ANALYSIS OF INSTRUMENTAL ERRORS OF IDENTIFIED MULTIPLYING-DIVIDING DEVICES

V. P. SHERSTOBITOV

(Presented by the Department of Mathematical and Computing Instruments and Devices)

The development of computing technology in our time is accompanied by the development of methods and means that make it possible to use elements and units with a certain spread of parameters without substantially reducing the accuracy of the mathematical operations performed. The method considered in the present work for constructing analog multiplying-dividing devices (MDDs) with artificial identification of channels corresponds to this direction of development. It is known [1—4] that MDDs with variable transmission coefficients (compensation MDDs) are, in principle, simple and accurate; however, their practical application is limited by the imperfection of existing controlled elements.

In the works cited, and also in [5], two methods are given for constructing MDDs with variable transmission coefficients.

First method. The use of two channels with identical characteristics of the control circuit (meaning the dependence of the channel transmission coefficients on the control voltage or another physical quantity). In this case one channel is used to set the transmission coefficients of both channels by means of a tracking system acting on the common control input. Obviously, this method requires careful adjustment and selection of the elements that carry out the change in transmission coefficients, which is its disadvantage.

Second method. The use of one channel with separation of two or more input variables by means of any type of modulation. In this case, for high-quality MDDs, complex compensation modulators are required. Of course, it is possible to construct a multiplication circuit [5] without a modulator if a high-frequency reference generator is used. In this case rigid stabilization of the generator amplitude is necessary, since its value enters directly into the basic operational relation. The circumstances mentioned are a disadvantage of the second method of constructing MDDs.

In the present work a new (third) method of constructing MDDs is considered. The essence of the method is that two controlled channels are used, to which, together with the analog variables, high-frequency auxiliary voltages are applied, intended for forming the signal of mismatch of the coefficients

transmission gains of the channels. In this case two servo systems are provided, one of which sets the transmission coefficient of one channel in the usual way, while the other adjusts the transmission coefficient of the second channel to the transmission coefficient of the first, using a differential high-frequency signal.

Let us call an MDD constructed by the method under consideration identified, since the method implements artificial identification of the channels. The block diagram of an identified MDD is shown in Fig. 1, where

\(K_1, K_2\)—controlled channels,

\(U_x, U_y, U_z\)—analog variables (voltages),

\(U_f\)—amplitude of the high-frequency sinusoidal voltage,

\(\Phi_1, \Phi_2\)—filters separating the low-frequency voltages \(e_{1H}\) and \(e_{2H}\) from the high-frequency voltages \(e_{1B}\) and \(e_{2B}\) at the outputs of the controlled channels \(K_1\) and \(K_2\), respectively;

\(Y_1, YM\)—dc amplifiers;

\(Y_2\)—ac amplifier;

\(\Phi D\)—phase detector.

Fig. 1. Block diagram of an identified MDD

The low-frequency component \(e_{1H}=K_1U_z\) is compared with the variable \(U_x\) by means of the servo amplifier \(Y_1\), which sets the required transmission coefficient of the first channel:

\[ K_1=\frac{U_x}{U_z}. \tag{1} \]

The second servo system \((Y_2,\Phi D,YM)\) ensures adjustment of the transmission coefficient \(K_2\) to the value \(K_1\). It should be noted that, in the circuit shown, the auxiliary voltages are applied to blocks \(K_1\) and \(K_2\) with opposite phases, which makes it possible to obtain the differential signal by simple summation of the high-frequency output voltages \(e_{1B}\) and \(e_{2B}\). Thus, with exact adjustment, the voltage \(e_{2H}\) is determined by:

\[ e_{2\mathrm{n}} = U_{\mathrm{out}} = \frac{U_x \cdot U_y}{U_z}. \tag{2} \]

The problem is to analyze the accuracy with which operation 2 is performed. It may be assumed that the methodological error is zero for all methods of constructing multiplying–dividing devices (MDDs) with variable transfer coefficients. Let the transformed error caused by the inaccuracy in setting the variables \(U_x, U_y, U_z\) also be zero, since it can always be taken into account during measurement.

Then the problem reduces to analyzing the static instrumental errors of the MDD. (Dynamic errors are not considered in this paper.) It may be assumed that, for small deviations of the element parameters, the sources of error are independent, which makes it possible to determine the errors of operation 2 for each source separately, and to determine the total error as the sum of the error components. Since the transformed errors are taken to be zero, the relative error of the operation can be determined through the transfer coefficients:

\[ \delta = \frac{K_2 - K_{2\mathrm{n}}}{K_{2\mathrm{n}}}, \tag{3} \]

where \(K_{2\mathrm{n}} = K_{1\mathrm{n}} = \dfrac{U_x}{U_z}\) is the exact value of the transfer coefficient of either channel.

In accordance with the above, let us consider the error components.

