Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 161 1967

Static Properties of MUS-D Drives with Various Methods of Introducing Current Feedback

V. A. Sevastyanov, A. P. Ineshin

(Submitted by the scientific seminar of the Department of Electric Drive and Automation of Industrial Installations, UPI)

It is shown in [1] that the static characteristics of a drive depend to a great extent on circuit solutions. One of the requirements imposed on a feed drive is constancy of the static error over the range [2], which must not exceed the permissible limits of \(\pm(5 \div 10)\%\).

The analytical expression for the static error \((\Delta\%)\) for a typical open-loop automatic control system of an MUS-D, in accordance with the circuit in Fig. 1 [3] and the block diagram in Fig. 1a, may be represented in the form:

\[ \Delta_p^\%=\frac{I_{\mathrm{ya\,n}}(R_{\mathrm{ya}}+R_{\phi})}{a U_p K_{\mathrm{mu}}}\,100, \tag{1} \]

where \(U_p\) is the supply voltage of the MUS control windings;
\(K_{\mathrm{mu}}\) is the voltage gain of the MUS;
\(a\) is the speed setting;
\(R_{\mathrm{ya}}, R_{\phi}\) are, respectively, the resistance of the motor armature and the fictitious resistance of the MUS;
\(I_{\mathrm{ya\,n}}\) is the rated value of the motor armature current.

Here, in the general case, \(R_{\phi}\) and \(K_{\mathrm{mu}}\) are complex functions of \(a\), and \(R_{\phi}\), in addition, depends on the load current \(I_{\mathrm{ya}}\).

Assuming for the time being, in order to simplify the analysis, that \(K_{\mathrm{mu}}(a)=\mathrm{const}\), \(R_{\phi}(a)=R_{\phi\,\mathrm{avg}}=\mathrm{const}\), and does not depend on \(I_{\mathrm{ya}}\), it may be considered that

\[ \Delta_p^\%=\frac{A}{a}, \]

where

\[ A=\frac{I_{\mathrm{ya\,n}}(R_{\mathrm{ya}}+R_{\phi})}{U_p K_{\mathrm{mu}}}\,100=\mathrm{const}. \]

We see that \(\Delta_p^\%(a)\) varies approximately according to a hyperbolic law. In Fig. 2 the dependence \(\Delta_p^\%(a)\) is plotted for the PMU9M. From analysis of this curve it follows that, in an open-loop automatic control system, taking into account the assumptions made, \(\Delta_p^\%(a)\) varies over the range within large limits and goes beyond the permissible limits.

In an actual drive \(R_{\phi}(\lambda) \ne \mathrm{const}\), but varies, according to the calculations performed, from 23 to 2 ohms, which leads to a redistribution of \(\Delta_p^{\%}(\alpha)\) over the range (Fig. 2, curve 1 shown by the dashed line): to its increase for \(\alpha < 0.5\) and decrease for \(\alpha > 0.5\). Moreover, at \(\alpha = 1\), \(\Delta_p^{\%} = 40\%\), while at \(\alpha = 0.1\), \(\Delta_o^{\%} = 400\%\).

Fig. 1. Block diagrams.

Obviously, such a drive does not provide the required range of speed regulation and does not meet the requirements imposed on the feed mechanism of machine tools.

To increase the stiffness of the mechanical characteristics and to expand the speed-control range in deeply regulated MUS-D drives, feedbacks by voltage, armature current, motor speed, etc., are used. Let us determine the influence of feedback by armature voltage \(U_{\mathrm{a}}\) on the static error.

The static error of an automatic control system closed by unit rigid negative feedback with respect to armature voltage (rigid negative voltage feedback), in accordance with the block diagram of Fig. 1b, will be:

\[ \Delta_{3b}^{0/0}= \frac{ I_{\mathrm{ya}}\left(R_{\mathrm{ya}}+\dfrac{R_{\phi}}{1+K_{\mathrm{mu}}}\right) }{ \alpha U_{\mathrm{p}}K_{\mathrm{mu}} }\,100 . \tag{2} \]

Here \(U_{\mathrm{p}}=\dfrac{U'_{\mathrm{p}}}{1+K_{\mathrm{mu}}}\), where \(U'_{\mathrm{p}}\) is the supply voltage in the closed automatic control system.

