NONLINEAR STRUCTURAL CONSTRAINTS OF MUS-D ELECTRIC DRIVES WITH PPU

A. P. INESHIN, V. A. SEVASTYANOV

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

It is shown in [1] that, in drives based on the MUS-D system with a PPU, the stabilizing \(RC\) circuit transmits from the converter output (the motor armature) to the PPU input a high level of interference at a frequency of 300 Hz, which leads to a reduction in the gain of the automatic-control system (ACS), a limitation of the speed-control range, and other static characteristics of the drive. Below we consider possible structural constraints of MUS-D electric drives with a PPU in dynamics.

The structural diagram of a deeply controlled MUS-D electric drive is shown in Fig. 1. Here the outer loop (1) includes: a semiconductor–magnetic converter consisting of intermediate semiconductor (PPU) and power magnetic (MUS) amplifiers, and a direct-current actuator motor with tachometric feedback. The transfer functions of these elements of the ACS may be represented in the form:

\[ W_{\mathrm{p}}(p)=W_{\mathrm{PPU}}W_{\mathrm{MU}} =\frac{K_{\mathrm{p}}e^{-p\tau}}{1+T_{\mathrm{MU}}p}, \]

\[ W_{\mathrm{d}}(p)= \frac{K_{\mathrm{d}}}{1+T_{\mathrm{m}}p+T_{\mathrm{m}}T_{\mathrm{e}}p^{2}}, \tag{1} \]

where: \(K_{\mathrm{p}}\) is the converter gain;

\(T_{\mathrm{MU}}, \tau\) are the time constant of the MUS and the half-period delay;

\(K_{\mathrm{d}}\) is the resultant transfer coefficient of the motor–tachogenerator;

\(T_{\mathrm{m}}, T_{\mathrm{e}}\) are the equivalent electromechanical and electromagnetic time constants of the electric motor, taking into account \(R_{\phi}\) of the MUS.

Fig. 2. Structural diagram of the MUS-D electric drive.

The static accuracy and control range of the ACS are ensured by large values of \(K_{\mathrm{p}}\). At the same time, according to the stability condition of the outer loop (1), the converter, which includes elements with the largest and unstable coefficients and parameters, must be...

encompass it with a standard correction circuit (an \(RC\) circuit or differentiating transformer) with transfer function:

\[ W_{\mathrm{k}}(p)=\frac{K_{\mathrm{k}}pT_{\mathrm{k}}}{1+T_{\mathrm{k}}p}. \]

Realization of this drive structure with the specified parameters is possible provided that the necessary additional check [3] of the stability of the inner loop (II) is performed by opening it at point 2 (Fig. 1).

However, in a real system with a high gain coefficient, designed in accordance with linear theory, self-oscillations [2] may arise, associated with the nonlinearity of the converter elements, the high level of disturbances, and the beak-shaped character of the amplitude-phase frequency characteristic (APFC) of the automatic control system.

A typical nonlinearity of the system under consideration is saturation of the PPU and MUS, caused by the limited value of the supply voltage of these elements. To take the effect of saturation into account, in accordance with [3] the structure of Fig. 1 should be opened not at the system output, at point 1, as is usually done in analyzing the stability of automatic control systems, but directly at point 3 at the converter input. Now both system loops, the outer and the inner, are open, and the transfer function of the automatic control system corresponding to the beak-shaped APFC has the form:

\[ W_{p}(p)=W_{\mathrm{p}}(p)\,[W_{\mathrm{d}}(p)+W_{\mathrm{k}}(p)]. \tag{2} \]

Figure 2 presents the APFCs corresponding to the transfer functions: 1 — of the open automatic control system; 2 — of the open outer feedback loop; 3 — of the open inner feedback loop; 4 — of the open system without taking into account the delay of the MUS.

Fig. 2. Amplitude-phase frequency characteristics of the system.

The characteristics are plotted for a real electric drive [1] with the adopted parameters: \(K_{\mathrm{p}}=2000\); \(T_{\mathrm{mu}}=0.45\ \mathrm{sec}\); \(T_{\mathrm{m}}=0.2\ \mathrm{sec}\); \(T_{\mathrm{e}}=0.02\ \mathrm{sec}\); \(T_{\mathrm{k}}=0.05\ \mathrm{sec}\); \(\tau=0.01\ \mathrm{sec}\); \(K_{\mathrm{d}}=0.067\); \(K_{\mathrm{k}}=0.04\).

