Chapter 6
RESISTANCE
THERMOMETRY
6.1 Principles
A resistance the thermometer is a temperature-measuring
instrument consisting of a sensor , an electrical circuit
element whose resistance varies with temperature; a
framework on which to support the sensor；a sheath by which
the sensor is protected；and wires by which the sensor is
connected to a measuring instrument(usually a bridge)，which
is used to indicate the effect of variations in the sensor
resistance.
Sir Humphry Davy had noted as early as 1821 that the
conductivity of various metals was always lower in some
inverse ratio as the temperature was higher.
Sir William Siemens, in 1871 first outlined the method of
temperature measurement by means of a platinum resistance
thermometer. It was not until 1887,however,that precision
resistance thermometry as we know it today began.
This was when Hugh Callendar’s published his famous
paper on resistance thermometers [1]．Callendar’s equation
describing the variation in platinum resistance with
temperature is still pertinent; it is now used in part to define
temperature on the IPTS（see Chapter 4）from the ice point
to the antimony point.
Resistance thermometers provide absolute temperatures in
the sense that no reference junctions are involved, and no
special extension wires are needed to connect the sensor with
the measuring instrument．
The three s that characterize resistance the thermometers are
simplicity of circuits sensitivity of measurements, and stability
of sensors.
6.2 Sensors
The sensors [2],[3] can be divided conveniently into two
basic groups: resistance temperature detectors (RTDs) and
thermistors RTDs are electrical circuit elements formed of
solid conductors(usually in wire form) that are characterized
by a positive coefficient of resistivity.
It is interesting to note in this regard that there is a direct
relation between the temperature coefficient of resistivity of
metals and the coefficient of expansion of gases .
The RTDs of general usage are of platinum, nickel and
copper.
Platinum sensors
Platinum being a noble metal , is used exclusively for
precision resistance thermometers .Platinum is stable (i.e. it is
relatively indifferent to its environment, it resists corrosion
and chemical attack, and it is not readily oxidized ), is easily
workable (i.e., it can be drawn into fine wires), has a high
melting point (i.e. it shows little volatilization below 1000℃),
and can be obtained to a high degree of purity (i.e. it has
reproducible electrical and chemical characteristics).
All this is evidenced by a simple and stable resistancetemperatures (R-T) relationship that characterizes the platinum
sensor over a wide range of temperatures.
These desirable and necessary features. however, do not
came without effort.
Platinum is actually extremely sensitive to minute
contaminating impurities and to strains (to avoid the strain
gage effect whereby extraneous changes in resistance occur ),
and the thermometer must be manufactured in a manner that
ensures freedom from contaminants.
These two effects are guarded against by elaborate
precautions.
Since the time of Callendar, the sensors have been
constructed by forming a coil of very pure platinum wire
around two mica strips that are joined to form a cross , Meyers
[4]，at the National Bureau of Standards , first wound the
wire in a fine helical coil around a steel wire mandrel, and
then wound this coil helically around a mica cross framework
( see Figure 6.1) .
 Such procedures leave the wire relatively free from
mechanical constraints as the wire experience differential
thermal expansions .
The sensor is also annealed to relieve winding strains . The
diameter of the platinum wire is on the order of 0.1 mm , and
it takes a length of about 2m to yield a satisfactory 25.5Ω
resistance at the ice point.
The resulting precision platinum sensors are usually about
1 in. long。
The sensors are then assembled in a protecting sheath
( usually a tube of pyrex or quartz ) about 18 in. long with four
gold wires joining the sensor at the bottom of the tube to four
copper connecting wires outside the tube .
The tube assembly is evacuated and filled with dry air at
about one-half atmosphere and hermetically sealed to exclude
contaminants .
Mica disks separate the internal connecting wires and also
serve to break up any convection currents in the tube [ 5 ] .
Construction is such that δ（the characteristic coefficient of
the particular thermometer as determined at the zinc point) is
between 1.488 and 1.498 .
The purity of the platinum wires is assured by testing at the
ice and steam point, where R100/R0 must be greater than
1.3925 , where the higher this value , the greater the purity of
the platinum .
The reasons for the temperature limitation of present-day