It follows from (2) that rigid negative voltage feedback in the MUS-D system reduces \(R_{\phi}\) by a factor of \((1+K_{\mathrm{mu}})\), and consequently also \(\Delta_{3b}^{0/0}\). Since all the quantities entering into (2) can have only positive values, for all values of \(\alpha\) the condition \(\Delta_{3b}^{0/0}>0\) is satisfied. Adopting the same assumptions, Fig. 2 shows the dependence \(\Delta_{3b}^{0/0}(\alpha)\) (curve 2), calculated according to equation (2) for the same other equal conditions. It follows from them that, when rigid negative voltage feedback is introduced into the control law, \(\Delta_{3b}^{0/0}\) also changes as a function of \(\alpha\) according to an approximately decreasing hyperbolic law; however, at one and the same \(\alpha\), \(\Delta_{3b}^{0/0}\) has a smaller value than in the open-loop automatic control system. Here, for comparison, this dependence is also shown by a dashed line, taking into account the variation of \(R_{\phi}(\alpha)\) over the range. It follows from it that the reduction of \(\Delta_{3b}^{0/0}\) has a substantial effect only at high speeds at \(\alpha=1\), that is, at small \(R_{\phi}\) (here \(\Delta=20\%\)), whereas at the lower end of the range, because of the small \(K_{\mathrm{mu}}\) and large \(R_{\phi}(\alpha)\gg R_{\mathrm{ya}}\), the action of rigid negative voltage feedback proves insufficient and \(\Delta_{3b}^{0/0}\) remains impermissibly large. Consequently, in MUS-D systems, because of the nonlinearity of \(R_{\phi}(\alpha)\) and \(K_{\mathrm{mu}}(\alpha)\), rigid negative voltage feedback can provide the required rigidity of the mechanical characteristics only at high speeds, while its action at the lower end of the range proves insufficient.

Fig. 2. Dependence \(\Delta_{\rho}^{0/0}=f(\alpha)\) and \(\Delta_{3b}^{0/0}=f(\alpha)\):
1 — for an open-loop automatic control system;
2 — for an automatic control system with voltage feedback.

The static error of an automatic control system in the presence of rigid positive current feedback (rigid positive current feedback), in accordance with the block diagram of Fig. 1, has the form:

\[ \Delta_{3B}^{0/0}= \frac{ I_{\mathrm{ya}}\left(R_{\mathrm{ya}}+R_{\phi}-K_{\mathrm{ot}}R_{\mathrm{t}}\right) }{ \alpha U_{\mathrm{p}}K_{\mathrm{mu}} }\,100, \tag{3} \]

where \(K_{\mathrm{ot}}=0.82\,\dfrac{W_1}{W_2}\) is the current-feedback coefficient,

\(R_{\tau}\) is the resistance for picking off the current-feedback signal.

From analysis of (3) it follows that here \(\Delta_{\mathrm{fb}}^{0/0}\) may take both positive and negative values. Moreover, for \(R_{\phi}=\mathrm{const}\), the sign of \(\Delta_{\mathrm{fb}}^{0/0}\) is determined by the value of \(K_{\mathrm{ot}}R_{\tau}\) and does not depend on the speed setting \(\alpha\). Indeed, for \(K_{\mathrm{ot}}R_{\tau}>R_{\phi}\), \(\Delta_{\mathrm{fb}}^{0/0}\) has a positive value for all values of \(\alpha\), and conversely, for \(K_{\mathrm{ot}}R_{\tau}<R_{\phi}\), \(\Delta_{\mathrm{fb}}^{0/0}\) takes a negative value. The decrease of \(\Delta_{\mathrm{fb}}^{0/0}\) over the range occurs inversely proportionally to \(\alpha\); therefore a substantial reduction of \(\Delta_{\mathrm{fb}}^{0/0}\) is observed at high speeds \((\alpha \simeq 1)\).

In an actual drive, as \(\alpha \to 1\), \(R_{\phi}(\alpha)\) decreases sharply. This leads to an increase of \(\Delta_{\mathrm{fb}}^{0/0}\) at the lower end and a decrease at the upper end of the range. Moreover, since \(K_{\mathrm{ot}}R_{\tau}=\mathrm{const}\), \(\alpha U_{\mathrm{p}}K_{\mathrm{mu}}\ne\infty\), and \(I_{\mathrm{ya.n}}\ne0\), \(\Delta_{\mathrm{fb}}^{0/0}\) changes sign when \(R_{\mathrm{ya}}+R_{\phi}(\alpha)=K_{\mathrm{ot}}R_{\tau}\), as follows from the family of curves \(\Delta_{\mathrm{fb}}^{0/0}\), Fig. 3 (dashed), calculated according to equation (3) for different values of \(K_{\mathrm{ot}}\).

