ERROR OF PHOTOELECTRIC CONVERTERS AT HIGH FREQUENCIES

Yu. M. FOMICHEV, M. S. ROITMAN

(Presented by the scientific seminar of the Department of Radio Engineering)

The error of a comparator is determined by its sensitivity, the degree of influence of external disturbing factors, the temporal instability of the parameters of the measuring converter, its frequency characteristic, asymmetry, and transition error.

When measuring voltage in the low- or high-frequency range, the frequency error \((\delta_f)\) of the converter becomes of primary importance. The remaining errors, in comparison with it, are quantities of the second order of smallness, and their influence, in the first approximation, may be neglected.

In this connection, the expression for the frequency error of comparison may be written in the following form:

\[ \delta_f = \frac{U_0 - U_f}{U_0}, \]

where \(U_f\) is the rms value of the measured alternating voltage, and
\(U_0\) is the direct voltage equivalent in effect to the alternating one.

The present work is devoted to a theoretical and experimental study of the error in the high-frequency range of one of the most promising types of converters—the photoelectric converter.

The high-frequency error is caused by the reactances of the elements forming part of the converter and by effects arising in the circuits when high-frequency currents flow.

For a photoelectric converter (PC), the sources of such error are:

1) the skin effect;
2) nonuniformity of the distribution of alternating current along the length of the heater;
3) reactances of the lead-in traverses and the filament of the incandescent lamp (IL).

The skin effect manifests itself in two aspects: in an increase in the active resistance of the IL circuit and in an increase in the inductance of the leads and the filament due to the internal magnetic flux. As studies [2] have shown, these components of the frequency error, for the geometrical dimensions of production lamps suitable for comparison purposes, are sufficiently small and, in the frequency range up to \(200 \div 300\) MHz, are not determining factors.

The nonuniform magnitude of the high-frequency current along the filament body causes nonuniformity in the temperature distribution and, consequently, in the luminous flux and its spectral composition. The reactances of the elements of the FL circuit lead to the fact that equal magnitudes of the effective values of alternating and direct voltages create different currents in the primary circuit of the converter. This leads to different responses of the light-sensitive element to direct and alternating currents, i.e., to a frequency error.

Fig. 1.

To determine the magnitude of these components of the error, it is necessary to consider the primary circuit of the PC as a long line. In the general case the line will be nonuniform, since the physical dimensions and parameters along its length do not remain constant. In connection with this, the analysis is carried out as follows: the FL circuit is divided into uniform sections (Fig. 1), the first of which is short-circuited, while each of the subsequent ones is loaded by the input resistance of the preceding one. Then, using the equations of a long line, an expression is determined for the distribution of the current \((I_1)\) along the FL filament, as a function of frequency \((\omega)\) and the per-unit-length parameters of the line \((L, C, r)\), for a given input voltage \((U_f)\), in the form:

\[ i_1 = U_f \cdot \varphi(\xi,\omega,L,C,r), \]

where \(\xi\) is the current coordinate.

Using this expression, one can find the average value of the current flowing over the entire length of the filament:

\[ I_{\mathrm{av}} = \frac{1}{\xi}\int_{0}^{\xi} U_f |\varphi(\xi,\omega,L,C,r)|\,d\xi . \]

Further, assuming that the photodetector responds identically to \(I_{\mathrm{av}}\) and to a direct current equal to it in magnitude, the expressions for the frequency error of the PC (1) can be reduced to the form:

\[ \delta_f = 1 - \frac{1}{ R_l \cdot \frac{1}{\xi}\int_{0}^{\xi} |\varphi(\xi,\omega,L,C,r)|\,d\xi }, \tag{2} \]

where \(R_l\) is the resistance of the FL to direct current at the operating point.

