Linearizing Transducer Characteristics
D. PATRANABIS, S. GHOSH, AND C. BAKSHI
Abstract-Most of the practical transducers fall into two categories
regarding their inpvt-output characteristics: type I-whose character-
istic is exponentially rising and type II-whose characteristic is expo-
nentially decaying. While type I is easily linearized using a logarithmic
converter, type I1 needs additional inverting means for the purpose of
linearization. This paper is intended to present a technique of linear-
izing type I1 transducer characteristics using a log-converter and a
FET, and to compare its performance with that of a simple 'single-
stroke' digital technique. Examples with experimental results are pre-
sented along with pertinent discussions.
I. INTRODUCTION
A LL PHYSICALLY STABLE systems can basically
be modelled as integrators and their response char-
acteristics to static inputs are, therefore, like charging and
discharging characteristics of an uncharged or a charged
condenser, respectively. Transducers also are no excep-
tions. These basic natures of transducers are shown in Fig.
l(a) and (b), the corresponding response curves being %@-+ X
designated as type I and type 11, respectively. These lead (b)
to two common specific nonlinear responses of the variety Fig. I . (a) Type I Input-output characteristic. (b) Type 11 Input-output
of transducers. They are not necessarily exponentially ris- characteristic.
ing and falling, as suggested, over,the entire range. In
fact, often they are to be modelled by series type expres- be improved upon with time and increased analytical
sions. However, for most of the transducers, for a given complexity.
operation range, an exponential form seems to be ade-
quate. In such a situation, associated analog circuits have 11. THE ANALOG LINEARIZATION TECHNIQUE
been developed and are being developed for linearizing The type I nonlinearity is modelled by the equation
the characteristics, and the trans'ducer along with this cir-
cuit appears as the linear module of the transducer ele- y = yo( 1 -
ment of a measurement system. where a determines the quality of the response nature.
Similar to this, digital linearizing modules are being de- From (1) one transforms
veloped for application in computer/processor-basedsys-
tems where linearization is performed through software yo - y - e-orx =2
packages. Because of the drastic reduction in the use of (la>
Yo
digital hardware, this technique is worth considering.
While nonlinearity of type I characteristics has now where 2 represents the unit incomplete response [l] and
been well .taken care of as far as linearization is con- is labelled as a new transformed variable. A log amplifier
cerned, lots of problems still persist for the type 11, par- serves to linearize this response easily. Thus (la) changes
to
ticularly when y has a very large value with x 0, such
-+
as conductive moisture sensors, thermistors, etc. It is in- In 2 = - a x (lb)
tended here to discuss en elegant analog method, with an
example, to linearize a transducer response of type I1 and If 2 is fed to a log converter, we have converted 2 linearly
compare its performance with a typical digital method, making it proportional to x, the original variable, and the
also with an example. However, it must be remembered output 2, of the log converter is given by
that a digital method, being software based, is likely to -kTa
z, = -X
Manuscript received March 23, 1987; revised July 8, 1987. 4
The authors are with the Department of Electronics and Telecommuni-
cations Engineering, Jadavpur University, Calcutta, 700032 India. where k, T, q are parameters of the log converter. If ( yo
IEEE Log Number 8717640. - y ) is considered in place of 2, from (1) one gets at the
0018-9456/88/0300-0066$01.OO O 1988 IEEE
PATRANABIS et al. : LINEARIZING TRANSDUCER CHARACTERISTICS 67
output of the log converter
kT
yc = - (In yo - a x )
4
=A+&.
(3) fitting now
With the nonlinearity of type 11, the linearization is not
so simple. The matter is further complicated by the as- -
q-v ~ In A, = 0
sumed empirical relation 6f the decaying exponential type kT
for its coefficients and compensation constants. However, one gets
a generalized approach can be followed for the type rep-
resented by 'the epoation & Gfi --r6
x = XoeS/". (4) kT 2 1 ~ ~X ~
The exponential equation requires a log conversion lead- giving finally
ing finally to
kT
X, = - B / X .
