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Development of a Circuit-equivalent Model for Thermoelectric Devices
Christopher D. Meyer
This paper highlights the development and verification of a SPICE-compatible circuit-equivalent model for use
in modeling thermoelectric devices. Thermoelectric phenomena are first described in terms of fundamental
concepts of physics and thermodynamics. These concepts are then applied to the thermocouple, which is the
building block of thermoelectric devices. After exploring general concepts of circuit theory, a circuit model
is developed for the thermocouple. Scaling of the model for simulation of more complicated devices is then
discussed. The circuit model is then adapted to simulate the cooling and power generation modes of
commercially available thermoelectric modules. Simulation results compared favorably to the actual
experimental results, proving the extensibility of the model.
The thermoelectric effect enables direct conversion between heat and electric energy streams. By heating one end
of a thermoelectric material with respect to the other, cooler end, an electromotive force (voltage potential)
is generated within the material that may be harnessed for powering electrical systems. Conversely, if an
electric current is passed through the material, a temperature difference is induced between the ends of the
device. This effect may be used to create heating or cooling devices.
Devices operating on the thermoelectric effect have many attractive features for both power generation and
heating/cooling applications. These devices operate with no moving parts, release no exhaust gases, have no risk
of leaks, may be designed in any number of shapes, and can function properly for years with little to no
maintenance. However, poor energy conversion efficiency has meant that thermoelectric devices have
been historically relegated to niche applications, such as cooling for lasers and portable drink coolers.
Novel approaches to leverage the thermoelectric effect include miniaturization and implementation
in microelectromechanical systems (MEMS). Given the reduced dimensions of these devices, there is now a
greater need for accurate modeling and computer simulation. Computer simulation allows for analysis of
device characteristics that would otherwise be difficult to measure at such small scales. Modeling also allows
for optimization of potential device designs to reduce the number of design cycles.
This paper follows the development of a circuit model for thermoelectric devices, starting with first principles
of physics and progresses through to verification of the model against measurements taken from a large-scale device.
The thermoelectric effect arises from fundamental observations that electronic charge carriers in a conductor
diffuse from locations of higher temperature to locations of lower temperature and that these charge carriers
also conduct heat .
The first of these phenomena is known as the Seebeck effect. In the presence of a temperature gradient VT,
charge carriers in a conductor with conductivity a diffuse from warmer to cooler regions such to produce an
external current density 
E = -oaVT
where a is a material property known as the Seebeck coefficient. As a result, for a material that is
electrically isolated, an induced electrical current density
jE = crE
is established to maintain equilibrium such that there is no net current density,
jE JE E =0
Rearranging Equation with the expressions of Equations and , the electric field induced by the Seebeck effect is thus
E - aVT . (1.4)
The second fundamental phenomenon is known as the Peltier effect. Because of the heat capacity of electrons,
a current density will conduct heat energy with flux proportional to the current density , such that
JT - IIJE
i , (1.5)
where n is known as the Peltier coefficient and is related to the Seebeck coefficient by
Although any conductor will experience thermoelectric effects, the term "thermoelectric material" is used in this
paper to denote any material having a Seebeck coefficient greater than that of most metals.
The development of the circuit-equivalent model will first begin with an analysis of a single thermocouple such as
the one depicted in Figure 1. A thermocouple consists of two legs of thermoelectric materials with differing
Seebeck coefficients and connected at one end. For high-performance devices, the materials are typically p- and
n-type semiconductors having positive and negative Seebeck coefficients, respectively. One end of the
thermocouple analyzed here is connected by a metal tab. Metal tabs are also found at the other end of
the thermocouple for external connections. Ceramic plates having high thermal conductivity are affixed at the ends
of the thermocouple for heat spreading to ensure a uniform temperature at either end. The ceramic plates and
metal interconnects are assumed to have uniform temperatures TH and TL at the top and bottom,
respectively. Current density is also assumed to be constant at JE over the area of the junctions between
Figure 1. Example of a thermocouple formed by legs of p- and n-type semiconductor legs joined at
one end by metal.
