On the Structural Analysis of MUS-D Systems with PPU

V. A. Sevastyanov, A. P. Ineshin

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

In [1] it was shown that the static and dynamic characteristics of deeply regulated electric drives using the MUS-D system over a range of more than \(1:100\) depend, in particular, on their method of stabilization. It was also noted that, for the stabilization of such systems, the most noise-immune electromagnetic variant of correction deserves attention, in which a free control winding of the MUS is used to obtain the correcting signal. Below is a comparative analysis of possible structural solutions for the coordinate from which the correction signal is taken.

Fig. 1.

Figure 1 shows structural diagrams of the electric drives of MUS-D:

a — for the generally accepted method of taking the stabilizing signal in the form of a flexible negative feedback with respect to the armature voltage of the electric motor (RC circuits),

b — for the electromagnetic variant of correction used in [2].

In principle, these structures differ in the coordinate from which the differentiating signal is taken, which, owing to the presence in the MUS of a constant delay \(\tau\), leads to different degrees of stability of the internal closed loop of the SAR, to different speed of response and quality of the transient processes. Let us determine the influence of \(\tau\) on the dynamic indices of the SAR for the structural solutions shown in Fig. 1.

a) Influence of \(\tau\) on the stability of operation of the internal loop.

The transfer function of the closed internal loop of the SAR, in accordance with the structure of Fig. 1, a, may be represented in the form:

\[ W(p)= \frac{K_{\text{PPU}}K_{\text{MU}}(T_kp+1)e^{-p\tau}} {(T_{\text{MU}}p+1)(T_kp+1)+K_{\text{PPU}}K_{\text{MU}}K_kT_kpe^{-p\tau}} . \tag{1} \]

Here: \(K_{\text{MU}}, K_{\text{PPU}}\) are the gain coefficients of the power magnetic (MUS) and intermediate semiconductor (PPU) amplifiers;

\[ T_{\text{MU}},\ T_k \]
are the time constants of the MUS and of the correction circuit;

\[ K_k \]
is the transmission coefficient of the correction circuit;

\[ p \]
is the differentiation operator \(p\);

\[ e \]
is the base of natural logarithms.

The transfer function of the closed internal loop of the SAR of Fig. 1, b, reduced to equivalent stabilization conditions in comparison with the circuit of Fig. 1, a (an element with transfer coefficient \(K_{\text{MU}}\) is connected in series into the correction circuit), differs from (1) by the absence of the factor \(e^{-p\tau}\).

The \(D\)-partition equation with respect to the parameter \(K_{\text{PPU}}\) in accordance with (1) will be:

\[ K_{\text{PPU}}(j\omega)= -\frac{T_{\text{MU}}+T_k}{K_{\text{MU}}T_k}e^{j\omega\tau} -j\frac{T_{\text{MU}}T_k\omega-\frac{1}{\omega}}{K_{\text{MU}}T_k}e^{j\omega\tau}, \tag{2} \]

where, by Euler’s formula,

\[ e^{j\omega\tau}=\cos\omega\tau+j\sin\omega\tau . \]

After substituting \(e^{j\omega\tau}\) into (2) and transformations, we finally obtain:

\[ K_{\text{PPU}}(j\omega)= \frac{ \left(T_{\text{MU}}T_k\omega-\frac{1}{\omega}\right)\sin\omega\tau -(T_{\text{MU}}+T_k)\cos\omega\tau } {K_{\text{MU}}K_kT_k} - \]

\[ -j \frac{ \left(T_{\text{MU}}T_k-\frac{1}{\omega}\right)\cos\omega\tau +(T_{\text{MU}}+T_k)\sin\omega\tau } {K_{\text{MU}}K_kT_k}. \tag{3} \]

For variant “b” in (3) it is necessary to set \(\tau=0\); then

\[ K_{\text{PPU}}(j\omega)= -\frac{T_{\text{MU}}+T_k}{K_{\text{MU}}K_kT_k} -j\frac{T_{\text{MU}}T_k\omega-\frac{1}{\omega}}{K_{\text{MU}}K_kT_k}. \tag{4} \]

Figure 2 shows the \(D\)-partition curves calculated from equations (3) and (4) for the adopted parameter values:

\[ T_{\text{MU}}=0.45\ \text{s};\quad T_k=0.1\ \text{s};\quad K_k=1;\quad K_{\text{MU}}=20. \]

From the analysis of equations (3), (4) and of the curves it follows that, for \(\omega=0\), \(K_{\text{ppu}}\) does not depend on \(\tau\); the \(D\)-decomposition curves emerge from one characteristic point:

Fig. 2.

\[ K_{\text{ppu}}(0)=-\frac{T_{\text{mu}}+T_{\text{k}}}{K_{\text{mu}}K_{\text{k}}T_{\text{k}}}, \]

that is, at low frequencies the systems behave approximately identically.

