ON THE DESIGN OF TEST EQUIPMENT FOR ELECTRICAL AND RADIO MEASURING INSTRUMENTS

E. N. Ruzaev

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

Test equipment intended for monitoring the parameters of measuring instruments and for ensuring the uniformity of measurements in the country must have characteristics 3–5 times better than those of working instruments. Measuring technology in the field of electrical and radio measurements has achieved very high accuracies, which are realized by means of fairly complex technical solutions. Therefore, the scientific design of the corresponding test equipment is a very urgent and difficult task.

In the design process, the development engineer must be guided by two basic principles:

1) implementation of the specified technical parameters;
2) creation, for the given conditions, of the optimal variant.

The design of test equipment may be represented in the following sequence:

1) collection of information on modern measuring instruments of one functional group, for whose verification it is necessary to design precision equipment;

2) determination of the best characteristics attainable by these measuring instruments and formulation of the requirements imposed on the test equipment;

3) final definition of the technical assignment for designing the test setup;

4) selection of the principle on which the equipment is to be built, using which the technical assignment can be implemented;

5) selection of the optimal:

a) structural diagram;
b) circuit solution;
c) engineering design.

The solution of each of the optimization problems listed above is possible when an appropriate criterion is available. The criterion (objective function) must satisfy the following basic requirements: it must be representative, critical with respect to the parameters under study, and, if possible, simple.

Representativeness of the criterion means an assessment of the principal task solved by the device.

Criticality with respect to the parameters under study consists in significant changes in the numerical value of the criterion under comparatively small changes in the parameters under study.

The search for the optimal variant is possible in two directions:

1) fulfillment of the assigned tasks at minimum cost;
2) maximum efficiency at fixed material costs.

The synthesis of the objective quality function is the most difficult question in solving the problem of optimal design of a device. There are several methods for obtaining this criterion.

The simplest problem is the search for the optimal variant under the condition that optimization is carried out with respect to one parameter, which is analytically related to other characteristics of the device on the basis of physical principles. The solution becomes considerably more complicated when optimization is performed with respect to several parameters.

In the absence of an analytical relationship between the parameters, a method is used for synthesizing the objective quality function based on applying the basic propositions of information theory to measurements and measuring apparatus. The criterion obtained in this way combines energy and accuracy characteristics and the parameters of the ranges of measured or generated quantities.

Of interest is the synthesis of an aggregate objective function using estimates obtained on the basis of the physical interrelation of parameters, information estimates, and also reliability, economic, operational, etc. estimates as particular criteria, whose weighting coefficients are determined by methods of mathematical statistics, without resorting to expert analysis.

A necessary condition for solving this problem is a body of statistical material that will include existing implementations of the given functional group and variants of the device being designed.

We shall define the objective quality function as

\[ V=\prod_{i=1}^{k} C_i^{P_i}, \tag{1} \]

where \(V\) is the aggregate criterion of operating quality;
\(C_i\) is the \(i\)-th particular criterion;
\(P_i\) is the weight of the \(i\)-th criterion;
\(k\) is the number of particular criteria.

Each particular criterion combines a group of characteristics of the instrument

\[ C_i=\xi(x_{i1},\,x_{i2},\ldots x_{ik}), \]

where \(x_{i1}, x_{i2}\) are parameters of the device.

The objective function (1) combines indicators of the capabilities of the device, but any product is also characterized by costs.

Such characteristics may include: cost, number of discrete elements, power consumed, time for performing a specific functional task, etc.

It is known that the indicator of capabilities and the cost indicator are related to each other. It is evident that, by increasing costs, we can also increase capabilities. Experience in using information criteria and a number of other works [1, 2] show that the connecting function is a power function. Solving the problem of synthesizing the objective quality function without initial conditions, which are determined from previous works or set in practice, is practically impossible. As the first such condition, we specify the form of the relationship between the capability indicator and the cost characteristic in the form of a power function

\[ V=A\cdot n^{b}, \tag{2} \]

where \(A\) is a constant coefficient,
\(n\) is the cost index,
\(b\) is an unknown exponent.

