Transcript chapter 7 thermoelectric thermometry

Chapter 7
THERMOELECTRIC
THERMOMETRY
Thomson refers to this as the specific heat of electricity
because of an apparent analogy between and the usual
specific heat c of thermodynamics.
Note that represents the rate at which heat is absorbed , or
evolved , per unit temperature difference per unit current ,
whereas c represents the heat transfer per unit temperature
difference per unit mass .
The Thomson coefficient is also seen to represent an
emf per unit difference in temperature. Thus the total
Thomson voltage set up in a single conductor may be
expressed as
7.1 Historical Development of Basic Relations
Thomas Johann Seebeck , the German physicist , discovered
in 1821 the existence of thermoelectric currents while
observing electromagnetic effects associated with bismuthcopper and bismuth-antimony circuits [1],[2].
His experiments showed that , when the junctions of two
dissimilar metals forming a closed circuit are exposed to
different temperatures , a net thermal electromotive force is
generated that induces a continuous electric current .
The Seebeck effect concerns the net conversion of thermal
energy into electrical energy with the appearance of an electric
current.
The Seebeck voltage refers to the net thermal electromotive
force set up in a thermocouple under zero-current conditions.
The direction and magnitude of the Seebeck voltage E, depend
on the temperatures of the junctions and on the materials
making up the thermocouple.
For a particular combination of materials, A and B, and for
a small temperature difference, dt, the Seebeck voltage may be
written
dEs   A, B dt
(7.1)
where αA,Bis a coefficient of proportionality called the
Seebeck coefficient. (This is commonly called the
thermoelectric power).
Thus the Seebeck coefficient represents, for a given material
combination, the net change in thermal emf caused by a unit
temperature difference; that is ,
 A, B
Es dEs
 lim it

T 0 t
dt
(7.2)
Thus if E  at  1 bt 2
2
2
then   a  bt .
determined by calibration,
Note that, based on the validity of the experimental
relation,
t
t
t2
t2
t1
t1
Es    dT    dT    dT
(7.11)
where t1 < t2 < t ,
It follows that is entirely independent of the reference
temperature employed .
In other words, for a given combination of materials, the
Seebeck coefficient is a function of temperature level only.
Jean Charles Athanase Peltier, the French physicist,
discovered in 1834 peculiar thermal effects when he
introduced small, external electric currents in Seebeck’s
bismuth- antimony thermocouple .
His experiments show that when a small electric current
is passed across the junction of two dissimilar metals in one
direction, the junction is cooled (i.e., it acts as a heat sink)
and thus absorbs heat from its surroundings.
When the direction of the current is reversed, the junction
is heated (i.e., it acts as a heat source) and thus releases heat
to its surroundings.
The Peltier effect concerns the reversible evolution, or
absorption, of heat that usually takes place when an electric
current crosses a junction between two dissimilar metals. (In
certain combinations of metals, at certain temperatures,
there are thermoelectric neutral points where no Peltier
effect is apparent).
This Peltier effect takes place whether the current is
introduced externally or is induced by the thermocouple
itself.
The Peltier heat was early found to be proportional to the
current and may be written
dQP   Id
(7.12)
Where л is a coefficient of proportionality known as the
Peltier coefficient or the Peltier voltage.
Note thatл represents the reversible heat that is absorbed,
or evolved, at the junction when unit current passes across
the junction in unit time, and that it has the dimensions of
voltage.
The direction and magnitude of the Peltier voltage
depend on the temperature of the junction and on the
materials making up the junction ; however，л at one
junction is independent of the temperature of the other
junction .

External heating, or cooling, of the junctions results in
the converse of the Peltier effect.
Even in the absence of all other thermoelectric effects,
when the temperature of one junction (the reference
junction) is held constant and the temperature of the other
junction is increased by external heating , a net electric
current will be induced in one direction .
If the temperature of the latter junction is reduced below
the reference junction temperature by external cooling, the
direction of the electric current will be reversed.
Thus the Peltier effect is seen to be closely related to the
Seebeck effect.
Peltier himself observed that, for a given electric
current, the rate of absorption, or liberation, of heat at a
thermoelectric junction depends on the Seebeck
coefficient a of the two materials.
Note that the Peltier thermal effects build up a
potential difference that opposes the thermoelectric
current, thus negating the perpetual-motion question.
William Thomson (Lord Kelvin), the English
physicist, came to the remarkable conclusion in 1851
that an electric current produces different thermal
By applying the (then) new principles of thermodynamics
to the thermocouple and by disregarding (with tongue-incheek) the irreversible and conduction-heating processes,
Thomson reasoned that, if an electric current produced
only the reversible Peltier heating effects, the net Peltier
voltage would equal the Seebeck voltage and would be
linearly proportional to the temperature difference at the
junctions of the thermocouple .

This reasoning led to requirements at variance with
observed characteristics (i.e., dEs/dt constant).
Therefore, Thomson concluded that the net Peltier voltage
was not the only source of emf in a thermocouple circuit but
that the single conductor itself, whenever it is exposed to a
longitudinal temperature gradient, must also be a seat of
emf. A. C.
Becquerel had at that time already discovered a
thermoelectric neutral point, that is, Es = 0, for an
iron-copper couple at 280℃. Thomson agreed with
Becquerel’s conclusion and started his thermodynamic
reasoning from there.
The Thomson effect concerns the reversible evolution, or
absorption, of heat occurring whenever an electric current
traverses a single homogeneous conductor, across which a
temperature gradient is maintained, regardless of external
introduction of the current or its induction by the
thermocouple itself.
The Thomson heat absorbed, or generated, in a unit
volume of a conductor is proportional to the temperature
difference and to the current; that is,
T2