Fig. 3. Dependence of \(\Delta_{\mathrm{fb}}^{0/0}=f(\alpha)\) for different values of \(K_{\mathrm{ot}}\): solid lines are for \(R_{\phi}=R_{\phi,\mathrm{av}}\), dashed lines are for \(R_{\phi}=f(\alpha)\).

It also follows from them that, in MUS-D systems, because of the nonlinearity of \(R_{\phi}(\alpha)\), the introduction of rigid positive current feedback promotes a substantial reduction of the static error only at the upper limit of the speed-control range \((\alpha \simeq 1)\). Moreover, for the corresponding values of \(K_{\mathrm{ot}}\), the action of rigid positive current feedback may prove excessive, which leads to overcompensation at high speeds, insufficient effectiveness of the action at the lower end of the range at \(\alpha=0.1\), and an overall nonuniformity in the distribution of \(\Delta_{\mathrm{fb}}^{0/0}\) over the range.

Consequently, in MUS-D systems, rigid positive current feedback gives a positive effect at the lower end of the range only under the condition of overcompensation at high speeds. From comparison of the characteristics in Fig. 3 (dashed and solid), it is seen that they all intersect in one zone \((\alpha=0.5\div0.6)\), corresponding to the average fictitious resistance adopted in the calculation as constant and equal to \(14\ \Omega\).

To eliminate overcompensation at high speeds \((\alpha=1)\), it is necessary that the action of rigid positive current feedback be weakened as the speed setting increases. This can be achieved by the circuit solution shown in Fig. 1,b [3], when feedback is introduced in series with the potentiometer.

The equation of the static error in this case, in accordance with the block diagram of Fig. 1,e, has the form:

\[ \Delta_{\mathrm{zkh}}^{\%} = \frac{\alpha U_{\mathrm{p}} + I_{\mathrm{yan}}\left[R_{\mathrm{ya}} + R_{\mathrm{f}} - K_{\mathrm{ot}} R_{\mathrm{t}}(1-\alpha)\right]} {\alpha U_{\mathrm{p}}K_{\mathrm{mu}}}\,100. \tag{4} \]

Here, at \(\alpha=0\), expression (4), analogously to (3), and the effectiveness of the action of the rigid positive current feedback are constant and do not depend on the speed setting \(\alpha\). At \(\alpha=1\), \(\Delta_{\mathrm{zkh}}^{\%}=\Delta_{\mathrm{r}}^{\%}\), which is equivalent to the absence of feedback.

The sign of the error is still determined by the value of \(K_{\mathrm{ot}}\), but depends on \(\alpha\).

Figure 4 gives a family of curves \(\Delta_{\mathrm{zkh}}^{\%}\), calculated from equation (4) for various \(K_{\mathrm{ot}}\) and \(R_{\mathrm{f}}=R_{\mathrm{f\,av}}\). It follows from them that, at \(\alpha<0.5\), they do not differ in any way from those shown in Fig. 3. At \(\alpha=1\) they all intersect at one characteristic point “a,” corresponding to the open-loop automatic-control system (Fig. 2, curve 1):

\[ \Delta_{\mathrm{zkh}}^{\%}(1)=\Delta_{\mathrm{r}}^{\%}= \frac{I_{\mathrm{yan}}\left(R_{\mathrm{ya}}+R_{\mathrm{f}}\right)} {\alpha U_{\mathrm{p}}K_{\mathrm{mu}}}\,100. \]

In a real drive, because of the nonlinearity of \(R_{\mathrm{f}}(\alpha)\), point “a” shifts downward (point \(a'\)), which causes a reduction of the error at \(\alpha=1\) owing to the smallness of \(R_{\mathrm{f}}(\alpha)\).

At \(\alpha<0.5\), because of the large \(R_{\mathrm{f}}\), the error correspondingly increases. This leads to a deterioration in the nonuniformity of the distribution of \(\Delta_{\mathrm{zkh}}^{\%}\) over the range.

Fig. 4. Dependence \(\Delta_{\mathrm{zkh}}^{\%}=f(\alpha)\) for various values of \(K_{\mathrm{ot}}\): solid lines—for \(R_{\mathrm{f}}=R_{\mathrm{f\,av}}\), dashed lines—for \(R_{\mathrm{f}}=f(\alpha)\).

Fig. 5. Dependence \(\Delta_{\mathrm{zg}}^{\%}=f(\alpha)\) for various values of \(K_{\mathrm{ot}}\): solid lines—for \(R_{\mathrm{f}}=R_{\mathrm{f\,av}}\), dashed lines—for \(R_{\mathrm{f}}=f(\alpha)\).

