Method for Calculating the Static Characteristics of Electric Drives with Thyristor Reversible Pulse-Width Converters with a Variable Commutation Law

V. A. Beinarovich, T. S. Udut, V. S. Fadeev, A. E. Deev

(Presented by the Scientific and Technical Seminar of the AEM Research Institute)

The basic expressions for calculating and constructing the static electromechanical \(\Omega(I_{\mathrm{я}})\) or mechanical \(\Omega(M)\) characteristics of DC electric drives with thyristor reversible pulse-width converters with a variable commutation law [1] are determined on the basis of an analysis of quasi-steady electromagnetic processes [2] in various operating modes (Fig. 1). The following assumptions are adopted: the valves are ideal, the parameters of the converter–motor circuit are constant, the resistance of the power supply source is equal to zero, and the motor emf does not change during the commutation period. The results of the analysis are conveniently represented in relative units (see the Appendix).

Fig. 1.

Electromechanical Characteristics of the Electric Drive

Symmetric commutation law

The initial expressions describing the electromagnetic processes over the time intervals \(\tau_1\) and \(\tau_2\) (Fig. 1a, c) have the form

\[ \pm 1 = \tau_{\mathrm{я}} \frac{d i_{\mathrm{я}}^*}{dt} + i_{\mathrm{я}}^* + e^* . \tag{1} \]

In the continuous-current zone (Fig. 1a), the electromechanical characteristics are determined by the expression

\[ \Omega = (2\tau_1 - 1) - I_{\mathrm{я}}^*, \tag{2} \]

while in the discontinuous-current zone (Fig. 1c) they are constructed by the simultaneous solution of the following equations:

\[ \Omega^* = E^* = 1 - \frac{2 I_{\mathrm{a}}^* \cdot \tau_{\mathrm{a}}}{\tau_1^2 + 2\tau_1 \cdot \tau_2 - \tau_2^2}; \tag{3} \]

\[ I_{\mathrm{a}}^* = \frac{1 - E^*}{2\tau_{\mathrm{a}}}\tau_1^2 + \frac{1 - E^*}{\tau_{\mathrm{a}}}\tau_1\tau_2 - \frac{1 + E^*}{2\tau_{\mathrm{a}}}\tau_2^2; \tag{4} \]

\[ \tau_2 = \tau_{\mathrm{a}} \ln \left[ 1 + \frac{1 - E^*}{1 + E^*} \left(1 - e^{-\frac{\tau_1}{\tau_2}}\right) \right]. \tag{5} \]

The boundary of the modes is determined by the following values of the limiting rotational speed and motor current:

\[ \Omega_{\mathrm{gr}1}^* = - \Omega_{\mathrm{gr}2}^* = 2\tau_1\left(1 + \frac{\tau_1 - 1}{2\tau_{\mathrm{a}} - 1}\right) - 1; \tag{6} \]

\[ I_{\mathrm{agr}1}^* = - I_{\mathrm{agr}2} = \frac{4\tau_1 - 2\tau_1^2 - 1}{2\tau_{\mathrm{a}}} - \frac{\Omega_{\mathrm{gr}1}^*}{2\tau_{\mathrm{a}}}. \tag{7} \]

Asymmetrical commutation law

The initial expressions for the electromagnetic processes in the armature circuit over the time intervals \(\tau_1\) and \(\tau_2\) (Fig. 1b, c) have the form

\[ 1 = \tau_{\mathrm{a}} i_{\mathrm{a}}^{*'} + i_{\mathrm{a}}^* + e^*; \]

\[ 0 = \tau_{\mathrm{a}} i_{\mathrm{a}}^{*'} + i_{\mathrm{a}}^* + e^*. \tag{8} \]

Then the electromechanical characteristics in the continuous-current zone (Fig. 1b) are described by the expression

\[ \Omega^* = \tau_1 - I_{\mathrm{a}}^*, \tag{9} \]

and in the discontinuous-current zone (Fig. 1c) they are determined by the simultaneous solution of the equations

\[ \Omega^* = E^* = \frac{(\tau_1^2 + 2\tau_1\tau_2) - 2 I_{\mathrm{a}}^* \tau_{\mathrm{a}}} {\tau_1^2 + 2\tau_1\tau_2 - \tau_2^2}; \tag{10} \]

\[ I_{\mathrm{a}}^* = \frac{1 - E^*}{2\tau_{\mathrm{a}}}\tau_1^2 + \frac{1 - E^*}{\tau_{\mathrm{a}}}\tau_1\tau_2 - \frac{E^*}{2\tau_{\mathrm{a}}}\tau_2^2; \tag{11} \]

\[ \tau_2 = \tau_{\mathrm{a}} \ln \left[ 1 + \frac{1 - E^*}{E^*} \left(1 - e^{-\frac{\tau_1}{\tau_2}}\right) \right]. \tag{12} \]

