ON THE ANALYSIS OF CORRECTION CIRCUITS OF MUS-D DRIVES

V. A. Sevastyanov, A. P. Ineshin, A. I. Esin

(Recommended by the scientific seminar of the Department of EPP of Tomsk Polytechnic Institute)

The development and production of complete drives with a wide range of speed regulation, using power magnetic amplifiers with self-saturation (MUS), together with their laboratory investigation and industrial testing, now make it possible to draw a number of conclusions regarding the features of correction of drives of the indicated type. This question is also of considerable interest because, in the known circuits of analogous drives made by foreign firms such as AEG and others, where negative feedback with respect to motor speed is applied by means of a tachogenerator, and standard transistor direct-current amplifiers are used as a series amplification stage, the correction circuits are not shown [1].

In domestically produced drives [2], the problems of correction and stabilization are solved by applying structures widely tested in regulated drives with electromechanical amplifiers (EMU).

However, the MUS differs fundamentally from the EMU in its transfer function, which includes, in particular, a delay element, the discrete character of operation, and non-reversibility due to the presence of valves in the power circuit. Therefore the question of developing stabilization as applied to systems with MUS deserves attention.

The dynamic properties of the uncorrected system are determined from analysis of the transfer function of the open-loop drive system (Fig. 1a or b)

\[ W(p)= \frac{K_{\mathrm{ppu}}\cdot K_{\mathrm{mu}}\cdot e^{-p\tau}\cdot K_{\mathrm{d}}\cdot \gamma} {(1+pT_{\mathrm{mu}})(1+pT_{\mathrm{m}}+p^{2}T_{\mathrm{m}}T_{\mathrm{ya}})}, \tag{1} \]

where \(K_{\mathrm{mu}}, K_{\mathrm{ppu}}\) are, respectively, the gain coefficients of the MUS and the PPU;

\[ K_{\mathrm{d}}=K_{\mathrm{d1}}\cdot K_{\mathrm{d2}},\quad \gamma \]

are the transfer coefficients of the electric motor and the TG;

\(T_{\mathrm{mu}};\ T_{\mathrm{m}};\ T_{\mathrm{ya}}\) are time constants characterizing the MUS and the motor;

\(\tau\) is the constant delay over a half-period of the carrier frequency, which cannot be neglected in closed-loop automatic-control systems.

The cutoff frequency \(\omega_c\) and the static gain coefficient possible under the stability condition for such a system are determined from an analysis of the expressions for the modulus \(A(\omega)\) and the phase \(\varphi(\omega)\), i.e., its amplitude-phase frequency characteristic. The system has a lag angle of \(180^\circ\) at the frequency for which:

\[ \varphi(\omega)=\pi=\omega_c \tau+\arctan \omega_c T_{\mathrm{mu}}+\arctan \frac{\omega_c T_{\mathrm{m}}}{1-T_{\mathrm{m}}T_{\mathrm{ya}}\omega_c^2} \tag{2} \]

Fig. 1

In this case, according to the stability condition, the analytical form of the Nyquist criterion will be:

\[ A(\omega)= \frac{K_{\mathrm{pmu}}\cdot K_{\mathrm{mu}}\cdot K_{\mathrm{d}}\cdot \gamma} {\sqrt{1+(\omega_c T_{\mathrm{mu}})^2}\cdot \sqrt{(1-\omega_c^2 T_{\mathrm{m}}T_{\mathrm{ya}})^2+(T_{\mathrm{m}}\omega_c)^2}} \leqslant 1 . \tag{3} \]

Assuming that the cutoff frequency of the open-loop system does not exceed the resonance frequency of the motor being used, at which the real part of its frequency characteristic is equal to zero, expression (3) can be transformed into the form:

\[ \frac{K_{\mathrm{pmu}}\cdot K_{\mathrm{mu}}\cdot K_{\mathrm{d}}\cdot \gamma} {\omega_c\cdot T_{\mathrm{m}}\cdot \sqrt{(\omega_c T_{\mathrm{mu}})^2}} \leqslant 1, \tag{4} \]

where \(\omega_c T_{\mathrm{mu}}\gg 1\), and

\[ \omega_c \cong \omega_{180}\cong \frac{1}{\sqrt{T_{\mathrm{m}}T_{\mathrm{ya}}}} . \]

