ON THE STUDY OF THE DYNAMICS OF VALVE ELECTRIC DRIVES BASED ON FULLY CONTROLLED CONVERTERS

A. I. Zaitsev, V. N. Mishin

Recommended by the scientific seminar of the Electromechanical Faculty

In view of the nonlinearity of the characteristics of a valve electric drive in the region of discontinuous currents, it is advisable to study the dynamics in two stages: analysis of the behavior of the linearized system for small perturbations, and study in the large. In the present work, the linearization of the characteristics of a drive operating in various discontinuous-current modes is considered for the purpose of studying the dynamics of systems for small perturbations.

The calculations are carried out under the assumption that the motor parameters \(R\), \(L\), and the magnetic flux \(\Phi\) are constant; the voltage drops across the valves of the converter \(E_{\mathrm{B}}\) are equal and do not depend on the magnitude of the flowing current; and the system is supplied from an absolutely rigid network of infinite power.

Let us denote:

\(U_m\) — amplitude of the phase voltage of the network;
\(\omega\) — angular frequency of the network;
\(m\) — number of converter phases;
\(E\) — motor e.m.f.;

\[ \varepsilon=\frac{E \pm E_{\mathrm{B}}}{U_m} \]
— relative e.m.f.;

\[ j=\frac{iR}{U_m} \]
— relative current;

\(\gamma_{\mathrm{B}}, \lambda\) — firing angle of the power valves and duration of conduction, respectively;

\[ \tg \Theta=\omega\frac{L}{R} \]
— motor parameter.

Under the accepted assumptions, the current in relative units corresponds to the relative electromagnetic torque of the motor, while the speed is determined by the value of the relative e.m.f. Thus, the averaged mechanical characteristic in relative units is specified by the function \(\varepsilon=f(j_{\mathrm{av}})\). The transition from relative speeds and torque to absolute quantities is made on the basis of the following formulas

\[ n=(\varepsilon-\varepsilon_{\mathrm{B}})n_0=(\varepsilon-\varepsilon_{\mathrm{B}})\frac{U_m}{C_e\Phi}, \tag{1} \]

\[ M=j_{\mathrm{cp}}M_0=j_{\mathrm{cp}}\frac{U_m}{R}C_M\Phi, \tag{2} \]

where \(\varepsilon_{\mathrm{B}}=\dfrac{E_{\mathrm{B}}}{U_m}\) is the relative voltage drop across the valves;

\(C_e, C_M\) are proportionality coefficients.

As shown in [1], the transient processes of a valve electric drive can be calculated with a sufficiently high degree of accuracy from averaged mechanical characteristics. In the general case, for controlled systems with zero valves, the mechanical characteristics are a nonlinear function \(\varepsilon=f(\gamma_{\mathrm{B}}, \lambda, j_{\mathrm{cp}})\); expanding it in a Taylor series and discarding the nonlinear remainder, we obtain the linearized equation in deviations:

\[ \varepsilon=\varepsilon_0+k_\gamma\Delta\gamma_{\mathrm{B}}+k_\lambda\Delta\lambda+k_j\Delta j, \tag{3} \]

where

\[ k_\gamma=\frac{\partial \varepsilon}{\partial \gamma_{\mathrm{B}}},\qquad k_\lambda=\frac{\partial \varepsilon}{\partial \lambda},\qquad k_j=\frac{\partial \varepsilon}{\partial j_{\mathrm{cp}}} \tag{4} \]

are, respectively, the slope coefficients of the regulating characteristics \(\varepsilon=f(\gamma_{\mathrm{B}})\) and \(\varepsilon=f(\lambda)\) at \(j_{\mathrm{cp}}=\mathrm{const}\), and of the mechanical characteristic \(\varepsilon=f(j_{\mathrm{cp}})\) at \(\gamma_{\mathrm{B}}=\mathrm{const}\), \(\lambda=\mathrm{const}\); at the operating point \(\gamma_{\mathrm{B}}=\gamma_{\mathrm{B}0}\), \(\lambda=\lambda_0\), \(j_{\mathrm{cp}}=j_{\mathrm{cp}0}\).

