ACCURACY ANALYSIS OF A SPECIALIZED COMPUTING DEVICE FOR OPTIMIZING AUTOCOMPENSATION CONDITIONS IN A SPECIAL TYPE OF REGULATOR

V. M. Razin, V. L. Ryabukhin, V. M. Khlopovskikh

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

The total error of the result obtained at the output of a specialized computing device is defined as

\[ a = a_{\mathrm{T}} + a_{\mathrm{P}} + a_{\mathrm{I}}, \tag{1} \]

where \(a_{\mathrm{T}}\) is the transformed error caused by error in the values of the input quantities of the computing device;

\(a_{\mathrm{P}}\) is the principal error, resulting from the approximate implementation by the computing device of the true dependences;

\(a_{\mathrm{I}}\) is the instrumental error, caused by the imperfection of the implementation of the computing device.

Let us consider these errors as applied to the developed device [1]. For the digital blocks (ROM and control unit), \(a_{\mathrm{I}}=0\). For the analog-to-digital blocks (ADC and DAC), as their experimental investigations show, it is likewise possible to take \(a_{\mathrm{I}}=0\).

The magnitude of the principal error \(a_{\mathrm{P}}\ne 0\), although the modeled and modeling formulas coincide. The value of this error in the present case is determined by the accuracy with which the analog input quantities are represented by their discrete equivalents, i.e., by the quantization error.

A complete characteristic of the random variable \(a_{\mathrm{T}}\) is the density of its distribution \(f(a_{\mathrm{T}})\). In practice it is sufficient to characterize its error by the mean value \(\overline{a_{\mathrm{T}}}\) and the variance \(D(a_{\mathrm{T}})\), or by the root-mean-square error \(\sigma(a_{\mathrm{T}})=\sqrt{D(a_{\mathrm{T}})}\).

The computing device for optimizing the injection phase implements a function that can be represented in general form as

\[ Z = Z(x_1, x_2, \ldots x_n). \]

The presence of input errors \(a_i\) in the values of the arguments \(x_i\) causes the appearance of a transformed error

\[ a_{\mathrm{T}} = \sum_{i=1}^{n} \frac{\partial Z}{\partial x_i}\, a_i . \tag{2} \]

The input errors are random variables; therefore, for them [2]:

\[ \bar a_{\mathrm{T}}=\sum_{i=1}^{n}\frac{\partial Z}{\partial x_i}\,\bar a_i; \tag{3} \]

\[ D(a_{\mathrm{T}})=\sum_{i=1}^{n}\left(\frac{\partial Z}{\partial x_i}\right)^2 D(a_i)+ \sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\partial Z}{\partial x_i}\frac{\partial Z}{\partial x_j}K(a_i,a_j), \tag{4} \]

where \(K(a_i,a_j)\) are the correlation moments for the errors \(a_i\) and \(a_j\).

As applied to the device under consideration, the analysis of errors is simplified. For the blocks of the computing device, the mean errors are equal to zero: \(\bar a_i=0\). The presence of \(\bar a_i\ne 0\) would indicate the action of a systematic error, whose influence can be eliminated by appropriate adjustment of the input quantity. With \(\bar a_i=0\), according to (3), we have \(\bar a_{\mathrm{T}}=0\).

The characteristic of the transformed error remains, in essence, only \(D(a_{\mathrm{T}})\) or \(\sigma(a_{\mathrm{T}})\). In addition, taking into account that all input errors are independent, we have

\[ K(a_i,a_j)=0; \tag{5} \]

\[ D(a_{\mathrm{T}})=\sum_{i=1}^{n}\left(\frac{\partial Z}{\partial x_i}\right)^2D(a_i); \tag{6} \]

\[ [\sigma(a_{\mathrm{T}})]^2=\sum_{i=1}^{n}\left(\frac{\partial Z}{\partial x_i}\right)^2[\sigma(a_i)]^2. \tag{7} \]

Let us find, according to (7), the characteristics of the output error of the specialized computing device (SCD), consisting of functional and nonfunctional blocks. For such blocks the error is determined, respectively, as

\[ a=a_{\mathrm{p}}+a_{\mathrm{T}}=a_{\mathrm{p}}+\sum_{i=1}^{n}\frac{\partial Z}{\partial x_i}a_i \tag{8} \]

and

\[ a=a_{\mathrm{p}}+\sum_{i=1}^{n}a_i. \tag{9} \]

In order to simplify the accuracy analysis, we shall divide the SCD under consideration into separate open-loop nodes of sequential type. Then the formula diagram of the SCD can be represented as in Fig. 1.

