Study of the Dynamic Properties of Thermoelectric Detectors by Modeling Heat Sources Using the Peltier Effect
Abstract Generated abstract
This paper proposes a method for studying the dynamic properties of non-armored thermoelectric detectors by replacing optical heating with a temperature gradient produced through the Peltier effect. The authors model stationary and transient heat conduction in thermopiles with lap-welded and butt-welded junctions, using heat-balance equations and Laplace transforms to derive the time dependence of the junction temperature difference after the heating current is switched off. Under simplifying assumptions, the derived thermoelectric emf decays exponentially, with parameters corresponding to the detector inertia index or time constant. The result supports a simple experimental procedure in which a thermopile is heated electrically, switched to a measuring instrument, and its emf decay is recorded to determine detector dynamics.
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UDC 681.327.66
STUDY OF THE DYNAMIC PROPERTIES OF THERMOELECTRIC DETECTORS BY MODELING HEAT SOURCES USING THE PELTIER EFFECT
E. M. BELOV, A. I. KUZNETSOV
(Presented by the scientific seminar of the Faculty of Physics and Technology)
The dynamic properties of non-armored thermoelectric detectors are approximated by a first-order inertial element; that is, the response of the detector to a simultaneous step change in the intensity of electromagnetic radiation is described by an exponential function of the change in thermo-emf with time
\[ E(\tau)=E_0\exp(-\mu\tau), \tag{1} \]
where \(E_0\) is the thermo-emf generated in the initial period;
\(\mu\) is a constant characterizing the dynamic properties.
Low-inertia, high-sensitivity thermoelectric detectors are thermopiles with a series connection of \(n\) thermoelements.
If the thermopile is connected to an external DC voltage source, then the action of the Peltier effect manifests itself in the establishment of a certain temperature gradient between the “hot” and “cold” junctions, which is equivalent to the action of electromagnetic radiation on a thermoelectric detector. By switching the thermopile to a measuring instrument, one can record the time variation of the generated thermo-emf caused by cooling of the thermopile. A theoretical proof of the exponential character of this dependence will serve as a justification for the possibility of using the proposed method to study the dynamic properties of thermoelectric detectors.
The material of the thermoelectrodes \((i)\) and junctions \((k)\) is characterized by:
\(\lambda_i, \lambda_k\) — thermal-conductivity coefficients;
\(c_i, c_k\) — specific heats;
\(\rho_i, \rho_k\) — specific resistances;
\(\gamma_i, \gamma_k\) — densities;
\(\varepsilon\) — specific thermo-emf of the pair, respectively.
It is assumed that these parameters, as well as the heat-transfer coefficients from the surfaces of the thermoelectrodes \(\alpha_i\) and junctions \(\alpha_k\), are independent of temperature.
The geometrical dimensions of the thermoelectrodes \((i)\) and junctions \((k)\) are: \(b_i, b_k\) — width; \(\delta_i, \delta_k\) — thickness; \(l_i, l_k\) — length; \(p_i\) and \(p_k\), \(f_i\) and \(f_k\) — perimeter and cross-sectional area, respectively; \(q_i, q_k\) — heat-release densities per unit length when a heating current is passed through the thermopile.
We choose the cross-sectional areas of the thermoelectrodes from the condition of their most advantageous ratio [1]:
\[ \frac{f_1}{f_2}=\sqrt{\frac{\rho_1\lambda_2}{\rho_2\lambda_1}}. \tag{2} \]
In order to simplify the mathematical expressions, we shall restrict ourselves to consideration of the case \(l_i \to \infty\) and \(q_{i=1}=q_{i=2}\). The latter equality, taking relation (2) into account, holds when
\[ \rho_1\lambda_1=\rho_2\lambda_2. \tag{3} \]
Equality (3) is satisfied with sufficient accuracy for thermoelectrodes made of chromel and alumel.
The stationary temperature distribution along the thermoelectrodes (the junctions are at the origins of the coordinates) is given by the solution of the Laplace equation
\[ \frac{\partial^2 T_i(x_i)}{\partial x_i^2} = m_i^2 T_i(x_i)-\frac{q_i}{\lambda_i f_i}, \qquad (0\le x_i\le \infty), \tag{4} \]
where
\[ m_i^2=\frac{\alpha_i\rho_i}{\lambda_i f_i}, \qquad (i=1,2). \tag{5} \]
The boundaries \(x_i\to\infty\) divide each thermoelectrode into two independent semi-infinite rods; therefore the assumption
\[ \left. \frac{\partial T_i(x_i)}{\partial x_i} \right|_{x_i\to\infty} =0 \tag{6} \]
is permissible.
