PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Volume 161 1967

ON IMPROVING STABILIZATION

OF THE MUS-D ELECTRIC DRIVE

V. A. SEVASTYANOV, A. P. INESHIN, A. P. RYBAKOVA

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

In [1] it is shown that the dynamics of the MUS-D depend on the adopted circuit solutions when introducing positive current feedback, CF, into the comparison circuit of the automatic control system (Fig. 1).

A feature of these circuit solutions, for the adopted parameters, is the failure to satisfy the condition corresponding to the most favorable course of the transient process, which, as applied to the circuit of Fig. 1, c, is written in the form:

\[ \frac{ T_{\mathrm{MU}}+T_{\mathrm{M}}+K_{\mathrm{MU}}T_{\mathrm{M}}+ \left(R_{\mathrm{a}}\sigma-K_{\mathrm{ot}}R_{\mathrm{t}}\right)(1-\alpha) \frac{K_{\mathrm{MU}}}{R_{\mathrm{e}}}T_{\mathrm{M}} }{ 2\sqrt{K_{\mathrm{MU}}\sigma(1-\alpha)+1} } \simeq 0.6 \div 0.7, \tag{1} \]

that is, the damping coefficient lies within the range \(0.6\)–\(0.7\).

Here: \(T_{\mathrm{MU}}, T_{\mathrm{M}}\) are, respectively, the time constant of the MUS and the electromechanical time constant of the electric motor;

\(K_{\mathrm{MU}}\) is the voltage gain of the MUS;

\(K_{\mathrm{ot}}\) is the current-feedback gain;

\(R_{\mathrm{b}}\) is the fictitious resistance of the MUS;

\(R_{\mathrm{a}}, R_{\mathrm{t}}\) are, respectively, the resistances of the motor armature and of the CF signal takeoff;

\(\alpha, \sigma\) are the speed setting coefficient and the coefficient of unity voltage feedback (VF).

Figure 2 shows the D-partition curves with respect to the parameter \(K_{\mathrm{ot}}\), calculated according to equation (2), obtained for the circuit of Fig. 1, c:

\[ K_{\mathrm{ot}}(j\omega)= \frac{ -\left\{ T_{\mathrm{MU}}T_{\mathrm{e}}T_{\mathrm{M}}\omega^{2} - \left[ T_{\mathrm{MU}}+T_{\mathrm{M}}+ \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}}K_{\mathrm{MU}}\sigma(1-\alpha)T_{\mathrm{M}} \right] \right\} }{ (1-\alpha)\frac{R_{\mathrm{t}}}{R_{\mathrm{e}}}K_{\mathrm{MU}}T_{\mathrm{M}} } + \]

\[ +j\, \frac{ \left\{ T_{\mathrm{MU}}T_{\mathrm{M}}+ \left[ 1+\frac{R_{\mathrm{a}}}{R_{\mathrm{e}}}K_{\mathrm{MU}}\sigma(1-\alpha) \right]T_{\mathrm{e}}T_{\mathrm{M}} \right\}\omega^{2} - \left[ K_{\mathrm{MU}}\sigma(1-\alpha)+1 \right] }{ (1-\alpha)\frac{R_{\mathrm{t}}}{R_{\mathrm{e}}}K_{\mathrm{MU}}T_{\mathrm{n}} }. \tag{2} \]

The equation of the \(D\)-partition for variant “a” differs by the absence of the multiplier \((1-a)\). For variant “b” this multiplier is retained before

\[ \frac{R_{\mathrm{t}}}{R_{\mathrm{e}}}\,K_{\mathrm{MU}}T_{\mathrm{m}}. \]

Fig. 1. Possible circuit solutions for introducing OT into the comparison circuit for PMU9M1, and circuits with differential connection of the potentiometer \(R_{\mathrm{p}}\).

It follows from the curves that, at the lower end of the range, for all circuit solutions the SAR is stable only at \(K_{\mathrm{ot}}\approx 0.03\), whereas, according to the requirements for the stiffness of the mechanical characteristics, it must be \(K_{\mathrm{ot}}\ge 0.042\). Thus, quite apart from the quality of the transient process, the considered—

...the circuit solutions themselves, without special stabilization measures, cannot ensure stable operation of the drive at the lower end of the range \(1:10\).