The boundary of the modes is determined by the following values of the limiting rotational speed and motor current:

\[ \Omega_{\mathrm{gr}1}^* = 1 - \Omega_{\mathrm{gr}2}^* = \frac{\tau_1(2\tau_{\mathrm{a}} - 2 + \tau_1)}{2\tau_{\mathrm{a}} - 1}; \tag{13} \]

\[ I_{\mathrm{gr}1}^* = - \tau_{\mathrm{gr}2}^* = \frac{\tau_1(1 - \tau_1)}{2\tau_{\mathrm{a}} - 1}. \tag{14} \]

Using the expressions obtained, the electromechanical characteristics of the open-loop electric-drive system RShIP-D with a variable commutation law at the numerical value \(\tau_{\mathrm{a}} = 5.5\) have been calculated and are shown in Fig. 5 (thin lines).

Control characteristics of the RShIP when operating

into a DC motor

It is expedient to represent the control characteristics of the converter in the form of dependences \(E_{\mathrm{pr}}^*(\tau_1)\) at \(I_{\mathrm{a}}^*=\mathrm{const}\), and the influence

of the internal resistance of the converter, take into account in the calculations the corresponding increase in the resistance of the motor armature circuit. The form of the characteristics depends on the commutation law and the operating mode of the converter.

Continuous-current mode (Fig. 1a, b). With continuous current, the control characteristics of the converter (Fig. 3) in the working region are linear, have a saturation zone, and, for the symmetric and asymmetric commutation laws, are determined respectively by the following expressions:

\[ E_{\text{pr}}^{*}=2\tau_{1}-1; \tag{15} \]

\[ E_{\text{pr}}^{*}=\tau_{1}. \tag{16} \]

Discontinuous-current mode (Fig. 1c, d). In this case the control characteristics of the converter in the working region are essentially nonlinear and depend on the armature current and the rotational speed (emf) of the motor, which, however, are related to each other uniquely by expressions (4), (5) and (11), (12). Taking into account that the condition

\[ E_{\text{pr}}^{*}=E^{*}+I_{\text{a}}^{*} \]

is always satisfied, on the basis of (3) and (10) the expression for the converter control characteristic in the discontinuous-current mode for the case of the symmetric commutation law (Fig. 1c) is determined by

\[ E_{\text{pr}}^{*} = 1- \frac{ I_{\text{a}}^{*}\left(2\tau_{\text{a}}-\tau_{1}^{2}-2\tau_{1}\tau_{2}+\tau_{2}^{2}\right) }{ \tau_{1}^{2}+2\tau_{1}\tau_{2}-\tau_{2}^{2} } \tag{17a} \]

or, in accordance with Fig. 1c,

\[ E_{\text{pr}}^{*}=(\tau_{1}-\tau_{2})+E^{*}\tau_{3}. \tag{17b} \]

With an asymmetric commutation law in the discontinuous-current mode (Fig. 1d), the converter control characteristic is determined by the expression

\[ U_{\text{pr}}^{*} = (\tau_{1}^{2}+2\tau_{1}\tau_{2}) - I_{\text{a}}^{*}\left(2\tau_{2}-\tau_{1}^{2}-2\tau_{1}\tau_{2}+\tau_{2}^{2}\right) \tag{18a} \]

or, in accordance with Fig. 1d,

\[ E_{\text{pr}}^{*}=\tau_{1}+E^{*}\tau_{3}. \tag{18b} \]

Figure 2 gives the control characteristics of the reversible pulse-width converter when operating with a dc motor, calculated from expressions (17) and (18).

In an analogous manner, using expressions (3)—(5) and (10)—(12), one can calculate and construct the external characteristics of the converter \(E_{\text{pr}}^{*}(I_{\text{a}}^{*})\) at \(\tau_{1}=\text{const}\) and the control characteristics of the RShIP-D system \(\Omega^{*}(\tau_{1})\) at \(I_{\text{a}}^{*}=\text{const}\), which will also be nonlinear in the discontinuous-current mode.

The control characteristic of an RShIP with a variable commutation law is determined by expressions (15), (16), (17), (18) and, even in the continuous-current mode (Fig. 3), proves to be essentially nonlinear—it has saturation and dead zones and a different slope of the linear part depending on the operating mode of the converter.