In the interval \(\omega=0 \div \omega_{1}\), the influence of the delay is negligible, and the APFC of the open automatic control system is determined mainly by the transfer function of the outer loop:

\[ W_{\mathrm{o}}(p)=W_{\mathrm{p}}(p)\cdot W_{\mathrm{d}}(p)= \frac{K_{\mathrm{p}}K_{\mathrm{d}}}{(1+T_{\mathrm{mu}}p)(1+T_{\mathrm{m}}p+T_{\mathrm{m}}T_{\mathrm{e}}p^{2})}. \tag{3} \]

At the frequency \(\omega=\omega_1\) the phase of the open-loop ACS will be:

\[ \varphi(\omega_1)=\pi-\operatorname{arc tg}\omega_1T_{\mathrm{MU}}+\varphi_{\mathrm{d}} . \tag{4} \]

In view of the evident \(\omega_1T_{\mathrm{MU}}\gg 1\), \(\operatorname{arc tg}\omega_1T_{\mathrm{MU}}\simeq \dfrac{\pi}{2}\); consequently \(\omega_1\) is close to the resonant frequency of the motor:

\[ \omega_1=\sqrt{\frac{1}{T_{\mathrm{m}}T_{\mathrm{e}}}} . \tag{5} \]

Hence the modulus of the AFC at the frequency \(\omega_1\) can be found from the relation:

\[ M(\omega_1)=\frac{K_{\mathrm{p}}K_{\mathrm{d}}}{\omega_1^2T_{\mathrm{MU}}T_{\mathrm{m}}} =K_{\mathrm{p}}K_{\mathrm{d}}\frac{T_{\mathrm{e}}}{T_{\mathrm{MU}}}. \tag{6} \]

In the interval \(\omega=\omega_3\div\infty\), the delay must also be taken into account, and the AFC is determined mainly by the transfer function of the open-loop inner loop:

\[ W_{\mathrm{p}}(p)\simeq W_{\mathrm{p}}(p)W_{\mathrm{k}}(p) =\frac{K_{\mathrm{p}}e^{-p\tau}}{(1+T_{\mathrm{MU}}p)}\cdot \frac{K_{\mathrm{k}}pT_{\mathrm{k}}}{(1+T_{\mathrm{k}}p)} . \tag{7} \]

At the frequency \(\omega=\omega_3\), in view of \(\omega_3T_{\mathrm{k}}>1\), the influence of the correction circuit on the phase of the open-loop ACS can be neglected; then:

\[ \varphi(\omega_3)=\pi-\operatorname{arc tg}\omega_3T_{\mathrm{MU}}+\omega_3\tau . \tag{8} \]

Hence

\[ \omega_3=\frac{\pi}{2\tau} \tag{9} \]

and the modulus of the AFC at the frequency \(\omega_3\) will be:

\[ M(\omega_3)=\frac{K_{\mathrm{p}}K_{\mathrm{k}}}{\omega_3T_{\mathrm{MU}}} =K_{\mathrm{p}}K_{\mathrm{k}}\frac{2\tau}{\pi T_{\mathrm{MU}}}<1, \tag{10} \]

which is the analytical form of the stability condition, according to Nyquist, for the inner loop [2].

The coordinates \(\omega_2\) and \(M(\omega_2)\) of the second intersection of the system AFC with the negative real semiaxis in Fig. 2 can be determined approximately without taking the delay into account, by the same method, taking:

\[ \varphi(\omega_2)=180^\circ=\varphi_{\mathrm{p}}+\varphi_{(W_{\mathrm{d}}+W_{\mathrm{k}})} \quad\text{and}\quad \varphi_{\mathrm{p}}(\omega_2)=\operatorname{arc tg}\omega_2T_{\mathrm{MU}}\simeq 90^\circ . \]

Hence \(\varphi_{(W_{\mathrm{d}}+W_{\mathrm{k}})}(\omega_2)\simeq 90^\circ\).