Fig. 1. Formula diagram of the specialized computing device

On the basis of formulas (7), (8), and (9), we shall carry out a sequential transformation of the input and intrinsic errors of the blocks.

Then

\[ \left. \begin{aligned} a_{1\mathrm{T}}&=a_{01}+a_1,\\ a_{2\mathrm{T}}&=a_2+\frac{\partial f_1}{\partial x_1}\,a_{1\mathrm{T}},\\ a_{3\mathrm{T}}&=a_{02}+a_3,\\ a_{4\mathrm{T}}&=a_4+\frac{\partial f_2}{\partial x_2}\,a_{3\mathrm{T}},\\ a_{5\mathrm{T}}&=a_5+\frac{\partial f_3}{\partial x_1}\,a_{2\mathrm{T}}+ \frac{\partial f_3}{\partial x_2}\,a_{4\mathrm{T}},\\ a_{6\mathrm{T}}&=a_6+a_{5\mathrm{T}}+a_{03}. \end{aligned} \right\} \tag{10} \]

\[ \left. \begin{aligned} \sigma^2_{1\mathrm{T}}&=\sigma^2_{01}+\sigma^2_1,\\ \sigma^2_{2\mathrm{T}}&=\sigma^2_2+\left(\frac{\partial f_1}{\partial x_1}\right)^2\sigma^2_{1\mathrm{T}},\\ \sigma^2_{3\mathrm{T}}&=\sigma^2_{02}+\sigma^2_3,\\ \sigma^2_{4\mathrm{T}}&=\sigma^2_4+\left(\frac{\partial f_2}{\partial x_2}\right)^2\sigma^2_{3\mathrm{T}},\\ \sigma^2_{5\mathrm{T}}&=\sigma^2_5+\left(\frac{\partial f_3}{\partial x_1}\right)^2\sigma^2_{2\mathrm{T}}+ \left(\frac{\partial f_3}{\partial x_2}\right)^2\sigma^2_{4\mathrm{T}},\\ \sigma^2_{6\mathrm{T}}&=\sigma^2_6+\sigma^2_{5\mathrm{T}}+\sigma^2_{03}. \end{aligned} \right\}. \tag{11} \]

Substituting specific data into formulas (10) and (11), one can obtain for them the values of the output errors. Since the error-transformation coefficients, i.e., the partial derivatives, are in the general case functions of the input variables, the calculation of the transformed errors must be performed repeatedly, varying the values of \(x_i\). Often, some single value is taken as the characteristic of the transformed error, for example, the maximum value of its variance \(D(a)_{\max}\). Since an analytical investigation of expression (7) for a maximum is difficult, we shall confine ourselves to finding the maximum error-transformation coefficients

\[ \left|\frac{\partial f}{\partial x_i}\right|_{\max} \]

for the expression

\[ t_i=\frac{k\sqrt{x_2}}{x_1}, \tag{12} \]

which represents the injection condition for the nonrelativistic case. Substitution of the found error-transformation coefficients into formula (10) gives an approximate value of \(D(a)_{\max}\), always overestimated relative to the true value of the greatest variance. The overestimation of the result is the known justification for such a method of calculation [2].

The transformation coefficients

\[ \frac{\partial f_1}{\partial x_1}=-\frac{k}{x_1^2};\quad \frac{\partial f_2}{\partial x_2}=\frac{1}{2\sqrt{x_2}};\quad \frac{\partial f_3}{\partial x_1}=-\frac{k\sqrt{x_2}}{x_1^2};\quad \frac{\partial f_3}{\partial x_2}=\frac{k}{2x_1\sqrt{x_2}}. \tag{13} \]

have their maximum value at \(x_{1\min}\) and \(x_{2\min}\), except for \(\frac{\partial f_3}{\partial x_1}\), but this coefficient may be neglected because of its small influence on the value of the output error; consequently, the calculation for the maximum errors may be carried out at \(x_{1\min}=4500\) oersted and \(x_{2\min}=37.5\) kV.