Solving (4) with respect to the value of the critical gain coefficient possible under the stability condition, we obtain:

\[ K=K_{\mathrm{pmu}}\cdot K_{\mathrm{mu}}\cdot K_{\mathrm{d}}\cdot \gamma \leqslant \frac{T_{\mathrm{mu}}}{T_{\mathrm{ya}}}. \tag{5} \]

which indicates the low dynamic and static properties of uncorrected MUS-D systems, which are capable of providing only a small range of speed control. One of the obligatory conditions in expanding the range of speed control of the actuator motor is the use of correction circuits [3], which makes it possible to realize a large gain of the ACS without loss of its stability. Thus, for example, in drives of the PPMU series, for a control range of \(100 \div 1\), in order to ensure stable operation of the drive, parallel correction is introduced into the circuit [2]—an \(RC\) circuit encompassing the converter—the MUS and the PPU. The block diagram of this ACS is shown in Fig. 1a.

The transfer function of the corrected open-loop ACS has the form

\[ W(p)=\frac{W_{\mathrm{k}}(p)}{1+W_{\mathrm{p}}(p)\cdot W_{\mathrm{k}}(p)}\cdot W_{\mathrm{d}}(p), \tag{6} \]

where \(W_{\mathrm{k}}(p)\), \(W_{\mathrm{p}}(p)\), \(W_{\mathrm{d}}(p)\) are, respectively, the transfer functions of the correcting element, the converter, and the electric motor.

For a stable system, when \(W_{\mathrm{p}}(p)\cdot W_{\mathrm{k}}(p)\gg 1\) in the pass band of the ACS frequencies, the unity in the denominator of expression (6) may be neglected, and then:

\[ W(p)=\frac{W_{\mathrm{d}}(p)}{W_{\mathrm{k}}(p)}, \]

i.e., the dynamic characteristic of the converter is stabilized by the correction circuit.

For the series introduction of the signal of the stabilizing element into the comparison circuit (Fig. 2), one may obtain an expression for the input current of the intermediate amplifier in operator form:

\[ I_{\mathrm{in}}(p)=\frac{U_{\mathrm{in}}(p)}{R_{\mathrm{in}}} = \frac{ U_{\mathrm{s}}(p)-E_{\mathrm{tg}}(p)-\dfrac{K_{\mathrm{k}}pT_{\mathrm{k}}U_{\mathrm{a}}(p)}{1+pT_{\mathrm{k}}} }{ r_{\mathrm{tg}}+R_{\mathrm{c}}+r_{\mathrm{in}}+R_{\mathrm{p}}\cdot a(1-a) }, \tag{7} \]

where

\[ T_{\mathrm{k}}=C_{\mathrm{k}}(R_{\mathrm{k}}+R_{\mathrm{c}});\qquad a=\frac{R_{1}}{R_{\mathrm{p}}};\qquad K_{\mathrm{k}}=\frac{R_{\mathrm{c}}}{R_{\mathrm{k}}+R_{\mathrm{c}}}<1. \]

Fig. 2.

Of interest is the method of introducing the correction signal in parallel with the input of the PPU, Fig. 2 (shown by the dashed line), adopted by a number of machine-tool-building organizations, for example [3]. In this case, in expression (7),

\[ T_k=T'_k=C_k R_k;\qquad K_k=K'_k=\frac{R'_{\text{in}}}{R_k}, \]

where

\[ R'_{\text{in}}=R_p\cdot \alpha(1-\alpha)+r_{\text{tg}}+R_c. \]

When the gain is regulated as a function of the speed setting, the correction-signal introduction methods considered are not equivalent, since \(K_k\), all other conditions being equal, is proportional to \(R_{\text{vx}}\) only in the second case. The resistance \(R'_{\text{in}}\) can be selected over wide limits; therefore \(K'_k\) may have a value both less than and greater than 1, which makes it possible to provide wider limits for adjustment of the correction tuning. The choice of the transfer coefficient of the correction circuit \(K_k\) and of the time constant \(T_k\) should be made in accordance with [3] from analysis diagrams of generalized structures. A feature of calculating the parameters of flexible feedback as applied to the MUS-D system is the dependence of \(T_m\) on the load current \(I_a\), owing to the nonlinear resistance of the power selenium rectifiers [2].

\[ K_k=\frac{1}{A\frac{T_m}{T_a}-1} \tag{8} \]

and

\[ T_k=\frac{1}{K_k\frac{T_m}{T_a}B}, \tag{9} \]

where \(A\) and \(B\) are generalized system parameters selected from the conditions of the permissible magnitude of overshoot \(\sigma\%\) and the minimum regulation time \(t_{p.\min}\).

