Budapest, Hungary, 17-19 September 2007
?EDA Publishing/THERMINIC 2007 -page- ISBN: 978-2-35500-002-7
Lumped and Distributed Parameter SPICE Models
of TE Devices Considering Temperature
Dependent Material Properties
D. Mitrani, J. Salazar, A. Tur?, M. J. Garc?a, and J. A. Ch?vez
Electrical Engineering Department, Universitat Polit?cnica de Catalunya,
Barcelona, Spain. Email: mitrani@eel.upc.edu
Abstract- Based on simplified one-dimensional steady-state
analysis of thermoelectric phenomena and on analogies between
thermal and electrical domains, we propose both lumped and
distributed parameter electrical models for thermoelectric
devices. For lumped parameter models, constant values for
material properties are extracted from polynomial fit curves
evaluated at different module temperatures (hot side, cold side,
average, and mean module temperature). For the case of
distributed parameter models, material properties are
calculated according to the mean temperature at each segment
of a sectioned device. A couple of important advantages of the
presented models are that temperature dependence of material
properties is considered and that they can be easily simulated
using an electronic simulation tool such as SPICE.
Comparisons are made between SPICE simulations for a
single-pellet module using the proposed models and with
numerical simulations carried out with Mathematica software.
Results illustrate accuracy of the distributed parameter models
and show how inappropriate is to assume, in some cases,
constant material parameters for an entire thermoelectric
element.
I. INTRODUCTION
Thermoelectric modules (TEM) are solid state devices
capable of use either in Peltier mode for transporting heat or
in Seebeck mode for electrical power generation [1]-[3].
Despite their low efficiency with respect to traditional
devices, TEM?s present distinct advantages as far as
compactness, precision, simplicity and reliability.
Applications of thermoelectric (TE) devices cover a wide
spectrum of product areas. These include equipment used in
military, aerospace, medical, industrial, consumer, and
scientific institutions.
As applications for TE devices increase, both
manufacturers and users are facing the problem of
developing simple yet accurate models for them. Simulations
are usually performed with mathematical software by means
of numerical methods [4]-[7] or with electronic and thermal
simulators that separately solve the electrical and thermal
parts of the model. Alternatively, another methodology is to
make use of the analogies between electrical and thermal
domains and perform the simulation of the device with an
electronic simulation tool such as SPICE. An important
benefit of such approach is that both electrical and thermal
phenomena can be simulated with a common tool, thereby
simplifying simulation of the overall system performance,
including control electronics as well as thermal elements.
In this work, we propose both lumped and distributed
parameter electrical models for TE devices. A significant
novelty of the presented models is that temperature
dependence of material parameters is considered. To
compare the different models, simulations were performed
for a TEM working in Peltier mode for the two more
extreme cases, i.e., maximum temperature difference and
maximum cooling power. To validate the models,
mathematical software was used to obtain a numerical
solution to the thermoelectric problem taking into account
temperature dependence of material parameters.
II. TEM DESCRIPTION AND FORMULAE
The basic unit of a TEM is a thermocouple. As illustrated
in Fig. 1(a), a thermocouple consists of a p-type
semiconductor pellet and an n-type semiconductor pellet
joined by metal interconnects. The two pellets of each
couple and the many couples in a thermoelectric device are
connected electrically in series but thermally in parallel and
sandwiched between two ceramic plates, as seen in Fig. 1(b).
Nevertheless, if the contribution of the metal interconnects is
ignored, there is no loss of generality in analyzing a single
couple or a single pellet.
When operated in Peltier mode, in order to pump heat
from one side of the TEM to the other by means of an
electrical current, four energy conversion processes take
place in the pellets: Joule heating, Seebeck power
Fig. 1. Schematic of a thermoelectric module, (a) basic unit,
and (b) multi-thermocouple module.
Budapest, Hungary, 17-19 September 2007
?EDA Publishing/THERMINIC 2007 -page- ISBN: 978-2-35500-002-7
generation, Peltier effect, and Thomson effect. These
processes, in conjunction with thermal conduction,
determine the performance of the module and are governed
by temperature dependent material parameters: Seebeck
coefficient, s, thermal conductivity, ?, and electrical
resistivity, ?. However, by assuming these parameters
constant, one-dimensional (1-D) steady-state analysis leads
to the widely used equations for TEM?s that we review next.
Heat absorbed at the cold junction of the module, Q
c
, and
heat released at hot junction, Q
h
, are respectively given by
( )
2
1
2
,
cc hc
QIST IRKTT=? ? ? (1)
( )
2
1
2
.
hh hc
QIST IRKTT=+ ? ? (2)
Where I is the electrical current, T
c
and T
h
are the module
cold and hot side temperatures. For a module made up of N
thermocouples with pellets of length L and cross-sectional
area A, the total Seebeck coefficient, S, serial electrical
resistance, R, and parallel thermal conductance, K, are given
by
2, 2, 2.
