Accounting for the Effect of Saturation in the Operation of a Cross-Field EMA on Alternating Current

A. I. Skorospelkin, B. I. Kostylev, Sh. S. Roiz

(Recommended by the scientific seminar of the departments of electrical machines and general electrical engineering.)

The problem of creating electric-machine amplifiers (commutatorless and commutator types) operating on alternating current has, in recent years and at present, been advanced by the ever-increasing need for them in connection with the development of various AC control systems, regulated electric drives, systems with parametric generators, and electric-machine amplifiers for frequency investigations of various electromechanical and tracking systems. In addition, in many cases the operation of electric-machine amplifiers is associated with frequently recurring transient processes, i.e., with currents and voltages that vary with time.

The problem of an adjustable AC electric drive using a squirrel-cage induction motor has in recent years acquired great practical importance. As is known, the most high-quality and economical method of regulating the speed of an induction motor is regulation by varying the frequency of the voltage supplied to the motor. In such systems it is possible to use a standard cross-field electric-machine amplifier as a single-phase generator of current of variable frequency.

When a cross-field amplifier operates on alternating current, its own special features appear, which affect the amplification factor, compensation adjustment, and commutation. Up to the present time, the operation of a cross-field amplifier on alternating current and the method for calculating it have been insufficiently investigated and developed.

In calculating an EMA, the no-load characteristic of the output stage is of great importance. It serves as the basis for choosing the operating point and reflects the degree of utilization of the magnetic material of the machine. Therefore, practically always the full characteristic is calculated, including the saturation region, although the operating point may also lie in the unsaturated region, as is the case in amplifiers. Calculation of this characteristic on direct current presents no difficulties and is carried out according to the method of [1].

When calculating an AC EMA it is necessary to take into account the following feature.

Since the calculation of the magnetic circuit of an amplifier, as of any electrical machine, is performed according to the amplitude value of the induction or magnetic flux, the magnetization curve of the machine will be the same on direct and alternating current. However, in the final result, what is of interest is the no-load characteristic expressed in effective values of the output emf and the magnetizing current. In this case, for an unsaturated machine the no-load characteristic will be

differ from the characteristic in maximum values only by the scale, which changes along both axes by a factor of \(\sqrt{2}\). In the case of saturation of the machine, this regularity is disturbed because of the nonlinearity of the magnetization curve, which leads to a nonsinusoidal current in the transverse circuit, with distinctly pronounced third and fifth higher harmonics. The magnetic flux remains sinusoidal, since a sinusoidal voltage is obtained in the transverse circuit.

As a result of the different effect of saturation under direct and alternating current, the no-load characteristics are different in the saturation region. Under alternating current it must lie somewhat lower than under direct current, since the rms value of the current is less than the amplitude value by more than a factor of \(\sqrt{2}\). Therefore, for calculating the no-load characteristic under alternating current it is necessary to use the expansion into the fundamental and higher harmonics of the nonsinusoidal current curve obtained from the magnetization curve of the amplifier along the transverse axis, in order to find the rms value of the current. In most cases it is sufficient to determine the fundamental harmonic, since its rms value differs little from the rms value of the transverse-circuit current, equal to

\[ I_z = \sqrt{I_{z1}^{\,2} + I_{z3}^{\,2} + I_{z5}^{\,2}}, \]

where \(I_{z1}; I_{z3}; I_{z5}\) are the rms values of the current of the first, third, and fifth harmonics.

Fig. 1. Graphical method for decomposing the current curve of the transverse circuit.

Determination of the current curve and its expansion can be carried out graphically, as shown in Fig. 1, and analytically.

Indeed, the magnetization curve of the amplifier along the transverse axis \(\Phi_z = f(i_z)\), obtained as a result of calculation of the magnetic circuit,

can be represented, with some approximation, by the expression [2]

\[ i_2 = a\Phi_2 + b\Phi_2^\beta, \tag{1} \]

where the coefficients \(a\) and \(b\) and the exponent \(\beta\) can be found from the initial slope and two points of the characteristic (Fig. 2).

