PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION
AND THE ORDER OF THE RED BANNER OF LABOR
named after S. M. Kirov

Volume 262                                                                                                                     1973

CALCULATION OF THE MECHANICAL CHARACTERISTICS OF ELECTRIC MOTORS SUPPLIED BY THYRISTOR PWM CONVERTERS WITH SYMMETRICAL CONTROL

L. S. UDUT, V. S. FADEEV, V. A. BEINAROVICH

(Presented by the scientific and technical seminar of the Research Institute of AEM)

The properties of reversible electric drives with separately excited motors supplied by reversible pulse-width converters (PWCs) with a symmetrical control law are analogous to the properties of a generator—motor system, which accounts for their wide application. The use of thyristors as power switches in a PWC makes it necessary to introduce a time delay \(t_z\) between the switching operations of the thyristors in the bridge arms. Fig. 1 shows the circuit of a thyristor PWC with transformer devices for forced commutation; from it it is clear that, for example, switching on thyristor T4 is permissible only after complete restoration of the blocking properties of thyristor T1. The introduction of a time delay substantially changes the operating modes of the PWC and the mechanical characteristics of the electric drive, and introduces differences into the method of calculating them in comparison with [1].

Fig. 1

We shall analyze the quasi-steady electromagnetic operating modes of thyristor PWCs feeding the armature circuit of a separately excited motor under the following assumptions: the valves are ideal (their resistance is equal to zero or infinity); the parameters of the armature circuit \(R_a, L_a\) are constant; the supply source with voltage has an internal resistance equal to zero and permits current to flow in any direction; the motor emf \(E\) does not change during the commutation period \(T\); the forced-commutation device, at the moment the thyristors are turned off, instantaneously reduces the motor supply voltage to zero. Fig. 2 presents diagrams illustrating, under the accepted assumptions, the various operating modes of a thyristor PWC at a constant pulse duty factor and different average currents and emf of the motor armature, taking into account the possibility of motor current flowing through the return diodes (Fig. 1) and the supply source independently of the state of the thyristors in the bridge arms. Fig. 2a shows the output voltage. Fig. 2b shows the armature-current diagrams for limiting cases,

dividing the current modes into a number of characteristic zones: the continuous-current mode—zone I; discontinuous unipolar current pulses—zone II; discontinuous bipolar current pulses—zone III; and the continuous alternating-current mode (with a possible dc component)—zone IV. The differential equations for the armature-circuit current on the separate intervals of the commutation period have the form

\[ \pm U = L_a \frac{d i_a}{d t} + R_a i_a + E. \tag{1} \]

Solving equation (1), followed by integration, makes it possible to determine the mechanical characteristics for the various operating modes (the zones in Fig. 2,b). In the plane of the mechanical characteristics, the four possible current modes will correspond to four regions separated by boundary curves.

Fig. 2

Let us determine the boundary between the continuous-current mode (zone I) and the mode of discontinuous unipolar current pulses (zone II) from the condition that, at the end of the period, the current value decreases to zero (curve 1--1 in Fig. 2,b). Taking the commutation period as the unit of time, under the stated conditions we obtain

\[ U\left(2 e^{-\frac{1-\tau_1}{\tau_c}} - e^{-\frac{1}{\tau_a}} - 1\right) - E_{\mathrm{gr}}\left(1 - e^{-\frac{1}{\tau_a}}\right) = 0. \tag{2} \]

Applying expansion in a power series, retaining the first three terms of the series, we obtain from (2) a simplified equation for the boundary speed at positive average values of the motor current (right-hand boundary)

\[ \frac{\omega_{\mathrm{pgr}1}}{\omega_0} = \frac{E_{\mathrm{ogr}1}}{U} = 2\tau_1 \left( 1 + \frac{\tau_2 - 1}{2\tau_a - 1} \right) - 1 \tag{3} \]

and, for negative average values of the current (left-hand boundary),

\[ \frac{\omega_{\mathrm{ogr}1}}{\omega_0} = \frac{E_{\mathrm{ogr}1}}{U} = 1 - 2\tau_2 \left( 1 + \frac{\tau_2 - 1}{2\tau_a - 1} \right). \tag{4} \]

