On the Circuit of a Three-Phase Ferromagnetic Frequency Tripler

Ya. V. Petrov, A. I. Zaitsev

(Presented by the scientific seminar of the Electromechanical Faculty)

In a number of branches of industry, low-power three-phase asynchronous motors are used, supplied with current at a frequency of 100–200 cycles/sec, which is usually obtained from an electric-machine generator or a frequency converter. As a source of 150 cycles/sec current for this purpose, a ferromagnetic frequency tripler is also of interest, having the following advantages in comparison with an electric-machine frequency converter (generator):

1) simplicity and low cost of construction;
2) reliability in operation owing to the absence of rotating parts, and minimal maintenance in service;
3) lower weight and higher efficiency (0.8–0.9 as against 0.6–0.75).

At the same time, the ferromagnetic tripler has major disadvantages that hinder its practical application:

1) the bulkiness of the installation when obtaining three-phase current of tripled frequency \((3f)\);
2) an excessively soft external characteristic \(U_3 = f(I_3)\);
3) low power factor, 0.2–0.3;
4) considerable heating of the construction during operation, since for economic reasons it is advantageous to adopt high values of induction in the magnetic-core limbs \((1.8 \div 2.0\ \text{Wb}/\text{m}^2)\) and a sufficiently high current density in the windings.

Without dwelling on the question of heating, which can be satisfactorily solved without reducing the utilization of active materials if heat-resistant insulation of the silicon-organic type is used for the windings, we shall indicate possible ways of eliminating the remaining shortcomings of the tripler. A number of propositions require theoretical development and experiment, which it is intended to carry out in the near future.

All known tripler circuits, according to the nature of the magnetizing current, may be divided into two groups: single-phase and three-phase current. Triplers with three-phase magnetizing current are more advantageous than the former, since they have a lower specific consumption of active materials, a better waveform of voltages and currents at \(3f\), better operating characteristics, and create a symmetrical load on the supply network. Therefore, when choosing a circuit for a three-phase tripler (i.e., a tripler, ge-

of the nonregulating three-phase symmetrical system of e.m.f. of triple frequency) only circuits with a three-phase magnetizing current are meant.

A single-phase current of triple frequency can be obtained by means of three single-phase transformers connected into a three-phase group, one three-phase transformer with a spatial or plane four-legged core, or, finally, one three-phase transformer with a five-legged core. The works of E. A. Ivanov and A. A. Kotariya [1, 2] have established that, in terms of the consumption of active materials and the waveform of the e.m.f. of frequency \(3f\), a tripler based on a five-legged magnetic core is inferior to the first two triplers. Obtaining a practically sinusoidal e.m.f. \(3f\) requires strengthening the yokes of the five-legged transformer. The small saving in steel losses and in the consumption of reactive materials is outweighed by the consumption of transformer steel, the greater total weight, and the greater complexity of manufacture and repair of the installation. An experimental investigation of such a tripler for single-phase current \(3f\) shows that, because of the asymmetry of the magnetic core, the voltage curve \(3f\) actually contains the fundamental harmonic of the voltage, which is the cause of the appearance of a noticeable opposing torque at certain speeds of the electric motor. A decrease in the voltage of triple frequency during motor starting was also noted, down to 50% of the no-load voltage of the tripler; the efficiency of the tripler reached 75–80% [3].

We have theoretically established that if, in a tripler with a spatial four-legged symmetrical magnetic core, the secondary windings are placed on the outer legs, then such a tripler, in its most important indicators (specific consumption of primary power and of active materials per unit power of triple frequency, efficiency, and \(\cos \varphi\)), is practically equivalent to a tripler made of three single-phase transformers, while being inferior to the latter in the manufacturability of its construction. When the secondary winding is placed on the central leg (or on the fourth leg of a plane magnetic core, the yokes of which must be strengthened by 40–50% for magnetic symmetry of the phases), the indicators of the tripler deteriorate considerably. Table 1 gives experimental data confirming the ineffectiveness of placing the secondary winding on the central (fourth) leg.

Table 1

Here \(B_{1m}\) is the amplitude of the induction of the fundamental wave; \(E_{30}\), \(I_{3k}\), \(P_{3m}\) are, respectively, the no-load e.m.f., the short-circuit current, and the maximum active power of triple frequency; \(x_3\) is the inductive reactance of the secondary winding.

Quantities with the prime sign belong to the tripler with the secondary ...

winding on the fourth core, and without the “prime” mark—to a tripler with secondary windings placed together with the primary windings on three cores. Thus, the simplest and most economical tripler for single-phase current \(3f\) is a tripler made of three single-phase transformers.

Three-phase current \(3f\) can be obtained in various ways:

1) convert single-phase current \(3f\) into three-phase current by means of a phase converter;

2) obtain two-phase current \(3f\) with a phase shift of \(90^\circ\) by means of two three-phase groups of single-phase transformers, and then, applying one of the known circuits, convert the two-phase current into a symmetrical three-phase current;

3) directly obtain three-phase current \(3f\) in the tripler by means of three three-phase groups.

