On the Issue of Powering Electric Motors from a Static Frequency Tripler

Ya. V. Petrov, A. I. Zaitsev

(Presented by the scientific seminar of the Electromechanical Faculty)

It was established in [1] that, in order to obtain three-phase current of triple frequency by means of magnetically saturated transformers, it is most advisable to use six single-phase transformers connected into two three-phase groups, and to stabilize the voltage at the motor terminals by means of longitudinal capacitive compensation of the tripler’s inductive resistance (Fig. 1). A two-group tripler has a more compact design and a lower consumption of active materials in comparison with a three-group tripler (of nine single-phase transformers). Indeed, assuming that the single-phase transformers of both triplers are geometrically similar and consume the same power from the supply network in order to produce the calculated power $S$ of triple frequency, according to [2] we obtain the ratio of the weights of the active materials of the three- and two-group triplers

Fig. 1

\[ \frac{G_{\mathrm{III}}}{G_{\mathrm{II}}} \simeq 9\sqrt[4]{\frac{S^3}{9^3}} : 6\sqrt[4]{\frac{S^3}{6^3}} \simeq 1.1 . \]

Their power losses and the cost of the active materials are also approximately in the same ratio.

At the output of the two-group tripler an asymmetrical three-beam star of triple-frequency emf is created; therefore, the selection of capacitor capacitances by phases has certain special features. The calculated power of triple frequency during independent operation of the windings $ao$ and $bc$ (without capacitors) is

\[ S = S_{ao} + S_{bc} = E_{ao} \cdot I_{ao} + E_{bc} \cdot I_{bc}, \]

where $E_{ao}$, $E_{bc}$ and $I_{ao}$, $I_{bc}$ are the corresponding no-load emf’s and short-circuit currents.

We shall assume that, when the windings operate in a connected mode, the line no-load emf’s $E_{ab}$, $E_{bc}$, $E_{ca}$ and the phase short-circuit currents

$I_{ab}, I_{bc}, I_{ca}$ form three-phase symmetrical systems (like the currents of a three-phase symmetrical load connected in delta, with $Z_n = 0$). From the condition of conservation of power when transforming the number of phases,

\[ S = E_{ab} I_{ab} = E_{bc} I_{bc} + E_{ca} I_{ca} = 3 E_l I_{k\phi}. \]

The inductive reactances of the delta phases are

\[ x_{ab} = x_{bc} = x_{ca} = \frac{E_l}{I_{k\phi}} = \frac{3 E_l^2}{S}. \]

For the equivalent symmetrical star,

\[ x_a = x_b = x_c = x_3 = \frac{x_{ab}}{3} = \frac{E_l^2}{S}. \]

It is not difficult to prove that the resistance $x_3$ is related to the inductive reactances of the independently loaded windings $ao$ and $bc$ by the relations

\[ x_3 = \frac{x_{ao}}{1.5} = \frac{x_{bc}}{2}. \]

The reactance $x_3$ is calculated. The capacitance of the capacitors per phase for the steady-state operating mode of the motor is chosen as $x_c = x_3$ or $x_c = x_3 + x_l$ (in the latter case the inductive reactance of the line between the tripler and the motor, $x_l$, is also compensated, which provides an increased voltage level at the motor terminals).

Fig. 2

Legend: with compensation; without compensation. Axes: $U_2/U_l$ and $I_l/I_{kl}$.

However, systems of line e.m.f.’s, as well as of line and phase short-circuit currents, in practical designs of two-group triplers cannot be strictly symmetrical, since it is difficult to obtain a phase angle between the no-load e.m.f.’s of the triple frequency equal to $90^\circ$, and the Scott circuit, on the basis of which the phase-number transformation is carried out here, introduces additional asymmetry of voltages and currents under load. In this case it may turn out that the connection of identical capacitances into the motor stator circuit causes inadmissible asymmetry of voltages and currents. Symmetrization of voltages and currents can be achieved by selecting the capacitances of the individual phases; it is recommended that this be done in the following order:

a) average capacitance of a phase

\[ C = 1061 \cdot \frac{S}{E_l^2} \quad \text{or} \quad C = 1061 \cdot \frac{S}{E_l^2 + S \cdot x_l} \; [\mu F], \]

where

\[ E_l = \frac{E_{ab} + E_{bc} + E_{ca}}{3}. \]

b) the average line short-circuit current of the tripler

\[ I_{k.\ell}=\frac{I_a+I_b+I_c}{3}, \]

where the currents \(I_a, I_b, I_c\) are measured by ammeters during a three-phase short circuit of the tripler;

c) the capacitance of the capacitors of the individual phases

\[ C_a \simeq C\frac{I_a}{I_{k.\ell}},\qquad C_b \simeq C\frac{I_b}{I_{k.\ell}},\qquad C_c \simeq C\frac{I_c}{I_{k.\ell}}. \]