platinum res stance thermometers are threefold:
1. The pyrex tube yields to stresses above 1000℉.
2. The water of crystallization in the mica insulators may
distort the mica above 1000 ℉.
3. The platinum wire may become, sensibly thinner ( by
evaporation )at the higher temperatures , and this would cause
an increase in resistance .
The National Bureau of Standards hopes to extend the range
of the platinum resistance thermometer standard from the
antimony to the gold point , in accord with proposals of
H.Moser[6] of Germany , by use of special sheaths and special
insulators .
Several papers describing studies on long-term stability and
performance of gold point resistance- thermometer have been
published [7] ,[8] . Their conclusion is that high temperature
resistance thermometers can lead to a better practical
temperature scale up to the gold point just as prognosticated in
section 4.4.2 .
Meantime , good precision platinum resistance thermometers
are commercially available with NBS certificates up to
1000 ℉, so that construction details need not be discussed
further here .
Nickel and Copper Sensors
These are much cheaper than the platinum sensors and are
useful in many special applications. Nickel is appreciably
nonlinear , however , and has an upper temperature limit of
about 600 ℉ .
Copper is quite linear , but is limited to about 250 ℉ , and
has such a low resistance .that very accurate measurements are
required. All in all, the platinum RTDs continue to dominate
the field in present resistance thermometry.
Thermistors
Temperature-sensitive resistor materials (such as metallic
oxides) have been known to science for some time.
However ,the chief stumbling block to the widespread use of
these materials was their lack of electrical and chemical
stability.
In 1946, Becker, Green, and Pearson , at the Bell
Laboratories , reported [9] on success in producing metallic
oxides that had high negative coefficients of resistivity and
exhibited the stable characteristics necessary for
manufacturing to reproducible, specifications.
Thus thermistors ( a contraction for "thermally sensitive
resistors") are electrical circuit elements formed of solid semiconducting materials that are characterized by a high negative
co- efficient of resistivity .
At any fixed temperature , a thermistor acts exactly as any
ohmic conductor . If its temperature is permitted to change ,
however , either as a result of a change in ambient conditions
or because of a dissipation of electrical power in it , the
resistance of the thermistor is a definite , reproducible function
of its temperature .
Typical thermistor resistance variations are function of its
temperature. Typical thermistor resistance variation are from
50,000Ω at 100℉ to 200Ω at 500℉. Their characteristic
temperature-resistance relationship can approximated by a
power function of the form
R=aeb/T
(6.1)
where R is the thermistor resistance at its absolute ambient
temperature T , and a and b are constants for the particular
thermistor under consideration , with typical values of 0.06
and 8000 , respectively.
Note that the logarithm of R plots as a straight line of
intercept log a and slope b, when charted versus (l/T). Their
use for temperature measurement is based on the direct or
indirect determination of the resistance of a thermistor
immersed in the environment whose temperature is to be
measured .
Because we are dealing with an ohmic circuit ，ordinary
copper wires suffice throughout the thermistor circuit; thus
special extension wires，reference junctions , and
bothersome thermal emfs are avoided .
Contrary to common belief , thermistor are quite stable when
they are properly aged before use ( less than 0.1％ drift in
resistance during periods of months).
Thermistors exhibit great temperature sensitivity (up to ten
times the sensitivity of the usual base-metal thermocouples),
whereas thermistor response can be in the order of
milliseconds .
The practical range for which the thermistor are useful is
from the ice point to about 6000 ℉[10]. Routine temperature
measurements to 0.1oF are not unusual if the current through a
thermistor is limited to a value that does not increase the
thermistor temperature (by I2R heating) above ambient by an
amount greater than that consistent with the measuring
accuracy required .
6.3 Circuits and Bridges
The conventional Wheatstone bridge circuit( Figure6 .2a) is
not very satisfactory for platinum resistance thermometry
because
(a) the slide-wire contact resistance is a variable depending
on the condition of the slide ;
(b) the resistance of the extension wires to the sensor is a
variable depending on the temperature gradient along them ;
(c) the supply current itself causes variable self-heating
depending on the resistance of the sensor [11] .
