SELECTION OF ADDITIONAL RESISTANCES FOR ELECTRICAL MEASURING INSTRUMENTS BASED ON THEIR POWER AND CASE CHARACTERISTICS

A. S. LYALIKOV, G. I. FUKS, G. D. LONINOV

Let us write the equations of steady-state heat exchange of an additional resistance located in the instrument case [1, 2] in the form

\[ P=\alpha''(t_3-t_f)F_k''=\alpha''\vartheta_3F_k'', \tag{1} \]

\[ P=\frac{\lambda_k}{\Delta_k}(t_2-t_3)\frac{F_k'+F_k''}{2} =\frac{\lambda_k}{\Delta_k}\vartheta_2F_{k.\mathrm{av}}, \tag{2} \]

\[ P=\alpha_{1-k}(t_1-t_2)F_k'=\alpha_{1-k}\vartheta_{1-2}F_{k.\mathrm{av}}. \tag{3} \]

The overheating of the surface of the winding relative to the surrounding medium,

\[ \vartheta=t_1-t_f, \]

is equal to the sum of the temperature differences

\[ \vartheta=\vartheta_{1-2}+\vartheta_2+\vartheta_3. \tag{4} \]

It has been established [2] that, for determining \(\alpha_{1-k}\), it is permissible to use the expression

\[ \alpha_{1-k}=\frac{F_1}{F_k'}\cdot \alpha_{\mathrm{can}}. \tag{5} \]

Substituting (4) and (5) into (3), we have

\[ P=\alpha_{\mathrm{can}}F_1(\vartheta-\vartheta_2-\vartheta_3). \tag{6} \]

For the case of cooling of the same coil under conditions of free convection, its dissipated power \(P_{\mathrm{sv}}\) at the same overheating \(\vartheta\) will be higher than in the case, i.e. \([P_{\mathrm{sv}}>P]_{\vartheta=\mathrm{idem}}\), and the heat balance is written as

\[ P_{\mathrm{sv}}=\alpha_{\mathrm{can}}F_1\vartheta. \tag{7} \]

Dividing (7) by (6), we obtain

\[ P_{\mathrm{sv}}=\frac{\vartheta}{\vartheta-\vartheta_2-\vartheta_3}\cdot P. \]

Let us denote

\[ \varphi=\frac{\vartheta}{\vartheta-\vartheta_2-\vartheta_3};\qquad \varphi>1. \tag{8} \]

Then, under the condition \(\vartheta=\mathrm{idem}\),

\[ P_{\mathrm{sv}}=\varphi\cdot P. \tag{9} \]

Equality (9) means that, at operating power \(P\), the additional resistance, being in a case with characteristic \(\varphi\) and cool-

...being under conditions of free convection at \(P_{\mathrm{sv}} > P\), must have identical temperature rises.

\(\vartheta_2\) and \(\vartheta_3\), required for determining \(\varphi\), can be calculated from (1) and (2) from the power and the characteristics of the case; \(\vartheta\) is specified depending on the requirements imposed on the instrument.

If one has a table of dissipated powers under conditions of free convection at specified temperature rises for various coils, then, after calculating \(\varphi\) and \(P_{\mathrm{sv}}\), from such a table one can select a coil that, in the case, ensures the specified temperature rise \(\vartheta\) at power \(P\).

On the basis of the considerations presented, the selection of a resistance coil, when its power and the case characteristics are known, can be carried out in the following order:

1) from (2) determine

\[ \vartheta_2 = \frac{P}{\dfrac{\lambda_k}{\Delta_k} F''_{k.\mathrm{sr}}} \simeq \frac{\Delta_k}{\lambda_k} P''_k; \]

2) from (1)

\[ \vartheta_3 = \frac{P}{\alpha'' F''_k} \simeq \frac{P''_k}{\alpha''} \]

with the aid of nomograms \(\vartheta_3 = f(P''_k D)\) [1, 2].

As \(d\) or \(D\), the vertical dimension of the case is taken;

3) according to (8), \(\varphi\) is calculated;

4) according to (9), \(P_{\mathrm{sv}}\) is calculated;

5) according to \(P_{\mathrm{sv}}\), from the table of powers in free convection at the specified temperature rise \(\vartheta\), a coil with the nearest dissipated power is selected.

Using the method described, examples were solved for selecting coils at a specified resistance power, known case characteristics, and permissible winding temperature rise. For the solved examples, an experimental check of the winding temperature rise was carried out; the results are given in Table 1.

Table 1

As can be seen from the table, the agreement between the calculated overheating values and the experimental ones is satisfactory.

At the same time, a relationship is observed between the power and the error of the calculations: at low powers the calculation gives positive errors; as the power increases, its errors pass into the negative region. A quantitative description of this relationship may lead to a refinement of the method presented here for selecting additional resistances.

References

1. G. I. Fuks, A. S. Lyalikov. Calculation of overheating of the surface of the winding of additional resistances of electrical measuring instruments. In the present collection.

2. A. S. Lyalikov, G. I. Fuks. A shortened method for calculating the overheating of the surface of the winding of additional resistances of electrical measuring instruments. In the present collection.