The sign of \(\Delta_{\mathrm{zkh}}^{\%}\) is determined from the condition:

\[ R_{\mathrm{ya}}+R_{\mathrm{f}}(\alpha) \gtreqless K_{\mathrm{ot}}R_{\mathrm{t}}(1-\alpha) \tag{5} \]

and depends on the intensity of variation of the terms \(R_{\mathrm{f}}(\alpha)\) and \(K_{\mathrm{ot}}R_{\mathrm{t}}(1-\alpha)\) of inequality (5) over the range. Moreover, since in a real drive, as \(\alpha\to1\), \(K_{\mathrm{ot}}R_{\mathrm{t}}(1-\alpha)\) usually decreases faster than \(R_{\mathrm{ya}}+R_{\mathrm{f}}(\alpha)\), then

the condition \(\Delta_{\mathrm{st}}^{\%} > 0\) is always satisfied. From the curves in Fig. 4 it is also evident that the introduction of a single rigid negative current feedback with attenuation as \(\alpha\) increases, in view of \(R_{\phi} \gg R_{\mathrm{a}}\), can provide the required value of \(\Delta_{\mathrm{st}}^{\%}\) only at the upper end of the range and is usually insufficient at \(\alpha = 0.1\). Therefore, the use in MUS-D drives of a single rigid negative current feedback with attenuation when the speed changes is not advisable.

A positive effect is obtained by using unit rigid negative speed feedback in combination with rigid negative current feedback. In this case, the expression for the static error, in accordance with the circuit of Fig. 1,a [3] and the block diagram of Fig. 1,c, will be:

\[ \Delta_{\mathrm{st}}^{\%} = \frac{ I_{\mathrm{an}} \left( R_{\mathrm{a}} + \dfrac{R_{\phi}}{1+K_{\mathrm{mu}}} - \dfrac{K_{\mathrm{ot}} R_{\mathrm{t}} K_{\mathrm{mu}}}{1+K_{\mathrm{mu}}} \right) }{ a U_{\mathrm{n}} K_{\mathrm{mu}} } \,100 . \tag{6} \]

It follows from (6) that in a closed-loop automatic-control system, in the presence of rigid negative speed feedback and rigid negative current feedback, the sign of the error is determined by the algebraic sum of the terms entering the numerator and depends on their magnitudes. Moreover, since \(K_{\mathrm{mu}} \gg 1\), the unity in the denominator may be neglected. Then, for \(R_{\phi}=\mathrm{const}\), the sign of the error is also determined by the magnitude \(K_{\mathrm{ot}}R_{\mathrm{t}}\) and does not depend on the speed setting \(\alpha\). However, the presence of rigid negative speed feedback reduces the term containing \(R_{\phi}\) by a factor of \(K_{\mathrm{mu}}\); therefore, a reduction of the error to an admissible value is now achieved at smaller values of \(K_{\mathrm{ot}}\) in comparison with the preceding variant. This reduces the nonuniformity of the distribution of \(\Delta_{\mathrm{st}}^{\%}\) over the range, as is seen from the dependences \(\Delta_{\mathrm{st}}^{\%}(\alpha)\), calculated from equation (6) for different values of \(K_{\mathrm{ot}}\) and shown in Fig. 5. In an actual drive, because of the nonlinearity \(R_{\phi}(\alpha)\), a redistribution of \(\Delta_{\mathrm{st}}^{\%}\) over the range occurs, since these characteristics intersect, respectively, in the zone \(R_{\phi}=R_{\phi\,\mathrm{av}}\). However, now the direction of redistribution of \(\Delta_{\mathrm{st}}^{\%}\) over the range, relative to its mean value \((R_{\phi}=14_{\mathrm{ohm}})\), depends on the sign of the error, which is determined by the magnitudes \(K_{\mathrm{ot}}\) and \(R_{\phi}(\alpha)\). Thus, for example, at small \(K_{\mathrm{ot}}\) (the region of positive \(\Delta_{\mathrm{st}}^{\%}\)), for values \(\alpha > 0.5\), \(\Delta_{\mathrm{st}}^{\%}\) increases, which leads to an increase in the nonuniformity of the distribution of \(\Delta_{\mathrm{st}}^{\%}\) over the range, requires its reduction at \(\alpha=0.1\), and an increase at high speeds. At large \(K_{\mathrm{ot}}\) (the region of negative \(\Delta_{\mathrm{st}}^{\%}\)), for \(\alpha < 0.5\) the error decreases, which leads, as before, to a certain equalization of it over the range; this could be useful, but in practice operation of the drive under overcompensation is not employed.