When a temperature difference exists between the two ceramic plates, the Seebeck effect results in a
voltage difference between the open ends of the thermocouple,
V=- E'd, (2.1)
J- , (2.1)
where E is given by Equation . This line integral can be broken into segments for each segment of
homogeneous material as
t' - VT-il -A V'dl - A f ,iVTdA - f a VI *A l-f (VT A.
Assuming no temperature gradient in the metal regions, the line integral reduces to
V = -f aVT- aVT-d
By the fundamental theorem of calculus for line integrals,
V = -, [T(c)- T(d)]- a [T(a) - T(b)]
Using the assumption of constant temperatures across the ceramic plates and metal interconnects,
V--an[TH-T]-a [T -TH]
which simplifies to
V = (aP-a ) (TH-T )
When a source is connected such that current flows through the thermocouple, the Peltier effect will result in
heat being released or absorbed at the junctions between dissimilar materials. When current flows from the
metal into the n-type semiconductor at junction "a" an amount of heat is released at a rate per unit area
q" = J - J~ = (a ap)TLJE
l = m- =p m p L
Integrating over the area of the junction, the heat released at junction "a" due to the Peltier effect is
qP -=(a -a )TLI
where I is the current flowing through the thermocouple. Similar expressions can be found for the
remaining junctions assuming the same direction of current flow. The rate of heat released at the upper portion
of the thermocouple due to the Peltier effect is then
qb , =ap-a m)TH I+(a-an)THI
q q = (ap - . )TH . (2.3)
Similarly, the heat released at the lower portion of the thermocouple is
q' = + (a - -a +)TLI+ (a - a,)TLI
qL =-(a -a L LI
L p nLI
The flow of electronic current through the thermocouple will additionally cause resistive heating of the structure.
If the sides of the legs are perfectly insulated, this resistive heat will flow equally out of each end of the legs
such that 
qH 2R (2.5)
q = I RE
where RE is the internal electrical resistance of the thermocouple.
Summing the terms for Peltier and resistive heat, the total rate of heat released at the upper and lower portions
of the thermocouple are, respectively,
q = q +q = I2R + ( - ) TH
L =q = I2RE-( -a )TLI
qL + qL -2 (2.8)
Although computer programs for circuit analysis are primarily designed to solve electrical circuit problems, as in
the case of SPICE, these tools can be equally useful for solving circuits in other energy domains with
certain adaptations. To create a circuit model for the thermal domain, the various thermal relationships must
be realizable in terms of one-dimensional circuit elements.
To understand how problems in the thermal energy domain can be solved using circuit theory, a circuit must
be considered in generalized terms. A circuit is a collection of nodes, wires, and elements all connected together
for analysis of two variables, effort and flow. Nodes are positions in the circuit where multiple flows connect
together, and they also act as markers that reference the effort potential at a particular position in the circuit.
Wires connect nodes and elements together with no loss in effort potential. Elements can be any entity that
provides a relationship between effort and flow. A difference in potential between nodes will push the flow
through the elements that connect the nodes. In electric circuits, the effort variable is voltage, V, with units of
volts (V) and the flow variable is current, I, with units of amperes (A). In thermal circuits, the effort variable
is temperature, T, with units of kelvin (K) and the flow variable is the heat transfer rate, q, with units of watts
(W). Modeling thermoelectric devices will require both electric and thermal circuits with interactions between the two.
---------------- --/------------------ --
Figure 2. Element with one-dimensional heat and electric flows.