The values of the coefficients of the linearized mechanical characteristic can be determined graphically directly from the characteristics or found by means of the corresponding formulas.

Equation (3), in absolute units, has the form

\[ n=n_*+k_\gamma'\Delta\gamma_{\mathrm{B}}+k_\lambda'\Delta\lambda-k_M\Delta M, \tag{5} \]

where:

\[ k_\gamma'=\frac{\partial n}{\partial \gamma_{\mathrm{B}}} =k_\gamma\frac{U_m}{C_e\Phi}=k_\gamma n_0; \tag{6} \]

\[ k_\lambda'=\frac{\partial n}{\partial \lambda}=k_\lambda n_0; \tag{7} \]

\[ k_M=\frac{\partial n}{\partial M}=-k_j\frac{R}{C_eC_M\Phi^2}. \tag{8} \]

In accordance with expressions (4), let us calculate formulas for determining the coefficients of the mechanical characteristic of a motor supplied from a fully controlled converter in various operating modes.

1. The motor current decreases to zero, flowing through the power valves. The mechanical characteristic in relative units is here specified in parametric form by the equation for the average current [1].

\[ j_{\mathrm{cp}}=\frac{m}{\pi}\sin\left(\gamma_{\mathrm{B}}+\frac{\lambda}{2}\right)\sin\frac{\lambda}{2} -\frac{m}{2\pi}\lambda\varepsilon \tag{9} \]

or by the equation for the counter-emf [1]:

\[ \varepsilon=\cos\Theta\, \frac{\sin(\gamma_{\mathrm{B}}-\Theta)-\sin(\gamma_{\mathrm{B}}-\Theta+\lambda)e^{\lambda\operatorname{ctg}\Theta}} {1-e^{\lambda\operatorname{ctg}\Theta}}. \tag{10} \]

The expression for the slope coefficient of the regulating characteristic \(\varepsilon=f(\gamma_{\mathrm{B}})\) is found from (10) [1]:

\[ k_\gamma=\frac{\cos\Theta}{1-e^{\lambda\operatorname{ctg}\Theta}} \left[\cos(\gamma_{\mathrm{B}}-\Theta)-\cos(\gamma_{\mathrm{B}}-\Theta+\lambda)e^{\lambda\operatorname{ctg}\Theta}\right]. \tag{11} \]

The slope coefficient at \(\Delta\lambda\) is equal to zero, since in the considered-

mode, the duration of switching-on of the power valves cannot be an independent variable.

The angular coefficient of the mechanical characteristic is found from (9), after first determining from (10) the partial derivative \(\frac{\partial \varepsilon}{\partial \lambda}\), and finally we obtain

\[ k_j=\frac{-2\pi}{m\{\lambda-(e^{-\lambda \operatorname{ctg}\Theta}-1)\operatorname{tg}\Theta\}} . \tag{12} \]

For the mode under consideration, uncontrolled operation of the converter is characteristic; this occurs if the command to switch on the power valve arrives when its anode voltage is less than the counter-emf in the dc circuit. In this case the actual switching-on angle of the valves is found as

\[ \gamma_{\mathrm{B}}=\arcsin \varepsilon, \tag{13} \]

the coefficients of the control characteristics are equal to zero, and the angular coefficient of the mechanical characteristic is equal to

\[ k_j=\frac{2\pi}{m}\sqrt{1-\varepsilon^2}:\left\{\sqrt{1-\varepsilon^2}\left[(1-e^{-\lambda \operatorname{ctg}\Theta})\operatorname{tg}\Theta-\lambda\right]+\right. \]

\[ \left.+\sin\Theta\left[e^{-\lambda \operatorname{ctg}\Theta}\cos(\arcsin\varepsilon-\Theta)-\cos(\arcsin\varepsilon-\Theta+\lambda)\right]+\right. \]

\[ \left.+2\sin\frac{\lambda}{2}\cos\left(\operatorname{ars}\sin\varepsilon+\frac{\lambda}{2}\right)\right\}. \tag{14} \]