The condition for a positive error is:

\[ R_{\mathrm{a}} + \frac{R_{\phi}}{K_{\mathrm{mu}}} > K_{\mathrm{ot}} R_{\mathrm{t}} . \tag{7} \]

Comparing (5) and (7), we see that, other conditions being equal, overcompensation is now obtained at smaller values of \(K_{\mathrm{ot}}R_{\mathrm{t}}\), since

\[ \frac{R_{\phi}}{K_{\mathrm{mu}}} < R_{\phi}. \]

It also follows from the curves of Fig. 6 that, at small values of \(K_{\mathrm{ot}}\) \((K_{\mathrm{ot}} \simeq 0.5)\), it is possible to obtain a sufficiently uniform distribution of the error and, consequently, rigid mechanical characteristics over a small range \((5 \div 8)\). In this case \(\Delta_{\mathrm{st}}^{\%}\) does not exceed the admissible limits \((\Delta=\pm 10)\). To obtain a wider range, it is necessary to increase \(K_{\mathrm{ot}}\). However, at large \(K_{\mathrm{ot}}\) (0.1 and above), one observes

overcompensation is produced at the upper end of the range, which leads to an increasing character of the mechanical characteristics \((- \Delta_{zg}^{0/0})\).

A more uniform distribution of the error over the range can be obtained for the block diagram of Fig. 1,d, composed in accordance with the diagram of Fig. 1,b [3] and equation (8), when the rigid positive current feedback, as the speed setting increases, is weakened by the factor \((1-\alpha)\), while the action of the rigid negative speed feedback remains constant. In this case, under the adopted assumptions, the static error may be represented in the form:

\[ \Delta_{zd}^{0/0} = \frac{ I_{an} \left[ R_a + \frac{R_f}{1+K_{mu}} - \frac{K_{ot} R_g K_{mu}}{1+K_{mu}}(1-\alpha) \right] }{ \alpha U_p K_{mu} } \,100. \tag{8} \]

Here, for \(\alpha = 1\), expression (8) does not differ in any way from (2), which is equivalent to the absence of rigid positive current feedback; and for \(\alpha = 0\) it takes the form of equation (6), with all the features following from it.

Therefore the structure of such an automatic-control system, in the sense of variation of its parameters, is variable over the range and occupies an intermediate position between the variants of Fig. 1a and b [3]. The sign of the error, for a fixed value of \(R_f\), is still determined by the magnitude \(K_{ot}\) and, in addition, depends on \(\alpha\).

Figure 6 shows a family of curves \(\Delta_{zd}^{0/0}(\alpha)\), calculated according to equation (8) for different values of \(K_{ot}\) and \(R_f = \mathrm{const}\). For small values of \(\alpha\) they do not differ in any way from those shown in Fig. 5, and the sign of the error is determined by the magnitude \(K_{ot}\). However, at \(\alpha = 1\), they all intersect at one characteristic point “b,” independently of the value of \(K_{ot}\), corresponding to:

\[ \Delta_{zd}^{0/0}(1) = \Delta_{zb}^{0/0} = \]

\[ = \frac{ I_{an}\left(R_a + \dfrac{R_f}{1+K_{mu}}\right) }{ U_p K_{mu} } \,100, \]

which occurred in the absence of rigid positive current feedback in Fig. 2 (curve 2).

In a real drive, owing to the nonlinearity \(R_f(\alpha)\), point “b” shifts downward (point \(b'\)); that is, \(\Delta_{zd}^{0/0}\) decreases, and for \(\alpha < 0.5\), owing to the larger \(R_f \gg R_a\), it increases. As before, the characteristics intersect, respectively, in the region \(R_f = R_{f,\mathrm{av}} = 14\ \Omega\), which causes an increase in the nonuniformity of the distribution of \(\Delta_{zd}^{0/0}\) over the range owing to the nonlinearity \(R_f(\alpha)\). However, in this case the introduction of rigid negative speed feedback leads to a reduction of \(R_f\) by a factor of \((1+K_{mu})\), which, with the corresponding choice of \(K_{ot}\), ensures a quite permissible error at \(\alpha = 1\); and rigid positive current feedback, owing to the effectiveness of its action at \(\alpha = 0.1\), somewhat compensates for the increase of \(\Delta_{zd}^{0/0}\) at the lower end of the range (where, owing to the smallness of \(\alpha\) in the denominator of (8) and \(R_f \gg R_a\),

Fig. 6. Dependence \(\Delta_{zd}^{0/0} = f(\alpha)\), for different values of \(K_{ot}\): solid lines—for \(R_f = R_{f,\mathrm{av}}\); dashed lines—for \(R_f = f(\alpha)\).