1. Finiteness of the values of the gain coefficients of the tracking systems

Let us denote the gain coefficients of amplifiers \(Y_1, Y_2, YM\) respectively by \(A_1, A_2, A_3\). Let the functions inverse to the characteristics of the control circuits have the form:

\[ U_1 = K_1 \sum_{i=1}^{n} a_i (K_1)^{i-1}, \tag{4} \]

\[ U_2 = K_2 \sum_{i=1}^{n} b_i (K_2)^{i-1}. \tag{5} \]

Expressions (4) and (5) are sufficiently general for many systems with controlled coefficients. In what follows, we everywhere assume that the transfer coefficients of the channels do not depend on the input variables, except for the control variables \(U_1\) and \(U_2\).

The control voltages \(U_1\) and \(U_2\) are found, according to the block diagram (Fig. 1), as:

\[ U_1 = A_1 (K_1 U_z - U_x), \tag{6} \]

\[ U_2 = A_2 \cdot A_3 U_f (K_1 - K_2) = A U_f (K_1 - K_2). \tag{7} \]

Comparing expressions (4) and (5) with (6) and (7), we obtain a system of two equations with two unknowns:

\[ \left. \begin{aligned} A_1 (K_1 U_z - U_x) &= K_1 \sum_{i=1}^{n} a_i (K_1)^{i-1},\\ A U_f (K_1 - K_2) &= K \sum_{i=1}^{n} b_i (K_2)^{i-1}. \end{aligned} \right\} \tag{8} \]

For monotonic dependences (4) and (5), the series is decreasing; therefore \(a_i\) and \(b_i\) decrease continuously as the index \(i\) increases. This makes it possible to simplify the system of equations (8) by substituting under the summation signs the approximate solutions \(K_1\) and \(K_2\) for infinitely large coefficients \(A_1\) and \(A_2=A_2\cdot A_3\), which coincide with expression (1),

\[ \left. \begin{aligned} A_1(\dot K_1U_z-U_x)&=K_1\sum_{i=1}^{n}a_i\left(\frac{U_x}{U_z}\right)^{i-1} \\[6pt] AU_f(K_1-K_2)&=K_2\sum_{i=1}^{n}b_i\left(\frac{U_x}{U_z}\right)^{i-1} \end{aligned} \right\} \tag{9} \]

The validity of the adopted simplification is justified for many real channels. Otherwise, one can retain a certain number of the first terms and solve more complicated equations of higher orders. For system (9) the following solutions will be obtained:

\[ K_1=\frac{U_x}{U_z}\cdot \frac{1}{ 1-\frac{1}{A_1U_z}\sum_{i=1}^{n}a_i\left(\frac{U_x}{U_z}\right)^{i-1} }, \tag{10} \]

\[ K_2=K_1\cdot \frac{1}{ 1+\frac{1}{A\cdot U_f}\sum_{i=1}^{n}b_i\left(\frac{U_x}{U_z}\right)^{i-1} }. \tag{11} \]

Substituting expressions (1), (10), (11) into (3) and neglecting quantities of small order, we obtain the error caused by the finiteness of the gain coefficients of the amplifiers,

\[ \delta_1 \simeq \frac{1}{A_1U_z}\sum_{i=1}^{n}a_i\left(\frac{U_x}{U_z}\right)^{i-1} - \frac{1}{A\cdot U_f}\sum_{i=1}^{n}b_i\left(\frac{U_x}{U_z}\right)^{i-1}. \tag{12} \]

It is evident from formula (12) that the operation error is the smaller, the larger the coefficients \(A_1\), \(A_2\), \(A_3\) and the voltages \(U_z\) and \(U_f\). The greatest error will occur in division, when \(U_z=U_{z\min}\).

2. Instability of the amplitude of the auxiliary voltage

Let us differentiate the value \(K_2\) (11) with respect to \(U_f\), pass to increments, and divide both sides of the resulting expression by \(K_2\). If quantities of small order are neglected, the following value of the error caused by instability of the amplitude of the auxiliary voltage can be obtained:

\[ \delta_2=\frac{\Delta K_2}{K_2}= \frac{\Delta U_f}{U_f}\cdot \frac{1}{A\cdot U_f}\sum_{i=1}^{n}b_i\left(\frac{U_x}{U_z}\right)^{i-1}. \tag{13} \]

Formula (13) shows that a highly stable source of sinusoidal voltage is not required, since the error can be reduced by increasing the coefficient \(A=A_2\cdot A_3\).

3. Nonidentity of the characteristics of the control circuits

The error due to nonidentity of the characteristics of the control circuits of the channels is taken into account by formula (12) and can be significant

reduced by increasing the gains of the tracking systems.