Figure 2 shows an element with uniform cross-sectional area A, thermal conductivity K, and electrical resistivity
pe. Assuming constant material properties and perfect insulation along the length of the element results in
one-dimensional heat and electric flows. This element will be used for comparison of electrical and
Determination of the resistance of the element begins with Ohm's law,
Integrating the current density over the cross-sectional area of the element yields
PE . (3.2)
The electric field can be integrated over the length of the element to obtain the voltage difference
-V, -PEL I = RE
A , (3.3)
where RE is defined as the electrical resistance of the element. Equation can be rearranged to the form
R , (3.4)
Conduction of heat can also be modeled with a resistor. Fourier's law of conduction for one-dimensional flow is 
L . (3.5)
By analogy with Equation , the equivalent thermal resistance is
KA z . (3.6)
Convection of heat occurs at the surface between a solid and a liquid or gas. The complex interaction between
heat and the flow of the liquid/gas medium can be simplified through the lumped heat transfer coefficient h. For
an object with surface area A and temperature Ts, the rate of heat flow out of the surface can be expressed as 
SO--amb ) (3.7)
where h is the average convection coefficient of the surface and Tamb is the ambient temperature far away from
the surface. By comparison with Equation (3.4), the equivalent thermal resistance is
hA . (3.8)
CIRCUIT MODEL OF A THERMOCOUPLE
Figure 3 shows the complete circuit model for a single thermocouple. The voltage resulting from the Seebeck
effect, Equation (2.2), is placed as a temperature-controlled voltage source in series with the internal
electrical resistance RE of the thermocouple.
In the thermal circuit, the nodes marked as TH and TLrepresent, respectively, the hot and cold junctions of
the thermocouple. Conduction of heat between these two nodes occurs through each of the legs of the
thermocouple, modeled as RTIeg.
Heat sources are connected to these two nodes to model the Peltier effect of Equations (2.3) and (2.4) as well as
the resistive heating given by Equations (2.5) and (2.6). These sources are connected to ground to
provide discontinuity from the thermal energy domain as the energy from these sources comes from the
Thermal resistances RTplate and RTconv are connected at the hot and cold side nodes and provide a paths for heat
loss by conduction through the ceramic and by convection to ambient, which is modeled as a constant
TB k2 2R
TH 1/2 1 2R
Rplte T b
Figure 3. Complete thermoelectric circuit of a thermocouple.
The major limitation of the circuit model is that it assumes one-dimensional heat flow. Experimental results
will determine whether this limitation will result in significant deviation of the simulated results.
In order to achieve higher voltage levels for generators and higher internal resistance for coolers,
thermoelectric devices typically employ many thermocouples connected together in what is called a thermopile.
The model, which has been developed for a single thermocouple, can be adapted for simulation of these
thermopiles via scaling. Connecting n thermocouples electrically in series effectively scales the electrically
resistive path by a factor of n. The generated voltage also scales by a factor of n. Since current must be
conserved, the current through each junction of a thermopile is the same as that of a single thermocouple. The
heat sources associated with the Peltier effect thus scale proportionately with n. With the thermocouples
connected thermally in parallel, the cross-sectional area of the thermal path scales by a factor of n, and
thermal resistance is thus inversely proportional to n.
To verify the accuracy of the model, experiments were conducted using thermoelectric modules, such as the
one shown in Figure 4 (model no. TE-65-0.6-1.5), which were purchased from TE Technology, Inc. Arranged
within the module are 130 elements (65 couples) of alternating p- and n-type Bismuth Telluride (Bi2Te3)
connected electrically in series using thin tabs of copper. Square plates of ceramic at the top and bottom hold all
the elements in place and provide support to the module.
Figure 4. TE-65-0.6-1.5 thermoelectric module purchased from TE Technology, Inc.
The material properties for Bi2Te3, ceramic plates, and copper tabs were taken from the manufacturer  and
are summarized in Table 1. Additionally, the Seebeck coefficient was provided as the average for the p- and n-
type Bi2Te3 ap = -aOn = 203.7 pV/K. The dimensions of each component were measured using calipers with
values tabulated in Table 2. The measurements are categorized such that, for modeling purposes with
one-dimensional flow, each component has a cross-sectional area A = Length x Width and spans a length L = Height.