\(\Delta_{33}^{\%}\) has large values) without exerting a practical effect at \(\alpha=1\). As a result, some leveling of \(\Delta_{33}^{\%}\) occurs in the range \(1:10\), which is quite acceptable for feed drives of heavy machine tools.

A quite satisfactory distribution of the error in the range \(1:30\) is given by the circuit-design variant of Fig. 1, e [3]. The static error for this case, according to the block diagram of Fig. 1, z, can be represented in the form:

\[ \Delta_{33}^{\%}= \frac{ I_{\mathrm{ya\,n}} \left[ R_{\mathrm{ya}}+ \dfrac{R_{\mathrm{f}}}{1+(1-\alpha)K_{\mathrm{mu}}} - \dfrac{K_{\mathrm{ot}}R_{\mathrm{t}}(1-\alpha)K_{\mathrm{mu}}}{1+(1-\alpha)K_{\mathrm{mu}}} \right] }{ aU_{\mathrm{p}}K_{\mathrm{mu}} }\,100. \tag{9} \]

Neglecting the unity in the denominator, in view of \(K_{\mathrm{mu}}\gg 1\), we obtain:

\[ \Delta_{33}^{\%}\simeq \frac{ I_{\mathrm{ya\,n}} \left[ R_{\mathrm{ya}}+ \dfrac{R_{\mathrm{f}}}{(1-\alpha)K_{\mathrm{mu}}} - K_{\mathrm{ot}}R_{\mathrm{t}} \right] }{ aU_{\mathrm{p}}K_{\mathrm{mu}} }\,100. \]

Here, as \(\alpha\to 1\), in view of \(R_{\mathrm{f}}\gg R_{\mathrm{ya}}\), the term

\[ \frac{R_{\mathrm{f}}}{(1-\alpha)K_{\mathrm{mu}}} \]

assumes a large numerical value, which, however, when \(K_{\mathrm{ot}}\) is selected from the conditions of the lower limit of the range, can be compensated to the desired value by \(K_{\mathrm{ot}}R_{\mathrm{t}}\).

In view of the fact that, under the adopted assumptions, the subtracted term \(K_{\mathrm{ot}}R_{\mathrm{t}}=\mathrm{const}\), while the term

\[ \frac{R_{\mathrm{f}}}{(1-\alpha)K_{\mathrm{mu}}} \]

decreases as \(\alpha\to 0\) because of the nonlinearity of \(R_{\mathrm{f}}\) with increasing \(\alpha\), \(\Delta_{33}^{\%}\to 0\) and, for the corresponding \(K_{\mathrm{ot}}\), may assume negative values.

The condition \(\Delta_{33}^{\%}=0\) is:

\[ R_{\mathrm{ya}}+\frac{R_{\mathrm{f}}}{(1-\alpha)K_{\mathrm{mu}}}=K_{\mathrm{ot}}R_{\mathrm{t}}. \]

For \(\alpha>0.8\), \(R_{\mathrm{f}}\simeq \mathrm{const}\); therefore, a further decrease of the term \((1-\alpha)K_{\mathrm{mu}}\) leads to an increase of \(\Delta_{33}^{\%}\). Finally, at \(\alpha=1\), \((1-\alpha)K_{\mathrm{mu}}\to 0\), and \(\Delta_{33}^{\%}\to\infty\). However, because of the presence of unity in the denominator, which we neglected earlier, \(\Delta_{33}^{\%}(1)=\Delta_{\mathrm{r}}^{\%}\) (unity may be neglected only at small \(\alpha\)).

As a result, the error fluctuates over the range within permissible limits of \(\pm 10\%\), and at \(\alpha=1\), \(\Delta_{33}^{\%}=\Delta_{\mathrm{r}}^{\%}\).