As the second boundary conditions for the rods we consider the heat balances of the junctions \((x_i=0)\)
\[ Q_\kappa+Q_{\lambda\kappa}-Q_{\alpha\kappa}+(-1)^\kappa Q_{\Pi\kappa}=0, \tag{7} \]
where
\[ Q_\kappa=\frac{q_\kappa}{\alpha_\kappa\rho_\kappa}, \qquad Q_{\lambda\kappa} = \lambda_i f_i \left. \frac{\partial T_i(x_i)}{\partial x_i} \right|_{x_i=0}, \]
\[ Q_{\alpha\kappa}=\alpha_\kappa\rho_\kappa T_i(x_i=0), \qquad Q_{\Pi\kappa}=\varepsilon I T_i(x_i=0), \qquad (\kappa=1,2). \tag{8} \]
The solution of equation (4) under the boundary conditions (6), (7) is
\[ T_i(x_i)=A+B_\kappa\exp(-m_i x_i), \tag{9} \]
where
\[ A=\frac{q_i}{\alpha_i\rho_i}; \tag{10} \]
\[ B_\kappa = \frac{ Q_\kappa-A\left[\alpha_\kappa\rho_\kappa l_\kappa-(-1)^\kappa\varepsilon I\right] }{ \sqrt{\alpha_i\rho_i\lambda_i f_i}+\alpha_\kappa\rho_\kappa l_\kappa-(-1)^\kappa\varepsilon I }. \tag{11} \]
The stationary temperature distribution along the thermoelectrodes depends on the manner in which the junctions are made.
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When the junctions are made by welding the thermoelectrodes with overlap, then \(\rho_i>\rho_\kappa\), \(f_\kappa \ge f_1+f_2\), and \(q_i>q_\kappa\). The temperature field has a maximum on the thermoelectrodes \(T_i(x_i\to\infty)=A\), and a minimum at the junctions \(T_i(x_i=0)=A+B_\kappa\).
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When the junctions are made by butt-welding the thermoelectrodes, then in the heat balance (7) the components \(Q_\kappa\) and \(Q_{\alpha\kappa}\) may be neglected. In this case the temperature field
\[ T_i^0(x_i)=A+B_\kappa^0\exp(-m_i x_i), \tag{12} \]
where
\[ B_\kappa^0 = \frac{(-1)^\kappa\varepsilon I} { \sqrt{\alpha_i\rho_i\lambda_i f_i}-(-1)^\kappa\varepsilon I } \tag{13} \]
has experimental points at the junctions \((x_i=0,\ i=1,2)\).
The dynamics of the change in the temperature field after switching off the heating current is determined by the solution of the nonstationary differential equations of heat conduction for the thermoelectrodes
\[ \frac{\partial T_i(x_i,\tau)}{\partial \tau} = a_i^2 \frac{\partial^2 T_i(x_i,\tau)}{\partial x_i^2} - \nu_i^2 T_i(x_i,\tau) \tag{14} \]
and for the junctions in accordance with their design:
\[ 1.\quad \frac{\partial T_k(\tau)}{\partial \tau} + \nu_k^2 T_k(\tau) + \psi \frac{\lambda_i f_i}{c_k}\cdot \left. \frac{\partial T_i(x_i,\tau)}{\partial x_i} \right|_{x_i=0} =0; \tag{15} \]
\[ 2.\quad \frac{\partial T_k^0(\tau)}{\partial \tau} + \psi \frac{\lambda_i f_i}{c_k}\cdot \left. \frac{\partial T_i(x_i,\tau)}{\partial x_i} \right|_{x_i=0} =0, \tag{16} \]
where
\[ a_i^2=\frac{\lambda_i}{c_i\gamma_i}, \qquad \nu_i^2=\frac{\alpha_i p_i}{c_i\gamma_i f_i}, \qquad \nu_k^2=\frac{\alpha_k p_k l_k}{c_k} \tag{17} \]
with the initial temperature distribution specified by equations (9—11) or (12), (13), depending on the design of the junctions, and the boundary conditions
\[ \left. \frac{\partial T_i(x_i,\tau)}{\partial x_i} \right|_{x_i\to\infty} =0, \tag{18} \]
\[ T_i(x_i=0,\tau)=T_k(\tau). \tag{19} \]
In equations (15) and (16) the factor \(\psi\) is equal to \(-1\), \(0\), or \(+1\), respectively, if
\[ \left. \frac{\partial T_i(x_i,\tau)}{\partial x_i} \right|_{x_i=0} \]
is greater than, equal to, or less than zero, respectively.