4
or
Tha matter imroves wry little because X, remains in-
b variable requiredto be mea-
2to& t
om ta think of an 'in-
able & that A, is
where
(6)
where Ki is a function of K , T, 0, and q. But the inverter
is not a commonly available module to be easily coupled This Rds can be converted to a voltaGe through a simple
to the bg cc&mertw€or* &sired lhuarization. Instead voltage deriving circuit and the o.wtput can thus be lit-sear
a @mewise ttechnique 'b been proposed with respect to x .
cwer.small w g d o € : x&aut@ Taylor's t h m m and se- Incidentally, it may be mentioned that some sensors
have a transfer charaqteristic of the form
y = k/x (13)
over I h e - d d s i d - j s ~ e i as in some conducting type moisture sensors. Such sen-
A technique that is quite simple, covers a wider (theo- sors can be linearizd easily using FET linearizers as
retically entirei) ranrgs, aul:inaunr less cost is, however, demonstrated above. An example of linearization of a
to utilize the m n k m r mh@& x3f asemiconductor device, sensor having a characteristic giv by (4) is presented in
a FBT for uxamph, r o t c w t e for the nonlinearity Section I11 with experimehtal rgsults.
shown by (51, %thg mplifler QBJ,- When a transducer
, am tie mtrinen as 111. EXAMPLES OF AN ANALOGLINEARIZATION
TECHNIQUE
A. Setup of Example
The example of a thermispr with the response charac-
E ylnb
h
+ p). (7) teristic of Fig. l(b) and guided by ( 4 ) is considered for
the linearization scheme using a log-converter and a FET
The FET characteristic is given by the dation inverter. The thermistor resistance-temperature relation is
given by
giving
The thermistor is used in a log-converter scheme preceded
by a resistance-volta8e converter as shown in Fig. 2 such
(8b) that its output is givqn by
ificance as found in VR kT,P
, we get from (7) and
v,=-
(In-+-
RIRTozsa ;j --
qT
(15)
68 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 37, NO. I , MARCH 1988
LIJ7J
Fig. 2 . The scheme of thermistor-coupled log amplifier.
Fig. 3. The practical scheme of the linear temperature to voltage converter
where using thermistor.
I, = reverse saturation current of the transistor
CY = common base current gain of the transistor output Vo is shown to be directly proportional to T by the
Tu = ambient temperature in degrees Kelvin. relation
If V2 is now fed to the gate-source of an FET, the FET Vu = cT (22)
drain-source resistance would be given by 181 (cf Sa), where
Voltage VR and resistance R, are the only parameters
where which can be arbitrarily controlled for controlling the sen-
Vg, is used for V2 sitivity, the others are either device-dependent or are con-
strained because of (21). For covering the desired span of
and the characteristic, (17) and (21) can be simultaneously
Ro = Drain-source resistance with Vg, = 0 satisfied by adequate selection of R and a corresponding
Vp = Pinchoff voltage of the FET. R8.
For proper operation of the FET as a voltage variable re- B. Results
sistor, however, v,, I$Vp; and V2, therefore, is condi- The circuit of Fig. 3 was assembled with FET input
tioned with an additional supply input V,. Moreover, the (LF 356) OA's and a log converter in turn with a FET
FET is coupled to a voltage-deriving circuit such that Rds input OA and a transistor (CIL 231) having large I,. A
is linearly converted to a suitable voltage Vu. The com- commonly available FET was used for the voltage vari-
plete scheme is shown in Fig. 3. Noting that Vgs is now able resistor and a thermistor (EUP 5k) having R , = 5k
given by at 25°C was the sensor. The experimental plot between
v2R7 I vxR7 30°C and 95°C is shown in Fig. 4. It has a slope of 4.4
vgs= - mV/"C.
R6 R8
We can find the standard error in the slope following
and if one adjusts to satisfy the method of sequential differences of the curve fitting
technique [9]. Accordingly a table was prepared with the
data (Fig. 4), and from that the mean, the mean deviation,
and standard deviation of the slope was found as 4.4
the output voltage is given by mV/"K, 0.38 mV/"K, and 0.47 mV/"K, respectively,
when the 'best' value of the slope is 4.4 f 0.17 mV/ "K
v = vR Ro / Rc with a standard error less than +4 percent.
