ON THE ASYMMETRY AND TRANSFER ERRORS OF THERMOELECTRIC CONVERTERS

E. I. Tsimbalist

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

At the present time, the principal element of comparators that compare an unknown alternating-current quantity with a direct-current quantity equivalent to it in effect is an electrothermal converter, in particular a thermoelectric converter (TC).

It is known that among the components of the comparison error of an electrothermal converter there are errors that do not depend on frequency and that ultimately determine its properties. These errors are caused by the manifestation of the Thomson and Peltier effects. The errors in question are the asymmetry error, due to the dependence of the converter output quantity on the polarity of the direct current, and the transfer error when comparing direct- and alternating-current quantities. The physical nature of the occurrence of these errors is set forth in sufficient detail in [1, 2]; in [2] a calculation of the errors is given for a homogeneous filament, heat removal from which takes place only through the current-carrying leads—the traverses.

Below an example is given of an error calculation that takes into account, in addition to heat losses due to thermal conductivity, heat losses from the heater due to radiation.

For a time-established process, the power balance for an element of length \(dl\) of a conductor carrying current \(i\) may be written as the following equation:

\[ dP_1+dP_2+dP_3+dP_4+dP_5=0. \tag{1} \]

Here

\[ dP_1=\frac{i^2\rho_0(1+\alpha\Theta)}{q}\cdot dl \]

is the power generated by the element \(dl\) by the current flowing through it;

\[ dP_2=\lambda q\frac{d^2\Theta}{dx^2}\cdot dl \]

is the power received by the element \(dl\) from neighboring sections (along the axis);

\(dP_3\) is the power dissipated by the element \(dl\) by convection. For vacuum TCs, \(dP_3=0\),

\[ dP_4=-b\sigma p\left(T^4-T_0^4\right)\cdot dl \simeq -b\sigma p k\Theta\,dl, \]

\(dP_4\) is the power given off by the element \(dl\) by radiation;

\[ dP_5=\pm S i\frac{d\Theta}{dx}\cdot dl \]

is the power generated or absorbed by the ele-

... by the element \(dl\) of the filament as a result of the Thomson effect (only when direct current flows).

In the indicated relations:

\(\rho_0\) is the specific resistance of the filament material at a temperature equal to \(T_0^\circ\)K; \(\alpha\) is the temperature coefficient of electrical resistance; \(q\) is the cross section, \(p\) the perimeter of the filament;

\(\lambda\) is the coefficient of thermal conductivity of the filament material;

\(\sigma\) is the Stefan–Boltzmann constant;

\(b\) is the coefficient of incomplete radiation (for a TC \(b=0.6\));

\[ k=\frac{T_{\max}^{4}-T_{0}^{4}}{T_{\max}-T_0} \]

is the coefficient of proportionality;

\(S\) is the Thomson coefficient, whose value and sign depend on the material;

\(T\) and \(T_0\) are the absolute temperatures of the filament and of the surrounding medium;

\(\Theta=T-T_0\) is the temperature rise of a section of the filament.

For alternating current, equation (1) takes the form:

\[ \Delta^2 \frac{d^2\Theta}{dx^2}-\Theta+\Theta_\infty=0, \tag{2} \]

where

\[ \Delta^2=\frac{\lambda q}{b\sigma pk-i^2\alpha\rho_0/q}; \qquad \Theta_\infty=\frac{i^2\rho_0/q}{b\sigma pk-i^2\alpha\rho_0/q}. \]

The solution of equation (2) under the boundary conditions

\[ \Theta\big|_{x=0}=\Theta\big|_{x=l}=\Theta_1 \]

is

\[ \Theta=\Theta_\infty\left\{1-\frac{\operatorname{ch}[(2x-l)/2\Delta]}{\operatorname{ch} l/2\Delta}\right\}+\Theta_1. \]

In this case the temperature at the middle of the filament, where the thermocouple is usually attached, reaches the value:

\[ \Theta\big|_{x=\frac{l}{2}} = \Theta_\infty\left[1-\frac{1}{\operatorname{ch} l/2\Delta}\right]+\Theta_1. \tag{3} \]

For direct current, owing to the manifestation of the Thomson and Peltier effects, we have

\[ \Delta^2\frac{d^2\Theta}{dx^2}\pm 2\delta\frac{d\Theta}{dx}-\Theta+\Theta_\infty=0, \tag{4} \]

where

\[ 2\delta=\frac{S\cdot i}{b\sigma pk-i^2\alpha\rho_0/q}. \]

The boundary conditions (the temperature at the ends of the filament) for direct current will differ from the conditions for alternating current because of the manifestation of the Peltier effect at the heater–traverse junction.