The correction parameters selected according to (8, 9) must be checked by the stability condition of the converter, taking into account the constant delay \(\tau\) when it is enclosed by a differentiating element [4]. For the block diagram of the open-loop internal enclosure of the SAR, shown in Fig. 1a, the AFC has the form:

\[ W_{\text{vo}}(j\omega)=A_{\text{vo}}(\omega)\cdot e^{-j\varphi_{\text{vo}}(\omega)} =K_{\text{ppu}}\, \frac{K_{\text{mu}}\cdot e^{-j\omega\tau}}{(1+\omega T_{\text{mu}})} \cdot \frac{K_k\cdot j\omega\cdot T_k}{(1+j\omega T_k)}, \]

where \(A_{\text{vo}}(\omega)\) is the modulus of the AFC of the internal enclosure, and \(\varphi_{\text{vo}}(\omega)\) is the phase.

The Nyquist stability condition for the internal enclosure, when

\[ \omega_{180}\cdot T_{\text{mu}}\gg 1 \]

and

\[ \frac{K\cdot j\omega_{180}\cdot T_k}{1+j\omega_{180}\cdot T_k}\simeq K_k \]

will be

\[ A_{\text{vo}}(\omega_{180})= \frac{K_{\text{ppu}}\cdot K_{\text{mu}}\cdot K_k} {\sqrt{(\omega_{180}\cdot T_{\text{mu}})^2}} = \frac{K_{\text{ppu}}\cdot K_{\text{mu}}\cdot K_k} {\omega_{180}\cdot T_{\text{mu}}} \ll 1 \tag{10} \]

and

\[ \varphi_{\text{vo}}(\omega_{180})=\pi =\arctg \omega_{180}\cdot T_{\text{mu}}+\omega_{180}\cdot \tau \simeq \frac{\pi}{2}+\omega_{180}\tau. \]

Hence

\[ \omega_{180}=\frac{\pi}{2\tau}. \]

The value of the critical gain coefficient of the converter is

\[ K_{\mathrm{ppu}} \cdot K_{\mathrm{mu}} = \frac{\tau \cdot T_{\mathrm{mu}}}{2 \tau K_k}. \tag{11} \]

Expansion of the control range of the speed of the actuator motor in drives with an MUS when using a parallel correcting RC circuit, as follows from (11), can be ensured only by increasing the time constant of the converter \(T_{\mathrm{mu}}\). Thus, the requirements for increasing the control range and the static accuracy of the system are in contradiction with the stability condition of the internal loop of the automatic-control system and with its speed of response. Therefore, application of the parallel correction considered here in a range greater than 100:1 is impractical. In addition, because of the substantial ripple of the MUS output voltage, a very high level of interference is transmitted through the RC circuit to the PPU input (Fig. 2).

In this connection, the question of developing and studying, for the MUS-D electric-drive system, another, more interference-resistant variant of correction—for example, an electromagnetic one—deserves attention. In [5] it is shown that, in order to obtain the correcting signal electromagnetically, one can use a free control winding of the magnetic amplifier, which is equivalent to the use of a differentiating transformer.

At the Department of “Electric Drive and Automation of Industrial Installations” of the Ulyanovsk Polytechnic Institute, an MUS-D electric drive with a class-D PPU and a range of 200:1 was developed. The block diagram of this electric drive is shown in Fig. 1b. Here: ID is the actuator electric motor; MUS is a self-saturating magnetic amplifier; PPU is an intermediate semiconductor amplifier of class D; TG is a tachogenerator; US is a comparison unit with variable transmission coefficient \(m\); GOS is flexible feedback with respect to the resultant magnetic flux of the MUS; TO is the current-limiting circuit.

The magnetic amplifier with the derived electromagnetic-correction signal is, in this case, structurally divided into two elements, of which the first—an aperiodic element of the form

\[ \frac{K_{\mathrm{mu}1}}{1 + p \cdot T_{\mathrm{mu}}} \]

is enclosed by the correction circuit (flexible negative feedback), while the second, a pure-delay element of the form \(K_{\mathrm{mu}2} e^{-p\tau}\), is included in the external-loop contour [5].

The internal loop, which includes the inertia-free PPU, the control circuit—MUS, and the electromagnetic-correction element proper, now remains stable for any values of the gain coefficients of the enclosed elements, which follows from (11) for \(\tau = 0\).

The electromagnetic variant of correction in the drive was investigated according to the circuit of Fig. 3.