'0
VR As long as (18) and (21) are satisified, the error, par-
I----
vpR8
'IR7 kTaR7
qvpR6 [ + In -] PR7 kT,
+-qVpR6T ticularly the external estimate of it in c, would depend on
the setting of V, and Rc Le., R 2 / ( 1 + R 5 / R 4 ) .In fact, it
where ac 2 --e-2 - (2) (2)
(19) can be shown that [9]
2 + 2
and VRis the reference voltage which can be controlled to
a certain extent for sensitivity adjustment. Letting
Thus a 0.1-percent change in each quantity on the right-
hand side of the above equation would produce an error
PATRANABIS et 01.: LINEARIZING TRANSDUCER CHARACTERISTICS 69
priate hardware. For comparison, the numerical search
technique was implemented which yielded the minimum
1.4
value of maximum deviation of 26.9"C occuring at a tem-
perature of 100°C for a range of Oo-lOO"Cwhen a ther-
mistor with initial resistance (at OOC) of lo5 Q and /3 =
3500°K was $sed, the optimal straight line having passed
through (OOC, 100 KQ) and (73°C 7.3 KQ).
Obviously, such a single stroke linearization is quite a
crude approximatien. A aumbjx@f impmvd W k s t m k e
- Theoreticol curve approximations are being considered along with a piece-
Experimental points wise linear splines technique with optimization for ob-
taining the results within acceptable deviation over the
0.9t
0.81 I I I I
entire range. The details of these techniques will be the
subject inaner ofa s&sequmt mmw*&n.
20 40 60 80 100
Tdmp. 1%)
Fig. 4. Plot of magnitude voltage I V, I versus temperature. Attempts are rna&-do li&e a patticular type of
transducer nonlinearity both through amkloghtWam and
a single stroke digital technique-while the former ap-
i? of not more than &%-6?i$ = 0.2 percent which ob- l ~ for the ~ x g p s fora
proach appeaq to b p , w ~syited e given
viously i s well within the limit of Nceptance.
IV. DIGJTAL LINEARIZATION TECHNIQUE
With the f a c i l e of comptation aow available through
digital cornpwters and nticqmessors, the problem of
transducer nonlinearity is being increasingly tackled by
them through so&m packages, However, for an arbi-
trary nonlinear transducer mponm.ehracteristic, a soft-
ware solution depends on the 'paper approach' through
mathema.tica1 meddling ef t k respame curve. It may be
mentioned that what h lkarieed with analogue hardware
using eimpler t&&iques Phd \kihh.hs cost sbould not be
D. Patranabis, Principles of Instrqmentatwn. New Delhi:
put to a digital f t y z n s - h i p m s which involves in-
creased comphxity .and/wleasacouhoy. Specifically, the 'alntost 14-t ORC ther-
two- or three-pint calibratiam techniqw [101 may be con- mistor temmlatltae tm&mq,'"IEEE Barn dn#ram. Meas., YO$.
IM-20
sidered as an iWtanc?eof mfmnce: Further, the problem D. Pat y con-
of the nodineachy af type E ( b)) may be consid- verter,
ered here to s h h simpkidigitd techniques may often A. A. K h t&yQftag4 m-
verkr using IEEE Tmns. In-
lead to unacceptable results. A single stroke linearization strum. Meas.
can be seen to have a lwge deviation over even such a J. M.Diamond, "LiAearizatlan Of IgJl$%& tkr#lomewrs and other
small range as w i d e r e d here. The approach is ta search
for fhe best fit linear resp43mse chmcteristic over the full
range and then WJthis stmigbt line as the input-output
tramfer charactcrieitie l h e , One mhad of this search is
to keep the initial paint &xed and alter the slope, simul- uit Design. Bristol: Adam
taneously finding the maxiglum dmkbtion and the value of
x where ma-m deviatiow OWP. This can be done
either (i) through numerical search or (ii) through closed of transducers,"
form solution. Both, however, can be coupled to appro-