1. The current decreases to zero while flowing through the zero valve.

In this mode the mechanical characteristics are determined by equations [2], [3].

\[ j_{\mathrm{cp}}=\frac{m}{\pi}\sin\frac{\lambda}{2}\sin\left(\gamma_{\mathrm{B}}+\frac{\lambda}{2}\right)-\frac{m}{2\pi}(\lambda+\lambda_{\mathrm{p}})\varepsilon, \tag{15} \]

\[ \varepsilon=\cos\Theta\, \frac{e^{\lambda \operatorname{ctg}\Theta}\sin(\gamma_{\mathrm{B}}-\Theta+\lambda)-\sin(\gamma_{\mathrm{B}}-\Theta)} {e^{(\lambda+\lambda_{\mathrm{p}})\operatorname{ctg}\Theta}-1}, \tag{16} \]

where \(\lambda_{\mathrm{p}}\) is the duration of operation of the zero valve.

By analogy with the above, the angular coefficients of the drive characteristics in accordance with (4) are found with the aid of equations (14 and (15).

\[ k_v=\cos\Theta\, \frac{\cos(\gamma_{\mathrm{B}}-\Theta)-\cos(\gamma_{\mathrm{B}}-\Theta+\lambda)e^{\lambda \operatorname{ctg}\Theta}} {1-e^{(\lambda+\lambda_{\mathrm{p}})\operatorname{ctg}\Theta}}, \tag{17} \]

\[ k_\lambda= \frac{\operatorname{ctg}\Theta}{1-e^{-(\lambda+\lambda_{\mathrm{p}})\operatorname{ctg}\Theta}} \left[e^{-\lambda_{\mathrm{p}}\operatorname{ctg}\Theta}\sin(\gamma_{\mathrm{B}}+\lambda)-\varepsilon\right], \tag{18} \]

\[ k_j=\frac{2\pi} {m\{[1-e^{-(\lambda+\lambda_{\mathrm{p}})\operatorname{ctg}\Theta}]\operatorname{tg}\Theta-(\lambda+\lambda_{\mathrm{p}})\}} . \tag{19} \]

In the discontinuous-current mode under consideration, a case is also possible in which the actual switching-on angles of the valves correspond to expression (13); then the actual angles of the duration of switching-on of the power valves can be found through those specified by the control circuit as

\[ \lambda'=\gamma_{\mathrm{B}}+\lambda-\arcsin\varepsilon. \tag{20} \]

Replacing \(\lambda\) in formula (16) by \(\lambda'\) from (20) and differentiating the expression obtained according to (4), we obtain the coefficients of the control characteristics.

\[ k_{\nu}=k_{\lambda}= \left[\varepsilon e^{\lambda_{\mathrm{p}}\operatorname{ctg}\Theta} -\sin(\nu_{\mathrm{B}}+\lambda)\right]: \left\{\left[e^{-\lambda'\operatorname{ctg}\Theta} -e^{\lambda_{\mathrm{p}}\operatorname{ctg}\Theta}\right]\operatorname{tg}\Theta -\frac{1}{\sqrt{1-\varepsilon^{2}}}\left[ e^{-\lambda'\operatorname{ctg}\Theta}\sin\Theta\cos(\arcsin\varepsilon-\Theta)+ \cos\Theta\sin(\nu_{\mathrm{B}}-\Theta+\lambda) -\varepsilon e^{\lambda_{\mathrm{p}}\operatorname{ctg}\Theta} \right]\right\}, \tag{21} \]

