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where V is open-circuit voltage, r el is the resistance of the
thermopile and load, and α is Seebeck coefficient. The
Figure Of Merit, Z, that quantifies thermoelectric quality of
the material is defined as
Z = α 2 σ / k = α 2 R th /r el , (2)
where σ is conductivity, k is thermal conductivity, and R th is
thermal resistance. By substituting (2) in (1) and introducing
a heat flow as W = ΔT/R th , (1) can be rewritten as
P = WZΔT/4 , (3)
where ZΔT/4 is thermodynamic efficiency. The best
thermoelectric materials for low-temperature energy
harvesting show a Z of about 0.003 K -1 . Therefore,
conversion efficiency of thermoelectric generators is at least
by a factor of 4 less than the Carnot efficiency. Equation (3)
is useful for the analysis of two basic regimes of operation of
a thermopile, Figs. 1a, 1b. In Fig. 1a, W is constant, so the
only ΔT in (3) is variable. As a result, the power is
proportional to R th and there is no optimum unless R th → ∞.
When ΔT is constant, Fig. 1b, the power is proportional to
R -1 th . Again, no optimum can be found except R th → 0. The
two limits contradict to each other therefore there must be a
kind of transitional regime between the regimes illustrated in
Figs. 1a, 1b. Because no optimum can be found, the
practical aspects affect the design choice instead of pure
theory, i.e., technological limitations, material properties,
required and feasible dimensions of the device, irreversible
losses and parasitic effects.
While optimization is conducted in case of W = const.
(Fig. 1a), thermal resistance can be increased either by
increasing the length of thermopile legs, l, or decreasing
their lateral dimension, t, Fig. 2. In both cases, the
technology becomes the barrier. In case of optimization at
ΔT = const. (Fig. 1b), thermal resistance can be decreased
either by decreasing the length of thermopile legs or
increasing their lateral dimension. In both cases, the
interfaces start to dominate, e.g., the contact resistance
between metal interconnects and semiconductor legs, and the
thermal resistance of interfaces to the heat source and sink.
7-9 October 2009, Leuven, Belgium
Furthermore, certain variations are observed on practice at
“constant” either heat flow or temperature difference due to
(i) above practical limitations and (ii) non-perfect both heat
sources and heat sinks. The effect is qualitatively shown in
Figs. 1a, 1b as dashed lines. Suspecting that a smooth
transition must take place from Fig. 1a to Fig. 1b, one may
qualitatively connect these Figs. as shown in Fig. 1 (dotted
lines). This area between Figs. 1a and 1b seems to be the
most interesting regime. Indeed, the optimization at W =
const. takes place to the right of Fig. 1a while optimization at
ΔT = const. takes place to the left of Fig. 1b. The reader
may also pay an attention at (3), which states that power is
proportional to the product of W by ΔT. Therefore, not
much power is produced at maximum thermodynamic
efficiency, Fig. 1b, while maximum power is not produced at
maximum heat flow either.
III. THERMOPILE AT VARIABLE BOTH HEAT FLOW AND
TEMPERATURE DIFFERENCE
Basing on above discussion, Figs. 1a and 1b are combined
in one Fig. 3. X-axis is replaced with the ratio of the thermal
resistance of thermoelectric generator (TEG), R TEG
, to the one
of the environment, R env
. Indeed, not the absolute value of
the thermal resistance of thermopile allows constant either
heat flow or temperature difference, but the ratio R TEG
/R env
.
The heat flow is constant in Fig. 1a because it is fully limited
by R env
(in case of energy harvesters). In Fig. 1b, constant
ΔT is provided by the environment with very low thermal
resistance. Once R TEG
approaches from either side to R env
,
neither W nor ΔT can remain constant and change as shown
by dashed lines in Figs. 1a, 1b. Using (3), the reader can
already qualitatively plot the dependence of power, Fig. 3.
According to the thermal circuit, Fig. 2, the ΔT observed
between the heat source and heat sink cannot appear on the
thermopile because of non-zero value of R env
. The task of
this Section is to find the temperature difference on the
thermopile, ΔT tp , corresponding to the power maximum.
Based on electro-thermal analogy, Fig. 2, R TEG
must be
comparable to or larger than the one of the serial thermal
resistor R env
. (This is to obtain large voltage on R tp
in an
electrical circuit, or to obtain large ΔT tp in a thermal circuit.)
This also means that while optimizing the device, i.e.,
changing its thermal resistance, neither constant temperature
metal
n
l
R cold R TEG R hot
p
t
Fig. 2. Thermoelectric generator (TEG) and its thermal circuit. The thermal
resistance of the environment is composed of a thermal resistance between
(i) the heat sink or (ii) the point where heat is generated and corresponding
metal-semiconductor p-n junction plane of a thermopile. Any additional
TEG elements (such as screws or a sealing sidewall on the perimeter of
thermopile) thermally interconnecting the two plates, as well as air between
them and radiation, are denoted as R pp (parallel parasitic thermal resistance).
ΔT/ΔTmax; W/W max; P/Pmax
1
0.8
0.6
0.4
0.2
0
W tp P
ΔT tp
0.01 0.1 1 10
R TEG /R env
Fig. 3. Normalized heat flow through a thermopile (R tp), the temperature
difference on it, and the power. Location of maxima can be at the other
R TEG/R env. The curves end on the left at complete filling of a TEG (fixed
size) with thermocouples, and on the right, with the only one thermocouple.
©EDA Publishing/THERMINIC 2009 96
ISBN: 978-2-35500-010-2