Principles of Constructing Frequency-Controlled Systems “Semiconductor Converter—Asynchronous Gyroelectric Motor”

I. F. Kalinin, B. P. Soustin, Yu. A. Shurygin

(Presented by the scientific-technical seminar of the Research Institute of AEM at TPI)

In frequency control it is difficult to create a converter with an ideal sinusoidal current waveform. Even an electromechanical converter (a synchronous generator whose rotor rotates according to a prescribed law) cannot ensure this condition, since the converter parameters change when the frequency changes, which leads to nonconstancy of the harmonic coefficient \((K_{\mathrm{r}} = 3 \div 10\%)\).

The use of the method of pulse-width modulation according to a sinusoidal law cannot be regarded as a solution to the problem, since even with the same \(K_{\mathrm{r}}\) at low and high frequencies the action of the higher harmonics is not the same, which ultimately leads to additional overheating of the rotor [1].

Frequency converters implemented on switching elements are of interest; these may have an arbitrary, in particular inverse, dependence on the stator frequency. In such circuits the formation of the amplitude and waveform of the current is carried out by means of a two-position regulator included in the feedback circuit of the closed loop [2, 3, 4], Fig. 1a. In this case the converter tracks the input signal with a prescribed degree of accuracy. To explain the operating principle, an electrical circuit on transistors and diodes for one phase is shown in Fig. 1b.

Initially no current flows through the load; then, owing to the excess of the input signal, the switches, for example T1 and T2, are turned on. The voltage of the supply source is applied to the load, and current begins to flow through it. When the current reaches the value \(i_{t1}\), the current sensor DT (implemented on the principle of a magnetic amplifier) issues a signal to turn off switches T1 and T2. Owing to the self-induction emf, the load is connected through diodes D1 and D2 to the source in the opposite direction, and the current begins to decrease. When the current reaches the value \(i_{t2}\), the current sensor again issues a signal to turn on switches T1 and T2; the current again begins to increase, then to decrease, and so on.

The frequency and depth of the pulsation will be determined by the difference between the input-signal voltage and the voltage proportional to the load current

\[ \Delta i = \frac{\Delta U_{\mathrm{in}}}{R_i} = \frac{U_{\mathrm{in}} - U_{\mathrm{fb}}}{R_i} = \frac{U_{\mathrm{in}} - Z_{\mathrm{DT}} i_{\mathrm{n}}}{R_i}, \tag{1} \]

The load current pulsates between the upper and lower sinusoid. At point 1 \(i = I_m \sin \omega t + \frac{\Delta i}{2}\), at point 2 \(i = J_m \sin \omega t - \frac{\Delta i}{2}\). Obviously, the current sensor and the mismatch-signal amplifier must be adjusted so that,

Fig. 1a. Formation of the load current.

Fig. 1b. Schematic circuit for one phase.

so that the switching-on of the keys occurs at a definite value of the input signal

\[ \Delta i_{\mathrm{in}} \geq \Delta i \tag{2} \]

The dead zone ensures minimum no-load losses in the absence of an input signal, since all the inverter keys are then closed, and provides the necessary time delay when switching on the keys that form the negative half-wave of the current. The magnitude of the current ripple can be made arbitrarily small; however, in this case the switching frequency increases and the dynamic losses in the converter increase. With an increase in the value of \(\Delta i\), the effective value of the current rises and the copper losses of the motor increase. In [2, 4] an optimal value of the current ripple is indicated, not exceeding \((20 \div 30)\%\) of the rated value.

To reduce the average switching frequency of the keys, it is advisable to connect additional inductances in series with the stator windings.

When a three-phase asynchronous gyromotor is connected as the load, three circuits of Fig. 1,b are required; they must be controlled by a sinusoidal voltage with a phase shift of \(\pm \dfrac{2\Pi}{3}\).

The equivalent input resistance of the asynchronous motor under investigation, operating with constant absolute slip \((\beta_{\mathrm{opt}})\) and flux \((\Phi)\) not exceeding \((0.7—0.8)\) of the rated value, depends linearly on frequency.