dQT      dT  Id
 T1

(7.13)
where σ is a coefficient of proportionality called the
Thomson coefficient.
T2
ET    dT
(7.14)
T1
where its direction and magnitude depend on temperature
level, temperature difference, and material considered.
Note that the Thomson voltage alone cannot sustain a
current in a single homogeneous conductor forming a closed
circuit, since equal and opposite emfs will be set up in the
two paths from heated to cooled parts.
Soon after his heuristic reasoning, Thomson succeeded in
demonstrating indirectly the existence of the predicted
Thomson emfs (see Figure 7. 1).
He sent an external electric current through a closed
circuit formed of a single homogeneous conductor that was
subjected to a temperature gradient and found the heat to be
slightly augmented or diminished by the reversible
Thomson heat in the paths from
Figure7.1 Thomson experiment to demonstrate existence
of reversible emfs in a single conductor.
(1) Points A and B can be found, when the switch is open,
having the same temperature.
(2)When an electric current is passed through the copper
bar, point A becomes cooler while point B becomes hotter.
(3) That is, Joule heating is slightly affected by the
reversible Thomson emf , which either opposes or adds to
the external emf when the copper bar is preferentially
heated.
(4) Thomson concluded that hot copper was electrically
positive with respect to cold copper, cold to hot or from hot
to cold, depending on the direction of the current and the
material under test.
In summary, thermoelectric currents may exist whenever
the junctions of a circuit formed of at least two dissimilar
materials are exposed to different temperatures.
This temperature difference is always accompanied by
irreversible Fourier heat conduction;
The passage of electric currents is always accompanied
by irreversible Joule heating effects.
At the same time, the passage of electric currents is
always accompanied by reversible Peltier heating or cooling
effects at the junctions of the dissimilar metals;
The combined temperature difference and passage of
electric current is always accompanied by reversible
Thomson heating or cooling effects along the conductors.
The two reversible heating-cooling effects are
manifestations of four distinct emfs that make up the net
Seebeck emf:
Es   A, B t2   A, B t1    A dT    B dT    A, B dt
(7.17)
where the three coefficients, α,π,σ are related by the Kelvin
relations.
7.2 The Kelvin Relations
Assuming that the irreversible and heat-conduction
effects can be completely disregarded (actually, they can
only be minimized, since, if thermal conductivity is
decreased , electrical resistivity usually is increased, and
vice versa , see Figure 7.2) , the net rate of absorption of
heat required by the thermocouple to maintain equilibrium
in the presence of an electric current is
2
Qnet 
q
  1   2   ( A   B )dT  I  Es I

1
 
(7.15)
This is in accord with the first law of
thermodynamics, according to which heat and work
are mutually convertible.
Thus the net heat absorbed must equal the electric
work accomplished or, in terms of a unit charge of
electricity, must equal the Seebeck emf Es, which
may be expressed in the differential form
dEs  d  ( A   B )dT
(7.16)
The second law of thermodynamics may also be applied
to the thermocouple cycle (the unit charge of electricity
again being considered) as
Sreversible  
Q
Tabsolute
0
(7.18)
where ΔQ represents the various components of the net
heat absorbed (i.e., the components of Es), and Tabsolute is
the temperature at which the heat is transferred across the
system boundaries.
Equation 7.10 can be expressed in the differential form
dSreversible