In calculating the output errors of the computing device, we make use of the values of the intrinsic errors of the blocks. Then, for the circuit of Fig. 1, we have

\[ a_1=a_3=a_6=10^{-3};\qquad a_2=a_4=a_5=0.5\cdot 10^{-3}. \tag{14} \]

In accordance with the laws of distribution of the block errors, we have

\[ \sigma_1=\sigma_3=\frac{a_1}{\sqrt{6}};\qquad \sigma_2=\sigma_4=\sigma_5=\frac{a_1}{\sqrt{12}}\quad\text{and}\quad \sigma_6=\frac{a_1}{\sqrt{3}}. \tag{15} \]

Let us now determine the physical meaning and the value of the input errors \(a_{0i}\). The mathematical expectation of the relative rate of change of the mains voltage over one period is \(m(\Delta U_c)=5\cdot 10^{-5}\), and \(\sigma(\Delta U_c)=1.4\cdot 10^{-5}\).

Analysis of the dynamic properties of the betatron elements shows that these changes in the mains voltage lead practically instantaneously to the corresponding changes in \(U_m\) and \(U_i\). Taking into account that the information is introduced into the computing device with a delay of no more than one period, it may be assumed that, in the worst case, \(\sigma_{01}=\sigma_{02}=\sigma_{03}\). In practice, all input errors of the computing devices, as random quantities, obey normal distribution laws. In this case

\[ a_{01}=a_{02}=m(\Delta U_c)+3\sigma_{01}. \tag{16} \]

It may be assumed that \(a_{03}=3\cdot 10^{-4}\).

Substituting the obtained error values into formulas (10) and (11), we obtain \(a_{\text{out max}}=0.98\), \(\sigma_{\text{out max}}=0.37\). As was already noted, the errors found in this way are overestimated. Taking into account that \(U_m\) and \(U_i\) in most cases prove to be mutually correlated random quantities, one may expect, according to (12), a reduction in the limiting errors.

It should be noted that, when characterizing the output error by the greatest value of the root-mean-square error, it is desirable to know the probability of its occurrence. Knowing the distribution law of the system of random quantities, one can calculate the value of \(\sigma\), averaged over all possible values of \(x_i\). However, this is difficult to do, and there is no need for it, since the value \(\sigma_{\text{out max}}\), calculated with an error on the high side, satisfies the stated conditions. Moreover, if it is possible to indicate or establish, for each input quantity, its value most frequently encountered in practice, then one confines oneself to calculating the variance only at these values of the arguments \(x_i\). Therefore it is important to determine the value of the variance of the error of the computing device at the nominal value of the mains voltage. The calculation carried out for this case shows that \(a=0.76\) and \(\sigma=0.21\).

On the basis of the errors obtained in determining the optimal injection instant, an approximate estimate is found for the magnitude of the losses in the betatron radiation intensity when it is controlled by the computing device.

In the region of small deviations of the current value of the injection phase from its optimal value, the dependence \(I=f(t_i)\) can be represented in the form of a quadratic parabola

\[ I=-kx^2 \tag{17} \]

or, in relative coordinates, as

\[ \delta I_{\text{rel}}=1-k(\delta t_i)^2, \tag{18} \]

where \(k(\delta t)^2\) is the magnitude of the relative losses in radiation intensity; \(k\) is the proportionality coefficient.

When the output characteristic of the betatron is approximated by expression (17), the mathematical expectation of the loss magnitude and the variance of the radiation intensity are determined as

\[ m(I)=k\sigma^2(\delta t_i), \tag{19} \]

\[ D(I)=2k^2\sigma^4(\delta t_i). \tag{20} \]

Figure 2 presents the dependence of the magnitude of the losses, from which it follows that, for the values of the computing-device errors found above,

of the device, for \(k = 0.012\), the value \(\delta I < 5\%\). The analysis shows that in most cases the magnitude of the radiation-intensity losses \(\delta I\) arising from the intrinsic errors of the computing device does not exceed \(1 \div 3\%\). Consequently, one may expect high efficiency from the use of the computing device on a betatron.

Fig. 2. Dependence of the magnitude of the radiation-intensity losses \(\delta I\) on the error in determining the injection phase.

It is evident that the calculation of the effectiveness of using the specialized computing device for cases of significant, frequent, and nonperiodic oscillations of the line voltage should be refined in an experimental study of the operation of the computing device on a real object.

REFERENCES

1. V. M. Razin, V. L. Ryabukhin. Computing device for regulating the injection phase of a betatron. Izv. TPI, vol. 184, 1970.

2. Design and calculation of continuous-action computing machines. Edited by Doctor of Technical Sciences A. N. Lebedev and Doctor of Technical Sciences V. B. Smolov. Mashinostroenie Publishing House, Moscow, 1966.