4. Inertia of the Channels for Auxiliary Signals

Since real MDD channels are inertial, the auxiliary signals passing through them acquire phase shifts and change in modulus. This makes it necessary to take into account the frequency properties of the channels not only in the dynamic mode, but also in the static mode. For convenience of analysis, a complex transfer coefficient is introduced for each channel:

\[ \begin{aligned} \dot K_1 &= K_1 \cdot B_1 \cdot e^{j\Delta\varphi_1},\\ \dot K_2 &= K_2 \cdot B_2 \cdot e^{j\Delta\varphi_2}, \end{aligned} \tag{14,15} \]

\(B_1\) and \(B_2\) are coefficients that take into account the change in the transfer coefficients at the frequency of the auxiliary generator;

\(\Delta\varphi_1,\ \Delta\varphi_2\) are the phase shifts of the auxiliary signals.

The voltage at the output of the multiplier amplifier is determined by:

\[ \begin{aligned} U_2 = A_3 \Bigl[& \sqrt{\left[U_f + A_2 U_f\left(K_1 \cdot B_1 \cdot \cos\Delta\varphi_1 - K_2 \cdot B_2 \cos\Delta\varphi_2\right)\right]^2 + A_2^2\left(K_1B_1\sin\Delta\varphi_1 - K_2B_2\sin\Delta\varphi_2\right)^2} \\ &-\sqrt{\left[U_f - A_2 U_f\left(K_1 \cdot B_1 \cdot \cos\Delta\varphi_1 - K_2 \cdot B_2 \cos\Delta\varphi_2\right)\right]^2 + A_2^2\left(K_1B_1\sin\Delta\varphi_1 - K_2B_2\sin\Delta\varphi_2\right)^2} \Bigr]. \end{aligned} \tag{16} \]

Substituting expression (16) into (5) and taking into account that \(A_3 \gg 1\), one can obtain the solution in the form:

\[ K_2 = K_1 \frac{B_1 \cdot \cos\Delta\varphi_1}{B_2 \cdot \cos\Delta\varphi_2}. \tag{17} \]

Formula (17) is convenient to use if \(B_2\) and \(\Delta\varphi_2\) do not depend on \(K_2\), and \(B_1\) and \(\Delta\varphi_1\) do not depend on \(K_1\). The relative error caused by the inertia of the channels has the form:

\[ \delta_3 = \frac{B_1 \cdot \cos\Delta\varphi_1 - B_2 \cos\Delta\varphi_2} {B_2 \cdot \cos\Delta\varphi_2}. \tag{18} \]

Formula (18) shows that it is necessary to take into account the phase and amplitude imbalance for the auxiliary signals. The error \(\delta_3\) includes the error caused by the asymmetry of the channels for the auxiliary signals.

5. Instability of the Phase Detector

Here it makes sense to take into account only the zero drift of the phase detector FD, caused by the temperature effect on the characteristics of the diodes. The change in the transfer coefficients of the detector arms can be taken into account in (18).

Thus, the control voltage of the second channel has the form:

\[ U_2 = [A_2U_f(K_1 - K_2) + U_D]\cdot A_3. \tag{19} \]

Substituting [19] into [5], making the same assumptions as in the solution of the system of equations (8), we obtain an expression for \(K_2\), from which

from which it is easy to proceed to the error \(\delta_4\), caused by drift of the detector zero:

\[ \delta_4=\frac{1}{A_2}\frac{U_D}{U_f}\cdot\frac{U_z}{U_x}. \tag{20} \]

Formula (20) shows that the zero drift of the detector is taken into account analogously to the zero drift in the operational amplifier. Hence follows the important conclusion that, in order to use a less stable phase detector, it must be connected after the amplifier, since the influence of the detector zero drift is substantially reduced. Formula (20) may include the error due to zero drift of the MD amplifier if, instead of \(U_D\), one substitutes \(U_D+\dfrac{U_{\mathrm{dr}.3}}{A_3}\), where \(U_{\mathrm{dr}.3}\) is the drift voltage of the named amplifier. The simple analysis presented makes it possible to formulate the following.

Identified MDUs make it possible to use:

a) channels with nonidentical characteristics of the control circuits;

b) generators with low amplitude and frequency stability.

This is possible because the influence of the factors noted can be significantly reduced by increasing the gains of the tracking systems.

REFERENCES

1. B. Ya. Kogan, Electronic Modeling Devices and Their Application to the Study of Automatic-Control Systems. Fizmatgiz, 1963.

2. V. B. Smolov, A. N. Lebedev et al. Continuous-Action Computing Machines. Vysshaya Shkola, 1964.

3. A. A. Maslov. Review and classification of multiplying devices. Automation and Remote Control, vol. XXI, no. 10, 1960.

4. I. V. Latenko. Analog Multiplying Devices. Gostekhizdat, Ukrainian SSR, 1963.

5. S. Nyudo. Principle of operation and principal characteristics of a broadband multiplier with variable transfer coefficients, translation no. 39 624/4 from Japanese, journal Denki Gakkai Zasshi, vol. 83, no. 3, 1963.