which prove to be equal because of the equivalence, in this case, of speed regulation by changing the firing angles or the duration of conduction, since either method ultimately leads to a change in the value of the actual conduction duration \(\lambda'\). Taking (20) into account, we find, with the aid of (15) and (16), the angular coefficient of the mechanical characteristics

\[ k_j=\frac{2\pi}{m}: \left\{\left[1-e^{-(\lambda'+\lambda_{\mathrm{p}})\operatorname{ctg}\Theta}\right]\operatorname{tg}\Theta +\frac{e^{-\lambda_{\mathrm{p}}\operatorname{ctg}\Theta}}{\sqrt{1-\varepsilon^{2}}} \left[e^{-\lambda'\operatorname{ctg}\Theta}\sin\Theta \cos(\arcsin\varepsilon-\Theta)+ \cos\Theta\sin(\nu_{\mathrm{B}}-\Theta+\lambda) -\varepsilon e^{\lambda\operatorname{ctg}\Theta} \right] -(\lambda'+\lambda_{\mathrm{p}})\right\}/m . \tag{22} \]

3. Regime of discontinuous currents falling to zero during operation of the power valves in the range of angles
\[ \frac{\pi}{2}>\nu_{\mathrm{p}}<\arc\sin\varepsilon . \]

The last regime of discontinuous currents is characterized by double operation of each power valve in the cycle [2], [3].

The mechanical characteristics here are specified by the equations [3]

\[ j_{\mathrm{cp}}=\frac{m}{\pi}\left[ \sin\!\left(\nu_{\mathrm{B}}+\frac{\lambda_1}{2}\right)\sin\frac{\lambda_1}{2} +\sin\frac{\nu_{\mathrm{B}}+\lambda-\nu_{\mathrm{B}2}}{2} \sin\frac{\nu_{\mathrm{B}}+\lambda+\nu_{\mathrm{B}2}}{2} \right] -\frac{m}{2\pi}\left(\frac{2\pi}{m}-\nu_{\mathrm{B}2}+\nu_{\mathrm{B}}+\lambda_1\right)\varepsilon, \tag{23} \]

\[ \varepsilon=\cos\Theta\, \frac{ e^{\lambda_1\operatorname{ctg}\Theta}\sin(\nu_{\mathrm{B}}-\Theta+\lambda_1) -\sin(\nu_{\mathrm{B}}-\Theta) +\sin(\nu_{\mathrm{B}}-\Theta+\lambda)e^{-\left(\frac{2\pi}{m}-\lambda\right)\operatorname{ctg}\Theta} -\sin(\nu_{\mathrm{B}2}-\Theta)e^{-\left(\frac{2\pi}{m}+\nu_{\mathrm{B}}-\nu_{\mathrm{B}2}\right)\operatorname{ctg}\Theta} }{ e^{\lambda_1\operatorname{ctg}\Theta} -e^{-\left(\frac{2\pi}{m}+\nu_{\mathrm{B}}-\nu_{\mathrm{B}2}\right)\operatorname{ctg}\Theta} }, \tag{24} \]

where \(\lambda\) and \(\lambda_1\) are the specified conduction duration and the duration of the first conduction of the power valves, respectively;

\[ \nu_{\mathrm{B}2}=\arc\sin\varepsilon \]
is the phase angle of the second firing.

The angular coefficients of the characteristics are found from (23)—(24) in accordance with (4)

\[ k_{\nu}=\cos\Theta\left\{ e^{\lambda_1\operatorname{ctg}\Theta}\cos(\nu_{\mathrm{B}}-\Theta+\lambda_1) -\cos(\nu_{\mathrm{B}}-\Theta) +\cos(\nu_{\mathrm{B}}-\Theta+\lambda)e^{-\left(\frac{2\pi}{m}-\lambda\right)\operatorname{ctg}\Theta} +\left[\sin(\nu_{\mathrm{B}2}-\Theta)\operatorname{ctg}\Theta -\frac{\varepsilon}{\sin\Theta}\right] e^{-\left(\frac{2\pi}{m}+\nu_{\mathrm{B}}-\nu_{\mathrm{B}2}\right)\operatorname{ctg}\Theta} \right\}: \left[ e^{\lambda\operatorname{ctg}\Theta} -e^{-\left(\frac{2\pi}{m}+\nu_{\mathrm{B}}-\nu_{\mathrm{B}2}\right)\operatorname{ctg}\Theta} \right]. \tag{25} \]