The three extension-wire Callendar - Griffiths bridge circuit,
Figure 6.2b, while alleviating some of the faults of the
Wheatstone bridge, still is not entirely satisfactory .
The four-extension-wire Mueller bridge circuit , Figure 6.2c,
is almost always used to determine the platinum sensor
resistance .
Most of the important measuring resistors are protected from
ambient temperature changes by incorporating them in a
thermostatically controlled constant-temperature chamber .
The range of resistance for which these bridge are intended
is from 0.0 to 422.llllΩ. The temperature rise caused by the
bridge current must be kept small so that no reduction in the
current produces an observable change in the indication or
resistance .
In direct opposition to this requirement of low self-heating
current is the fact that for maximum bridge sensitivity the
bridge current should be as large as possible .
Since the time of his 1939 paper [12] , Mueller has designed
a new bridge for precision resistance thermometry. H. F.
Stimson, at the NBS, gave a brief description of this bridge
somewhat as follows [13].
The intention of the design was to make measurements
possible within an
Figure 6.2
(a) Conventional Wheatstone bridge circuit. When G is
zeroed by moving Rslide , IsensorRsensor=IslideRslide and
IsensorRA=IslideRB .Dividing one by the other , Rsensor=(RA/RB)
Rslide ; therefore Rsensoris easily determined by fixed ratio
(RA/RB) and the variable Rslide which is obtained from the
calibrated scale .
(b) The Callendar-Griffiths bridge circuit ,in which slide-wire
contact has no effect of extension-lead resistance variation is
reduced , and the effect of self-heating of the sensor is reduced
if RA= Rsensor .
(c) The Mueller bridge circuit . With the switch in one
position (as shown), R1+RT=Rsensor+ RC.
With the switch in the other position, extension leads C and
T and CD and T are interchanged. respectively ,and
RD2+RC = Rsensor+RT . then, by addition, Rsensor=(RD1+RD2)/2
and the effects of the extension leads have been eliminated.
Uncertainty not exceeding 2 or 3μΩ . This bridge has a
seventh decade that makes one step in the last decade the
equivalent of 10μΩ.
One uncertainty that was recognized was that of the contact
resistance in the dial switches of the decades . Experience
shows that with care this uncertainty can be kept down to the
order of 0.0001Ω .
In Mueller's design , the ends of the equal-resistance ratio
arms are at the dial-switch contacts on the lΩ and the 10Ω
decades .
He has made the resistances of the ratio arm 3000Ω; thus the
uncertainty of contact resistance , 0.0001Ω, produces an
uncertainty of only one part in 30, 000,000 .
With 30Ω,for example , in each of the other arms of the
bridge , these two ratio-arm contacts should produce an
uncertainty of resistance measurement of 1.41μΩ .
The 10Ω decade has mercury contacts and the commutator
switch has large mercury contacts . one new feature in the
commutator switch is the addition of an extra pair of mercurycontact links that serve to reverse the ratio coils
simultaneously with the thermometer leads .
This feature automatically makes the combined error of the
normal and reverse readings only one in a million , for
example, when the ratio arms are unequal by as much as one
part in a thousand , and thus almost eliminates any error from
lack of balance in the ratio arms .
It does not eliminate any systematic error in the contact
resistance at the end of the ratio arms on the decades ,
however , because the mercury reversing switch must be in the
arms . The commutators on these bridges are made so as to
open the battery circuit before breaking contacts in the
resistance leads to the thermometer .
This makes it possible to lift the commutator , rotate it , and
set it down in the reverse position in less than a second and
thus not disturb the galvanometer or the steady heating of the
resistor by a very significant amount .
In balancing the bridge , snap switches are used to reverse
the current so the heating is interrupted only a small fraction
of a second .
This practice of reversing the current has proved very
valuable . It not only keeps the current flowing almost
continuously in the resistor but also gives double the signal of
the bridge unbalances .
Basic thermistor circuits are given in Figure 6 .3a and b. one
is the familiar Wheatstone bridge , and the other is a simple
series circuit that is quite effective in determining thermistor
temperature as a function of the emf drop across a fixed
resistor .
6 .4
Equation and Solutions
In Chapter 4 the Callendar interpolating equation for the
platinum resistance thermometer , which was used in part to
define temperature on the IPTS between the ice and antimony
points , was given by（4.1) as
 R1  R0 
 t
 t 
t 
 1
100   