Consequently, setting equal to zero the real components of the transfer functions \(W_{\mathrm{d}}(\omega_2)\) and \(W_{\mathrm{k}}(\omega_2)\) makes it possible to find the value \(\omega_2\), and the sum of the imaginary components of \(W_{\mathrm{d}}(\omega_2)\) and \(W_{\mathrm{k}}(\omega_2)\), after substituting the found \(\omega_2\), determines the modulus of the system AFC at this frequency:

\[ M(\omega_2)=\frac{K_{\mathrm{p}}}{\omega_2T_{\mathrm{MU}}} \left[ -\frac{\omega_2K_{\mathrm{d}}(T_{\mathrm{m}}+T_{\mathrm{e}})} {(1+\omega_2^2T_{\mathrm{m}}^2)(1+\omega_2^2T_{\mathrm{e}}^2)} + \frac{\omega_2K_{\mathrm{k}}T_{\mathrm{k}}} {(1+\omega_2^2T_{\mathrm{k}}^2)} \right]. \]

Taking into account the characteristic ratios of the above-given parameters of the ACS leads to simpler approximate expressions for the frequency and modulus:

\[ \omega_2\simeq \sqrt{\frac{K_{\mathrm{d}}}{K_{\mathrm{k}}T_{\mathrm{m}}T_{\mathrm{e}}}} =\omega_1\sqrt{\frac{K_{\mathrm{d}}}{K_{\mathrm{k}}}} \tag{11} \]

and

\[ M(\omega_2)\simeq \frac{K_{\mathrm{p}}K_{\mathrm{k}}^2T_{\mathrm{m}}}{K_{\mathrm{d}}T_{\mathrm{MU}}}. \tag{12} \]

The relations found, (5), (6), (9), (10), (11), (12), make it possible quickly to find the characteristic points of the key-shaped AFChX of the system, without resorting to its exact calculation and construction.

As a result of the action of disturbances and saturation of the converter, its gain coefficient \(K_{\mathrm{p}}\) decreases [1]; in this case the AFChX of the system in Fig. 2 contracts toward the origin. If the action of disturbances leads to a decrease of \(K_{\mathrm{p}}\) by \(M(\omega_2)\) times, then the AFChX of the open-loop system will encompass the critical point with coordinates \(-1, j0\), and self-oscillations will arise in the system with a frequency close to \(\omega_1\). Such a visual representation should be used in designing drives when it is necessary to check whether the parameters of the automatic-control system, chosen on the basis of linear considerations, satisfy the condition of absence of nonlinear self-oscillations due to the action of disturbances and saturation of the system. In this case the initial value of the modulus \(M(\omega_2)\) of the AFChX of the system is equivalent to a kind of initial margin of disturbance immunity of the automatic-control system. In practice \(M(\omega_2)=2 \div 6\), that is, it is very small.

In [2] a simple method is proposed for eliminating possible self-oscillations in similar systems, if the decrease in the gain coefficient of the automatic-control system occurs as a result of saturation of the converter at sufficiently large amplitudes of the input signal arriving through the tachometric-feedback circuit. In this case, in the system between the summing points of the signals of the main feedback and the correction circuit, a nonlinear element of the saturation type, NE, is inserted (Fig. 3), limiting the amplitude of the input signals of the system along the external loop by the value \(X_m\), which is sufficient for normal operation of the drive, i.e., \(X_m > X_{\mathrm{nom}}\), Fig. 3, but at the same time excludes the possibility of reducing \(M(\omega_2)\) to a value less than 1.

Fig. 3. Connection diagram of the nonlinear element.

However, this method, as applied to MUS-D systems, only partially eliminates the possibility of self-oscillations arising, since in these systems a high level of disturbances is transmitted by the correction circuit [1]. Therefore, the use of other stabilization methods deserves attention, for example electromagnetic [1], characterized by a lower level of disturbances.

LITERATURE

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

2. V. G. Kagan. The influence of saturations on self-oscillations of high-accuracy systems. Automation and Telemechanics, vol. XXVI, No. 1, 1965.

3. M. V. Meerov. Synthesis of structures of automatic-control systems of high accuracy. Fizmatgiz, 1959.