In our experiments with a tripler according to the circuit of Fig. 1, with power \(S_n=500\) VA, the systems of line voltages and currents remained practically symmetrical (the degree of asymmetry of the line voltages was not more than 3%) when the tripler load was varied from no-load to rated and above, in the range of inductions of the fundamental harmonic \(B_{1m}=16000\div20000\) Gs (Table 1); in this case the capacitances of the phase capacitors were taken to be identical. The loads were three-phase asynchronous motors of industrial frequency and a VChS-10 ore electric drill at 185 rev/sec.

Table 1

In Fig. 2, using the data of the table, the external characteristics of the tripler in relative units,

\[ \frac{U_\ell}{E_\ell}=f\!\left(\frac{I_\ell}{I_{k.\ell}}\right), \]

are plotted for the average values of the quantities.

The curves of Fig. 2 show that, with the aid of longitudinal capacitive compensation, sufficiently stiff external characteristics can be obtained and the power output can be significantly increased. As calculations and experiments show, when \(x_c=x_3\), as the load power factor increases the falling slope of the external characteristics increases. Stiffer external characteristics, up to rising ones, are achieved by overcompensation of the inductive resistance of the tripler, \(x_c>x_3\); however, the power of the capacitor bank then increases, and the load current in any normal operating mode of the motor must be less than

\[ I_{\max}= \frac{E_\ell} {\sqrt{3}\left[(r_3-r_1)\sin\varphi_n+(x_3+x_1-x_c)\cos\varphi_n\right]}. \]

in order to avoid unstable operation of the tripler (here \(r_3\) and \(r_\ell\) are the active resistances of the secondary windings of the tripler and of the tripler–motor line).

The specified voltage at the motor terminals \(u_n=\dfrac{U_n}{E_\ell}\) will be obtained if the capacitance reactance per phase is taken as equal to

\[ x_c=x_3+x_\ell+\frac{E_\ell}{\sqrt{3} I_n}\sin\varphi_n \]

\[ -\sqrt{\left(\frac{E_\ell}{\sqrt{3} I_n}\right)^2(2-u_n^2) -\left(\frac{E_\ell}{\sqrt{3} I_n}\cos\varphi_n+r_3+r_\ell\right)^2}. \]

The relationships indicated above have been established under the assumption that the equivalent circuit of the tripler phase on the 150 cycles/sec side can be represented by a source of e.m.f. \(\dfrac{E_\ell}{\sqrt{3}}\) with an internal linear resistance \(x_3\).

In transient operating conditions of the motor (starting; braking by a resisting torque exceeding the maximum torque of the motor), with longitudinal capacitive compensation, self-oscillations may arise in the tripler–motor circuit, manifested in the form of self-excitation and self-oscillation of the motor and subharmonic oscillations of currents and voltages of the beating type. The motor “sticks” on the starting portion of the mechanical characteristic. Self-oscillations were observed especially often by us at the moment of starting or braking of industrial-frequency asynchronous motors possessing a large uncompensated inductive reactance at a frequency of 150 cycles/sec; the VChS-10 electric drill did not produce oscillations.

These undesirable phenomena, which are sufficiently complex, have been partially described in the literature and require further investigation. The best solution to the problem is to obtain an installation with parameters that preclude the excitation of self-oscillations, since any of the known methods of suppressing the self-oscillations that appear leads, as a rule, to a significant reduction in the effectiveness of longitudinal capacitive compensation and noticeably complicates the installation.

REFERENCES

1. Ya. V. Petrov, A. I. Zaitsev. On the circuit of a three-phase ferromagnetic frequency tripler, Proceedings of the Tomsk Polytechnic Institute, vol. 97, 1959.

2. G. N. Petrov. Electrical Machines, part 1, Gosenergoizdat, 1956.

3. L. L. Rozhanskii. Static Electromagnetic Frequency Converters, Gosenergoizdat, 1959.

4. M. S. Mikhailov-Mikulinskii, P. M. Tikhomirov. Frequency Triplers for Supplying Electric Motors, Moscow, 1958.