In Fig. 5, \(a\), an oscillogram is shown of the transient process \(n(t)\) and \(I_a(t)\), recorded for the variant of Fig. 1, \(g\). It follows from it that the transient process in this case is characterized by oscillation, large overshoot, and a poor damping coefficient.

Fig. 2. Curves of the D-partition with respect to the parameter \(K_{\mathrm{ot}}\) for different speed settings \(\alpha\) at \(T_c = 0\):
\(a, b\) — solid lines for the circuit variant \(a\),
\(a\) — dashed lines for the circuit variant \(b\),
\(b\) — dashed-and-dotted lines for the circuit variant \(v\),
\(b\) — dashed lines for the circuit variant \(g\).

Increasing the stability of drive operation and improving the quality of transient processes can be achieved, in particular, by integrating the voltage \(U_{\mathrm{ot}}\), which is equivalent to introducing a time constant \(T_c\) into the rigid positive-feedback circuit and is accomplished practically by shunting the resistance \(R_t\) with a capacitance \(C = 2000\ \mu\mathrm{F}\) [1]. The time constant of the feedback in this case will be: \(T_c = R_t C = 35 \cdot 2000 \cdot 10^{-6} = 0.07\ \mathrm{s}\), which only slightly worsens the speed of response.

The transfer function for the feedback in this case is written as:

\[ W_{\mathrm{ot}}(p)=\frac{K_{\mathrm{ot}}R_t}{T_c p+1}(1-\alpha). \tag{3} \]

The transfer function of the automatic-control system with respect to the disturbing action, under the same other conditions [1], will be:

\[ W_{\mathrm{v}}(p)=\frac{\Delta n(p)}{\Delta M_c(p)} = \frac{\dfrac{R_e}{C_c C_m}\left(b_0p^3+b_1p^2+b_2p+b_3\right)} {a_0p^4+a_1p^3+a_2p^2+a_3p+a_4}. \tag{4} \]

Here:

\[ a_0 = T_{\mathrm{mu}} T_{\mathrm{e}} T_{\mathrm{m}} T_{\mathrm{c}}, \]

\[ a_1 = T_{\mathrm{mu}} T_{\mathrm{e}} T_{\mathrm{m}} + T_{\mathrm{mu}} T_{\mathrm{m}} T_{\mathrm{c}} + \left[1 + \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}} K_{\mathrm{mu}} \sigma (1-\alpha)\right] T_{\mathrm{e}} T_{\mathrm{m}} T_{\mathrm{c}}, \]

\[ a_2 = T_{\mathrm{mu}} T_{\mathrm{m}} + T_{\mathrm{mu}} T_{\mathrm{c}} + \left[1 + \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}} K_{\mathrm{mu}} \sigma (1-\alpha)\right] T_{\mathrm{e}} T_{\mathrm{m}} + \]

\[ +\left[1 + \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}} K_{\mathrm{mu}} \sigma (1-\alpha)\right] T_{\mathrm{c}} T_{\mathrm{m}}. \]

\[ a_3 = T_{\mathrm{mu}} + T_{\mathrm{m}} + [1 + K_{\mathrm{mu}} \sigma (1-\alpha)]T_{\mathrm{c}} + \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}} K_{\mathrm{mu}}\sigma(1-\alpha) T_{\mathrm{m}}, \]

\[ a_4 = K_{\mathrm{mu}} \sigma(1-\alpha) + 1, \]

\[ b_0 = T_{\mathrm{mu}} T_{\mathrm{e}} T_{\mathrm{m}}, \]

\[ b_1 = T_{\mathrm{mu}}T_{\mathrm{e}} + T_{\mathrm{mu}}T_{\mathrm{c}} + \left[1 + \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}} K_{\mathrm{mu}}\sigma(1-\alpha)\right] T_{\mathrm{e}}T_{\mathrm{c}}, \]