Summary of material properties
Material Electrical Resistivity
Bi2Te3 PETE = 1.08 x 10-5 *. m
KTE = 1.67W/ m * K
Kplate = 34.6 W/ m K
Copper pEc = 1 .7 x 10 8 ~ * m 0 * m Kcu = 400 W/ m * K
Summary of dimensions, categorized with respect to energy flow through element.
Component Length Width Height
Bi2Te3 Legs 0.60 mm 0.60 mm 1.50 mm
Copper Tabs 0.60 mm 0.36 mm 1.50 mm
Ceramic Plates 12.00 mm 13.00 mm 0.50 mm
Copper Blocks 50.00 mm 63.00 mm 6.50 mm
A/C RESISTANCE MEASUREMENT
Measurement of the internal electrical resistance of thermoelectric materials must always account for the
voltage resulting from any difference in temperature between the ends of the material. When measuring resistance,
a small but significant current flows through the subject material. In thermoelectric materials, this current will
cause heating and cooling at the ends of the material through the Peltier effect. This effect can be minimized if
the direction of current flow alternates quickly so that not enough heat is distributed in one half cycle to create
a significant temperature difference within the tested device .
An HP 4294A impedance analyzer was used to measure the internal resistance of the module. The analyzer was
set to source 20 mA and to sweep the oscillating frequency from 10 kHz to 100 kHz. The machine
simultaneously measured the voltage at the input of the device. The ratio of the amplitude of the voltage to that
of the current yields the resistance of the device. Averaging the measurement over 32 sweeps of the frequency at
an ambient temperature of 22.5 OC yielded a resistance of 5.866 Q.
CONVECTION COEFFICIENT MEASUREMENT
Owing to the complexities in accurately modeling the mechanisms by which convection occurs, the heat
transfer coefficient h is well suited for empirical determination . An experiment was devised to determine the
heat transfer coefficient by heating a copper block to an elevated temperature, removing the heat source, and
then observing the rate at which the block cooled to room temperature.
The copper block was first prepared by wrapping it in a layer of masking tape so that the entire surface of the
block was the same as the surfaces that dissipated heat in the generator experiment. It was then placed on a
hot plate and heated to 60 OC. The block was removed from the hot plate and allowed to cool by suspending it in air.
Measurements of the block temperature were taken every 5 seconds using the thermocouple that was embedded
PECu = 1.7 x 10-8 f * mQ * m
Kcu,= 400 W/m K
into the block.
This experiment can be modeled as a capacitor discharging through a resistor.
Figure 5. Convective cooling thermal circuit.
The heat capacity C of the block is defined as the ratio of a small amount of heat added to the block and
its corresponding temperature change
By rearrangement and time differentiation, the rate of heat leaving the block is
dt . (8.2)
The temperature Ts is also a function of the heat flowing through the convective resistor
T -T - q
S amb hA (8.3)
Combining Equations (8.2) and (8.3) yields the differential equation
dT hA hA
dt C- --TC a.(8.4)
dt C " C m. (8.4)
The solution to this differential equation is
T(t) = [T(t -0)-T- e c +C mb
This curve was fit to the data in a least-squares manner, with the result h = 10.6 as shown in Figure 6.
335 , I
330 T - (To - T "4c+To
305 ---*--- *-------* ***
0 200 400 600 800 1000 1200
Figure 6. Convective cooling of a copper block.
Figure 7. Experiment setup to verify model accuracy.
The experiment depicted in Figure 7 was constructed to test the ability of thermoelectric materials to supply power
to a load for comparison with simulated values. Two TE-65-0.6-1.5 modules were placed next to each other
and sandwiched between blocks of copper. Thermal interface paste was applied to each side of the modules
to improve the thermal contact between the ceramic plates of the modules and the copper blocks. A
thermocouple was embedded in the center of each block for temperature measurement. One module was
connected to a power supply to create a temperature difference between the separate blocks of copper. The
second module was connected to a load resistor and the output voltage was measured. A Keithley model
2400 multimeter measured the load resistance at REL = 15.7 2. The input current was increased in steps,
allowing the measured temperatures and output voltage to stabilize at each step for recording.
