Discontinuous Current Mode in Pulse Speed Control of Wound-Rotor Induction Motors

Yu. P. Kostyukov

Recommended by the scientific seminar of the Electromechanical Faculty.

The pulse method of speed control for wound-rotor induction motors has recently been becoming increasingly widespread.

The idea of pulse speed control of an induction motor is that, in the DC rotor circuit (Fig. 1), a circuit for artificial turn-off, \(IV\), using thyristors is introduced in parallel with the converter device \(P\).

Fig. 1

Such a circuit makes it possible, by changing the relative duration of conduction of the power thyristor, to regulate the rotational speed of the motor. The relative conduction duration of the power thyristor is

\[ \gamma = \frac{T - t_u}{T}, \tag{1} \]

where \(T\) is the pulse repetition period;
\(t_u\) is the pulse duration, corresponding to the time during which the converter conducts current.

A commutation law is adopted with a constant time of the off state of the power thyristor \((t_u = \mathrm{const})\) and a variable frequency. Rectifier \(V\) is assembled according to a three-phase bridge circuit; the smoothing

reactor \(P\) reduces the current pulsations. To avoid a short circuit of the converter, a protective diode \(D\) is included in its circuit.

In pulse speed control of wound-rotor induction motors, at small loads in the rotor circuit a discontinuous-current mode is possible. It is of interest to find the boundary of transition from continuous current to discontinuous current. Knowledge of the boundary curves in drives with pulse control facilitates the construction of mechanical characteristics in the region of discontinuous currents [1].

In deriving the basic relationships, the following assumptions are adopted: the steady-state mode is considered, in which the stator active resistance, the current pulsations with slip frequency, and the valve overlap angle are neglected; the artificial-switching circuit is equivalent to an ideal switch.

In the rectified-current circuit, the average value of the internal e.m.f. of the induction motor \((E s)\), the active resistance of two phases of the rotor winding \((R)\), the inductance of the reactor, and the leakage inductance of two phases of the induction motor \((L)\) are taken into account.

We shall assume that the forward voltage drop in the semiconductor valves does not depend on the magnitude of the current, while the voltage drops across the protective diode and the power thyristor are equal and are referred to the common dc circuit \((\Delta U)\).

In the converter circuit we take into account only the counter-e.m.f. of constant magnitude \((E_u)\) and the equivalent ohmic resistance \((R_u)\).

The equivalent circuit, taking into account the indicated considerations, is shown in Fig. 2. The initial equations characterizing the operation of the induction motor at an arbitrary constant slip \(s\) have the form:

1) for the time interval during which the power thyristor is switched on, \(0 \leq t \leq \gamma T\),

Fig. 2.

\[ i + T_1 \frac{di}{dt} = I', \tag{2} \]

2) for the time interval during which the current passes through the converter, \(\gamma T \leq t \leq T\),

\[ i + T_2 \frac{di}{dt} = I'', \tag{3} \]

where

\[ I'=\frac{E s-\Delta U}{R}, \qquad I''=\frac{E s-\Delta U-E_u}{R+R_u} \]
— limiting values of the currents respectively at \(\gamma=1\) and \(\gamma=0\);

\[ T_1=\frac{L}{R}, \qquad T_2=\frac{L}{R+R_u} \tag{5} \]

— time constants for the first and second commutation intervals. At the boundary point between the continuous and discontinuous modes, the minimum value of the current of the established periodic

the process becomes zero (Fig. 3). Introducing a new variable—the relative time [2]—and solving equation (2) with a zero initial condition, we find

Fig. 3.

\[ I'\left(1-e^{\beta\gamma/(1-\gamma)}\right)e^{-\alpha}+ I''\left(1-e^{-\alpha}\right)=0, \tag{6} \]

where

\[ \beta=-\frac{T_2}{T_1}, \tag{7} \]

\[ \alpha=\frac{t_u}{T_2}. \tag{8} \]

For the mean value of the current one can obtain the following expression

\[ I_{\mathrm{av}}=I'\gamma+I''(1-\gamma)- \frac{(1-\gamma)(1-\beta)}{\alpha\beta} \left(1-e^{-\alpha\beta\gamma/(1-\gamma)}\right)I'. \tag{9} \]

The dependence of the torque of an asynchronous motor on the mean value of the current is approximately determined by the formula [3]

\[ M_{\mathrm{av}}=\frac{I_{\mathrm{av}}\cdot E}{\omega_0}, \tag{10} \]

where \(\omega_0\) is the synchronous angular speed of rotation of the rotor.

Expression (6), taking into account (4), (5), and (8), can be rewritten as

\[ \frac{Es-\Delta U}{E_u}= \frac{1}{ 1+\dfrac{1-e^{-\alpha\beta\gamma/(1-\gamma)}}{(e^\alpha-1)\beta} }. \tag{11} \]

Obviously, the last equality is satisfied only when \(Es-\Delta U<E_u\). This is also physically clear, since only in this case does the current \(I''\) tend toward a negative value; however, owing to the valve action of the rectifiers, a reversal of the current direction is impossible, and at zero value it is interrupted.