Figure 7 shows the curves \(\Delta_{33}^{\%}(\alpha)\), constructed according to (9) for various values of \(K_{\mathrm{ot}}\): solid for \(R_{\mathrm{f}}=\mathrm{const}\), dashed for \(R_{\mathrm{f}}(\alpha)=\mathrm{var}\). All the curves intersect at points “a” and \((a')\) at \(\alpha=1\), corresponding to the open-loop automatic-control system.

\[ \Delta_{33}^{\%}(1)=\Delta_{\mathrm{r}}^{\%}= \frac{I_{\mathrm{ya\,n}}(R_{\mathrm{ya}}+R_{\mathrm{f}})}{aU_{\mathrm{p}}K_{\mathrm{mu}}}\,100. \]

The dashed and solid characteristics still intersect, respectively, in the zone \(R_{\mathrm{f}}=R_{\mathrm{f\,avg}}=14\ \Omega\). At small values of \(K_{\mathrm{ot}}\) \((0.05\div 0.1)\) it is positive. For \(K_{\mathrm{ot}}>0.1\), \(\Delta_{33}^{\%}\) is negative in the main range and assumes a positive value only at high speeds. By appropriate selection of \(K_{\mathrm{ot}}\), a quite permissible value of \(\Delta_{33}^{\%}\) over the range can be obtained.

The static error for the circuit-solution variant of Fig. 1, v in [3], in accordance with the block diagram of Fig. 1, e, has the form:

\[ \Delta_{3e}^{0/0} = \frac{ I_{\mathrm{yan}} \left[ R_{\mathrm{ya}}+ \frac{R_{\phi}'}{1+(1-\alpha)K_{\mathrm{mu}}} - \frac{K_{\mathrm{ot}}R_{\mathrm{t}}K_{\mathrm{mu}}}{1+(1-\alpha)K_{\mathrm{mu}}} \right] }{ \alpha U_{\mathrm{p}}K_{\mathrm{mu}} } \,100 . \tag{10} \]

Neglecting unity in view of \(K_{\mathrm{mu}}\gg 1\), we obtain:

\[ \Delta_{3e}^{0/0} = \frac{ I_{\mathrm{yan}} \left[ R_{\mathrm{ya}}+ \frac{R_{\phi}-K_{\mathrm{ot}}R_{\mathrm{t}}K_{\mathrm{mu}}}{(1-\alpha)K_{\mathrm{mu}}} \right] }{ \alpha U_{\mathrm{p}}K_{\mathrm{mu}} } \,100 . \]

Here, for small \(\alpha\), owing to the large value \(K_{\mathrm{mu}}\gg 1\) and the comparable value, at the bottom of the range, of \(K_{\mathrm{mu}}(1-\alpha)\), the sign of the error is determined by the magnitude \(K_{\mathrm{ot}}\), and, depending on its value, may be either positive or negative. As \(\alpha\to 1\), \(R_{\phi}(\alpha)\) decreases sharply, while \(K_{\mathrm{ot}}R_{\mathrm{t}}K_{\mathrm{mu}}=\mathrm{const}\); therefore \(\Delta_{3e}^{0/0}\) takes a positive value which, because of the simultaneous decrease \(K_{\mathrm{mu}}(1-\alpha)\to 0\), may reach a large magnitude. Therefore such a solution has found no practical application.

The distribution of \(\Delta\) over the range in the variant of Fig. 1, g [3] can be improved by the combined introduction of rigid negative current feedback, as shown in Fig. 1, g (dashed line) [3]. The static error in this case, in accordance with the block diagram of Fig. 1, z [3], will be:

\[ \Delta_{3zk}^{0/0} = \frac{ I_{\mathrm{yan}} \left\{ R_{\mathrm{ya}}+ \frac{R_{\phi}}{1+(1-\alpha)K_{\mathrm{mu}}} - \frac{K_{\mathrm{ot}}\left[R_{\mathrm{t}}''+R_{\mathrm{t}}'(1-\alpha)\right]K_{\mathrm{mu}}}{1+(1-\alpha)K_{\mathrm{mu}}} \right\} }{ \alpha U_{\mathrm{p}}K_{\mathrm{mu}} } \,100 . \tag{11} \]

Neglecting unity as before (at the bottom of the range), we obtain:

\[ \Delta_{3zk}^{0/0} = \frac{ I_{\mathrm{yan}} \left[ R_{\mathrm{ya}}+ \frac{R_{\phi}-K_{\mathrm{ot}}R_{\mathrm{t}}'(1-\alpha)K_{\mathrm{mu}}}{(1-\alpha)K_{\mathrm{mu}}} - K_{\mathrm{ot}}R_{\mathrm{t}}'' \right] }{ \alpha U_{\mathrm{p}}K_{\mathrm{mu}} } \,100 . \tag{12} \]

It follows from (12) that in this case there is a continuously acting component \(K_{\mathrm{ot}}R_{\mathrm{t}}''\), independent of \(\alpha\). This leads to a shift of the curves into the region of negative \(\Delta_{3}^{0/0}\) in proportion to the value of \(R_{\mathrm{t}}''\).