d
T
 ( A   B )
dT  0

T

(7.19)
Combining the differential expressions for the first and
second laws of thermodynamics, we obtain the Kelvin
relations
 dEs 
(7.20)
 A, B  Tabsolute 
  Tabsolute A, B
 dT 
 d 2 Es 
( A   B )  Tabsolute 
2 
 dT 
from which we can determine, α,π,Δσ , when
Es is obtained as a function of T.
(7.21)
1 2
(7.22)
Es  at  bt 
2
is taken to represent the thermoelectric characteristic of a
thermocouple whose reference junction is maintained at
0℃ in which the coefficients a and b are obtained, for
example , by the curve fitting of calibration data , then
  (a  bt  )
  Tabsolute (a  bt  )
  Tabsolute (b  )
(7.23)
(7.24)
(7.25)
An example of the use of these coefficients is given
in Figure 7.3.
Given the two constants, a and b, as determined with
respect to platinum,
Metal
a，μ V/℃
b, μ V/℃
Iron(I)
+16.7
-0.0297
Copper(Cu)
+2.7
+0.0079
Constantan(C)
-34.6
-0.0558
By way of illustration, consider the following
combinations of materials: iron-copper and
iron-constantan, with their measuring junctions at
200℃ and their reference junctions at 0℃:
Iron  copper
aI Cu  aI  aCu  16.7  2.7  14 V / C
bI Cu  bI  bCu  0.0297  0.0079  0.0376 V / C
Iron  cons tan tan
aI Cu  aI  aCu  16.7  (34.6)  51.3V / C
bI Cu  bI  bCu  0.0297  (0.0558)  0.0261V / C
Since Seebeck voltage
1 2
Es  at  bt
2
Iron  copper
1
Es  14(200)  ( 0.0376)(200) 2
2
Es  2048V
Iron  cons tan tan
1
Es  51.3(200)  (0.0261)(200)2
2
Es  10, 782V
Figure 7.3 Determination of various thermoelectric
quantities applied to thermocouples
Note how different combinations of materials give
widely different thermal emfs. Now we proceed to write
expressions for α,π and Δσ to note how the separate emfs
combine to give the ( net ) Seebeck emf. Since
Seebeck coefficient,
 A,B  a A,B  bA,BT 
Iron  copper
 0  14  (0.0376)(0)  14 V / C
 200  14  (0.0376)(200)  6048V / C
Iron  cons tan tan
 0  51.3  0.0261(0)  51.3V / C
 200  51.3  0.0261(200)  56.52 V / C
Note that it is the great difference in Seebeck coefficients
(thermoelectric powers) for the two combinations that
accounts for the difference in thermal emfs:
T
Es    A, B dT
TR
Since
 A,B  TabsoluteA,B
=Peltier coefficient
=Peltier voltage
Iron  copper
 0  273(14)  3822V
 200  473(6.48)  3065V
Iron-const an tan
 0  273(51.3)  14, 005V
 200  473(56.52)  26, 734V
Note that, in the case of the iron-copper ( I-Cu )
couple,  cold   hot , where as in the more usual I-C
couple,  hot   cold
.
Since  A,B  bA,BTabsolute =Thomson coefficient ,
T
and
1
ET    dT  bA, B (TR2  T 2 )
2
TR
＝ Thomson coefficient
Figure 7.3 continued
Iron  copper
0.0376
ET  
(2732  4732 )  2805V
2
Iron  cons tan tan
0.0261
ET 
(2732  4732 )  1947 V
2
We sum the various components
2
Es   2   1    dT  SeebeckVoltage
1
Iron  copper
Es  3065  3822  2805  2048V
Iron  cons tan tan
Es  26, 734  14, 005  1947  10, 782V
These figures, of course, check with the original
calculations. Note that, in the I-Cu case, the net Thomson
emf far outweighs in importance the net Peltier emf,
whereas in the I-C case , the converse is true .
Figure 7.3 (concluded)
7. 3 Microscopic Viewpoint of Thermoelectricity
What gives rise to these thermoelectric effects ? Recall
from your chemistry that when the loosely bound outer
(valence) electrons of a material absorb enough energy from
external sources, they may become essentially free from the
influence of their nuclei.
Once free, these electrons can absorb any amount of
energy supplied to them, and it is usual to suppose that the
free electrons in a metal act collectively as an idealized gas
(see Chapter2)。
Even at a common temperature, however, the energies
and densities of the free electrons in different materials need
not be the same.
Thus when two different materials in thermal
equilibrium with each other are brought in contact, there
usually will be a tendency for electrons to diffuse across the
interface.
The electric potential of the material accepting electrons
would become more negative at the interface, whereas that
of the material emitting electrons would become more
positive. In other words, an electric field would be set up by
the electron displacement that opposes the osmotic process.
When the difference in potential across the interface just
balances the thermoelectric (diffusion) force, equilibrium
with respect to a transfer of electrons would be established.
If two different homogeneous materials are formed into
a closed circuit and the two junctions are maintained at the
same temperature, the resultant electric fields would
exactly oppose each other, and there would be no net
electron flow.
However, if these two junctions are maintained at
different temperatures , a net diffusion current will be
induced in coincidence with a net electric field (the
random motion of the free electrons is on the average in
the direction of the net potential gradient , and this gives
rise to the electric current).
From conservation principles, the power to drive this
electric current could only come from a net absorption of
thermal energy from the surroundings by the free
electrons of the materials, since there is no observable
change in the nature or composition of the thermoelectric
materials.
But what accounts for this net absorption of heat?
Why does the thermocouple act as a heat engine (i.e., a
device that makes available as work some portion of the
thermal energy acquired from a source)?
Consider first a closed circuit of a single material (thus
avoiding for a time the thermoelectric effects). Under the
influence of a temperature gradient alone, the material will
conduct a thermal current. But all the thermal energy
absorbed by the circuit at one zone is rejected by the circuit
at another zone.
Thus the Fourier effect exhibits no accumulation of heat
in the steady state, and cannot account for a thermally
induced electric current in a thermocouple circuit.
Again, under the influence of a voltage gradient alone, this
same single material closed circuit will con- duct an electric
current.
But the only thermal effect associated with this current is
the inevitable heat generation. Thus the Joule effect
exhibits no absorption of heat from outside the circuit and
cannot account for a thermally induced electric current in a
thermocouple circuit.
Evidently, when examining the mechanism by which a
closed circuit formed of two unlike materials acts as a heat
engine, we need concern ourselves only with the reversible
thermoelectric effects. (Perhaps similar reasoning guided
Thomson when he successfully obtained the valid Thomson
relations by intuitively disregarding the irreversible Joule
and Fourier processes.)
From this viewpoint, the resultant electric current in the
thermocouple circuit having its junctions maintained at
different temperatures agrees in direction with the natural
(Peltier) potential gradient at the hot junction , and thus
tends to wipe out ( do away with the need for ) this field .
At the cold junction, this same thermally induced
current must cross the interface against the natural
potential gradient, and this tends to build up a stronger
field of opposition there.
In terms of the idealized gas, the free electrons
expand isothermally across the hot junction interface
because the flow is in the direction of the Peltier
potential gradient there.
This expansion tends to cool the hot junction but, in
face of the isothermal restriction and in accord with the
first law of thermodynamics, the hot junction absorbs
just enough heat from its surroundings to maintain its
temperature.
These Peltier (junction) effects, although usually
predominating, do not tell the complete story. As
Thomson first pointed out, there will be thermoelectric
heating effects along each of the materials making up the
circuit whenever an electric current and a temperature
gradient exist.
The Thomson effect may be visualized as follows. If a
single conductor is preferentially heated, electrons
usually tend to leave the hot end more frequently than
the cold.
Just as we found at the junctions, an opposing electric
field would be set up along the single conductor by this
diffusion of electrons.
Thus whenever an electric current occurs in a closed
thermocouple circuit, it must either agree with or oppose
these Thomson (material) potential gradients, and this
accounts for heating and cooling effects in addition to the
Peltier (junction) effects discussed previously.
Hence the thermocouple in a temperature gradient does
qualify as a heat engine in that there will always be a net
absorption of heat from its surroundings per unit time , and
we see once more that it is just this excess of thermal power
arising entirely from reversible thermoelectric effects that
sustains the thermoelectric current in the circuit .
7 .4 Macroscopic Viewpoint of Thermoelectricity
The historical viewpoint presented thus far has
avoided the real irreversible I2R and heat conduction
effects in order to arrive at the useful and
experimentally confirmed Kelvin relations. We now
discuss
how
the
present-day
irreversible
thermodynamic viewpoint removes this flaw in our
reasoning.
Basically, we judge whether a given process is
reversible or irreversible by noting the change in entropy
accompanying a given change in the thermodynamic
state. Thus, if ds  Qq / Tabsolute , we say the process is
irreversible ; or , stated in a more useful manner ,
dSsysterm  dSacross  dS produced
boundary
or
dSs  dSo  dSi 
 Qq
Tabsolute
inside