For a pulse-controlled resistance \(E_u=0\); then

\[ 1+\beta(e^\alpha-1)-e^{-\alpha\beta\gamma/(1-\gamma)}=0. \tag{12} \]

This equality is never satisfied; that is, with pulse-controlled resistance, a discontinuous-current mode is impossible. Physically this is explained by the fact that in the given case the limiting value of the current \(I''\) is always positive.

Expression (9) for the mean value of the current, taking into account (4), (5), and (8), is reduced to the form

\[ \frac{I_{\mathrm{av}}\cdot R}{E_u} = \frac{Es-\Delta U}{E_u} \left[ \beta+\gamma(1-\beta)- \frac{(1-\gamma)\left(1-\beta\left(1-e^{-\alpha\beta\gamma/(1-\gamma)}\right)\right)}{\alpha\beta} \right] -\beta(1-\gamma). \tag{13} \]

On the basis of (10), (11), and (13) we finally obtain

\[ \varphi= \frac{1}{ 1+\dfrac{1-e^{-\alpha\beta\gamma/(1-\gamma)}}{(e^\alpha-1)\beta} }; \tag{14} \]

\[ M=\varphi \left[\beta+\gamma(1-\beta)-\frac{(1-\gamma)(1-\beta)\left(1-e^{-\alpha\beta\gamma/(1-\gamma)}\right)}{\alpha\beta}\right]-\beta(1-\gamma), \tag{15} \]

where

\[ \varphi=\frac{E_s-\Delta U}{E_u} \]
is the slip in relative units,
\tag{16}

\[ M=\frac{M_{\mathrm{av}}R\omega_0}{E E_u} \]
is the value of the motor torque in relative units.
\tag{17}

Equation (14), together with equation (15), makes it possible to determine curves in the \(\varphi, M\) plane that bound the region of continuous current. Using the expressions given, it is easy to construct the mechanical characteristics in the regions of continuous current and discontinuous current (at small loads).

For example, suppose it is required to construct the mechanical characteristic for the relative on-time \(\gamma\) and given \(\alpha, \beta\). In the continuous-current mode, the torque–speed characteristic is expressed by equation (15). To determine a point of the mechanical characteristic in the discontinuous-current region for an arbitrary relative on-time \(\gamma_n\) \((\gamma_n>\gamma)\), we determine the system parameter \(\alpha_n\) from the formula

\[ \alpha_n=\alpha-\frac{\gamma(1-\gamma_n)}{\gamma_n(1-\gamma)} . \tag{18} \]

In this case, the slip \(\varphi_n\) and the torque \(M_n\) correspond to the coordinates of the point of the boundary mode of the motor operating at \(\alpha_n\) and \(\gamma_n\).

Expanding the exponential terms in a power series, we obtain approximate equations for constructing the mechanical characteristics in the discontinuous-current region:

\[ \varphi_n=\varphi\frac{1-\gamma_n}{1-\gamma}; \tag{19} \]

\[ M_n=(\varphi_n-1+\gamma_n)\beta . \tag{20} \]

where \(\varphi\) is the slip in relative units on the boundary curve for given \(\alpha, \beta, \gamma\).

In pulse speed control of wound-rotor induction motors, the value of the coefficient \(\beta\) is usually in the range \(0.5\div 1.0\); the value of the parameter \(\alpha\) can always be made less than \(0.4\).

To facilitate calculations, it is desirable to construct universal diagrams as functions of the similarity criteria \(\alpha\) and \(\beta\). Figure 4 gives a diagram of the dependence of \(\varphi\) on \(M\) for various values of \(\alpha\) at \(\beta=1\). These curves were constructed from the equations

\[ \varphi=\frac{1-e^{-\alpha}}{1-e^{-\alpha/(1-\gamma)}} , \tag{21} \]

\[ M=\varphi-1+\gamma . \tag{22} \]

To the left of the boundary curves lies the discontinuous-current region, and to the right, the continuous-current region. Figure 4 also shows a family of mechanical characteristics for \(\alpha=0.2\). At \(\beta=1\) the characteristics are close to straight lines; at \(\beta<1\) the nonlinearity of the mechanical characteristics appears more distinctly. The mechanical characteristics in the discontinuous-current region have considerable steepness (especially at low-

small \(\gamma\)) and are not suitable for drives requiring constant speed under a varying load. Knowing the limits of variation of the load torque and using the expressions given, it is not very difficult to find the required value of the parameter \(\alpha\) at which the motor operates in the discontinuous-current mode. For \(\alpha < 0.05\), the discontinuous-current mode is practically excluded.

Fig. 4.

Experimental verification of the calculation formulas was carried out on a \(4.0\ \mathrm{kW}\) induction motor and a machine-type converter unit. The discrepancy between the experimental and calculated data for slip and the average current value does not exceed \(10 \div 20\%\).

In this case, lower accuracy is obtained in the absence of a smoothing reactor, since the current pulsations at the slip frequency have a noticeable effect.

REFERENCES

1. Yu. P. Petrov. Calculation of transient processes of DC electric motors using universal diagrams in similarity criteria, Gosenergoizdat, 1963.

2. Ya. Z. Tsypkin. Theory of linear pulse systems. Fizmatgiz, 1963.

3. F. I. Butaev, E. L. Ettinger. Valve electric drive, Gosenergoizdat, 1951.