AL
K NRNSNs
L A
??
=== (3)
Electrical power consumed by the TEM is not simply
Joule power. The external current source must also work
against Seebeck voltage, and is equal to the difference
between heat flow at the hot side and heat flow at the cold
side,
()
2
.
ehc hc
PQQ IRISTT=?= + ? (4)
Finally, from the corresponding 1-D expression [8] for
temperature distribution along the pellets, T(x), given by
22
2
2
() ,
22
hc
c
TTIR IR
Tx x x T
LKLKL
???
=? + + +
??
??
(5)
The mean temperature, T
m
, of the pellets is calculated as
2
0
11
() .
212
L
hc
m
TT I R
TTxdx
LK
+
==
?
(6)
This temperature will serve for some of the models described
later, where temperature dependent parameters are calculated
according to it.
III. STEADY STATE ELECTRICAL MODELS
Equations (1)-(4) are widely used as building blocks for a
variety of thermoelectric device models, including electrical
models [9]-[14]. For these particular types of models, all
thermal processes are described in electrical terms using the
well-known analogies between electrical and thermal
domains described in Table I. According to these analogies,
the thermo-electrical behavior of a TEM can be modeled as
an electrical network composed of electrical current sources,
voltage sources, resistors, and capacitors. The resulting
network can then be simulated by means of electronic circuit
simulators such as SPICE.
We shall next describe a steady-state lumped parameter
electrical model of a TEM assuming constant material
properties, and then proceed to report a distributed parameter
electrical model with discrete temperature dependent
material properties.
A. Lumped Parameter Model
Thermoelectric devices can be modeled by a three-port
system consisting of two thermal ports and one electrical
port, see Fig. 2. Where, referring to the analogies of Table I,
thermal ports voltages T
c
and T
h
correspond to temperature at
the cold and hot junctions, while currents Q
c
and Q
h
represent absorbed and released heat at the cold and hot
junctions. At the electrical port, V is the total voltage across
the TEM?s terminals, and I is the supplied electrical current.
According to expressions for Q
c
, Q
h
, and P
e
, see (1)-(4),
Chavez et a1 [9] have proposed the electrical three-port
model of a thermoelectric device shown in Fig. 3 (with
P
e
= SIT
c
? ? I
2
R ). This model clearly illustrates the
thermoelectric behavior of a TEM. Within the thermal ports,
cooling power and input electrical power are easily readable
through current sources P
x
and P
e
, while heat conduction is
observed through the corresponding thermal resistance,
R
th
= K
-1
. In the electrical port, overall device voltage is
composed of Seebeck voltage, V
s
, and voltage drop due to
module?s electrical resistance, R.
If boundary conditions of the first kind are applied
(temperatures at both ends of the module are known) when
simulating the electrical three-port model of Fig. 3, voltage
sources have to be connected to thermal ports T
c
and T
h
. If
mixed boundary conditions are required (temperature and
heat flow at the same or opposite sides are known), one of
TABLE I
ANALOGIES BETWEEN THERMAL AND ELECTRICAL VARIABLES
Thermal variable Electrical variable
Heat Flow, Q (W) Current flow, I (A)
Temperature, T (K) Voltage, V (V)
Thermal resistance R
th
(W
-1
K) Electrical resistance, R (?)
Thermal mass, C
th
(J?K
-1
) Electrical capacity, C (F)
Fig. 2. Three-port block model of a TEM, consisting of two thermal
ports and one electrical port.
Budapest, Hungary, 17-19 September 2007
?EDA Publishing/THERMINIC 2007 -page- ISBN: 978-2-35500-002-7
the voltage sources (T
c
or T
h
) must be changed for the
appropriate current source (Q
c
or Q
h
). Electrical power
supply is included by connecting either a current or a voltage
source to the electrical port.
Values used for the constant material properties of the
model can be chosen according to different criteria. The
simplest method consists in assuming all parameters equal to
values at known temperature T
h
. A more accurate method
determines the parameters from mean module temperature,
T
m
. For this method, modifications must be made to the
electrical model of Fig. 3. A SPICE voltage-controlled-
voltage-source (VCVS) is added in order to determine T
m
according to (6). Electrical resistance, R, and thermal
conductance, K = R
th
-1
, are substituted for VCVS?s V
r
and
T
con
to simulate the corresponding voltage drop by means of
expressions that include polynomial approximations for
R(T
m
) and K(T
m
). These approximations, together with the
corresponding approximation of Seebeck coefficient, S(T
m
),
are also used in expressions for P
e
and P
x
. The resulting
electrical model is shown in Fig. 4, where VCVS?s S(T
m
),
R(T
m
), and K(T
m
) are added to monitor the values of these
parameters. Similarly, material parameters can also be
determined from cold side temperature, T
c
, or average
temperature, T
avg
= ?(T
h
+T
c
).