In Fig. 8 are given the curves \(\Delta_{3}^{0/0}\), constructed from equations (8 and 9) for \(K_{\mathrm{ot}}=0.115\) for the circuit variants of Fig. 1, b and g [3]. Also shown here (by dashed lines), for the combined variant, are the values \(R_{\mathrm{t}}'=1, 2, 3\ \Omega\). From comparison of the curves it follows that the variant with combined introduction of rigid negative current feedback helps to equalize \(\Delta_{3}^{0/0}\) at \(\alpha=1\) over the range in the circuit of Fig. 1, g [3] (curve 2), and worsens it in the variant of Fig. 1, b [3] (curve 1). For the remaining values of \(\alpha\), the circuits are approximately equivalent.

In Fig. 9 (by dashed lines) are shown the curves \(\Delta_{3}^{0/0}(\alpha)\) for all the cases considered of feedback introduction, calculated for \(K_{\mathrm{ot}}=0.1\) and otherwise equal conditions. It follows from them that the most favorable distribution of the error over the range is observed in the circuit of Fig. 1 b and Fig. 1 g [3]. For comparison, also plotted here (with solid lines) are the curves \(\Delta_{3}^{0/0}(\alpha)\) for the variants of Fig. 1, g [3] (curve 9) and Fig. 1, b [3] (curve 7), calculated at \(K_{\mathrm{ot}}=0.115\) and \(0.125\). From comparison of the curves

Fig. 7. Dependences \(\Delta_{з}^{0}\big|_{I_0}=f(\cdot)\) for various values of \(K_{\mathrm{ot}}\): solid curves—for \(R_{\phi}=R_{\phi\,\mathrm{sr}}\); dashed curves—for \(R_{\phi}=f(\varepsilon)\).

Fig. 8. Dependence \(\Delta_{з}^{0}\big|_{I_0}=f(\alpha)\) for various values of \(K_{\mathrm{ot}}\): solid curves—for \(R_{t}=0\); dashed curves—for \(R_{t}'=1;\ 2;\ 3\ \Omega\).

it is evident that variant (7), while ensuring a more uniform distribution of the speed error \(\Delta_{3d}^{0}/_{0}\), has a positive value within the specified limits and is inevitably accompanied by overcompensation at low

Fig. 9. Dependences \(\Delta_3^{0}/_{0}=f(\alpha)\), for various circuit solutions.

speeds \((\alpha \simeq 0.1)\). Variant (9) is free of this shortcoming; however, in the middle of the range \((\alpha = 0.5 \div 0.8)\), in variant (9) \(\Delta_{33}^{0}/_{0}\) changes sign, and at \(\alpha = 1\), \(\Delta_{33}^{0}/_{0} = \Delta_{\rho}^{0}/_{0}\), which cannot be considered a shortcoming, but is not necessary.

Here, for comparison, curve (5) presents the dependence \(\Delta_{33k}^{0}/_{0}\) for the variant of Fig. 1, \(g\) [3] with combined introduction of flexible positive current feedback, for \(K_{\mathrm{ot}} = 0.115\), \(R_{\mathrm{t}} = 32\ \Omega\), and \(R'_{\mathrm{t}} = 3\ \Omega\). It is evident from it that \(\Delta_{33k}^{0}/_{0}\) does not exceed \(\pm 7\%\), with slight overcompensation at \(0.8 > \alpha > 0.5\).

Thus, by means of the circuit solution and the corresponding selection of parameters in variants (7), (9), and the combined variant, it is possible to achieve a quite satisfactory distribution of the error over the range. Moreover, in our opinion, preference should be given to variant (9) with combined introduction of flexible positive current feedback.

References

1. V. A. Sevastyanov, A. P. Ineshin. Electric-drive systems with magnetic-semiconductor converters (MUS-D with PPU), Privolzhskoye Book Publishing House, 1966.

2. M. E. Verkholat, Yu. Ya. Morgovskii. An improved electric drive for the feed of metal-cutting machine tools with the D-2000 speed-control range, Leningrad, LDNTP, 1964.

3. V. A. Sevastyanov, A. P. Ineshin, A. P. Rybakova. On the question of improving the stabilization of the MUS-D electric drive, Proceedings of TPI, vol. 161, 1966.