\[ k_{\lambda}= \frac{\operatorname{ctg}\Theta\,\sin(\nu_{\mathrm{B}}+\lambda)} { e^{\left(\frac{2\pi}{m}+\lambda_1-\lambda\right)\operatorname{ctg}\Theta} -e^{(\nu_{\mathrm{B}}+\lambda-\nu_{\mathrm{B}2})\operatorname{ctg}\Theta} }, \tag{26} \]

\[ k_j= \frac{2\pi} { m\left\{ \left[1-e^{-\left(\frac{2\pi}{m}+\nu_{\mathrm{B}}-\nu_{\mathrm{B}2}+\lambda_1\right)\operatorname{ctg}\Theta}\right]\operatorname{tg}\Theta +\nu_{\mathrm{B}2}-\lambda_1-\frac{2\pi}{m} \right\} }. \tag{27} \]

4. Continuous-current mode

In the region of continuous current, the mechanical characteristics of the drive are straight lines and have the form [1]

\[ j_{\mathrm{cp}}=\frac{m}{\pi}\sin\frac{\lambda}{2}\sin\left(\vartheta_{\mathrm{B}}+\frac{\lambda}{2}\right)-e. \tag{28} \]

Taking the corresponding partial derivatives, from (28) we obtain the angular coefficients in the form

\[ k_{\vartheta}=\frac{m}{\pi}\sin\frac{\lambda}{2}\cos\left(\vartheta_{\mathrm{B}}+\frac{\lambda}{2}\right), \tag{29} \]

\[ k_{\lambda}=\frac{m}{2\pi}\sin(\vartheta_{\mathrm{B}}+\lambda). \tag{30} \]

\[ k_j=-1. \tag{31} \]

In the region of continuous current, the slope of the mechanical characteristics of a separately excited motor is proportional to the ohmic resistance of the armature circuit (8)

\[ k_{\mathrm{M}}=\frac{R}{C_e C_{\mathrm{M}}\Phi^2}. \tag{32} \]

The slope of the averaged characteristics in the region of discontinuous currents can also be estimated by means of a certain fictitious ohmic resistance [1]

\[ k_{\mathrm{M}}=\frac{R_{\Phi}}{C_e C_{\mathrm{M}}\Phi^2}, \tag{33} \]

which, for the given operating point, comparing (8) and (33), we express as

\[ R_{\Phi}=-k_jR. \tag{34} \]

The calculation of the dynamics on the basis of averaged mechanical characteristics presupposes finding the useful, so-called “smooth” component of the transient processes. In this case, in the region of discontinuous current it is not necessary to take into account the inductance of the armature circuit, since its influence is fully accounted for in calculating the averaged characteristics. The influence of the inductance is reflected by the fictitious ohmic resistance.

The dynamics of the averaged process in the region of discontinuous currents can be calculated on the basis of the averaged linearized mechanical characteristic (5) and the equation of motion of the drive in deviations

\[ \left. \begin{aligned} \Delta n &= k'_{\vartheta}\Delta\vartheta+k'_{\lambda}\Delta\lambda-k_{\mathrm{M}}\Delta M,\\ \Delta M &= I\,\frac{d(\Delta n)}{dt} \end{aligned} \right\}, \tag{35} \]

where \(I\) is the moment of inertia of the drive referred to the motor shaft. Solving system (35) with respect to the speed deviation, we obtain an equation characterizing the dynamics of the drive

\[ \Delta n=k'_{\vartheta}\Delta\vartheta+k'_{\lambda}\Delta\lambda-T_{\mathrm{M}}\frac{d(\Delta n)}{dt}, \tag{36} \]

where \(T_{\mathrm{M}}=Ik_{\mathrm{M}}\) is the electromechanical time constant.