Integrating, over the interval of the repetition period, the solution of equation (1) for the boundary conditions of transition from the continuous-current mode to the mode of discontinuous unipolar current pulses, and applying expansion in a power series, we determine the limiting average value of the motor current for positive currents (right-hand boundary)

\[ I_{\mathrm{lim}1} = \frac{U-E_{\mathrm{lim}1}}{2R_{\mathrm{a}}\tau_{\mathrm{a}}}\tau_1(2-\tau_1) - \frac{U+E_{\mathrm{lim}1}}{2R_{\mathrm{a}}\tau_{\mathrm{a}}}(1-\tau_1)^2 \tag{5} \]

and for negative currents (left-hand boundary)

\[ I_{\mathrm{lim}1} = - \frac{U-E_{\mathrm{lim}1}}{2R_{\mathrm{a}}\tau_{\mathrm{a}}}\tau_2(2-\tau_2) + \frac{U+E_{\mathrm{lim}1}}{2R_{\mathrm{a}}\tau_{\mathrm{a}}}(1-\tau_2)^2 . \tag{6} \]

Fig. 3

According to equations (3)--(6), Fig. 3 shows the boundaries 1 of the region of the continuous-current mode (zone I) in the plane of the mechanical characteristics.

The boundary conditions between the modes of discontinuous unipolar current pulses are determined by the absence of current in the motor over the interval of one time delay \((\tau_3)\) in each period (curve 2--2 in Fig. 2,b). Using the same method, we obtain an expression for the limiting emf speed and the limiting average current for positive currents (right-hand boundary).

\[ \frac{\omega_{\mathrm{pgr}2}}{\omega_0}= \frac{E_{\mathrm{pgr}2}}{U}= \frac{2\tau_1(2\tau_{\mathrm{ya}}+2\tau_3+\tau_1-2)} {2\tau_{\mathrm{ya}}(1-\tau_3)-(1-\tau_2)^2}-1 \tag{7} \]

\[ I_{\mathrm{pgr}2}= \frac{U-E_{\mathrm{pgr}2}}{2R_{\mathrm{ya}}\tau_{\mathrm{ya}}}\, \tau_1(2-\tau_1-2\tau_3) - \frac{U+E_{\mathrm{pgr}2}}{2R_{\mathrm{ya}}\tau_{\mathrm{ya}}} (1-\tau_1-\tau_3)^2 \tag{8} \]

and for negative currents (left boundary)

\[ \frac{\omega_{\mathrm{ogr}2}}{\omega_0}= \frac{E_{\mathrm{ogr}2}}{U} = 1- \frac{2\tau_2(2\tau_{\mathrm{ya}}+2\tau_3+\tau_2-2)} {2\tau_{\mathrm{ya}}(1-\tau_3)-(1-\tau_3)^2}, \tag{9} \]

\[ I_{\mathrm{ogr}2}= -\frac{U-E_{\mathrm{ogr}2}}{2R_{\mathrm{ya}}\tau_{\mathrm{ya}}}\, \tau_2(2-\tau_2-2\tau_3) + \frac{U+E_{\mathrm{ogr}2}}{2R_{\mathrm{ya}}\tau_{\mathrm{ya}}} (1-\tau_2-\tau_3)^2. \tag{10} \]

In Fig. 3 this boundary is denoted by the number 2.

The boundary conditions between the mode of intermittent bipolar current pulses and the mode of continuous alternating current are determined by the equality to zero of the instantaneous current value at the end of each period when the current changes direction within the period (curve 3--3 in Fig. 2,b). Accordingly, the expressions for the limiting values of speed, emf, and average current that determine the right boundary of the alternating-current region in the plane of the mechanical characteristics have the form

\[ \frac{\omega_{\mathrm{pgr}3}}{\omega_0} = \frac{\omega_{\mathrm{pgr}1}}{\omega_0} + \frac{2\tau_3(2\tau_{\mathrm{ya}}-\tau_3)}{2\tau_{\mathrm{ya}}-1}, \tag{11} \]

\[ I_{\mathrm{pgr}3} = I_{\mathrm{pgr}1} - \frac{eU\tau_3}{R_{\mathrm{ya}}(2\tau_{\mathrm{ya}}-1)}, \tag{12} \]

where \(\omega_{\mathrm{pgr}1}\), \(I_{\mathrm{pgr}1}\) are the relative speed (3) and the average current (5) at the boundary of the continuous mode (curve 1 in Fig. 3) for the corresponding values of \(\tau_1\). The left boundary of the alternating-current region is determined from the equations

\[ \frac{\omega_{\mathrm{ogr}3}}{\omega_0} = \frac{E_{\mathrm{ogr}3}}{U} = \frac{2(\tau_2+\tau_3-1)(1-\tau_2-2\tau_{\mathrm{ya}}+\tau_3)} {2\tau_{\mathrm{ya}}-1} -1, \tag{13} \]

\[ I_{\mathrm{ogr}3} = -\frac{U-E_{\mathrm{ogr}3}}{2R_{\mathrm{ya}}\tau_{\mathrm{ya}}} \left[(\tau_2+\tau_3)^2+2\tau_2(1-\tau_2-\tau_3)\right] + \frac{U+E_{\mathrm{ogr}3}}{2R_{\mathrm{ya}}\tau_{\mathrm{ya}}} (1-2\tau_2+\tau_2^2+\tau_3^2). \tag{14} \]