Fig. 2. Magnetization curve along the transverse axis of the amplifier.

Since the magnetic flux is sinusoidal, \(\Phi_2 = \Phi_m \sin \omega t\), and according to equation (1)

\[ i_2 = a\Phi_m \sin \omega t + b\Phi_m^\beta \sin^\beta \omega t = \]

\[ = (a\Phi_m + a_1)\sin \omega t + a_3 \sin 3\omega t + a_5 \sin 5\omega t + \ldots, \]

where the coefficients \(a_1, a_3, a_5\) are determined by expanding the function \(\sin^\beta \omega t\) into a harmonic series.

The general formula for these coefficients has the form

\[ a_k = \frac{2}{\pi}\int_0^\pi F(\alpha)\sin k\alpha\,d\alpha, \]

where \(\alpha = \omega t\).

For the function under consideration,

\[ a_k = \frac{2}{\pi}\int_0^\pi b\Phi_m^\beta \sin^\beta \alpha \sin k\alpha\,d\alpha. \]

The integral obtained for \(\beta > 0\) can be expressed in terms of the beta function \(B\)

\[ \int_{0}^{\pi} \sin^{\beta}\alpha \, \sin \kappa\alpha \, d\alpha = \frac{\pi \cdot \sin \dfrac{\kappa\pi}{2}} {2^{\nu-1}\cdot \nu \cdot B\left(\dfrac{\nu+\kappa+1}{2}\cdot \dfrac{\nu-\kappa+1}{2}\right)}, \]

where \(\nu=\beta+1\)

\[ B\left(\frac{\nu+\kappa+1}{2}\cdot \frac{\nu-\kappa+1}{2}\right) = \frac{ \Gamma\left(\dfrac{\nu+\kappa+1}{2}\right)\cdot \Gamma\left(\dfrac{\nu-\kappa+1}{2}\right) } {\Gamma(\nu+1)}. \]

Here \(\Gamma\) is the gamma function.

Figure 3 presents the no-load characteristics of the EMU-12A on direct current (1) and on alternating current (2, 3, 4), obtained respectively graphically, analytically by the method described, and experimentally.

Fig. 3. No-load characteristics of the output stage of the EMU-12A amplifier on direct and alternating current.
1—direct current; 2, 3, 4—alternating current.

The curves show good agreement between the calculation and the experiment, and a considerable difference between the no-load characteristics on direct and alternating current. This difference leads to the fact that, when seeking to obtain from the amplifier on alternating current the same voltage as on direct current, it is necessary to increase the current of the cross-field circuit considerably, since the operating point falls in the saturated part of the no-load characteristic.

An increase in the current of the cross-field circuit, with the same delivered power, leads to overloading of the armature winding and to deterioration of the commuta-

tion of the transverse brushes and to a decrease in the amplification factor. In addition, taking into account that the amplifier must operate with an unsaturated magnetic system, when it is supplied with alternating current the output voltage and, consequently, the power must be lower than with direct current.

Conclusions

1. Saturation of the steel of the magnetic circuit of the electric machine amplifier affects the amplifier characteristics differently in operation on direct and alternating current. In operation on alternating current, saturation does not affect the waveform of the output voltage, but it distorts the current curve of the transverse circuit and reduces the output voltage.

2. To ensure normal operation of the amplifier on alternating current, it is necessary to reduce the output e.m.f. and power in comparison with direct current.

3. Calculation of the no-load characteristic of the output stage of the electric machine amplifier must be carried out taking into account the nonlinearity of the magnetization curve and the nonsinusoidal nature of the transverse-circuit current.

References

1. F. A. Goryainov. Electric-machine amplifiers. GEI, 1962.
2. G. N. Petrov. Electric machines. Part 1, GEI, 1956.
3. I. S. Gradshtein and I. M. Ryzhik. Tables of integrals, sums, series, and products. Fizmatgiz, 1963.