Figure 8. Complete circuit model of experiment setup.
The circuit model for the single thermocouple was adapted for simulation of this experiment as shown in Figure
8 . This model assumes that the copper blocks are able to create a constant temperature across the two
ceramic surfaces. The outside surfaces of the ceramic plates were thus connected together as a temperature
node. Thermal resistances representing the copper blocks were placed between these nodes and the
convection resistances. The material parameters and dimensions listed in Table I and Table 2 were used to
calculate the values for the various resistances listed in Table 3 . Of particular interest, the calculated value
of REintsmatches the AC resistance measurement to within 0.2% error.
Table 3. Summary of resistances calculated for simulation of TE-65-0.6-1.5 device.
Ri, = ( pEHg p~H = 5.865 0
_ H, = 2500 KW
R = Hpe = 0.093 7w
plate" tate Lplate
-= H =C 0005 K/
Kc W'__I La
h(Wc.Lc + 2WfHo, +2LcH, )
= 20.42 K,/
(t :,.l,. j ino f,,.i,
. o o.2t
01 0.2 0,3 04 0 01
Input ctmen< (A)
z 2 0...
02 03 04 0
Input (curn (A)
Input cumrrt (A)
Figure 9. Model simulation results (solid line) and experiment results (dots).
The model was simulated in the PSPICE software package. Simulated and experimentally measured values for
the input voltage, output voltage, hot side temperature, and cold side temperature are plotted in Figure 9
for increasing values of input current. The slope of the I-V curves of Figure 9a indicates an effective device
resistance of approximately REeff = 7.5 n for this particular setup in both the simulation and the actual
experiment. This effective DC resistance differs from the actual internal resistance measured at ac due to
the temperature difference created by the Peltier effect and the resultant voltage produced by the Seebeck
effect. The simulated results for output voltage and hot side temperature in Figure 9b and 9c, respectively, follow
the trends of the curves of the actual measured values but increase at a greater rate. This indicates that the
actual value of the Seebeck coefficient is likely slightly lower than the value provided by the manufacturer. The
model was also able to capture the effect measured in Figure 9d in which the cold side temperature decreases
for moderate input currents before curving upward as resistive heating begins to dominate.
The circuit model is able to reasonably simulate the behavior of thermoelectric devices used for both cooling
and power generation. In addition to resistive heating, the Seebeck and Peltier effects were captured, as tested
by the experimental setup. The model will allow for analysis and optimization of thermoelectric device designs
when an analytical solution would prove to be too complex. Importantly, material properties and parameters can
be independently measured by separate experiments and used in the model with reasonable accuracy.
InJput cunart A)
1. A.F. loffe, Semiconductor Thermoelements and Thermoelectric Cooling, London: Infosearch, 1957, pp. 9-15.
2. S.D. Senturia, Microsystem Design, Boston: Kluwer, 2001, pp. 291-296.
3. G.S. Nolas, J. Sharp, and H.J. Goldsmid, Thermoelectrics: Basic Principles and New Materials Developments,
Berlin: Springer-Verlag, 2001, pp. 2-14.
4. R.E. Sonntag, C. Borgnakke, and G.J. van Wylen, Fundamentals of Thermodynamics, 6th. ed., New York:
Wiley, 2003, pp. 101-144.
5. P.G. Lau and R.J. Buist, "Temperature and Time Dependent Finite-Element Model of a Thermoelectric
Couple", presented at 15th International Conference on Thermoelectrics, Pasadena, California, March 25 - March
6. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 5th. Ed., New York: Wiley, 2002, pp.
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