For simplification, let us assume that the equivalent long line (of the primary circuit of the converter) consists of two uniform sections formed by the filament and the input traverses. In this case the following expression is obtained for the modulus of the current flowing through the heater:

\[ |I_1| = U_f \times \]

\[ \times \frac{ \sqrt{\operatorname{ch}^{2}\alpha_2 \xi_2 \cos^{2}\beta_2 \xi_2 + \operatorname{sh}^{2}\alpha_2 \xi_2 \sin^{2}\beta_2 \xi_2} }{ \sqrt{ \left[(E_2A_1+E_1A_2)-(F_2B_1+F_1B_2)\right]^2 + \left[(F_2A_1+F_1A_2)+(E_2B_1+E_1B_2)\right]^2 } }, \tag{3} \]

where

\[ A_1 = \operatorname{sh}\alpha_2 l_2 \cos\beta_2 l_2 \operatorname{ch}\alpha_1 l_1 \cos\beta_1 l_1 - \operatorname{ch}\alpha_2 l_2 \sin\beta_2 l_2 \operatorname{sh}\alpha_1 l_1 \sin\beta_1 l_1; \]

\[ \begin{aligned} B_1 &= \operatorname{ch}\alpha_2 l_2 \sin\beta_2 l_2 \operatorname{ch}\alpha_1 l_1 \cos\beta_1 l_1 + \operatorname{sh}\alpha_2 l_2 \cos\beta_2 l_2 \operatorname{sh}\alpha_1 l_1 \sin\beta_1 l_1;\\ A_2 &= \operatorname{ch}\alpha_2 l_2 \cos\beta_2 l_2 \operatorname{sh}\alpha_1 l_1 \cos\beta_1 l_1 - \operatorname{sh}\alpha_2 l_2 \sin\beta_2 l_2 \operatorname{ch}\alpha_1 l_1 \sin\beta_1 l_1;\\ B_2 &= \operatorname{ch}\alpha_2 l_2 \cos\beta_2 l_2 \operatorname{ch}\alpha_1 l_1 \sin\beta_1 l_1 + \operatorname{sh}\alpha_2 l_2 \sin\beta_2 l_2 \operatorname{sh}\alpha_1 l_1 \cos\beta_1 l_1; \end{aligned} \]

\(\alpha_1; \alpha_2; \beta_1; \beta_2\) are the attenuation and phase constants for the corresponding sections of the line;

\(E_1, F_1\) and \(E_2, F_2\) constitute the real and imaginary parts of the characteristic impedances of these sections.

An analysis of expression (3) makes it possible to conclude that the pattern of current distribution along the length of the heater is given by its numerator, while the denominator is the modulus of the total impedance of the heater circuit under alternating current. To estimate the quantitative relationships, let us consider the following specific example. The converter is made in the form of a section of coaxial line with outer-conductor diameter \(D = 2 \cdot 10^{-2}\,m\). The inner conductor consists of the input crosspieces of the incandescent lamp, of diameter \(d = 0.5 \cdot 10^{-3}\,m\), length \(l = 2 \cdot 10^{-2}\,m\), and the filament, whose diameter is \(d = 5 \cdot 10^{-6}\,m\), length \(10^{-2}\,m\), and linear resistance \(R = 10^{4}\,\Omega/m\). The calculated current values obtained (from expression 3) at the beginning and end of the filament at a frequency of \(300\,\text{Mc/s}\) differ by \(0.2 \cdot 10^{-2}\%\), which is such a negligible value that, even despite the high degree of dependence of the output parameter of the PC on the current, it may be disregarded. In view of this, expression (2) takes the simpler form:

\[ \delta_f = 1 - \frac{|Z_l|}{R_l}, \tag{4} \]

where \(|Z_l|\) is the denominator from expression (3), whose value is determined by the geometric dimensions and the material of the constituent elements of the primary circuit.

The comparison error, which depends on the magnitude of the reactances and is determined by formulas (3) and (4), for the converter parameters listed above at a frequency of \(300\,\text{Mc/s}\), is \(18\%\).

Thus, from all that has been set forth above, it follows that the error of a PC in the high-frequency region depends mainly on the reactances of the elements of the heater circuit.

Fig. 2.

Calculating the frequency error of comparison using expression (3) is a rather cumbersome task. A simpler and more illustrative solution is obtained by using equivalent circuits of the PC with lumped parameters.

Indeed, in a first approximation, a long line can be replaced by the equivalent circuit of Fig. 2 if \(L_l\) is taken as the total inductance of the leads and the lamp filament, \(C_l\) as their capacitance relative to the shield, and \(R_l\) as the resistance of the incandescent lamp at the operating point; then we obtain an equivalent circuit of the PC with lumped parameters.