B. Distributed Parameter Models
Lumped parameter models are only accurate as long as the
thermoelectric properties do not vary significantly over the
length of the pellets. Hence, if the pellets are divided into
many small segments, each segment would be closer to
meeting such criteria. Under this condition, material
properties for a given segment are assumed to be constant
over the small temperature gradient across it. Fig. 5
illustrates a distributed parameter electrical model of a TEM
divided for simulation into three segments, where material
properties are calculated according to the mean temperature
of each segment. To simplify the model, a sub-circuit has
been used for each finite element.
IV. SIMULATION SETUP AND RESULTS
In this section, comparisons are made between SPICE
simulations carried out with the proposed lumped and
distributed parameter electrical models for TE devices and
with a numerical simulation carried out with Mathematica
software [8]. To clarify the results presented hereafter, we
shall use the nomenclature presented in Table II to refer to
the different TEM models.
A. Simulation Setup
In order to estimate the value of the temperature
dependent parameters according to T
h
, T
m
, T
avg
, or T
c
for
lumped parameter models, or to mean temperature of each
segment in a distributed parameter model, as well as for
numerical simulation, polynomial approximations of
Fig. 3. Steady-state lumped parameter three-port electrical model
of a TEM with fixed material properties.
Fig. 4. Steady-state lumped parameter three-port electrical model of a
TEM with material properties calculated according
to mean temperature, T
m
.
Fig. 5. Steady-state distributed parameter electrical model of a
segmented TEM, divided for simulation into three segments
represented by the corresponding sub-circuits.
Budapest, Hungary, 17-19 September 2007
?EDA Publishing/THERMINIC 2007 -page- ISBN: 978-2-35500-002-7
temperature dependent material parameters are required.
In this work, we have used the experimentally measured
properties of (Bi
0.5
Sb
0.5
)
2
Te
3
presented in [15].
For all simulations presented here, device parameters
correspond to a single (Bi
0.5
Sb
0.5
)
2
Te
3
pellet of cross-
sectional area A=10 mm
2
and length L=1 mm. For
comparative reasons with [8] and to resemble a practically
relevant situation, all simulations where made for a fixed hot
side temperature T
h
=300 K.
B. Simulation Results
As mentioned before, of all the possible operating
conditions for a TEM, we will limit our discussion to the
particular cases of temperature difference, ?T, at zero heat
absorption (Q
c
=0 W), and cooling power, Q
c
, at zero
temperature difference (?T=0?C). Comparisons between the
different models for these two cases over a broad electrical
current range are shown in Figs. 6(a) and 7(a),
1
where the
secondary axis shows the relative error with respect to
numerical simulation (model H). As expected, at low
currents where the temperature profile is relatively flat, there
is not much difference between any of the models. However,
as electrical current increments and the temperature profile
becomes more pronounced, differences between models
become noticeable. As can be seen in the expanded graphs of
Figs. 6(b) and 7(b), there is an electrical current that
produces a maximum temperature difference, ?T
max
, and
maximum cooling power, Q
cmax
. Values of ?T
max
and Q
cmax
and corresponding electrical currents I
?Tmax
and I
Qcmax
for
each model are summarized in Table III. Clearly, the models
that best resemble the highly realistic numeric simulation
results are the distributed parameter models. Furthermore,
according to results for this particular single-pellet TEM,
using ten finite elements proves to be sufficient to produce
accurate results (0.03% relative error at I
?Tmax
and 0.01% at
I
Qcmax
). With concerns to lumped parameter models, using
material parameters evaluated at mean temperature T
m
(model B) is the most accurate (3.2% relative error) for the
case of zero temperature difference (where T
c
=T
h
=T
avg
), and
provides very similar results to model C for the case when
1
Do to the great similarity in predictions by all distributed parameter models
and numerical simulation, models E and F where excluded from Figs. 6-9 to
add clarity. However results for these models are included in Table III.
Q
c
=0 W (0.2% relative error). For both cases, model D has
the largest error.
To better understand the variations presented between the
different models, simulations where carried out to determine
the corresponding temperature distribution, T(x), for each of
the ?T
max
and Q
cmax
cases presented in Table III (an
equivalent distributed parameter model for models A, B, C
and D was used applying constant material properties to each
finite element).