F
Tabsolute
(7.26)
(7.27)
Hence only in the absence of entropy within the system
boundaries do we have the reversible case,
,
dsreversible   Qq / Tabsolute which may be handled adequately by
classical thermodynamics in the steady and quasi-steady
states .
Evidently the rate of production of entropy per unit
volumeξis an important quantity in irreversible
thermodynamics . It may be expressed as
 1  dSi  1   F
 



 Adx  d  Adx  Tabsolute
(7.28)
where Adx is the area times the differential length.
Another significant quantity, the product Tabsolute（called
the dissipation) , can always be split either into two terms or
into a sum of two terms , one associated with a flow J, and
the other associated with a force X.
Furthermore, in many simple cases a linear relation
is found (by experiment) to exist between the flow
and force terms so defined; for example, in the onedimensional, isothermal, steady flow of electric
charges, Qe / d , across a potential gradient-dE/dx , it
may be shown that
I  dE 
Tabsolute   
  Je X e
A  dx 
(7.29)
where Je and Xe represent, respectively, the electric
flow and force terms, as defined by the entropy
production method. (D.G. Miller gives an excellent
review of the thermodynamic theory of irreversible
processes .
We follow his development closely. The term Je
represents the electric current density and the term Xe the
electric field strength or the electromotive force.
These are, of course, related by the linear Ohm's law
(i.e., Je = LeXe, where Le represents the electrical
conductivity).
Again, in the one-dimensional, steady flow of
thermal charges Qq / d across a temperature gradient
- dT/dx, it may be shown that
 Q   1 dT 
Tabsolute    
  Jq X q
 A   Tabsolute dx 
(7.30)
where Jq and Xq represent, respectively, the thermal flow
and force, as defined by the entropy production method. The
term Jq represents the thermal current density, and the term
Xq represents the thermomotive force.
These are, of course, related by the linear Fourier’s law
(i.e., Jq = LqXq where Lq represents the product of the
thermal conductivity and the absolute temperature).
It has been found that even in complex situations it may
always be stated that
Tabsolute   Jk X k
(7.31)
When several irreversible transport processes occur
simultaneously (e.g., the electric and thermal conduction in a
thermocouple), they will usually interfere with each other.
Therefore, the linear relations must be generalized to
include the various possible interaction terms.
Thus for the combined electric and thermal effects we
would write
Je  Lee X e  Leq X q
(7.32)
J p  Lqe X e  Lqq X q
(7.33)
Ji   Lij X j
(7.34)
Or, in general
We have just seen that an entropy production necessarily
accompanies both the and heat conduction effects (i.e., they
are irreversible); therefore, the Kelvin relations could not
follow from reversible thermodynamic theory without
certain intuitive assumptions.
By reasoning that the electric and thermal currents were
independent, Thomson tacitly assumed that Leq＝ Lqe as
we shall show subsequently.
Experimentally, this reciprocal relationship was often
found to be true. The American chemist Lars Onsager
proved, in 1931, from a statistical-mechanics viewpoint,
that the assumption
(7.35)
L L
ij
ji
is always true when the linear relations between flows Jk
and forces Xk are valid . (For this work Onsager
subsequently was awarded a Nobel Prize.)
The Onsager reciprocal relation forms the basis of
irreversible thermodynamics .
By applying these concepts to the processes involved in
the thermocouple, we are led rationally and unambiguously
to the Kelvin relations .
Thus whenever the junctions of a thermocouple are
maintained at different temperatures, we expect that an
electric potential difference, an electric current, and a
thermal current will be present .
The dissipation for this thermoelectric process is simply
the sum of the electric and thermal terms previously given.
That is
I  dE  Q  1 dT 
Tabsolute   

 
A  dx  A  Tabsolute dx 
(7.36)
Leq  dT 
 dE 
J e  Lee  



 dT  Tabsolute  dx 
(7.37)
The generalized linear laws for this case have also
been given as
Lqq  dT 
 dE 
(7.38)
J q  Lqe  



 dT  Tabsolute  dx 
Recalling that the Seebeck emf is determined under
Leq
dE


s
(7.39)
conditions of zero
the
  electric
  current, the Seebeck
 dT  I 0 LeeTabsolute
coefficientsαmay be expressed in terms of the Onsager
Recalling that the Peltier coefficientπ represents the heat
absorbed or evolved with the passage of an electric current
across an isothermal junction, this too may be expressed in
terms of the Onsager coefficients as
Lqe
 Jq 
  

 J e  dT  0 Lee
(7.40)
Finally we recall that Thomson found experimentally (and
expressed in the Kelvin relations) that the Seebeck and
Peltier coefficients are related, Thus
 dEs 
  Tabsolute 

dT

 I 0
(7.41)
In terms of the Onsager coefficients, this requires that
 Leq 
 Tabsolute 

