Analysis of an Unbalanced Bridge Based on FS-K1 Photoresistors

A. K. Bolbot, K. N. Popov, V. N. Titov

(Presented by the scientific seminar of the Faculty of Physics and Technology)

Methods for calculating direct-current bridges are described in detail in [1], [2]. A graphoanalytical method for calculating unbalanced direct-current bridges with active resistances is described in [3].

In the present article an attempt is made to extend the graphoanalytical method for calculating direct-current bridges to the calculation of unbalanced bridges based on FS-K1 photoresistors.

The indicated bridge was used by the authors in a photoelectronic instrument for determining the carbon content in ash from thermal power plants [4].

Carbon, being the main component in the combustible elements of ash, has a high absorbing capacity for light waves of the visible range; therefore, an increase in the carbon content in a sample reduces the amount of light reflected from its surface.

The investigations were carried out on ash samples taken from high-pressure boilers of the Tomsk GRES-2, in which the carbon content was determined by the ignition method according to GOST—6382—52.

Fig. 1. Circuit for measuring \(R_{\mathrm{fs}}\) as a function of carbon content.
1—illuminating lamp 6.3 V \(\times\) 0.28 A; 2—sample under investigation; 3—FS-K1 photoresistor; 4—M 252 microammeter.

The change in the reflected luminous flux as a function of carbon content was read from the change in current through the photoresistor according to the circuit shown in Fig. 1.

The specimen being analyzed was smoothed and placed under lamp 1. The results of measurements of the dependence of the current through the photoresistor on the carbon content are presented in Table 1.

Table 1

It is seen from the table that changing the carbon content from 1 to 8% causes a change in the resistance of the photoresistor by $R = 1.06$ MΩ.

The circuit of the bridge under investigation is shown in Fig. 2. To determine the character of the scale of the unbalanced bridge (assuming that the scale of the microammeter is uniform), it is necessary to calculate the values $S_0$ and $R_a$.

Fig. 2. Diagram of the measuring bridge.
$R_1$, $R_2$—FS-K1 photoresistors; $R_3$, $R_4$—arms of the bridge; $R_\mathrm{g}$—resistance of the microammeter frame, equal to 100 Ω; $R_0$—internal resistance of the power source; 1—the specimen under investigation; 2—standard specimen; Л—illuminating lamp.

These values, in the graphical method of calculation, determine the position of the pole $P$ (Fig. 3). For the most commonly encountered case of bridge operation, when the internal resistance of the source is small, $S_0$ and $R_a$ are found from the formulas [3]:

$$ S_0 = \frac{U}{ R_1 \left[ R_1 + R_2 + R_3 + R_4 + R_y \left(2 + \frac{R_3}{R_4} + \frac{R_4}{R_3}\right) \right] }, \tag{1} $$

$$ R_a = \frac{ R_y (R_1 + R_2) + R_1 (R_2 + R_3) }{ R_y + R_2 + \dfrac{R_1 \cdot R_3}{R_1 + R_2} }, \tag{2} $$

where $S_0$ is the sensitivity of the bridge near the instant of balance, or the so-called sensitivity at the zero point;

$R_a$ is an auxiliary quantity.

As an indicator, an M 252 microammeter of class 0.5 was used, with internal resistance \(R_r=1000\) ohms.

\(S_0\) is determined under the condition

\[ R_1\cdot R_3=R_2\cdot R_4, \tag{3} \]

introducing the notation

\[ R_3=n\cdot R_1, \tag{4} \]

one can investigate the dependence of \(S_0\) on the value of \(n\), for the arms \(R_3\), \(R_4\), if expression (4) is substituted into (1).

The calculation results for various values of \(n\) are presented in Table 2.

It is seen from the table that, as \(n\) decreases, the sensitivity \(S_0\) increases.

Table 2

However, as \(n\) decreases, the resistance of the arms \(R_3\), \(R_4\) decreases, and the current \(I_n\) consumed from the power source through the indicated arms increases. Therefore, when selecting the arms \(R_3\), \(R_4\), it is necessary to take into account the value of \(I_n\), since an excessively large current will cause self-heating of the arms, which will lead to instability of the voltage in the indicator diagonal. Taking the above into account, the value of \(n\) was set by us equal to 0.115.