Let \(\Theta\big|_{x=0}=\Theta_2<\Theta_1\) and \(\Theta\big|_{x=l}=\Theta_3>\Theta_1\) for one direction of current, and \(\Theta\big|_{x=0}=\Theta_3,\ \Theta\big|_{x=l}=\Theta_2\) for the opposite direction.

Taking the boundary conditions into account, the solutions of equation (4) have the form:

\[ \Theta=\Theta_\infty- \frac{ (\Theta_\infty-\Theta_1)\,e^{\delta x/\Delta^2}\cdot \operatorname{sh}\left(\frac{l-x}{\Delta}\right) + (\Theta_\infty-\Theta_2)\,e^{-\delta(l-x)/\Delta^2}\cdot \operatorname{sh}\left(\frac{x}{\Delta}\right) }{ \operatorname{sh}\frac{l}{\Delta} } \tag{5} \]

\[ \Theta=\Theta_\infty-\frac{(\Theta_\infty-\Theta_2)\,l^{-\frac{\delta x}{\Delta^2}}\cdot \operatorname{sh}\left(\frac{l-x}{\Delta}\right)+(\Theta_\infty-\Theta_1)\,l^{\frac{\delta(l-x)}{\Delta^2}}\cdot \operatorname{sh}\left(\frac{x}{\Delta}\right)}{\operatorname{sh}\frac{l}{\Delta}} . \tag{6} \]

Calculating the values of \(\Theta\) at \(x=\frac{l}{2}\) from (5) and (6), we find that, when the direction of the current is changed, the temperature at the middle of the heater does not change and is equal to

\[ \Theta\bigg|_{x=\frac{l}{2}} = \Theta_\infty \left( \frac{\operatorname{ch}\frac{\delta l}{2\Delta^2}}{\operatorname{ch}\frac{l}{2\Delta}} \right) + \frac{1}{2}\, \frac{ \Theta_1 l^{\frac{\delta l}{2\Delta^2}}+\Theta_2\cdot l^{-\frac{\delta l}{2\Delta^2}} }{ \operatorname{ch}\frac{l}{2\Delta} }. \]

Thus, the asymmetry error observed in thermoelectric converters is determined entirely by the quality of their manufacture (the deviation of the thermocouple installation from the middle of the heater and its nonuniformity along the length).

In contrast to alternating current, under direct current the temperature maximum is shifted relative to the center of the heater by an amount proportional to the Thomson coefficient; i.e., there is a transfer error from direct to alternating current even when the thermocouple of the thermoelectric converter is correctly positioned \(\left(x=\frac{l}{2}\right)\):

\[ \delta_n= \left( \frac{\Theta_{=}-\Theta_{\sim}}{\Theta_{=}} \right)_{x=l/2} \approx - \frac{ \left(\dfrac{S\cdot il}{4\lambda_h q}\right)^2 }{ 2\left(\operatorname{ch}\frac{l}{2\Delta}-1\right) }. \tag{7} \]

The transfer error calculated from (7) is the maximum possible. Indeed, owing to the finite geometric dimensions of the thermocouple, the transfer error will decrease as a result of averaging the temperature by the thermocouple bead. At the same time, a reduction in the asymmetry error should also be observed.

The greatest reduction in errors can be achieved by using converters for which the effect of the temperature shift relative to the center of the filament does not play a substantial role. These include multielement thermoelectric converters, and emission and photoelectric converters based on incandescent lamps. The output quantity of such converters, determined by the integral value of the heater temperature rather than by the temperature of its midpoint, will depend on parasitic shifts of the temperature maximum to a considerably lesser degree. The latter is confirmed by a number of experimental data; for example, for multielement thermoelectric converters [1] and photoelectric converters, the asymmetry error does not exceed \(\pm(0.01 \div 0.02)\%\), whereas for ordinary thermoelectric converters this value reaches 1%.

In conclusion, the author thanks Docent M. S. Roitman for his attention to the work.

REFERENCES

1. T. B. Rozhdestvenskaya. Electrical Comparators for Precise Measurements of Current, Voltage, and Power. Standards Publishing House, 1964.

2. Hermach F. L. Thermal Converters as AC—DC Transfer Standards for Current and Voltage Measurements at Audio Frequencies. “Journal Research NBS,” v. 48, No. 2, February, 1952.