The frequency error is determined from the expression:

\[ \delta_f = \frac{2}{3}\omega^2 C_l L_l - \frac{1}{2} \frac{ \omega^2 \left[ \frac{3}{2}L_l + \frac{1}{2}C_l R_l^2 \right]^2 }{ \frac{9}{4}R_l^2 }, \tag{5} \]

where \(\omega\) is the frequency of the measured voltage.

The quantities \(C_\ell\) and \(L_\ell\) are determined from known formulas [3], taking into account the structural dimensions of the converter. To ascertain the validity of such a representation of the primary circuit of the PC, let us use the numerical example given earlier. The compensation error calculated by formula (5) is \(17.7\%\) at \(300\ \text{MHz}\). Thus, the difference between the result obtained earlier and the present one is quite admissible; moreover, when the frequency of the voltage being measured is reduced, this difference will decrease.

The presence of two components in the last expression, positive and negative, indicates the possibility of compensating the frequency error. Denoting \(\omega C_\ell R_\ell = a\), \(L_\ell/C_\ell R_\ell^2 = K\), expression (5) can be reduced to the form:

\[ \delta_f = -a^2 \left[\frac{1}{2}K^2 - \frac{1}{3}K + \frac{1}{18}\right]. \]

From this it is easy to find that \(\delta_f = 0\) when \(K = \frac{1}{3}\).

The obtained relation can be used to determine the optimum dimensions of the converter. Thus, for a coaxial design the inductance and capacitance of the inner conductor are equal to:

\[ L_\ell = \frac{\mu}{2\pi} l \cdot \ln \frac{D}{d}; \tag{6} \]

\[ C_\ell = \frac{2\pi \varepsilon \cdot l}{\ln D/d}; \tag{7} \]

where \(l\) is the length of the section;
\(D\) and \(d\) are the outer and inner diameters of the coaxial line;
\(\mu\) and \(\varepsilon\) are the magnetic and dielectric permeabilities.

Then

\[ K = \frac{L_\ell}{C_\ell R_\ell^2} = \frac{ \frac{\mu}{2\pi} l \cdot \ln \frac{D}{d} }{ \frac{2\pi\varepsilon l}{\ln \frac{D}{d}} \cdot R_\ell^2 }, \]

whence

\[ \ln \frac{D}{d} = 2\pi R_\ell \sqrt{\frac{\varepsilon}{\mu}} \cdot \sqrt{K}. \]

For

\[ K=\frac{1}{3},\quad \varepsilon=\varepsilon_0=\frac{1}{36\pi}10^{-9}\left[\frac{\text{F}}{\text{m}}\right]; \]

\[ \mu=\mu_0=4\pi\cdot10^{-7}\left[\frac{\text{H}}{\text{m}}\right]; \]

\[ \ln \frac{D}{d} \approx 10^{-2} R_{\ell x} \]

Studies of the frequency error of comparators on PCs were carried out at the D. I. Mendeleev All-Union Scientific Research Institute of Metrology by comparison with a standard compensation voltmeter of type V3-24.

For carrying out the experiment, a coaxial PC design was developed that made it possible to combine the connection points of the converter, the standard voltmeter, and the high-frequency voltage generator (Fig. 3).

In the work, PCs with experimental incandescent lamps of spotlight design (developed by A. I. Lysov) were investigated; these used filaments from standard lamps of the types NSM \(6.3 \times 20\), NSM \(6 \times 150\), NSM \(1.5 \times 50\). The functional diagram of the experiment is given in Fig. 4.

\[ \begin{aligned} l_1 &= 2.8 \cdot 10^{-3}\ \mathrm{m},\\ l_2 &= l_3 = 5.95 \cdot 10^{-3}\ \mathrm{m},\\ l_4 &= 6.8 \cdot 10^{-3}\ \mathrm{m},\\ D &= 22 \cdot 10^{-3}\ \mathrm{m},\\ d_2 &= d_3 = 0.25 \cdot 10^{-3}\ \mathrm{m},\\ d_4 &= 5 \cdot 10^{-3}\ \mathrm{m},\\ d_1 &= 3 \cdot 10^{-3}\ \mathrm{m}. \end{aligned} \]

1. Incandescent lamp
2. Photoresistor
3. Measuring head VZ-24
4. Plug for connecting the AC voltage source

Fig. 3.

Fig. 4.

Fig. 5.

