Pulse Multiplier-Divider Device

A. P. Zaitsev, R. K. Gachik

(Recommended by the scientific seminar of the Electromechanical Faculty)

In the automation of many production processes, electric traction, etc., it becomes necessary to form an electrical signal according to a control law that is determined by three independent variables related to one another by equations of the form

\[ F_1 = a \frac{xy}{z}, \tag{1} \]

or

\[ F_2 = y \left(1 - a \frac{x}{z}\right), \tag{2} \]

where: \(F_1, F_2\) are functions of three variables (the control law);
\(x, y, z\) are independent variables;
\(a\) is a constant coefficient.

Constant voltages \(U_1, U_2, U_3\), whose magnitudes are proportional to \(x, y, z\), respectively, are applied to the three inputs of the device; a constant voltage \(U_{\mathrm{cp}}\), proportional to \(F_1\) or \(F_2\), depending on the design version of the device, is taken from the output.

The principle of operation of the device is based on the fact that the average value of the voltage \(U_{\mathrm{c}}\) of a periodic sequence of pulse-width-modulated pulses is equal to the product of the pulse-voltage amplitude \(U_3\) and the relative duration, which is determined by the ratio of the two other input voltages \(U_2\) and \(U_1\).

The block diagram of the device is shown in Fig. 1a. A constant voltage \(U_1\) is applied to the sawtooth-voltage generator SVG; from the output of the SVG a periodically repeating sawtooth voltage \(U'_1\) is taken, whose amplitude is proportional to the magnitude of the voltage \(U_1\). This follows from consideration of the similar triangles \(oaf\) and \(ohm\) (Fig. 1c). A constant voltage \(U_2\) is applied to one of the inputs of the pulse-width modulator PWM “Division”; the voltage \(U'_1\) is applied to the second input of the PWM. At the output of the PWM, a periodic sequence of rectangular pulses is formed with the frequency of the sawtooth voltage and with relative duration

\[ \gamma=\frac{\tau}{T}\frac{U_2}{U_1}=k\frac{U_2}{U_1}. \tag{3} \]

This follows from consideration of the similar triangles \(ac\delta\) and \(aed\) (Fig. 1,b), as well as \(oqf\) and \(ohm\) (Fig. 1,c). The shaped rectangular pulses control the operation of the two-position semiconductor switch “Multiplier,” to whose input a constant voltage \(U_3\) is applied.

Fig. 1.

At the output of the switch a periodic sequence of rectangular pulses is formed with amplitude \(U_3\), with the frequency and relative duration of the rectangular pulses arriving from the PWM. The average voltage at the switch output is

\[ U_{\mathrm{av}}=U_3\gamma=\frac{kU_2U_3}{U_1}. \tag{4} \]

The schematic diagram of the device is presented in Fig. 2. The sawtooth-voltage generator includes a blocking oscillator on triode \(T_1\) and an \(RC\) circuit \((R_3C_3)\), shaped into a sawtooth voltage, with discharge triode \(T_2\); the pulse-width modulator contains two cathode followers on triodes \(T_3, T_4\) and a two-stage amplifier in relay mode on triodes \(T_5, T_6\). The semiconductor two-position switch is made on triodes \(T_7, T_8\) of different conductivity type. If the base of triode \(T_5\) is connected to resistor \(R_5\), and the emitter to resistor \(R_4\), then the average voltage at the switch output will be

\[ U_{\mathrm{av}}=U_3\left(1-k\frac{U_2}{U_1}\right). \tag{5} \]

Equations (4) and (5) have the form of equations (1) and (2), respectively.

In the real case, the voltage on the GPN capacitor rises nonlinearly, and the relay amplifier changes its state at a certain differential input voltage \(\Delta U\), which depends on the sensitivity of the amplifier. This leads to a difference between the calculated switching time \(t_p\) and the actual time \(t_d\) (Figs. 1, 2), and ultimately to an error in the solution of equation (1). Let us estimate this error.

Fig. 2.

From Figs. 1 and 2 it is evident that at \(t_d\)

\[ U_c = U_2 + \Delta U = U_1 \left(1 - e^{-\frac{t_d}{\tau}}\right). \tag{6} \]

From (6) we determine

\[ t_d = \tau \ln \frac{U_2}{U_1 - U_2 - \Delta U}. \tag{7} \]

The deviation of the relative duration of the rectangular pulses from the calculated value is

\[ \Delta \gamma = \frac{t_d - t_p}{T}. \tag{8} \]

We determine \(t_p\) from the similarity of triangles \(acb\) and \(aed\) (Fig. 1, b) as

\[ t_p = \frac{U_2 \cdot T}{U_1'} = \frac{\tau U_2}{U_1}. \tag{9} \]

From (7), (8), and (9) we obtain

\[ \Delta \gamma = \frac{\tau}{T} \left( \ln \frac{U_2}{U_1 - U_2 - \Delta U} - \frac{U_2}{U_1} \right). \tag{10} \]

The relative error is determined from (3) and (10):

\[ \delta\% = \frac{\Delta \gamma}{\gamma}\cdot 100 = \left( \frac{U_1}{U_2} \ln \frac{U_2}{U_1 - U_2 - \Delta U} - 1 \right)100. \tag{11} \]

The accuracy of solving equations (1) or (2), all other conditions being equal, will be higher the greater the input voltages are in absolute value. When the voltage \(U_2\) was varied from \(0.2\ \mathrm{V}\) to \(7\ \mathrm{V}\) and the amplitude \(U_1'\) from \(0.5\ \mathrm{V}\) to \(7\ \mathrm{V}\), the average error in solving the equations at \(U_3 = 10\ \mathrm{V}\) was \(\pm 1\%\). The maximum value of the voltage \(U_1\) was \(140\ \mathrm{V}\).