Lee
 LeeTabsolute 
Lqe
(7.42)
which indicates that the experimental results agree with
those which are predicted by the entropy-production linearlaw Onsager reciprocal relation approach;
Lqe  Leq
(7.43)
In other words, by irreversible thermodynamics, without
using any intuitive assumption. The Kelvin relations, also in
accord with experiment, must follow.
7.5 Laws of Thermoelectric Circuits
We have seen that, in 1821, T. J. Seebeck discovered
"thermomagnetism" when the junctions between two
different metals forming a closed circle were subjected
to a temperature difference.
Just a few years later, in 1826, A.C. Becquerel read a
paper to the Royal Academy of Sciences in Paris
describing the first recorded suggestion to make use of
Seebeck’s discovery as a means of measuring
temperature.
It was not until 1885, however, that the problem of
He determined experimentally that the use of a platinum
wire against a platinum rhodium alloy wire gave the most
satisfactory "thermo-electric” results . LeChatelier’s work is
still the basis of modern standard thermoelectric
thermometry.
Numerous investigations of thermoelectric circuits ,in
which accurate measurements were made of the current ,
resistance , and electromotive force have resulted in the
establishment of several basic laws.
These laws have been established experimentally beyond
a reasonable doubt and may be accepted in spite of any lack
of a theoretical development.
Law of Homogeneous Materials
A thermoelectric current cannot be sustained in a
circuit of a single homogeneous material however it
varies in cross section by the application of heat alone
A consequence of this law is that two different
materials are required for any thermocouple circuit.
Experiments have been reported suggesting that a
nonsymmetrical temperature gradient in a homogeneous
wire gives rise to a measurable thermoelectric emf.
A preponderance of evidence, however, indicates that
any emf observed in such a circuit arises from effects of
local inhomogeneities.
Furthermore, any current detected in such a circuit
when the wire is heated in any way whatever is taken as
evidence that the wire is inhomogeneous.
Law of Intermediate Materials
The algebraic sum of the thermoelectromotive forces
in a circuit composed of any number of dissimilar
materials is zero if all of the circuit is at a uniform
temperature.
A consequence of this law is that a third
homogeneous material can always be added in a circuit
with no effect on the net emf of the circuit as long as its
extremities are at the same temperature.
Therefore, it is evident that a device for measuring the
thermo-electromotive force may be introduced into a circuit
at any point without affecting the resultant emf, provided all
of the junctions that are added to the circuit by introducing
the device are all at the same temperature.
It also follows that any junction whose temperature is
uniform and that makes good electrical contact does not
affect the emf of the thermoelectric circuit regardless of the
method employed in forming the junction (see Figure 7.4).
Figure 7.4 The emf is unaffected by the third material C
Another consequence of this law may be stated as
follows. If the thermal emfs of any two metals with
respect to a reference metal (such as C) are known, the
emf of the combination of the two metals is the
algebraic sum of their emfs against the reference metal
(see Figure 7.5).
Law of Successive or Intermediate Temperature
If two dissimilar homogeneous materials produce a
thermal emf of E1: when the junctions are at
temperatures T1, and T2, and a thermal emf of E2 when
the junctions are at T2 and T3,the emf generated when
the junctions are at T1 and T3 will be E1 + E2 .
One consequence of this law permits a thermocouple
calibrated for a given reference temperature to be used with
any other reference temperature through the use of a
suitable correction (see Figure 7.6).
Another consequence of this law is that extension wires
having the same thermoelectric characteristics as those of
the thermocouple wires can be introduced in the
thermocouple circuit (say from region T2 to region T3 in
Figure 7.6) without affecting the net emf of the
thermocouple.
7.6 Elementary Thermoelectric Circuits
Two continuous, dissimilar thermocouple wires
extending from the measuring junction to the reference
junction, when used with copper connecting wires and a
potentiometer connected as shown in Figure 7.7, make up
the basic thermocouple circuit.
The ideal circuit given in Figure 7.8 is for use when
more than one thermocouple is involved. Note that each
thermocouple is made up of two continuous wires
between the measuring and reference junctions. This
circuit should be used for all precise work in
thermoelectric thermometry.
A circuit that requires only one reference junction is
known in Bureau of Standards publications as a zone-box
circuit. A diagram of one is shown in Figure 7.9.
Such a circuit should not be used when uncertainties
under 1F are required.
The usual thermocouple circuit, however, includes
measuring junctions, thermocouple extension wires,
reference junctions, copper connecting wires, a selector
switch, and a potentiometer, as indicated in Figure 7.10.
The uncertainties introduced by this circuit should be
expected to be greater than 1℉.
Many different circuit arrangements of these
components also are acceptable, depending on given
circumstances, and some are shown in Figure 7.11a, b,
and c. These circuits can be characterized briefly as
follows:
a. Thermopile, for sensing average temperature level.
Advantages: magnifies thermoelectric power by the
number of couples in series, enabling the detection of
small changes in temperature level; one observation
gives, by calibration, the arithmetic mean of
Precautions: the emf may be too great for the indicator;
a short circuit may escape detection (giving erroneous
readings); this thermopile gives no indication of
temperature distribution; and the temperature / emf
calibrations of all couples must be identical.
b. Parallel arrangement, for sensing average temperature
level.
Advantage: one observation gives, by calibration, the
arithmetic mean of the temperatures sensed by N
individual measuring junctions.
Precautions: gives no indication of temperature distribution;
the temperature/emf calibrations of all couples must be
identical and linear; and all couples must have equal
resistances.
c. Thermopile, for sensing average temperature differences.
Advantages: the signal is magnified by the number of
couples in series ( permits detection of very small
temperature differences), and one observation, by
calibration , gives the arithmetic mean of the temperature
differences sensed by two groups of N measuring junctions
when the total emf is divided by N.
Precautions: temperature level is not indicated; a short