The solution of the nonstationary problem (14—19) was carried out by an operational method based on Laplace transforms. The differential equations (14—16) and the boundary conditions (18), (19) in transforms have the form
\[ \frac{\partial^2 \Theta_i(x_i,s)}{\partial x_i^2} = \frac{s+\nu_i^2}{a_i^2}\Theta_i(x_i,s) - \frac{1}{a_i^2} \left[ A+B_k\exp(-m_i x_i) \right], \tag{14a} \]
\[ \Phi_k(s) = \frac{A+B_k}{s+\nu_k^2} + \psi \frac{\lambda_i f_i}{c_k(s+\nu_k^2)} \left. \frac{\partial \Theta_i(x_i,s)}{\partial x_i} \right|_{x_i=0}, \tag{15a} \]
\[ \Phi_k^0(s) = \frac{A+B_k^0}{s} + \psi \frac{\lambda_i f_i}{c_k s}\cdot \left. \frac{\partial \Theta_i(x_i,s)}{\partial x_i} \right|_{x_i=0}, \tag{16a} \]
\[ \left. \frac{\partial \Theta_i(x_i,s)}{\partial x_i} \right|_{x_i\to\infty} =0, \tag{18a} \]
\[ \Theta_i(x_i=0,s)=\Phi_k(s). \tag{19a} \]
The solution of equation (14a) is found from the known fundamental system of solutions of the homogeneous differential equation corresponding to the given nonhomogeneous equation [2]
\[ \Theta_i(x_i,s) = \frac{A}{s+\nu_i^2} + \frac{B_k}{s}\exp(-m_i x_i) + D(s)\exp[-\eta_i(s)x_i], \tag{14b} \]
where
\[ \eta_i^2(s)=\frac{s+\nu_i^2}{a_i^2}. \tag{20} \]
\(D(s)\) is a function of the parameter \(s\) to be determined.
In solution (14b), boundary condition (18a) and \(\nu_i^2 \equiv a_i^2 m_i^2\) have been taken into account. Solution (14b) makes it possible to determine \(\Phi_\kappa(s)\) and \(\Phi_\kappa^0(s)\).
\[ \Phi_\kappa(s)=\frac{A+B_\kappa}{s+\nu_\kappa^2} -\psi\left[ \frac{\lambda_i f_i m_i}{c_\kappa s(s+\nu_\kappa^2)}\,B_\kappa +\frac{\lambda_i f_i \eta_i(s)}{c_\kappa(s+\nu_\kappa^2)}\,D(s) \right], \tag{15б} \]
\[ \Phi_\kappa^0(s)=\frac{A+B_\kappa^0}{s} -\psi\left[ \frac{\lambda_i f_i m_i}{c_\kappa s^2}\,B_\kappa^0 +\frac{\lambda_i f_i \eta_i(s)}{c_\kappa s}\,D^0(s) \right]. \tag{16б} \]
We use boundary condition (19a) to determine \(D(s)\) and \(D^0(s)\).
\[ D(s)=\left[ A\frac{\nu_i^2-\nu_\kappa^2}{(s+\nu_i^2)(s+\nu_\kappa^2)} -B_\kappa\frac{\nu_\kappa^2+\psi\lambda_i f_i m_i c_\kappa^{-1}}{s(s+\nu_\kappa^2)} \right] \left[ 1+\psi\frac{\lambda_i f_i\eta_i(s)}{c_\kappa(s+\nu_\kappa^2)} \right]^{-1}, \tag{21} \]
\[ D^0(s)=\left[ A\frac{\nu_i^2}{s(s+\nu_i^2)} -B_\kappa^0\psi\frac{\lambda_i f_i m_i}{c_\kappa s^2} \right] \left[ 1+\psi\frac{\lambda_i f_i\eta_i(s)}{c_\kappa s} \right]^{-1}. \tag{22} \]
Substitution of (21) and (22) into (15б) and (16б) gives
\[ \Phi_\kappa(s)= A\frac{c_\kappa(s+\nu_i^2)+\psi\lambda_i f_i\eta_i(s)} {(s+\nu_i^2)\left[c_\kappa(s+\nu_\kappa^2)+\psi\lambda_i f_i\eta_i(s)\right]} + B_\kappa \frac{c_\kappa s+\psi\lambda_i f_i[\eta_i(s)-m_i]} {s\left(c_\kappa(s+\nu_\kappa^2)+\psi\lambda_i f_i\eta_i(s)\right)}, \tag{15в} \]
\[ \Phi_\kappa^0(s)= A\frac{c_\kappa(s+\nu_i^2)+\psi\lambda_i f_i\eta_i(s)} {(s+\nu_i^2)\left[c_\kappa s+\psi\lambda_i f_i\eta_i(s)\right]} + B_\kappa^0 \frac{c_\kappa s+\psi\lambda_i f_i[\eta_i(s)-m_i]} {s\left[c_\kappa s+\psi\lambda_i f_i\eta_i(s)\right]}. \tag{16в} \]
Here the property \(\psi^2=1\) has been used for either sign of
\[ \left.\frac{\partial\Theta_i(x_i,s)}{\partial x_i}\right|_{x_i\to 0}. \]
Thus, solutions (15в) and (16в) determine the dynamics of the temperature change of the “hot” (\(\kappa=1\)) and “cold” (\(\kappa=2\)) junctions.