At medium and high frequencies the inner loop of the automatic-control system in variant \(b\) \((\tau=0)\) does not depend on \(\tau\) and is stable for any \(K_{\text{ppu}}\), whereas in variant \(a\) \((\tau\ne0)\) the stability region is limited by the value \(K_{\text{ppu}}=36.4\), which reduces the range of speed regulation.

Since \(K_{\text{ppu}}\) is a real number and \(K_{\text{mu}}K_{\text{k}}T_{\text{k}}\ne\infty\), its value could in principle be obtained from relation (5):

\[ R(\omega)=K_{\text{ppu}}(\omega)= \frac{\left(T_{\text{mu}}T_{\text{k}}\omega-\frac{1}{\omega}\right)\sin\omega\tau -\left(T_{\text{mu}}+T_{\text{k}}\right)\cos\omega\tau} {K_{\text{mu}}K_{\text{k}}T_{\text{k}}} \tag{5} \]

after substituting into it \(\omega\), found from the condition

\[ \left(T_{\text{mu}}T_{\text{k}}-\frac{1}{\omega}\right)\cos\omega\tau+ \left(T_{\text{mu}}+T_{\text{k}}\right)\sin\omega\tau=0. \]

However, it is impossible to find \(\omega\) from the latter relation in explicit form; therefore one has to restrict oneself to a graphical construction of the \(D\)-decomposition curve and to finding from it \(K_{\text{ppu}}(\cdot)=R(\cdot)\).

Figure 3 presents frequency characteristics (curves 1 and 2), calculated according to equation (6) for the open inner loop of the automatic-control system in Fig. 1 \(a,b\), for \(\tau=0\), \(\tau=0.01\) sec and the same other conditions.

\[ W_{\text{p}}(j\omega)= \frac{K_{\text{ppu}}K_{\text{mu}}e^{-j\omega\tau}}{T_{\text{mu}}j\omega+1}\cdot \frac{K_{\text{k}}T_{\text{k}}j\omega}{T_{\text{k}}j\omega+1}. \tag{6} \]

From comparison of the characteristics it follows that the inner loop of the automatic-control system shown in Fig. 1, \(b\), has a large stability margin in modulus and phase, which is explained by the absence of the influence of delay.

In the general case the intermediate amplifier may have a time constant \(T_{\text{pu}}\).

Then the amplitude-phase frequency characteristics for the open inner loop of the automatic-control system of Fig. 1, \(a\), will be:

\[ W_{\text{p}}(j\omega)= \frac{K_{\text{pu}}}{T_{\text{pu}}j\omega+1}\cdot \frac{K_{\text{mu}}e^{-j\omega\tau}}{T_{\text{mu}}j\omega+1}\cdot \frac{K_{\text{k}}j\omega}{T_{\text{k}}j\omega+1}. \tag{7} \]

Figure 3 presents the amplitude-phase frequency characteristics calculated according to equation (7) for \(\tau=0\) (curve 3) and \(\tau=0.01\) sec (curve 4).

From comparison of these characteristics it follows that, when an inertial intermediate amplifier is used, the automatic-control system of Fig. 1, \(b\), because of the absence of delay \(\tau\) in it, has a more stable inner loop.

This advantageously distinguishes it from the circuit in Fig. 1, a, from the standpoint of circuit unification (attempts to use an intermediate magnetic amplifier).

b) The effect of the delay \(\tau\) on the stability of a closed, uncompensated automatic-control system under an external loop.