The quantities indicated in Fig. 3 were recorded by means of a loop oscillograph.

To vary the level of the stabilization signal, a control switch was introduced into the correction circuit (a four-position switch). The results of the experimental test are recorded on the oscillogram shown (Fig. 4). In the initial portion of the oscillograms, the curves \(I_a\), \(U_a\), \(n\), and \(U_k\) indicate fully stable operation of the drive in the steady-state mode \((n = \mathrm{Const})\). The curves \(I_a\) and \(U_a\) have ripples.

Fig. 3.

with frequencies of 300 and 100 Hz. The latter is a consequence of the operation of the pulse-width modulator of the PPU at this carrier frequency. At time \(t_1\) the switch is moved to position 2, and at time \(t_2\) a load impulse is applied to the shaft of the ID by means of the braking device. Owing to the action of the weakened correction, the oscillations of the drive are damped. The oscillation frequency is 6.3 Hz. At time \(t_3\) the switch was moved to position 3. Owing to the opening of the correction circuit, the drive passes into the regime of self-oscillations under rigid excitation with a frequency of 4 Hz. The oscillogram shows that the voltage of the correction signals \(U_k\) in all sections is proportional to the derivatives \(I_a\) and \(U_a\). The curves \(I_a\) and \(U_a\), because of the limiting switching of the PPU,

large signals at its input contain pulsations only of frequency 300 Hz. At time \(t_4\) the switch was moved to position 4. In view of the restoration of the correction circuit with the original parameters, the process of stabilizing the drive begins. From the oscillogram in this interval it is evident that switching on the correction circuit after 0.5 sec leads to stabilization and to steady-state motion of the drive at the specified speed; however, as in the case of RC correction, the response speed of the automatic control system is thereby worsened.

Fig. 4.

Fig. 5.

For a comparative assessment of the influence of disturbances in the drive circuit with electromagnetic correction (Fig. 3), as compared with the RC correction considered, an oscillogram was taken (Fig. 5).

The actuator motor, rotating at the minimum no-load speed, at time \(t_1\) was loaded with a torque one and a half times the nominal torque. The dynamic speed drop of the motor in this case was 100%, i.e., the motor stopped, and then, as the current and the torque on its shaft increased, it again reached the specified rotational speed. During this process, changes in the voltage values on the correction winding \(U_{\mathrm{ok}}\) and on the armature of the actuator motor \(U_{\mathrm{я}}\) were recorded. The magnitude of the disturbance voltage \(U_{\mathrm{ok}}\) increases by 25%, while the magnitude of the disturbance voltage \(U_{\mathrm{я}}\) correspondingly increases by 625% when the armature current is increased from 2 A to 25 A. Consequently, from the standpoint of the influence of disturbances on the static accuracy of the system, the electromagnetic method of correction differs advantageously from RC correction in that the magnitude of the disturbance voltages—and, consequently, their influence on the gain coefficient of the system—depends to a significantly lesser degree on the magnitude of the motor load. The action of the disturbance with this method of correction is reduced only to the initial decrease in the gain coefficient of the PPU, which can be taken into account when selecting the gain coefficient of the intermediate amplifier.

Conclusions

1. Because of its poor static and dynamic properties, the uncorrected MUS-D system can be used in electric drives only with a small speed-control range.

2. Implementation of a structure with parallel correction by an RC circuit for a control range greater than 100:1 is difficult because of poor noise immunity and the contradiction between the stability condition of the converter’s internal loop and the dynamics of the system.

3. The use in MUS-D electric-drive systems with a PPU of the electromagnetic-correction variant deserves attention; it is distinguished by noise immunity and, in connection with this, makes it possible to obtain a wider speed-control range with satisfactory dynamic performance.

LITERATURE

1. Contactless Control Systems of Leading Foreign Firms. Moscow, 1962.

2. A. Ya. Petrovsky, Ya. B. Rozman. Adjustable Electric Drive with Magnetic Amplifiers. Energiya Publishing House, 1964.

3. Ya. S. Brovman, V. G. Kagan, F. D. Kochubievsky. Electric Drives with Semiconductor Control. Systems with Electric-Machine Converters (PMK-G-D). Energiya, 1964.

4. M. V. Meerov. Synthesis of Structures of High-Precision Automatic-Control Systems. Fizmatgiz, 1960.

5. M. A. Rosenblat. Magnetic Amplifiers. Sovetskoye Radio Publishing House, 1960.