Expression (2) is a notation for the functional constraint

\[ V - A \cdot n^b = 0. \]

The condition of uniform power-law growth of the aggregate criterion of operating quality as a function of the base cost index can be written in the form

\[ \frac{d(\lg V)}{d(\lg n)}=\mathrm{const}. \tag{3} \]

Passing to finite increments, we obtain

\[ \frac{\Delta(\lg V)}{\Delta(\lg n)}=\mathrm{const}'. \]

Neglecting the constant coefficient, this condition may be written as

\[ \frac{\lg V}{\lg n}=\mathrm{const}''=b. \tag{4} \]

Substituting the values of the objective function (1) into expression (4), we obtain the equation

\[ \frac{ \lg \prod_{i=1}^{k} C_i^{P_i} }{\lg n}=b \]

or

\[ \frac{P_1 \lg C_1}{\lg n} + \frac{P_2 \lg C_2}{\lg n} +\ldots+ \frac{P_k \lg C_k}{\lg n} =b. \]

Then, for the group of implementations and variants of the device being designed, the following system of equations may be written:

\[ \left. \begin{aligned} \frac{P_1 \lg C_{11}}{\lg n_1} + \frac{P_2 \lg C_{12}}{\lg n_1} +\ldots+ \frac{P_k \lg C_{1k}}{\lg n_1} &= b,\\ \frac{P_1 \lg C_{21}}{\lg n_2} + \frac{P_2 \lg C_{22}}{\lg n_2} +\ldots+ \frac{P_k \lg C_{2k}}{\lg n_2} &= b,\\ \cdots\quad \cdots\quad \cdots\quad \cdots\quad \cdots\quad \cdots\\ \frac{P_1 \lg C_{m1}}{\lg n_m} + \frac{P_2 \lg C_{m2}}{\lg n_m} +\ldots+ \frac{P_k \lg C_{mk}}{\lg n_m} &= b \end{aligned} \right\} \tag{5} \]

where \(C_{ij}\) is the \(i\)-th cost index for the \(j\)-th device;
\(n_j\) is the base cost index for the \(j\)-th device;
\(m\) is the number of implementations and variants.

Denoting \(\dfrac{\lg C_{ij}}{\lg n_j}\) by \(a_{ij}\), we obtain a system of \(m\) linear homogeneous equations with \(k+1\) unknowns \((P_1 \div P_k\) and \(b)\):

\[ \left. \begin{aligned} a_{11}P_1+a_{12}P_2+\ldots+a_{1k}P_k&=b,\\ a_{21}P_1+a_{22}P_2+\ldots+a_{2k}P_k&=b,\\ \cdots\quad \cdots\quad \cdots\quad \cdots\quad \cdots\\ a_{m1}P_1+a_{m2}P_2+\ldots+a_{mk}P_k&=b \end{aligned} \right\}. \]

But regional constraints are imposed on the indicator \(b\) by the condition that capabilities increase with an increase in costs:

\[ b>0. \]

(In principle, regional constraints may also exist for other indicators.) From previous works one can determine an approximate range of variation of \(b\) \((0 \div 15)\). Moreover, solving a system of linear homogeneous equations presents a known difficulty even when implemented on a digital computer. Therefore the following method of solution is the most rational.

The values of the coefficient \(a_{ij}\) are calculated from the data of known implementations and variants of the device being designed. The value of the indicator \(b\) is specified. As a result, we obtain a system of \(m\) linear equations with \(k\) unknowns. It will have a unique solution when \(m=k\) and \(D=\det[a_{ij}]\ne 0\). If an overdetermined system is obtained, i.e., when \(m>k\), the number of solutions will be equal to the number of combinations of \(m\) taken \(k\) at a time. In such a variant the most probable values of the unknowns may be determined using, for example, the method of least squares.

Thus, by specifying the values \(b_{\min}\), \(b_{\max}\), and \(\Delta b\)—the step of variation of \(b\)—we can determine a series of solutions.

The next step is to find the best solution. A very objective evaluation criterion is the degree of scatter of points with coordinates \(n_j,\ V_j\) around the curve characterized by the indicator \(b_j\). From the minimum of the variance one can determine the most probable value of the coefficient \(b\) and, from the group of solutions, choose the solution corresponding to this value. This algorithm is easily implemented on a digital computer.

The criterion for evaluating the quality of operation of the device may be defined as

\[ V=\prod_{i=1}^{k} C_i^{p_i}=n. \]

(In the general case \(V=A\cdot n^b\), but when relative estimates rather than absolute ones are used, the constant coefficient may be omitted.) Then the expression

\[ \omega=\frac{V}{n^b}=\frac{\prod_{i=1}^{k} C_i^{p_i}}{n^b}=1 \]

is the condition that the variant for which it is satisfied, for specified \(C_i\) and \(n\), lies at the average level of technical implementation. A device having the maximum value of the coefficient \(\omega\) will possess the greatest operating capabilities per unit cost. Thus, we can obtain the optimum variant among those proposed for the design of the calibration device.

References

1. G. Strauber. Materials of the meeting of the CMEA and SFRY member countries on the exchange of experience in compiling scientific and technical forecasts. Prague, 1967.
2. P. V. Novitskii. Fundamentals of the Information Theory of Measuring Devices. Leningrad, “Energiya,” 1968.