The \(D\)-partition equations with respect to the parameter \(K_{\text{PPU}}\) for the external loop of an uncompensated closed automatic-control system, in accordance with the structural diagram of Fig. 1, a, can be represented in the form:

\[ K_{\text{PPU}}(j\omega)= \frac{\omega (a_0\omega^2-a_2)\sin \omega\tau+(a_1\omega^2-1)\cos \omega\tau}{K} - j\frac{\omega (a_0\omega^2-a_2)\cos \omega\tau-(a_1\omega^2-1)\sin \omega\tau}{K}, \tag{8} \]

where:

\[ \begin{aligned} a_0&=T_{\text{MU}}T_{\text{E}}T_{\text{M}};\\ a_1&=T_{\text{MU}}T_{\text{M}}+T_{\text{E}}T_{\text{M}};\\ a_2&=T_{\text{MU}}+T_{\text{M}}+T_{\text{E}};\\ K&=K_{\text{KE}}K_{\text{MU}}K_{\text{D}}\gamma. \end{aligned} \]

Figure 4 shows the \(D\)-partition curves constructed from equation (8).

Fig. 3.

Fig. 4.

From analysis of the curves and expression (8) it follows that, for \(\omega=0\),

\[ K_{\text{PPU}}(0)=\frac{a_1\omega^2-1}{K}=\text{const} \]

and does not depend on the delay \(\tau\); that is, at low frequencies the dynamic behavior of the systems is approximately the same. For \(\omega\ne 0\), because in the closed uncompensated automatic-control system the motor usually operates at a frequency close to the resonant one,

\[ \omega=\frac{1}{\sqrt{T_{\text{E}}T_{\text{M}}}}, \]

which varies within the limits \((15\div16)\ \frac{1}{\text{sec}}\), the delay \(\tau\) has practically no effect on the stability of operation of the automatic-control system, which is confirmed by the coincidence of the \(D\)-partition curves in Fig. 4. In this case the stability region is bounded by \(K_{\text{PPU}}=1.8\), which corresponds to \(D\approx12.7\).

c) The effect of the delay \(\tau\) on the stability of operation of a closed, compensated automatic-control system under an external loop.

The \(D\)-partition equations with respect to the parameter \(K_{\text{PPU}}\) for the external loop of a compensated closed automatic-control system, in accordance with the structure of Fig.

1, and taking into account the delay \(\tau\), it can be obtained from the characteristic equation (9) after substituting \(p=j\omega\) into it, which for variant \(a\) can be represented in the form:

\[ a_0'(j\omega)^4+a_1'(j\omega)^3+a_2'(j\omega)^2+a_3'(j\omega)+a_4'=0, \tag{9} \]

where

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

\[ a_1'=T_{\mathrm{mu}}T_{\mathrm{k}}T_{\mathrm{m}} +T_{\mathrm{mu}}T_{\mathrm{e}}T_{\mathrm{m}} +T_{\mathrm{k}}T_{\mathrm{e}}T_{\mathrm{m}} +\frac{R_{\mathrm{ya}}}{R_{\mathrm{e}}}K_{\mathrm{ppu}}K_{\mathrm{mu}}T_{\mathrm{k}}T_{\mathrm{e}}T_{\mathrm{m}}e^{-j\omega\tau}, \]

\[ a_2'=T_{\mathrm{mu}}T_{\mathrm{m}} +T_{\mathrm{e}}T_{\mathrm{m}} +T_{\mathrm{k}}T_{\mathrm{m}} +T_{\mathrm{mu}}T_{\mathrm{k}} +\frac{R_{\mathrm{ya}}}{R_{\mathrm{e}}}K_{\mathrm{ppu}}K_{\mathrm{mu}}K_{\mathrm{k}}T_{\mathrm{k}}T_{\mathrm{m}}e^{-j\omega\tau}, \]

\[ a_3'=T_{\mathrm{mu}}+T_{\mathrm{k}}+T_{\mathrm{m}} +K_{\mathrm{ppu}}K_{\mathrm{mu}}K_{\mathrm{k}}T_{\mathrm{k}}e^{-j\omega\tau} +K_{\mathrm{ppu}}K_{\mathrm{mu}}K_{\mathrm{d}}K_{\mathrm{esc}}T_{\mathrm{f}}e^{-j\omega\tau}, \]

\[ a_4'=1+K_{\mathrm{ppu}}K_{\mathrm{d}}K_{\mathrm{esc}}\gamma e^{-j\omega\tau}. \]

The equation for variant \(b\) has an analogous form, but differs by the absence of the factor \(e^{-j\omega\tau}\) in the coefficients \(b_1\) and \(b_2\)

\[ b_0(j\omega)^4+b_1(j\omega)^3+b_2(j\omega)^2+b_3(j\omega)+b_4=0. \tag{10} \]

After substituting the numerical values of the parameters and transformations, we obtain respectively for variant “a”:

\[ K_{\mathrm{ppu}}(j\omega)=\frac{A(\omega)}{B(\omega)}-j\frac{C(\omega)}{B(\omega)}, \tag{11} \]

for variant “b”:

\[ K_{\mathrm{ppu}}(j\omega)=\frac{D(\omega)}{E(\omega)}+j\frac{G(\omega)}{E(\omega)}, \tag{12} \]

where \(A(\omega)\), \(B(\omega)\), \(C(\omega)\), \(D(\omega)\), \(G(\omega)\), \(E(\omega)\) are complex functions of the frequency \(\omega\) and depend on the delay.