\[ b_2 = T_{\mathrm{mu}} + \left[1 + \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}} K_{\mathrm{mu}}\sigma(1-\alpha)\right]T_{\mathrm{e}} + \left[1 + \frac{R_{\mathrm{a}}}{R_{\mathrm{e}}} K_{\mathrm{mu}}\sigma(1-\alpha)\right]T_{\mathrm{c}}, \]

\[ b_3 = (R_{\mathrm{a}}\sigma - K_{\mathrm{fb}}R_{\mathrm{t}})(1-\alpha)\frac{K_{\mathrm{mu}}}{R_{\mathrm{e}}} + 1. \]

The Hurwitz stability condition for a third-order automatic-control system \((T_{\mathrm{e}} = 0)\), with \(a_1 > 0\), \(a_2 > 0\), \(a_3 > 0\), and \(a_4 > 0\), is written as

\[ a_2 a_3 > a_1 a_4. \tag{5} \]

Analyzing the last inequality, we see that the introduction of the constant \(T_{\mathrm{c}}\) increases the coefficients of the middle terms of the characteristic equation and decreases \(a_1\) \((T_{\mathrm{c}} < 1)\), independently of \(\alpha\), i.e., it helps improve the stability of the automatic-control system over the entire range. In addition, depending on the circuit solution, the coefficients containing \(R_{\mathrm{e}}\) change by the factor \((1-\alpha)\), which leads to an additional improvement in the stability of operation of the drive.

The equation of the D-decomposition curve with respect to the parameter \(K_{\mathrm{fb}}\) for the variant of Fig. 1, c, will be:

\[ K_{\mathrm{fb}}(j\omega) = \frac{\left(a_1\omega^2 - a_3 - \frac{a_4}{\omega}\right)} {(1-\alpha)\frac{R_{\mathrm{t}}}{R_{\mathrm{e}}}K_{\mathrm{mu}}T_{\mathrm{m}}} - j\frac{\omega(a_0\omega^2-a_2)} {(1-\alpha)\frac{R_{\mathrm{t}}}{R_{\mathrm{e}}}K_{\mathrm{mu}}T_{\mathrm{m}}}. \tag{5} \]

For variant “a” the factor \((1-\alpha)\) is absent from equation (5), while for variant “b” it appears in the denominator. It follows from (5) that \(K_{\mathrm{fb}}\) is a complex function of \(\alpha\).

Figure 3 shows D-decomposition curves constructed in accordance with equation (5) for speed settings \(\alpha = 0; 0.2; 0.4; 0.6; 0.8; 0.9\); and Table 1 summarizes comparative data for the values of \(K_{\mathrm{fb}}(\omega)\) by variants. From a comparison of the curves in Fig. 2, Fig. 3, and Table 1 it is seen that, when the resistance \(R_{\mathrm{t}}\) is shunted by the capacitance \(C\), the stability region expands—

...that is, the introduction of capacitance \(C\) contributes to improving the dynamics of the drive at the lower end of the range (oscillogram б in Fig. 5) of speed regulation.

Table 1

At high speeds, owing to the weakening of the negative feedback, the stabilizing action of capacitance \(C\), in the circuit variants of Fig. 1, б and г, proves to be excessive (the automatic-control system is stable for \(K_{\text{ot}}>0{,}042\)) and leads only to a prolongation of the transient process. Moreover, in this matter preference should be given to the circuit of Fig. 1, г, since, as follows from Fig. 3, for it the stability margin over the range is smaller and the capacitance \(C\) is used better.

Fig. 3. Curves of the \(D\)-partition with respect to the parameter \(K_{\text{ot}}\) for various speed settings \(\alpha\) at \(T_c = 0{,}7\) sec:
solid (bold) — for circuit variant а,
solid (thin) — for circuit variant б,
dashed — for circuit variant в,
dash-dotted — for circuit variant г.