The resulting temperature profiles are shown in Figs. 8 (in
practice, a TEM should operate between these two curves).
TABLE II
NOMENCLATURE USED TO REFER TO THE DIFFERENT TEM
MODELS EMPLOYED THROUGHOUT THIS WORK
Ref. Model Description
A Lumped param. SPICE model with param. eval. at T
h
B Lumped param. SPICE model with param. eval. at T
m
C Lumped param. SPICE model with param. eval. at T
avg
D Lumped param. SPICE model with param. eval. at T
c
E Distributed param. SPICE model with 3 finite elements
F Dist. param. SPICE model with 10 finite elements
G Dist. param. SPICE model with 20 finite elements
H Numerical Simulation
Fig. 6. Prediction comparisons between the different models in Table II
for temperature difference vs. electrical current at Q
c
=0 W, (a) broad
current range, including a secondary axis to show the relative error with
respect to model H, and (b) expanded data near maximum current.
TABLE III
COMPARISONS BETWEEN ?T
max
AND Q
cmax
VALUES
Model ?T
max
(?C) I
?Tmax
(A) Q
cmax
(W) I
?Tmax
(A)
A 59.83 30.21 1.237 37.74
B 60.51 31.34 1.230 36.95
C 60.63 32.38 1.237 37.74
D 59.65 35.68 1.237 37.74
E 62.14 32.77 1.231 37.04
F 62.34 32.97 1.232 37.14
G 62.35 32.99 1.232 37.15
H 62.36 32.99 1.232 37.15
Budapest, Hungary, 17-19 September 2007
?EDA Publishing/THERMINIC 2007 -page- ISBN: 978-2-35500-002-7
Finally, spatial profiles of material parameters for the
?T
max
case including the figure-of-merit of a thermoelectric
material, z = s
2
?
-1
?
-1
, are shown in Fig 9. These figures
clearly illustrate the errors produced when assuming constant
material parameters throughout the entire pellet.
Furthermore, analyzing Fig. 6(b) and the figure-of-merit
profile of Fig. 9(d), it is surprising to see that even though
the mean values of z across the pellet for numerical
simulation (model H) and for distributed parameter
(model F) are below the constant values of models B and C,
these models actually underestimate the maximum
temperature difference predicted by models H and F. In fact,
the value ?T
max
obtained with numerical simulation or
distributed parameter models is higher than one would
expect from even the highest figure-of-merit within the
pellet. The explanation is that numerical simulation and
distributed parameter models include the effect of Thomson
cooling, which is neglected by the lumped parameter models
[4,16].
CONCLUSIONS
Based on one-dimensional steady-state analysis of
thermoelectric phenomena and on analogies between thermal
and electrical domains, two types of electrical three-port
models for thermoelectric devices have been proposed:
lumped and distributed parameter. An important advantage
of these models is that both electrical and thermal behavior
is simulated using a common electronic circuit simulation
tool (e.g., SPICE simulation programs), thus allowing
analysis of the overall thermoelectric system performance,
including control electronics and thermal elements. Lumped
parameter models are easily implemented and, if carefully
used, can provide accurate results. Distributed parameter
models account for temperature dependence of material
properties, thus they are more accurate but slightly more
complicated to implement. Simulations made for a
(Bi
0.5
Sb
0.5
)
2
Te
3
pellet show that for the cases of maximum
temperature difference, ?T
max
,
and maximum cooling power,
Q
cmax
, a lumped parameter electrical model with material
Fig. 7. Prediction comparisons between the different models in Table II
for cooling power vs. electrical current at ?T=0?C, (a) broad electrical
current range, including a secondary axis to show the relative error with
respect to model H, and (b) expanded data near maximum current.
Fig. 8. Comparisons between temperature profile predictions of the
different models presented in Table II for the cases (a) when
?T=?T
max
., and (b) when Q
c
=Q
cmax
.
Budapest, Hungary, 17-19 September 2007
?EDA Publishing/THERMINIC 2007 -page- ISBN: 978-2-35500-002-7
parameters evaluated at mean pellet temperature has, with
respect to the highly realistic numeric simulation, a relative
error of less than 3.2% for ?T
max
and less than 0.2% for
Q
cmax
. By dividing the same pellet into ten segments, the
distributed parameter electrical model reduces the relative
error to 0.03% for ?T
max
and 0.01% for Q
cmax
.
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Fig. 9. Comparisons between the different models in Table II of spatial profiles of material parameters s(x), ?(x), ?(x), and z(x) for ?T
max
case
(see Fig. 6) according to material data given in [15].