In the general case, in systems with zero valves, control can be effected both by changing the firing angles and by changing the duration of conduction of the power valves. With sequential control, the transfer functions of the “rectifier–motor” element as a function of

the relationships from the regulated coordinate on the basis of (36) will have the form

\[ W(p)_v=\frac{\Delta n(p)}{\Delta v_{\mathrm{B}}(p)}=\frac{k'_v}{1+T_{\mathrm{M}}p}, \tag{37} \]

\[ W(p)_\lambda=\frac{\Delta n(p)}{\Delta \lambda(p)}=\frac{k'_\lambda}{1+T_{\mathrm{M}}p}. \tag{38} \]

When \(v_{\mathrm{B}}\) and \(\lambda\) vary simultaneously, it is more convenient to carry out the investigation by taking as the independent variable a certain generalized coordinate, for example, the voltage at the input of the control circuit \((\Delta U_y)\). Then, considering the control circuit to be inertialess, we rewrite (36) in the form

\[ (1+T_{\mathrm{M}}p)\Delta n=[k_{(v)}k'_v+k_{(\lambda)}k'_\lambda]\Delta U_y, \tag{39} \]

where

\[ k_{(v)}=\frac{\partial v_{\mathrm{B}}}{\partial U_y}, \qquad k_{(\lambda)}=\frac{\partial \lambda}{\partial U_y} \]

are the gain coefficients of the control circuit.

From (39) we obtain the transfer function as

\[ W(p)=\frac{\Delta n(p)}{\Delta U_y(p)}=\frac{k}{1+T_{\mathrm{M}}p}, \tag{40} \]

where \(k=k_{(v)}k'_v+k_{(\lambda)}k'_\lambda\) is the generalized gain coefficient of the links “control circuit—converter—motor.”

As follows from (37), (38), (40), the separately excited motor in intermittent-current modes is an aperiodic first-order element. The electromagnetic time constant does not enter into the expression for the transfer function in these modes, since the electromagnetic transient process ends with each current pulse, without in any way affecting the subsequent pulses.

In the region of continuous current, the transfer functions of the “rectifier—motor” element assume their usual form

\[ W(p)_v=\frac{k'_v}{T_{\mathrm{M}}T_{\mathrm{a}}p^2+T_{\mathrm{M}}p+1}, \tag{41} \]

\[ W(p)_\lambda=\frac{k'_\lambda}{T_{\mathrm{M}}T_{\mathrm{a}}p^2+T_{\mathrm{M}}p+1}, \tag{42} \]

\[ W(p)=\frac{k}{T_{\mathrm{M}}T_{\mathrm{a}}p^2+T_{\mathrm{M}}p+1}, \tag{43} \]

where

\[ T_{\mathrm{a}}=\frac{L}{R} \]

is the electromagnetic time constant of the armature circuit.

Having calculated, from the expressions obtained, the angular coefficients of the control and mechanical characteristics and the transfer functions in the neighborhoods of the selected operating point, one can proceed directly to the small-signal study of a specific system, using the generally known methods and procedures of the linear theory of automatic control.

REFERENCES

1. A. A. Bulgakov. Fundamentals of the Dynamics of Controlled Valve Systems. Publishing House of the Academy of Sciences of the USSR, 1963.

2. A. I. Zaitsev, V. N. Mishin. On the calculation of certain commutators using fully controlled elements. Izvestiya TPI, vol. 153, 1965.

3. A. I. Zaitsev, V. N. Mishin. Intermittent-current modes of fully controlled converters. Present volume.