circuit may escape detection; and the temperature/emf
calibrations of all couples must be identical. A typical
ice bath is shown in Figure 7.12.
Various circuits for using potentiometers,
compensated to indicate temperature directly and
uncompensated to indicate voltages directly, are shown
in Figure 7.13.
Grounded Thermocouple Circuits
Relatively little attention has been paid to electrical
effects that are extraneous to the basic thermocouple
circuit. Prime examples of external effects not always
considered are: effects of electrical and magnetic fields,
cross-talk effects, and effects connected with common
mode voltage rejection.
An excellent discussion of these noise effects in
general instrument circuits and of methods for reducing
them is given by Klipec . (On common mode voltage
1. Electric fields radiated from voltage sources are
capacitively coupled to thermocouple extension wires
and cause an alternating noise signal to be
superimposed on the desired signal.
Electric noise is minimized by shielding the
thermocouple extension wire and grounding the shield.
2. Magnetic fields radiated from current-carrying
conductors produce noise current and hence noise
voltage in the thermocouple circuit. Magnetic noise is
minimized by twisting the thermocouple extension
3.
Adjacent pairs of a multipair cable tend to pick up noise
when pulsating d.c. signals are transmitted. Cross-talk
noise is minimized by shielding the individual pairs of
thermocouple extension wires.
4.
Electrical connections made between a thermocouple and
a grounded instrument may introduce common mode
noise if different ground potentials exist along the wire
path.
Common mode noise is minimized by grounding the
thermocouple and its shielding at a single point as close
as practical to the measuring junction.
Several arrangements of thermocouple/extension
wire/shield/ground combinations, acceptable from the
noise viewpoint, are shown in Figure 7.14.
It is evident from the foregoing that the grounding of
thermocouple circuits is an important consideration. It
is a fact that grounding is often done improperly.
It is shown in Section 7.9, under circuit analysis, that
the introduction of multiple grounds in a thermocouple
circuit can lead to serious temperature errors, errors that
are over and above those associated with thermoelectromotive force generation, with static and dynamic
Circuits for Rotating Parts
Often in engineering the temperature of a rotating part
must be obtained. There are several methods for
accomplishing this task.
A rotating transformer may be used where in one coil
remains stationary while the other rotates, and hence the
signal is passed electrically from rotating to stationary parts.
Radio telemetry may be employed wherein a rotating
transmitter sends the temperature signal from a rotating
sensor to a stationary indicating device by means of a radio
frequency signal, antenna, and receiver system.
However, the most usual method is by a slip ring
arrangement wherein the electrical signal from a
rotating thermocouple is passed to the stationary world
by means of rotating slip rings and fixed brushes.
These three methods are illustrated in Figure 7.15.
The slip ring circuit that has proven to be useful for the
highest accuracy temperature measurement of a rotating
part is shown in Figure 7.16a.
This makes use of a separate rotating thermocouple to
sense the temperature in the zone where the necessary
In Figure 7.16b, the accuracy of the basic slip ring
circuit is compromised through the built-in assumption
that a common temperature exists at both the rotating
and the fixed terminals of the slip ring assembly.
If this can be safely posited, or if the temperature
difference across the slip ring can be tolerated in the
overall measurement accuracy (e.g., a 2°Δt across the
slip ring will introduce a 2°error in the measurement),
the simplification of circuit b over a may be warranted.
A third arrangement, wherein the rotating zone
thermocouple of Figure 7.16a is replaced by a fixed
zone thermocouple extending into the slip ring
assembly may sometimes be used if it is felt that such a
fixed zone thermocouple adequately proclaims the
temperature in the rotating portion of the slip ring
assembly.
A fourth arrangement, wherein a rotating resistance
thermometer (RTD, see Chapter 6) is used in place of
any of the above rotating or fixed zone thermocouple
junctions, can be used to advantage if it is important
7.7 Circuit Component Uncertainties
Typical uncertainties that can accompany the various
circuit components are discussed briefly below .
Measuring Junction
The junction of a thermocouple that is at the
temperature to be measured is referred to as the
measuring junction.
Each measuring junction has its own peculiar
characteristics depending on such factors as materials,
methods of joining, age, and history of junction. Yet the
measuring uncertainties introduced by this junction are
small;
They are accounted for in the overall circuit
Thermocouple Extension Wires
Extension wires are thermoelements inserted between
the measuring junction and the reference junction that
have approximately the same thermoelectric properties
as the thermocouple wire with which they are used (the
A' and B' wires of Figure 7.10).
The use of extension wires introduces at their
extremities at least four extraneous thermoelectric
junctions in each thermocouple circuit.
Even with normal usage, the resulting uncertainties can
amount to as much as, depending on the temperatures at the
ends of the extension wires, and on the particular wires used.
Such large uncertainties can be minimized by calibrating
with given extension wires and then using the same wires in
the field, maintaining (as nearly as possible) the same
temperatures at their extremities in both circumstances.
These uncertainties can be entirely avoided by not
introducing any extension wires in the thermocouple proper.
Selector Switch
When placed in any circuit other than the basic circuit
the switch can introduce uncertainties of ±1.5℃. It is
not enough to keep a switch at a uniform temperature,
nor even at a constant temperature.
To minimize uncertainties, the switch must be kept at
the same temperature in the field as it was during the
calibration. The uncertainties can be avoided entirely by
placing the switch in the copper extension Wires
leading from the reference junctions to the
Reference Junctions
The junction of a thermocouple that is maintained at a
known temperature is referred to as the reference junction.
In any but the basic circuit, reference junctions can
introduce uncertainties of ±1℉.
These can be minimized by using the same reference
junction in the field as was used in the calibration.
The uncertainties can be avoided entirely by using a
proper thermocouple, that is, one for which the measuring