Separating the real part of the input resistance \(R_{\mathrm{in}} = K\alpha\) (here \(K\) is a proportionality coefficient determined by the motor parameters, and \(\alpha = \dfrac{f_1}{f_{1N}}\) is the relative stator frequency), we find the law of variation of the current in the load circuit in the time interval from \(t_0\) to \(t_1\).

\[ i(t)=\frac{U_{\mathrm{p}}}{R_{\mathrm{in}}} \left[1-\exp\left(-\frac{t}{\tau}\right)\right] + I_0 \exp\left(-\frac{t}{\tau}\right). \tag{3} \]

Here \(\tau = \dfrac{L_{\mathrm{eq}}}{R_{\mathrm{in}}}\) is the equivalent electromagnetic time constant of the asynchronous gyromotor at constant slip.

Obviously, the turn-off instant at point 1 is determined when the exponential (3) and the sinusoid are equal:

\[ \frac{U_{\mathrm{p}}}{R_{\mathrm{in}}} \left[1-\exp\left(-\frac{t}{\tau}\right)\right] + I_0 \exp\left(-\frac{t}{\tau}\right) = I_m \sin \omega t + \frac{\Delta t}{2}. \tag{4} \]

In the section from point 1 to point 2 the keys are closed; the load current decreases according to the law

\[ i(t)=-\frac{U_{\mathrm{p}}}{R_{\mathrm{in}}} \left[1-\exp\left(-\frac{t}{\tau}\right)\right] + I_1 \exp\left(-\frac{t}{\tau}\right). \tag{5} \]

The switching-on instant at point 2 is found by solving

\[ -\frac{U_{\mathrm{p}}}{R_{\mathrm{in}}} \left[1-\exp\left(-\frac{t}{\tau}\right)\right] + I_1 \exp\left(-\frac{t}{\tau}\right) = I_m \sin \omega t - \frac{\Delta t}{2}. \tag{6} \]

The required current waveform is described in the general case by a system of transcendental equations:

for \(t > t_{2n-2}\)

\[ i(t_{2n-2})\exp\left(-\frac{t-t_{2n-2}}{\tau}\right) +\frac{U_{\mathrm{p}}}{R_{\mathrm{in}}} \left[ 1-\exp\left(-\frac{t-t_{2n-2}}{\tau}\right) \right] = \]

\[ = I_m \sin \omega t + \frac{\Delta i}{2}; \]

for \(t > t_{2n-1}\)

\[ i(t_{2n-1})\exp\left(-\frac{t-t_{2n-1}}{\tau}\right) -\frac{U_{\mathrm{p}}}{R_{\mathrm{in}}} \left[ 1-\exp\left(-\frac{t-t_{2n-1}}{\tau}\right) \right] = \]

\[ = I_m \sin \omega t + \frac{\Delta i}{2} \]

\[ \begin{gathered} i(t_{2n-1})=i(t_{2n-2})\exp\left(\frac{t_{2n-1}-t_{2n-2}}{\tau}\right)+ \frac{U_{\mathrm{p}}}{R_{\mathrm{in}}}\left[1-\exp\left(-\frac{t_{2n-1}-t_{2n-2}}{\tau}\right)\right],\\ i(t_{2n})=i(t_{2n-1})\exp\left(-\frac{t_{2n}-t_{2n-1}}{\tau}\right)- \frac{U_{\mathrm{p}}}{R_{\mathrm{in}}}\left[1-\exp\left(-\frac{t_{2n}-t_{2n-1}}{\tau}\right)\right],\\ i(t_{2n-2})=C, \end{gathered} \tag{7} \]

where \(i_{2n}\) are the required values of the currents at the instants when the switches are turned on; \(i_{2n-1}\) are the required values of the currents at the instants when the switches are turned off. \(t_{2n}\) and \(t_{2n-1}\) are the instants of turning the switches on and off.