R

R
100
100



0 
 100
(6.2)
Similarly , between the oxygen and ice points , the
Callendar - VanDusen equation was given by (4 .2). The point
here is that it is one thing to measure a resistance precisely
( no mean feat in itself , as indicated in Section6.3) and quite
another to convert the measurement readily to a temperature .
Hand calculations , involving quadratics and cubics , are
practical if only a few conversions are required .
Eggenberger [ 14 ] has Presented a solution to (6 .2) , based
in part on tables and in part on graphs .
With the advent of the computer , machine solution make
resistance-to-temperature conversions routine , precise ,and
nearly instantaneous[15].Note that (6 .2) , for example , also
can be expressed as
4
6
 Rt  R0 


10
10
2
2
t 
 10  t 

0
  R100  R0 
 

(6.3)
Solving the quadratic yields
t  A  B C  R0
(6.4)
where
A
50

  10 
2
1 2
50  400 
B


  R100  R0 
C   R100
1 25 
 
 R0  
 
  R0
 
 400 2
Equation 6 .3 can be solved readily by hand , whereas（6.4)
may be more amenable to machine solution .
However , (6.4) may be manipulated still further to obtain
Eggenberger’s tabular-graphical solution ; that is , to avoid
solution of (6 .4) for each measurement , the temperature also
can be expressed as
t  t1  t  correction
(6.5)
where
t = the empirical temperature corresponding to the total
resistance measurement ;
tl =that part of the temperature measurement corresponding to
the integral part of the resistance measurement ;
Δt = the approximate part of the temperature measurement
corresponding to the decimal part of the resistance
measurement (the approximation is that the slope⊿t/⊿R is
10℃/Ω) ;
correction = the adjustment necessary to account for the
approximation of a constant temperature-resistance slope .
Equation 6.5 is synthesized as follows . First, the resistance
(Rt) of ( 6 .4 ) is expressed in parts . Thus
t  A B
 C  Ri   R0
(6.6)
where Ri represents the integral part of the resistance, and △R
the decimal part . The temperature corresponding to the
integral part of the resistance is then
ti  A  B C  Ri
(6.7)
This expression can be tabulated for the individual resistance
thermometer ( e. g ., see Table 6.l ）. The approximate
temperature, including the integral part of the resistance plus
the approximate effect of the decimal part of the resistance is
ta  A  B S  10R
(6.8)
where S = ( C-Ri ) , and the slope ， △t/△R , is assumed to
be 10℃/Ω( or18℉/Ω). The complete expression for the
temperature is then
t  ta   A  B S  R  A  B S  10R 
(6.9)
Expressed as a correction ,(6.9) yields
12

 R  
12
t  ta  BS 1  1 
(6.10)
   10R
S  
 
This correction can be plotted for the individual resistance
thermometer (e.g., see Figurc 6 . 4 ) .
Table 6.1 Typical Platinum Resistance Thermometer Chart
( absolute ohms versus degrees Fahrenheit )
Example 1．
A platinum resistance thermometer having the constants
R0=25.517Ω, R100/ R0 =1.392719, and δ=1.493 , indicate
Ri=28.5547Ω.
What is the corresponding Callendar temperature on the
Fahrenheit scale ?
A . By hand calculation of ( 6.3)
4
6


10
10
 28.5547  25.517 
2
2
t 
 10  t 

0
 1.493
 1.493  35.538  25.517 
t  6797.924t  203036.449  0
2
t  29.998C  85.996F
B . By machine calculation of (6.4 )
t=30℃=86℉(see table 6.2)
C . By Eggenberger’s method of ( 6 .5)
R1=28Ω,t1=76.106℉(see Table 6.1)
ΔR=0.5547Ω,Δt=℉9.98℉
Correction= -0.084(see Figure 6.4)
Summing: t=76.106+9.985-0.084=86.007℉
6 .5 The Calendar Coefficients
In this section , we consider a practical method for
determining the required resistance thermometer interpolating
equations [16] , [17] .
According to (4.11), namely,
T68=t’+Δt (6.12)
We are assured that the Callendar temperature ,t', as defined
by (6.2), is still very important in determining T68 .
IPTS-68 calls for the platinum resistance thermometer to be
calibrated at the triple point of water (0.01℃), the steam point
(100℃),and the zinc point(419.58℃).
However, it is common practice to use the tin
point(231.9681℃) in place of the steam point. This
complicates the computation of the Callendar coefficients in
that R100 may not be available experimentally.
Hence , to evaluate the Callendar interpolation equation
used to obtain t68 one must determine δ and R100 analytically .
A detailed description for doing this is given next.
Example 3.
A certain resistance thermometer exhibits the following
resistance at the fixed point of interest:
Resisitance
temperature
R,Ω
Fixed point
t68℃
25.56600 Triple
0.01
48.39142 Tin
231.9681
65.67700 Zinc
419.58
These resistance represent the means of many
experimental determinations ,
DETERMINATION of CALLENDAR TEMPERATURE
of TIN .
Although the Callendar temperatures of the triple and
zinc points are identical with t68s , since △tS at these points
are zero ( see equation 4 .12) , this cannot be said of the tin
point .
The temperature correction at the tin point is △tSN = 0 .
038937℃ as determined by a short iterative procedure .
First, t is assumed equal to t68 and △t is calculated from
(4 .12). Then , t is adjusted for this △t according to (6 .12) .
The process is repeated until t does not change by more
than some acceptable value , say 1 in the fifth decimal
place . For the △t so determined ，△tSN = 231.92916℃ .
DETERMINATION OF CALLENDAR COEFFICIENTS
b AND c .
The Callendar equation can be written in the entirely equivalent form
Rt=R0+bt+ct2
(6.13)
where the two unknown coefficients can be determined from
the algebraic relations
c
R0  tZN  tSN   RSN tZN  RZN tSN
tSN tZN  tZN  t SN 
2
RZN  R0  ctZN
b
tZN
where tSN= Callendar temperature at the tin point
tZN= Carllendar temperature at the zinc point
RSN =Measured resistance at the tin point
RZN =Measured resistance at the zinc point
R0 =Measured resistance at the ice point ( this should be
about 0.001Ω less than the triple point resistance ).
For the value given in this example ,
b = 1.019045×10-1
c ＝- 1.502488 ×10-5
DETERMINATION OF R100． Although R100 is no longer
a measured quantity , it can be determined readily from (6.13)
written as
R100=R0+b(102)+c(104)
For the values given , R100＝35.6052Ω and
R100//R＝1.3927323 .
(6.16)
DETERMINATION OF δ
As indicated earlier ,the characteristic constant of the
thermometer can be determined from any fixed point data
other than the ice and steam points .For example :
tZN  ptZN
tSN  ptSN