The initial data for the calculation had the following values:

\(R_1=1.16\) MΩ, \(R_2=1.42\) MΩ—the resistances of FS-K1\(_1\), FS-K1\(_2\), when the carbon content in the samples is \(C=1\%\); \(R_3=0.133\) MΩ, \(R_4=0.110\) MΩ. The supply voltage \(U=200\) V.

Determined by formulas (1) and (2), \(S_0=59\) μA/MΩ, \(R_a=1.218\) MΩ, \(R_a\cdot S_0=72\) μA determined the position of the pole P in Fig. 3.

Rotating the half-line \(l\) about the pole in accordance with the specified \(R\)’s taken from Table 1, we find the corresponding values of the current \(I_y\).

The points of intersection of the abscissas of resistance and the ordinates of current are the points of the curve \(I_y=f(R)\), which directly represents the form of the scale of the unbalanced bridge.

If the angle of inclination of the half-line to the abscissa axis is denoted by \(\alpha\), then this angle will be determined exclusively by the sensitivity at the zero point.

\[ \tg \alpha=S_0=59\,\frac{\text{μA}}{\text{MΩ}}. \]

Obviously, \(\alpha_1=\alpha\).

The deviation of the curve \(I_y=f(R)\) from the straight-line law determines for us the degree of nonuniformity of the scale.

The equation for determining \(I_y\), taking into account the degree of nonuniformity, has the form

\[ I_y=\frac{R}{1+D}\cdot S_0, \tag{5} \]

where \(S_0\) is the slope at the zero point, \(R\) is the current value of the resistance of FS-K1\(_1\), \(D\) is the relative degree of nonuniformity, equal to

with respect to the ratio \(\dfrac{R}{R_a}\). At a carbon content in the sample of \(C = 8\%\), the change in resistance of the FS-K1\(_1\) was \(R = 1.06\) MΩ, \(D = 0.87\), \(I_y = 33.4\) μA.

In Fig. 3 are shown: 1—the curve \(I_y^{p} = f(C)\), constructed by the grapho-analytical method from the data of Table 1; 2—the experimental curve \(I_y^{e} = f(C)\), obtained by measurement according to the circuit of Fig. 2. The samples under study were placed under FS-K1\(_1\); the carbon content in the reference sample, \(C = 1\%\), was constant.

Fig. 3. Dependence of the current in the indicator \(I_y = f(C)\) for \(R_g = 1000\) Ω.
1—calculated; 2—experimental; 3—voltage in the indicator diagonal for \(R_g^{1} = 150\) kΩ.

The difference in \(I_y\) between the calculated and experimental curves at all points of the scale was no more than 0.5 μA; 3 is the experimental curve of the dependence of the voltage in the indicator diagonal on the carbon content, obtained by measurement with a voltmeter with input resistance \(R_i = 150\) kΩ on the 30 V scale.

In order, for given \(R_0\) and \(R_y\), to obtain the greatest sensitivity from this bridge circuit, the two smaller resistances must be connected in one arm of the bridge, and the two larger ones in the other [3].

Calculation of the current magnitude under bridge unbalance with the same carbon content but with different arrangement of the supply and indicator diagonals showed that the sensitivity for the version of the bridge circuit shown in Fig. 2 is higher.

Thus, from the investigations presented it is evident that measuring bridges using photoresistors, widely employed in electrical measurements of nonelectrical quantities, are readily amenable to calculation by existing methods for calculating direct-current bridges with active resistances.

In those cases where weight, dimensions, and compactness of the photoelectronic measuring device are of primary importance, a microammeter may be connected directly into the indicator diagonal of the bridge.

References

1. A. D. Nesterenko. Fundamentals of the Calculation of Balanced Electrical Measuring Circuits, Publishing House of the Academy of Sciences of the Ukrainian SSR, 1960.

2. A. M. Turichin. Electrical Measurements of Nonelectrical Quantities, GEI, Moscow, 1959.

3. K. B. Karandeev. Special Methods of Electrical Measurements, GEI, Moscow, 1963.

4. P. P. Elizarov. Operation of High-Pressure Boiler Installations at Electric Power Stations, GEI, 1961.