The system of equations (7) was solved on the “Promin-M” digital computer. The cyclic program for solving (7) was supplemented with commands making it possible to compute the values of cosines at the required points. The solution was carried out to an accuracy of four decimal places. Harmonic analysis is performed by expanding into Fourier series the current curve obtained on the digital computer for various control parameters.

The dependence of the switching (pulsation) frequency of the current on the parameters of the “inverter—motor” system is found from an approximate solution of equations (3) and (5),

\[ f_k=\frac{2U_{\mathrm{p}}+\Delta I R_{\mathrm{in}}}{4L_{\mathrm{eq}}\Delta I} \cdot \frac{I_m^2\left(2L_{\mathrm{eq}}\omega_{\mathrm{out}}+R_{\mathrm{in}}^2\right)} {2\Delta I_{\mathrm{eq}}\left(2U_{\mathrm{p}}+\Delta I R_{\mathrm{in}}\right)} \tag{8} \]

Here \(L_{\mathrm{eq}}=L_{\mathrm{in}}+L_{\mathrm{ct}}+L_{\mathrm{add}}\) is the inductance of the asynchronous motor, the current sensor, and the additional inductance. The functional diagram of the system, in addition to the power circuit implemented either with transistors or with thyristors, and the current sensors, includes devices for comparing the instantaneous current values in six triggers, amplifiers—shapers of the control pulses, and a sinusoidal stator-frequency setter.

Expression (7) is taken as the basis for constructing a gyroscopic drive with an asynchronous motor. In this case the switching frequency is selected from the condition of minimum losses in the steel and copper of the stator and rotor due to nonsinusoidal current.

To verify the above derivations, a mock-up using P210-type transistors was assembled and various tests were carried out. The nominal power of the mock-up is 1000 W, the nominal output voltage is 127/40 V, and the output frequency varies from 0 to 1000 Hz in both directions; the frequency stability at \(f_{\mathrm{out}}=500\) Hz is \(\pm 0.005\%\).

The law of frequency variation is linear or exponential; current stability in the nominal mode is 0.2%; the magnitude of nonlinear distortions without additional filters did not exceed 14%; the coefficient of nonlinear current distortion when 10 μF capacitors per phase were connected at the output did not exceed 2%, and for voltage, 2.5%. The mock-up is powered from a 220 V, 400 Hz mains supply; the efficiency in the nominal mode is 0.65. The load may vary from 0 to nominal; no-load operation of the converter is permissible. The weight of the experimental mock-up is less than 30 kg.

In addition to frequency starting, it is possible to carry out direct starting at any frequency with a load power of up to 600 W.

In Figs. 2,b,c, oscillograms are given of the converter current without filters at \(f=5\) and 500 Hz and of the voltage at \(f=50\) Hz.

Experimental investigations were carried out with a gyromotor \(J=2.6\ \mathrm{g\,cm\,s^2}\), \(f_{1N}=1000\) Hz, \(U_{1N}=36\) V in the reversible-control mode. The following parameters were measured:

a) reversal time (acceleration and braking) at frequencies \(\pm 500\) Hz;

b) motor overheating temperature;

c) coefficients of nonlinear distortion for current and voltage.

The results obtained in testing the “converter—gyromotor” system are summarized in Table 2. All reversible-control parameters were ...

Table 1

Fig. 2a. Oscillogram of the load current at \(f = 5\) Hz.

Fig. 2b. Oscillogram of the load current at \(f = 500\) Hz.

Fig. 2v. Oscillogram of the voltage across the load at \(f = 50\) Hz.

with regulation are given for the optimal frequency law with readjustment of the stator-current magnitude. The reversing time was measured as two components: regenerative braking from rated speed to zero, \(t_{\mathrm{t}}\), and starting in the opposite direction from zero to the rated value, \(t_{\mathrm{r}}\). In addition, the table gives the time of the first start, \(t_{\mathrm{p}}\), which in all cases is smaller, since the motor was not overheated by starting currents. Table 2 gives the time to reach steady state, \(t_{N}\), and the number of cycles, \(N_{\mathrm{c}}\).

Table 2

Fig. 3. Dependence of overheating on the coefficient of nonlinear distortion.