 tZN
 tZN   tSN
 tSN 
 1
 1

 

100
100
100
100


 


where pt=100 (Rt-R0)/(R100-R0)
and pt is known as the platinum temperature . For the values
given ,δ=1.496472 , and ptSN = 227.35022℃ and
ptZN＝399.51390℃ . Note that the platinum temperatures and
the resistances measured at the fixed points are independent of
the temperature scale being used .
GENERAL OBSERVATIONS .
(l) Since the IPTS-68 △t is zero at the ice , steam , zinc ,
and antimony points , it follows that the R –t68 points
representing these baths are all on the 68-Callendar parabola .
(2) It further follows that the tin and sulfur points are
not ． △tsN has already been given in this example , By the
same method ， △tsu＝-0.012174(See Figure 6.5).
(3) Given the Callendar constants , b and c . and the
68-Callendar temperatures at the sulfur and antimony
points ,the corresponding resistances can be determined from
(6.13) as R SU = 67.90940Ωand RSD = 83.8628Ω.
4. A contain resistance thermometer yields the following
resistance at the fixed points of interest:
 Resistance
temperature
 R,Ω
Fixed point t68℃
 6.2268
oxygen
-182.962
 25.5500
ice
0
 35.5808
steam
100
 65.67700
Zinc
419.58
RESISTANCE THERMOMITER COEFFICIENTS .
Platinum temperature at the zinc point :
ptZN=100(RZN-R0)/(R100-R0)=4007.46/10.0308=399.5158℃
Fundamental coefficient :
α=(R100-R0)/(100R0)=10.0308/2555.00=0.0039259591
Callendar constant :
tZN  ptZN
419.58  399.5158


＝1.496334
 tZN
 tZN   3.1958  4.1958 
 1


100
100



VanDusen Constant :

100  Rox  R0 
 tox
 tox 
tox 
 
 1

R
100

R
0
100
100

 


 tox
 tox 
 1


100
100



3
 0.1113
where tQX = Callendar - VanDusen temperature at the oxygen
point
ROX =measured resistance at the oxygen point .
Measured resistance ratios :
WM ox
Rox
6.2268


 0.24371037
Ro 25.5500
WM 100
R100 35.5808


 1.39259491
R0
25.5500
By equation 4.10 and Table 4.6 :
 W4 ox  0.24371037  0.24379909  0.00008872
 W4 100  1.39259491  1.39259668  0.00000177
Also , by Table4 .6 , the interpolating polynomial for
determining deviations in Range l , Part 4 is:
W4  A4t  C4t 3 t 100
By simultaneous solution at the oxygen and steam points ,
the constants , A4 and C4 , are determined to be :
A4  0.177  10－7
C4＝－0.5306  10－13
With this information about this resistance thermometer
available , we can ask for example : If the measured resistance
ratio is W4 =0.5 , what is t68?
To determine △W4 by (6 .18 ) , one must estimate t and
then show by iteration ( trial and error ) that this estimate is
close enough to satisfy accuracy requirements .
A useful estimate of t is given by the Callendar-VanDusen
equation (4 .2) , which can be rewritten as
tcvd
 t
 t 
 t
 t 
 WM  1   
 1
 1
 


100
100
100
100






1
3
With the thermometer constants given , and by standard
calculation procedures
tcvd= -122.8038℃
Using this value of temperature, △W4 is determined to be
ΔW4=-0.00001973
By (4 .10) , the reference resistance ratio is
Wref=0.5+0.00001973=0.50001973
By (4.9 ) , an estimate of t68 is found by computer
calculation to be
t68= -122.8128℃
Since △W does not change significantly for the small
temperature difference(t68 - tCVD) of -0.009 K , this estimate of
t68 can serve as t68．