and reference junctions are simply the extremities of the
thermocouple wires.
Copper Connecting Wires
Copper wires used to connect the reference junctions
of a thermocouple to the switch or potentiometer are
called copper connecting wires.
These wires cause no uncertainty when they are used
between the reference junctions and the potentiometer
Potentiometer
A potentiometer is used to measure the emf generated
by the thermocouple. A standardized current is passed
through a wire of fixed resistance. The wire is
calibrated by marking the corresponding voltage drop
along its length.
Thus any unknown external emf (such as that
encountered in the thermocouple circuit) can be
compared with the known voltage drop along the
calibrated wire.
A sliding contact on the calibrated wire facilitates the
comparison, whereas a galvanometer indicates the
balance (null) condition by absence of needle deflection.
With reference to Figure 7.17, operation of the
potentiometer is as follows.
The switch is placed on (1), and the current from the
working cells is adjusted by rheostat C until the
galvanometer indicates the null. Under this condition
the current through the calibrated wire is
"standardized."
The switch is then placed on (2), and the slide on the
calibrated wire is adjusted until the galvanometer again
indicates the balance condition.
The unknown voltage is now determined by the position
of the slide on the calibrated wire.
Since no current flows in the external circuit, the
resistance of that circuit introduces no error in the voltage
measurement.
An increase in external circuit resistance, however,
does decrease the sensitivity of the potential measurement.
This component introduces no uncertainty that is not
accounted for in the overall calibration.
7.8 Thermoelectric Reference Tables
Only a few of the large number of possible
combinations of materials are actually used in
thermoelectric thermometry.
These are chosen on the basis of their standing in the
Thermoelectric Series, their Seebeck coefficients (i.e.,
thermoelectric powers), and on their stability and
reproducibility as evidenced by the establishment of
Thermocouple Reference Tables. In the following
section, these three tabulations are presented.
Thermoelectric Series
The various conductors have been tabulated in an
order such that, at a specified temperature, each
material in the list is thermoelectrically negative with
respect to all above it and positive with respect to all
below it (see Table 7.1).
Although
this
tabulation
resembles
the
electrochemical electromotive force series, the position
of a material in one series bears no relation to that in the
other.
Seebeck Coefficients
Nominal thermoelectric powers of various
thermoelements with respect to Platinum 67 (a standard
maintained at the NBS), and of various common
thermocouple types, are presented in Tables 7.2 and 7.3.
These are quite useful in the analysis of circuits such
as those dealt with in Section 7.9. Curves to present this
same information graphically are given in Figures 7.18
and 7.19.
Thermocouple Reference Tables
Thermocouple reference values represent in both form
and magnitude the nominal characteristics of particular
thermocouple types.
Thermocouple wires are carefully selected and
matched to conform with these tables within prescribed
limits. The printed scales of all direct temperatureindicating potentiometers are based on such tables.
Calibration curves, in the form of voltage -or
temperature-difference plots, are obtained by comparing
A satisfactory table of reference values for use with
thermocouples must be capable of easy and unique
generation for a number of reasons:
(1) the values are used by many people;
(2) their storage in computer applications must not present a
problem; and
(3) since interpolation must be used, unique values at all
arbitrary points must be available.
The last requirement means that a mathematically
continuous functional relation must exist between the
temperatures and voltages with which the table is concerned.
Satisfactory reference values must also agree closely
with the characteristic of the thermocouple type being
considered so that differences in values change slowly
and smoothly.
The established reference tables adequately fulfill all
these requirements. Unique polynomials are available
for generating tables for each of the seven
thermocouples now in common use.
These are given in Tables 7.4 and 7.5 for easy
reference.
TYPE T (copper / constantan).
This type can be represented by an eighth-degree equation
as in Table 7.4a. This is used to generate exact values of
voltage as a function of temperature from 0℃ to 400℃.
The inverse relation, for temperature as a function of
voltage, is generated over the same temperature range by
another polynomial also given in the table.
This procedure is necessarily inexact, but will yield
reference temperatures to within known uncertainty bands.
For type T, temperatures are given to 0.2℃ from 0℃ to
400℃ .
TYPE J (iron/constantan).
Output of this type can be determined as a function of
temperature from by a seventh-degree equation in Table
7.4b.
The resulting values are exact by definition. This
shows the inverse relation for temperature as a function
of voltage over the same temperature range. For type J,
temperatures are given to 0.1℃ from 0℃ to 760℃.
TYPE E (chromel /constantan).
Thermocouples of this type can be represented by a
ninth-degree polynomial as in Table 7.4c.
Voltage generated as a function of temperature from
0℃ to 1000℃ can be calculated, as can temperature as a
function of voltage using the inverse relationship . For
the latter case , temperatures are given to 0.1℃ from
0℃ to 1000℃ .
TYPE K (chromel / alumel).
Outputs of these thermocouples can be represented by the
equation in Table 7.4d.
An eighth -degree equation with an exponential term is
used to generate values of voltage as a function of
temperature from 0℃ to 1372℃. The inverse relation
provides reference temperatures accurate to 0.02℃ from
400℃to 1100℃.
TYPE B (platinum-30% rhodium / platinum).
An eighth-degree equation, given in table 7.5a is used to
generate these voltages as a function of temperature from
0℃ to 1820℃.
Type R (platinum-13% rhodium / platinum).
Two third-degree equations are required to generate
voltage as a function of temperature from 630℃ to 1665℃,
as shown in Table 7.5b.
TYPE S (platinum-10% rhodium / platinum).
From the antimony point to the gold point, that is, from
630.74℃ to 1064.43℃, a second-degree equation is used to
generate voltages for this type thermocouple. From 1064℃
to 1665℃ , a third-degree equation is used. The required
coefficients of these polynomials are given in Table 7.5c.
A table generated by the polynomials given in Tables
7.4 and 7.5, for six of the most commonly used types of