If the total heat capacities of the junctions differ only slightly from one another, then, putting \(c_{\kappa=1}=c_{\kappa=2}=c_0\), \(\nu_{i,\kappa=1}=\nu_{i,\kappa=2}=\nu_0\), and using (15в) and (16в), we obtain
\[ \Delta\Phi_\kappa(s)=\Phi_1(s)-\Phi_2(s) =|B_1-B_2|\, \frac{c_0s+\psi\lambda_i f_i[\eta_i(s)-m_i]} {s\left[c_0(\nu_0^2+s)+\psi\lambda_i f_i\eta_i(s)\right]}, \tag{23} \]
\[ \Delta\Phi_\kappa^0(s)=\Phi_1^0(s)-\Phi_2^0(s) =|B_1^0-B_2^0|\, \frac{c_0s+\psi\lambda_i f_i[\eta_i(s)-m_i]} {s\left[c_0s+\psi\lambda_i f_i\eta_i(s)\right]}. \tag{24} \]
To pass from the images to the original, we expand the numerator and denominator of equations (23), (24) in a Maclaurin series in powers of \(s\) and retain only the linear terms of the expansion [3]. As a result we obtain
\[ \Delta\Phi_\kappa(s)=|B_1-B_2|\,\frac{1}{s+M}. \tag{23a} \]
\[ \Delta\Phi_\kappa^0(s)=|B_1^0-B_2^0|\,\frac{1}{s+M_0}, \tag{24a} \]
where
\[ M= \frac{\alpha_\kappa p_\kappa l_\kappa+\psi\sqrt{\alpha_i\rho_i\lambda_i f_i}} {c_\kappa+\psi\,0.5\,c_i\gamma_i f_i\sqrt{\dfrac{\lambda_i f_i}{\alpha_i\rho_i}}}, \tag{25} \]
\[ M_0= \frac{\sqrt{\alpha_i\rho_i\lambda_i\rho_i}} {\psi c_\kappa+0.5\,c_i\gamma_i f_i\sqrt{\dfrac{\lambda_i f_i}{\alpha_i\rho_i}}}. \tag{26} \]
The originals of images (23a) and (24a), respectively, are
\[ \Delta T_{\kappa}(\tau)=|B_1-B_2|\exp(-M\tau), \tag{23б} \]
\[ \Delta T_{\kappa}^{0}(\tau)=|B_1^{0}-B_2^{0}|\exp(-M_0\tau). \tag{24б} \]
In a certain temperature interval, the thermoe.m.f. generated by the thermobattery is proportional to the temperature gradient between the junctions,
\[ E=n\cdot\varepsilon\Delta T, \tag{27} \]
therefore, in this temperature interval, the dynamics of the change in thermoe.m.f. after switching off the heating current of the thermobattery, in the case of lap-welded junctions,
\[ E(\tau)=n\varepsilon|B_1-B_2|\exp(-M\tau) \tag{23в} \]
and in the case of butt-welded junctions
\[ E^{0}(\tau)=n\varepsilon|B_1^{0}-B_2^{0}|\exp(-M_0\tau) \tag{24в} \]
are described by exponential functions, similarly to relation (1).
Consequently, the dynamic properties of thermoelectric detectors can be investigated by the proposed method, since the correspondence of the parameters \(M\) and \(M^{0}\) to the parameter \(\mu\) has been proved.
The practical implementation of the method is characterized by its simplicity. Initially, a heating current is passed through the thermobattery, as a result of which a temperature gradient is created between the junctions. We switch the thermobattery from the power source to the measuring instrument and record the change in the generated thermoe.m.f. over time. Processing the experimental results makes it possible to determine the inertia index \(\mu\) (time constant) of the thermoelectric detector used.
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- E. Kamke. Handbook of Ordinary Differential Equations. Moscow, “Nauka,” 1965.
- N. A. Yaryshev. Izv. Vuzov SSSR. “Priborostroenie,” 1963, vol. 6, No. 4.