From equations (11)--(14), the boundaries 3 of the alternating-current region are plotted in Fig. 3.

The intersections of the right and left boundaries 3 (Fig. 3) of the alternating-current mode form region IV, within which the mechanical characteristics are rectilinear and are expressed by the equation

\[ \frac{\omega_{\mathrm{IV}}}{\omega_0} = (2\tau_1+\tau_3-1) - \frac{I_{\mathrm{ya}}}{I_{\mathrm{yakz}}}, \tag{15} \]

where \(I_{\mathrm{yakz}}=\dfrac{U}{R_{\mathrm{ya}}}\) is the short-circuit current.

In the region of continuous currents \(I_{\mathrm{ya}}\) (Fig. 3), the mechanical characteristics are also rectilinear, but in the right half-plane (for positive currents) they are expressed by the equation

\[ \frac{\omega_{\mathrm{I}}}{\omega_0} = (2\tau_1-1) - \frac{I_{\mathrm{ya}}}{I_{\mathrm{yakz}}}, \tag{16} \]

and in the left half-plane (for negative currents), by the equation

\[ \frac{\omega_{01}}{\omega_0}=(2\tau_1+2\tau_3-1)-\frac{I_{\mathrm{a}}}{I_{\mathrm{a}k3}} . \tag{17} \]

In the regions of discontinuous currents II and III (Fig. 3), the mechanical characteristics are nonlinear. However, in view of the narrowness of the regions of discontinuous currents, the nonlinear mechanical characteristics can be replaced by straight-line segments connecting the points of intersection of the mechanical characteristics for regions I and IV of continuous current with the boundaries of discontinuous-current modes 1 and 3 (Fig. 3).

The following method is recommended for calculating the mechanical characteristics of electric drives with thyristor PWM converters under symmetrical joint control.

1. From equations (3), (5) and (4), (6), boundary 1 (Fig. 3) of the transition from the continuous-current mode to the discontinuous-current mode is constructed, and region I is determined.

2. From equations (11), (12), and (13), (14), boundary 3 (Fig. 3) of the transition from the discontinuous-current mode to the alternating-current mode is constructed, and region IV is determined.

3. For given values of \(\tau_1\), \(\tau_2\), and \(\tau_3\), the mechanical characteristics for the continuous-current modes are constructed: in the right half-plane of region I (Fig. 3), according to (16); in the left half-plane of region I, according to (17); within region IV, according to (15).

4. The points of intersection of the mechanical characteristic with the mode boundaries (points \(a\), \(c\), and \(a'\), \(c'\) in Fig. 3) are connected by straight-line segments so that a continuous broken-line mechanical characteristic is obtained. If the straight line \(aa'\) does not meet region IV, then the alternating-current mode does not exist, and the mechanical characteristic in the zone of discontinuous currents is represented by the line \(aa'\).

If discontinuous-current modes are neglected, the mechanical characteristics can be constructed in the right half-plane from equation (16), and in the left half-plane from (17). In this case, at an average current equal to zero, there is a discontinuity in the mechanical characteristic, accompanied by a jump in speed, determined from (16) and (17):

\[ \frac{\Delta\omega}{\omega_0}=4\tau_3 . \tag{18} \]

For example, at a commutation frequency of \(500\ \mathrm{Hz}\) (\(T=2000\ \mu\mathrm{s}\)), for type T-150 thyristors a time delay \(t_3=150\ \mu\mathrm{s}\) is required, which leads to the appearance of a speed jump when the sign of the torque (current) changes:

\[ \frac{\Delta\omega}{\omega_0}=4\frac{150}{2000}=0.3 . \]

The experiments carried out confirmed the correctness of the proposed method for calculating mechanical characteristics and the unsuitability, for the systems under consideration, of the method [1] developed for systems with transistor PWM converters.

References

1. O. A. Kossov. Power Amplifiers Using Transistors in Switching Mode. “Energiya,” 1971.