thermocouples, is presented as Table 7.6. Prior to the
publication of the NBS polynomials of Tables 7.4 and
7.5, there was no unique , internationally accepted,
mathematical generation scheme available that would
reproduce the established thermocouple reference table
values .
One method that found wide acceptance in the ISA ,
ASME , ASTM , and industry in general , was based on the
trial-and-error establishment of a few key values of voltage
and temperature and a series of equations of specified
degree to interpolate voltages between the key points .
Today this generation scheme is no longer required and is
only of historical interest. NBS Monograph 125[29] is the
accepted standard for the nominal characteristic for all
conventional thermocouple types.
7.9 Thermoelectric Circuit Analysis
An important phase of each temperature measurement by
means of thermocouples is the thermoelectric circuit
analysis. This operation can be systematized by observing
the following set of rules.
1. Draw an equivalent circuit, numbering all
thermoelectric junctions, and show an assumed direction of
current.
2. Indicate the direction of the potential drop at each
junction, guided by the outputs of each material with respect
to platinum-67 (see Figure 7.18).
3. Compute the net emf in terms of the potentials at each
junction (assigning a plus sign if the potential drop is in the
same direction as the assumed current).
4. Combine the voltages algebraically to a form that
allows easy reference table look-up (i.e., combine in terms
of usual thermocouple combinations).
5. Compute the errors.
The examples that follow illustrate the application of
these rules, and have been chosen to represent common and
yet serious thermocouple circuit errors.
Example 1.
Use of Incorrect Extension Wires. A chromel-alumel
(Ch-Al) thermocouple was connected by mistake with
iron-constantan (I-C) extension wires to a potentiometer
in such a manner that the output emf was on scale.
The actual temperatures were at measuring junction ，
TM＝60℉; at reference junction，TR＝32℉; and at the
junction between the thermocouple and extension wires，
TJ＝130℉. What was the error in the measurement?
1. Draw an equivalent circuit (see Figure7.20).
Enet   E1  E2  E3  E4
E1  (Ch  A1)60 F  0.619mV
E2  (Ch  C )130 F  3.329mV
E3  ( I  C )32 F  0mV
E4  ( I  A1)130 F  (ch  A1)130 F  (Ch  C )130 F  ( I  C )130 F
 2.206  3.329  2.820mV
 1.697mV ( seeFigure7.21)
Enet  0.619  3.329  0  1.697  4.407mV
Ewhereas
 (Ch  the
Al )60correct
－(Chemf
 Alshould
)32 F＝0.619mV
have been:
F
5. Compute errors
net voltage error =4.407-0.619=3.788mV
net voltage error =225  60  165 F, (on Ch  A1scale)
net voltage error =183  60  123 F, (on I  Cscale)
It goes without saying that, on either thermocouple scale,
errors of this magnitude cannot be tolerated.
Example 2.
Use of Incorrect Polarity Wires. The extension wires of a
chromel-alumel
thermocouple
were
inadvertently
interchanged. The actual temperatures were at measuring
junction，TM＝980℉; at reference junction ，TR＝ 32℉;
and at the junction between the thermocouple and extension
wires，TJ＝100℉.
What was the error in the measurement?
1, 2. Draw an equivalent circuit (see Figure 7.20).
3, 4. Compute net emf:
Enet   E1  E2  E3  E4
E1  (Ch  A1)60 F  0.619mV
E2  (Ch  C )130 F  3.329mV
E3  ( I  C )32 F  0mV
E4  ( I  A1)130 F  (ch  A1)130 F  (Ch  C )130 F  ( I  C )130 F
 2.206  3.329  2.820mV
 1.697mV ( seeFigure7.21)
Enet  0.619  3.329  0  1.697  4.407mV
whereas the correct emf should have been
E  (Ch  A1)980 F  (Ch  A1)32 F  21.777mV
5. Compute errors
net voltage error =26.637-7.958=18.679mV
net temperature error =700-255  445 F
Once again, a most common thermocouple circuit error,
namely interchanging extension wires, is seen to lead to a
very serious temperature measuring error.
Example 3.
Use of Improper Grounding. A chromel-constantan
thermocouple, grounded at its measuring junction
according to Figure 7.14, was inadvertently grounded at
its reference junction also. The actual temperatures
were: at measuring junction, TM = 700℉; at reference
junction, TR =32℉.
What is the predicted error in the measurement?
1. Draw an equivalent circuit:
Figure7.22 illustrates the multigrounded circuit under
consideration. Figure7.23 depicts the equivalent circuit that
represents the multigrounded thermocouple.
2. In this example, the direction of all potential drops is
indicated by electrical cell symbols, where the drop is from
minus to plus at the junction.
3, 4. Compute net emf: A voltage summation is written
around the outer ground loop to obtain the current in the
loop as follows.
 E  (E
)  iRg  iRn  (Epn )M  0
pq M
(7.44)
Where（Epg）M signifies the emf set up by the positive
thermoelement and the ground wire at the measuring
junction temperature , and so on . Thus
where（Epg）M signifies the emf set up by the positive
thermoelement and the ground wire at the measuring
junction temperature , and so on . Thus
i
( E pn )M  ( E pg ) M
Rg  Rn
(7.45)
Note that chromel is the positive element with respect to
constantan and that copper is assumed to be the grounding
wire.
Given the ground loop current, the net voltage set up by
the doubly grounded thermocouple circuit can be
determined as follows.
Enet  iRg  (Epg )M  iRn  (Epn )M
The pertinent resistance values are:
for constantan Rn  2.902
Rg  0.0646
for copper
(7.46)
The pertinent voltage values are:
( E pn ) M  ( ECh C )700 F  26.637mV
( E pg ) M  ( ECh Cu )700 F  ( ECh C  ECu C )700 F
( E pg ) M  26.637  19.095  7.542mV
26.637  7.542
 6.437mV
0.0646  2.902
Enet  iRg  ( E pg ) M  6.437(0.0646)  7.542  7.958mV
i
5. The temperature error for this doubly grounded
thermocouple can be estimated as follows
net voltage error =26.637-7.958=18.679mV
net temperature error =700-255  445 F
This error is obviously much larger than any other error
associated with thermocouple. circuits, and it is clear that
doubly grounded thermocouple circuits are to be avoided at
all costs.
Example 4.
Rotating thermocouple circuit analysis. A rotating ironconstantan thermocouple is connected through a slip ring to
a stationary potentiometer and reference junction.
The unknown temperature at the transition between the
thermocouple wires and the copper slip ring wires is
accounted for by a special rotating iron-constantan
thermocouple connected as shown in Figure 7.16a.
The actual temperatures were: at measuring junction，
TM＝ 500℉; at reference junction，TR＝32℉; and at the slip
ring, TJ=150℉.What was the error in the measurement?
1, 2. Draw an equivalent circuit (see Figure7 .24）.
3, 4. Compute net emf:
Enet  E1  E2  E3  E4
E1  (Ch  A1)980 F  21.777 mV
E2  (Ch  A1)100 F  1.520mV
E3  (Ch  A1)32 F  0mV
E4  (Ch  A1)100 F  1.520
Enet  21.777  1.520  0  1.520  18.737 mV
Since this is the voltage required, there are no circuit
errors in this example.