Annotations in Fig. 5:

1. \(U_f = 1.31\ \mathrm{V}\), \(R_l = 137\ \Omega\)
2. \(U_f = 1.76\ \mathrm{V}\), \(R_l = 153\ \Omega\)
3. \(U_f = 2.7\ \mathrm{V}\), \(R_l = 169\ \Omega\)
4. Calculated curve at \(R_l = 153\ \Omega\)

Fig. 6.

Annotations in Fig. 6:

1. \(U_f = 0.63\ \mathrm{V}\), \(R_l = 24.2\ \Omega\)
2. \(U_f = 0.73\ \mathrm{V}\), \(R_l = 27.2\ \Omega\)
3. \(U_f = 1.03\ \mathrm{V}\), \(R_l = 33.6\ \Omega\)
4. Calculated curve at \(R_l = 27.2\ \Omega\)

Fig. 7.

Annotations in Fig. 7:

1. \(U_f = 2.05\ \mathrm{V}\), \(R_l = 51\ \Omega\)
2. Calculated curve at \(R_l = 51\ \Omega\)

The AC voltage \((U_f)\) from generator 1, through filter 2 and coaxial switch 3, was fed to the photoelectric comparator 7 and to the reference voltmeter 6, by which its prescribed value was set; then the comparator bridge was balanced with the aid of indicator 8 (indicator sensitivity \(0.01\%\) of a division). After this

by switch 3, instead of generator 1, a dc source 4 was connected, and the magnitude of the dc voltage was varied until indicator 8 gave zero readings. The magnitude of the dc voltage \((U_0)\) was measured with potentiometric setup 5. The frequency error was calculated by formula (1), taking into account the frequency correction of the voltmeter.

The results are given in Tables (1, 2, 3) and in the graphs of Figs. 5, 6, 7. The solid lines show the experimentally obtained frequency errors of the photoelectric converter at different lamp resistances at the operating point; the dashed line shows the calculated course of the converter frequency error for one of the operating points.

Table 1

Experimental lamp No. 1

Filament parameters as in NSM 6.3 × 20. Number of turns
\(W = 112,\ I_l = 10\ \mathrm{mA}\)

\(I_l = 11.5\ \mathrm{mA}\)

\(I_l = 13\ \mathrm{mA}\)

The calculations were carried out using formulas (5, 6, 7), taking into account the structural dimensions of the converter (see Fig. 3).

Conclusions

1. The main influence on the compensation error of the photoelectric converter in the high-frequency region is exerted by the reactances of the elements of the incandescent-lamp circuit.

Table 2

Test lamp No. 3

Straight filament \(d = 5 \cdot 10^{-6}\) m, \(l = 3.37\) mm
\(I_l = 26\) mA

\(I_l = 27\) mA

\(I_l = 30.7\) mA

Table 3

Test lamp No. 4

Filament parameters of NSM 6 × 150
\(I_l = 40.5\) mA

1. It has been shown theoretically and experimentally that the primary circuit of a photoelectric converter can be represented in the form of an equivalent circuit (Fig. 2) with lumped constants.

2. In calculations of the frequency error of the converter, it is necessary to take into account the reactances not only of the heater and the input traverses, but also of all current-carrying conductors included between the LN and the point at which the voltage is measured.

3. In a coaxial design of the converter, compensation of the commutation frequency error can be achieved by selecting the optimal dimensions of the outer conductor.

The authors express their gratitude to A. I. Lysov for assistance in the development and fabrication of experimental incandescent-lamp samples, and also to A. M. Fedorov for assistance in setting up the experiment.

REFERENCES

1. B. A. Perminov, M. S. Roitman, E. I. Tsimbalist. Comparator of alternating current using photoelectric converters. Avtometriya, No. 5, 1965.

2. M. S. Roitman, V. F. Anikin, Yu. M. Fomichev, V. R. Tsibulsky, E. I. Tsimbalist, A. I. Kramnyuk. Reference sources of alternating voltages. Research report “Vereteno,” “Vitamin,” inv. No. 502500, state registration Nos. 68021571, 68021575, VINITI, 1969.

3. L. A. Tseitlin. Inductances of Wires and Circuits. Gosenergoizdat, Moscow–Leningrad, 1950.

4. N. V. Zernov and V. G. Karpov. Theory of Radio-Engineering Circuits. Energiya Publishing House, Moscow–Leningrad, 1965.