent times. The linearizing
scheme gives acceptable linearity measures somewhat better
than those of [29]–[30]. This is true for all the thermistors
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SARKAR et al.: LINEARIZATION OF NTC THERMISTOR CHARACTERISTIC USING OP-AMP BASED INVERTING AMPLIFIER
TABLE II
4625
R EFERENCES
C OMPARATIVE S TUDY OF D IFFERENT L INEARIZING C IRCUITS
Method
Proposed
By
Temperature
Range
Covered
Deviation
from
Linearity
The authors
in the
present
paper
30 oC to
120 oC
± 1%
35 oC to
115 oC
± 0.5%
0 ºC to
120 ºC
±1.7%
0 ºC to
100 ºC
± 1%
Nenova et al
Rosa et al
40 ºC to 65 ºC
± 0.01%
0 ºC to 120 ºC
± 0.2%
30 ºC to 50 ºC
± 0.1%
Mohan et al
Comments
Decent linearity,
simple circuit
configuration.
Decent linearity,
moderate
complexity of
circuit.
Excellent linearity,
but obtained only
over a very narrow
range of
temperature;
complex circuit,
and hence less
reliability and
increased drift.
Very good
linearity, but
circuit is highly
complicated.
Therefore, least
reliability and
most drift prone.
with different nominal resistances (RT 0 ) and β values, with
which experiments have been performed.
IV. C ONCLUSION
A low-cost linearizing circuit has been developed for the
NTC thermistor, employing inverting amplifier circuit using
operational amplifier. A close examination of the system developed, together with experimentations, revealed the following
points in its favour:
1) Good linearity could be achieved with the proposed
circuit, a little better than those reported by some of the other
investigators [29]–[30] in recent times.
2) Simplicity of the proposed arrangement makes it a very
low-cost solution to tackle the problem of linearization of
thermistor characteristics.
3) The naivety of the circuit configuration ensures greater
reliability and also permits a higher degree of stable performance due to reduced aggregation of drifts in the characteristics of circuit elements.
4) The gain (sensitivity) of the system can be adjusted by
using a resistive pot, without affecting the linearity of the
transducer transfer curve.
5) Where processor based systems are used, the proposed
circuit may be conveniently used as a cheap first-stage analog
linearizer, and the linearity may be further augmented by some
software based method, after the output of this analog circuit
is digitized.
6) For applications in analog control systems, the output
of the circuit presented in this paper can be directly used.
This is not the case with the competing linearizing circuits
[29]–[31].
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Aloke Raj Sarkar received the M.S. in engineering
and electrical engineering from Jadavpur University,
Kolkata, India, in 2012. He is currently with the
Electrical Engineering Department, Camellia School
of Engineering and Technology, Kolkata, as an
Assistant Professor. His current research interests
include linearization of transducer characteristics,
field-programmable gate array based data acquisition
systems, and image processing.
Debangshu Dey (M’09) received the B.E.E.,
M.E.E., and Ph.D. degrees from Jadavpur University,
Kolkata, India, in 2003, 2005, and 2009, respectively, where he is currently a Faculty Member
with the Electrical Engineering Department. He has
published more than 40 research papers. His current research interests include condition monitoring of electrical equipment, intelligent instrumentation and measurements related to condition assessment techniques, applications of signal conditioning,
and processing tools in electrical and biomedical
systems.
Sugata Munshi received the B.E.E. and M.E.E.
degrees from Jadavpur University, Kolkata, India, in
1980 and 1985, respectively. He was an Engineer
with the Plasma Physics Division, Saha Institute of
Nuclear Physics, India, from 1985 to 1990. In 1986,
he had training on the “Tokamak” machine in the
Heavy Engineering Works of Toshiba Corporation,
Japan. In 1990, he joined the Electrical Engineering
Department, Jadavpur University, as a Faculty Member. He is currently a Professor with this department.
He has published more than 40 research papers in
refereed journals and over 20 papers in conference proceedings. He was the
joint recipient of The President of India’s Prize (English) from 1989 to 1990,
the Pandit Madan Mohan Malaviya Memorial Prize from 1989 to 1990, the
Sir Thomas Ward Memorial Prize from 1994 to 1995, the Tata Rao Medal,
by the Institution of Engineers, India, from 1996 to 1997, and a Certificate
of Merit from IE, India, from 1996 to 1997. His current research interests
include signal processing, surge phenomena in power equipment, and sensor
systems.
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