From Table 2 it is evident that the minimum overheating was established at the last cyclicity, with a steady-state time of 20 min and a reversal time of about 6 min. In this case the overheating temperature was \(16 \div 17^\circ\mathrm{C}\). When the steady-state time is increased, the overheating temperature decreases; however, increasing this time beyond 30 min is not advisable.

To determine the optimal control mode at different ambient temperatures \(T_{\mathrm{amb}}\), dependences were obtained of the gyromotor overheating temperature on the magnitude of the coefficient of nonlinear distortion in current and voltage, Fig. 3.

From Fig. 3 it is evident that, as the coefficient of nonlinear distortion increases from \(2 \div 11\%\), the overheating temperature increases approximately identically, irrespective of the ambient temperature. With a further increase in KNI, overheating grows exponentially. In this case, current distortion has less effect on overheating than voltage distortion.

In addition to the problem solved, the question of optimal starting under the condition of minimum time was also considered. For this purpose, calculations were carried out for forced starting by the method of graphical integration of the dynamic torque over time, and the losses were determined. The calculated data coincided with the experimental data. Moreover, the time for the second and third modes turned out to be shorter (respectively 59 sec instead of 82 sec and 35 sec instead of 47 sec); however, in this case the calculated overheating exceeded the experimental value by \(4 \div 5^\circ\mathrm{C}\).

The installed power of the converter was determined, and a comparative assessment was made of energy losses in various control modes. With optimal frequency starting, electrical losses decrease by a factor of 3.27, while the motor starting time is reduced by a factor of 10 in comparison with direct starting.

It should be borne in mind that in many practical cases the problem of minimizing the starting time is solved with a large number of constraints. Thus, the effect of the starting time on the state of readiness of the instrument, which depends on the established temperature, must be taken into account; this temperature, in turn, is determined by the thermal time constant and by the power consumed.

In specific cases such a problem can be solved under the condition that the time for establishing the thermal regime and the starting time are equal.

In this case the system “secondary power supply—gyromotor” is designed for a prescribed acceleration time. No less important is the question of the influence of the acceleration mode on the change in the initial position of the main axis of the gyroscope, and on the behavior of this axis during starting.

Constraints of this kind can be taken into account when studying the system “secondary power supply—gyromotor—gyroscope.”

Conclusions

1. A converter with output-current shaping makes it possible to obtain a minimum harmonic coefficient over the entire frequency range.

2. The most acceptable mode for reversing is the mode of optimal frequency control. With minimum reversal time and maximum stator current, the installed power of the converter is reduced by more than a factor of 3 in comparison with direct switching.

3. When the nonlinear distortion coefficient is varied upward from 2 to 11%, rotor overheating also increases (approximately \(0.8^\circ\text{C}\) per 1% of the nonlinear distortion coefficient), tending toward a steady-state value in the range of a nonlinear distortion coefficient of 15–32%.

LITERATURE

1. A. M. Azarov, V. I. Ivanchura, B. P. Soustin. Study of the harmonic composition of the inverter output voltage. Collection of reports of the scientific and technical conference of the Academy of Sciences of the Ukrainian SSR, Institute of Electrodynamics. Kiev, 1969.

2. I. F. Kalinin, B. P. Soustin. Closed-loop system for forming the inverter curve current. Collection “Converter Engineering Devices,” Kiev, 1969.

3. S. I. Volkov, I. F. Kalinin, B. P. Soustin. Study of a closed-loop system for speed regulation and torque of an asynchronous motor. Abstracts of reports of the XXII Scientific and Technical Conference. LIAP, Leningrad, 1969.

4. V. N. Brodskii, E. S. Ivanov. Contactless electric drive with frequency-current control for closed-loop regulation systems. Electricity, No. 10, 1967.

5. I. F. Kalinin, B. P. Soustin. Acceleration of an asynchronous hydroelectric motor optimal in terms of response speed. Proceedings of the V Scientific and Technical Conference on Problems of Production Automation. Tomsk, 1967.