In Fig. 4 the curves of the \(D\)-partition with respect to the parameter \(T_c\), constructed from equation (6) for various values of \(\alpha\), are presented.

\[ T_c(j\omega)= \frac{ T_{\mathrm{mu}}T_{\mathrm{e}}T_{\mathrm{m}}(j\omega)^3+ \left\{T_{\mathrm{mu}}T_{\mathrm{m}}+ \left[1+\frac{R_{\mathrm{ya}}}{R_{\mathrm{e}}}K_{\mathrm{mu}}\sigma(1-\alpha)\right]T_{\mathrm{e}}T_{\mathrm{m}}\right\}(j\omega)^2+ } { T_{\mathrm{mu}}T_{\mathrm{e}}T_{\mathrm{m}}(j\omega)^4+ \left\{T_{\mathrm{mu}}T_{\mathrm{m}}+ \left[1+\frac{R_{\mathrm{ya}}}{R_{\mathrm{e}}}K_{\mathrm{mu}}\sigma(1-\alpha)\right]T_{\mathrm{e}}T_{\mathrm{m}}\right\}(j\omega)^3+ } \to \]

\[ \to \frac{ +\left\{T_{\mathrm{mu}}+T_{\mathrm{m}}+(R_{\mathrm{ya}}\varepsilon-K_{\mathrm{ot}}R_{\mathrm{t}})(1-\alpha)\frac{K_{\mathrm{mu}}}{R_{\mathrm{e}}}T_{\mathrm{m}}\right\}(j\omega) +K_{\mathrm{mu}}\sigma(1-\alpha)+1 } { +\left[T_{\mathrm{mu}}+T_{\mathrm{m}}+\frac{R_{\mathrm{ya}}}{R_{\mathrm{e}}}K_{\mathrm{mu}}\sigma(1-\alpha)T_{\mathrm{m}}\right](j\omega)^2+ \left[K_{\mathrm{mu}}\sigma(1-\alpha)+1\right](j\omega) }. \tag{6} \]

It follows from them that at the lower end of the range \((\alpha \simeq 0.1)\), stable operation is possible only at \(T_c=0.01\) sec. As the speed increases \((\alpha = 1)\), the \(D\)-partition curves shift upward relative to the real axis \(R(\omega)\), and the automatic-control system is stable at smaller values of \(T_c\). At \(\alpha=1\) the automatic-control system is stable for any value of \(T_c\), including \(T_c=0\); that is, at the upper end of the range stabilization is not required, and introducing \(T_c\) merely leads to a prolongation of the transient process. Consequently, in order to obtain good dynamic characteristics as a whole, the variation of \(T_c\) over the range must be nonlinear.

Fig. 4. \(D\)-partition curves with respect to the parameter \(T_c\) at
\(K_{\mathrm{ot}}=0.1\) for different speed settings:
solid lines — for variants a and b,
dashed lines — for variant v,
dash-dotted lines — for variant g.

Figure 5 shows oscillograms recorded during application and removal of load for speeds of 300 and 3000 rpm: a, g — at \(C=0\); b, v — at \(C=2000\ \mu\mathrm{F}\), for the circuit variant of Fig. 1, g. It follows from them that effective stabilization of the automatic-control system by a capacitance \(C=2000\ \mu\mathrm{F}\) occurs at the lower limit of the range \((n=300\ \text{rpm})\). When the drive operates at a speed of 3000 rpm, stabilization by the capacitance \(C\) has practically no effect.

Fig. 5. Oscillograms of the transient process for application and removal of the load.

Conclusions

1. Introducing the time constant \(T_c\) into the control law improves the stability of the electric drive at the cost of degrading its speed of response.

2. Effective stabilization of the automatic control system is required at the lower limit of the speed-control range. At the upper end of the range, stability can be improved by weakening the negative feedback through a circuit solution; that is, stabilization by the capacitance \(C\) must be nonlinear over the range,

1. When choosing the capacitance $C$ from the condition of the lower limit of the speed-control range, its best use is observed in the variant with differential connection of the potentiometer.

References

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

2. M. A. Borovikov. Dynamic processes in an electric drive based on the single-phase magnetic amplifier—motor system. Elektromekhanika, No. 2, 1965.