Figure 5 shows the D-partition curves calculated and constructed from equations (11) and (12) for the adopted parameters:

\[ T_{\mathrm{mu}}=0.45\ \mathrm{sec};\quad T_{\mathrm{e}}=0.02\ \mathrm{sec};\quad T_{\mathrm{m}}=0.2\ \mathrm{sec};\quad \tau=0.01\ \mathrm{sec};\quad T_{\mathrm{k}}=0.01\ \mathrm{sec}; \]

\[ K_{\mathrm{mu}}=20;\quad K_{\mathrm{k}}=1;\quad R_{\mathrm{ya}}=1.36\ \Omega; \]

\[ R_{\mathrm{e}}=12\ \Omega;\quad K_{\mathrm{d}}=10;\quad \gamma=0.067. \]

From comparison of the curves it is evident that the stability region of operation

Fig. 5.

Fig. 6.

the ACS of Fig. 1, a is limited by the value \(K_{\text{PPU}}=140\), which corresponds to the speed-control range \(D \approx 75\), whereas in the ACS variant of Fig. 1, b it is practically unlimited, since the stability zone is determined by the intersection of the asymptotes of the \(D\)-partition curves at the point \(K_{\text{PPU}}=560\), Fig. 5 (dashed line).

The speed of response of the ACSs considered and the quality of regulation can be estimated from the real frequency characteristics, which, according to the method of [3], are constructed from the obtained \(D\)-partition curves and are shown in Fig. 6.

From a comparison of the characteristics \(R(\omega)\) for the ACSs considered it follows:

1. The transient processes of the ACSs considered are not monotonic and are distinguished by a certain oscillatory character, since their \(R(\omega)\) have clearly expressed maxima, respectively at the frequencies \(\omega=12\) and \(38\ \frac{1}{\text{s}}\) (curve 1), and \(\omega=8\) and \(85\ \frac{1}{\text{s}}\) (curve 2), which is characteristic of MUS-D systems. However, the absence of discontinuities of nonuniformity and of a tendency of \(R(\omega)\) to infinity indicates a quite satisfactory character of the transient processes, the absence of undamped harmonic oscillations, and the stability of operation and operability of the drive as a whole.

2. The characteristic \(R(\omega)\) of the ACS variant of Fig. 1, b (curve 2) is distinguished by more uniform damping, which indicates a lower oscillatory character and greater monotonicity in the course of the transient process.

3. Since the interval of positivity \(\omega_{\text{pb}}=45\) (variant b) is greater than \(\omega_{\text{pa}}=30\) (variant a) and \(R_b(\omega)\) is “wider” than \(R_a(\omega)\), the time of the transient process is

\[ t_{\text{pb}}=\frac{\pi}{45}<t_{\text{pa}}=\frac{\pi}{30}, \]

that is, variant (b), in view of the smaller influence of the delay \(\tau\), is the faster-acting one.

Conclusions

1. A constant delay \(\tau\) has a practical effect on the control range, the stability of operation of the drive, and the speed of response only of a corrected ACS, which depends, in particular, on the circuit solutions: the point at which the correction signal is taken off and its type.

2. The uncorrected MUS-D system, because of its low static and dynamic properties, can be used in an electric drive only with a small speed-control range.

3. The implementation of a structure with a parallel correcting RC circuit for a control range of more than \(1:100\) is difficult because of poor noise immunity and the contradiction between the condition of stability of the internal feedback loop of the converter and the dynamics of the system.

References

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

2. V. A. Sevastyanov, A. P. Ineshin, A. P. Rybakova. “Development and study of the MUS-D electric drive with a class-D PPU.” Proceedings of TPI, vol. 153, 1965.

3. M. V. Meerov. Synthesis of Structures of High-Accuracy Automatic Control Systems. Fizmatgiz, 1959.