Static characteristics of the closed-loop RShIP-D system

When the converter is represented as a linear element, which is possible only in the region of continuous currents, the selection of the parameters of the closed-loop speed-stabilization system and the calculation of its static characteristics are carried out by conventional methods and cause no difficulties.

Fig. 2.

Fig. 3.

Taking into account all the indicated nonlinearities of the RShIP-D system can be carried out using a graphoanalytic method for calculating the static characteristics of the closed-loop system. For this purpose, according to the method set forth above, it is necessary first to calculate and construct the static characteristics of the open-loop system \(\Omega^*(I^*_{\mathrm{a}})\) at \(\tau_1=\mathrm{const}\) (Fig. 5), the control characteristics of the PWM at the two inputs \(\tau_1(U^*_{y1})\) and \(\tau_2(U^*_{y2})\) (Fig. 4б), the characteristic of the BOS \(U^*_{y1}(\Omega^*)\) at \(U_{\mathrm{set}}=\mathrm{const}\) (Fig. 4а), the characteristic

Fig. 4.

of the BTO \(U^*_{y2}(I^*_{\mathrm{a}})\)—when using the cutoff circuit, or \(U^*_{y2}(\Omega)\)—when using anticipatory current limiting (Fig. 4в). The procedure for constructing the static characteristics of the closed-loop system for a specified value of \(U_{\mathrm{set}}\) is clear from Figs. 4 and 5. The characteristics (Fig. 5, thick lines) turn out to be nonlinear in the region of small loads and in the region of operation of current cutoff (thick solid lines) or anticipatory current limiting (thick dashed lines).

The proposed method makes it possible, for the required form of the static characteristic of the closed-loop system, to synthesize the characteristics and parameters of the PWM, BOS, and BTO while taking into account the nonlinearities present in the system, which is especially important in the design of high-precision systems. It should be noted that introducing the threshold \(U_{y20}\) for operation through the \(U_{y2}\) channel of control of the modulators makes it possible to eliminate the influence of the nonlinearity of the initial segment of the characteristic of the nonlinear current element—

of the cut-off determined by the value \(U_{y20}\) corresponding to the beginning of the cut-off action.

Fig. 5.

The letters A and B in Figs. 4 and 5 denote the instants of switching of the commutation law in accordance with the exact condition found (solid lines) and the simplified condition (dashed lines) for its change.

Appendix

\(U_p, E_{pr}, R_{pr}, E, I_a, M, \Omega, R_a, L_a, U_{set}, U_y\) are the mean values of the supply-source voltage; the emf and internal resistance of the converter; the emf, current, torque, rotational speed, resistance, and inductance of the motor armature circuit; the set voltage and the control voltage.

\(e, i_a\) are the instantaneous values of the motor emf and current;

\[ U_b = U_p \]
is the base value of voltage;

\[ R_b = R_a + R_{pr} \]
is the base resistance;

\[ I_b = \frac{U_b}{R_b} \]
is the base value of current;

\[ \Omega_b = \frac{U_b}{C_e} \]
is the base value of speed;

\[ T_b = T_k \]
is the base value of time, equal to the commutation period of the converter;

\[ T_a = \frac{L_{ay}}{R_{ay}} \]
is the time constant of the motor armature-winding circuit.

Notation of quantities in relative units:

\[ U^* = \frac{U}{U_b} \]
is voltage,

\[ E^* = \frac{E}{U_b} \]
is emf,

\[ I_a^* = \frac{I_a}{I_b} \]
is armature current,

\[ \Omega^* = \frac{\Omega}{\Omega_b} \]
is rotational speed,

\[ R^*=\frac{R}{R_{\mathrm{b}}} \]
— resistance,

\[ \tau=\frac{t}{T_{\mathrm{b}}} \]
— time,

\[ U_y^*=\frac{U_y}{U_{y\,\max}} \]
— control voltage.

\(\Omega^*_{\mathrm{gr}1},\ I_{\mathrm{a}\,\mathrm{gr}1}^*,\ \Omega^*_{\mathrm{gr}2},\ I_{\mathrm{a}\,\mathrm{gr}2}^*\) — limiting values of speed and current, respectively, in quadrants I and IV and in quadrants II and III.

REFERENCES

1. V. A. Beinarovich, L. S. Udut, V. S. Fadeev, A. E. Deev. Principles of constructing high-speed reversible electric drives with thyristor pulse-width converters with a variable commutation law. In the present collection.

2. L. S. Udut, V. S. Fadeev, V. A. Beinarovich. Calculation of the mechanical characteristics of electric motors when supplied from thyristor RPWCs with symmetrical control. Izvestiya TPI, vol. 262, TSU Publishing House, 1972.