Conversion of single-phase current into three-phase current can be carried out by a static or rotary phase converter. However, such conversion is associated with great difficulties, since an unbalanced system is transformed into a balanced system: either the conversion is performed with a low efficiency—in which case the symmetry of the three-phase system is not disturbed when the motor load changes—or the conversion is performed with a sufficiently high efficiency, but the three-phase system remains symmetrical only at a definite motor load. The latter drawback is eliminated if adjustable capacitance and inductance are used, but this reduces the efficiency of the phase converter and considerably complicates the overall circuit of the tripler. With a rotary phase converter (for example, a three-phase induction motor with a squirrel-cage rotor), moreover, the attractiveness of the ferromagnetic tripler as a static device is lost.

The method of directly obtaining three-phase current \(3f\) by means of three three-phase groups of single-phase transformers must also be regarded as unacceptable, since the installation proves too bulky and heavy.

Let us consider the circuit for obtaining two-phase current \(3f\) with a phase shift of \(90^\circ\) and the subsequent conversion of this current system into a symmetrical three-phase system of currents \(3f\) fairly simply (for example, by the Scott connection) and with high efficiency. In the known circuits, in order to obtain three-phase current \(3f\), six single-phase transformers are required and, for carrying out the Scott connection, two single-phase transformers or one autotransformer rated for power \(3f\). In order to reduce the weight and dimensions of the tripler, increase the efficiency of the phase converter and the overall efficiency of the installation, it is proposed to carry out the Scott connection without using special transformers or autotransformers (Fig. 1).

Fig. 1.

The primary windings of one three-phase group are connected in \(Y\), those of the other in \(Z\); the ratio of the numbers of turns of the primary windings is

\[ W_1 : W'_1 : W''_1 = 1 : 0.578 : 0.578. \]

Owing to this, the vectors of the magnetic fluxes of the fundamental harmonic of the phases of group II lag by \(30^\circ\) behind the corresponding vectors...

of the fluxes of group I, which corresponds to a mutual displacement of the flux vectors of the third harmonic of the groups by \(90^\circ\) (Fig. 2a). For example, for phases A we have

\[ \Phi_{AI}=\Phi_1 \cos \omega t+\Phi_3 \cos 3\omega t; \]

\[ \Phi_{AII}=\Phi_1 \cos(\omega t-30^\circ)+\Phi_3 \cos 3(\omega t-30^\circ)= \]

\[ =\Phi_1 \cos(\omega t-30^\circ)+\Phi_3 \cos(3\omega t-90^\circ). \]

Figure 2b gives a potential diagram explaining the principle of converting a two-phase e.m.f. system of \(3f\) with a phase displacement of \(90^\circ\) into a three-phase symmetrical e.m.f. system of the same frequency. The sinusoidal nature of the e.m.f. of the three-phase system is ensured in the case when the sum of the e.m.f.’s of the fundamental frequency at the terminals \(ob\) and \(oc\), and also \(oa\), is equal to 0. For this it is necessary to divide the secondary windings of one group into two parts and then connect the parts to one another and the secondary windings as a whole in a known manner. The ratio of the numbers of turns of the secondary windings is

\[ W_2:W'_2:W''_2=0.865:0.5:0.5. \]

Fig. 2.

For normal operation of a motor supplied from the tripler, it is necessary to stabilize the voltage of triple frequency. The stabilizer must not appreciably reduce the efficiency and \(\cos \varphi\) of the tripler (preferably, on the contrary, voltage stabilization should be combined with an increase in the power factor and in the utilization of the tripler materials) and at the same time be simple and reliable in operation, have small overall dimensions and weight, acceptable inertia, and not significantly distort the voltages and currents of the triple frequency. All these requirements are met to the necessary degree by a stabilizing device consisting either only of capacitors (when they are connected in series with the load) or of capacitors and adjustable chokes (when the capacitors and load are connected in parallel). Figure 3 shows possible circuits for stabilizing the load voltage: with series (a) and parallel (b) connection of the capacitors and load, and according to the circuit of A. D. Kratirov (c), proposed for a tripler to single-phase current \(3f\) [4].

In the latter case, additional windings must be placed on the transformer cores of each group, and the adjustable choke connected between the neutral points \(Y\) of these windings and \(Y\) of the capacitors. A fundamental feature of A. D. Kratirov’s circuit is the use of capacitors not only for stabilization and for increasing the output power of triple frequency (this is also achieved in

in two other circuits), but also for partial compensation of the phase shift between the voltage and the current of the fundamental wave, since currents of odd harmonics, including the fundamental, flow through the capacitors. The compensating effect of the capacitors for the third harmonic of the current may be equated to the effect of the circuit with parallel connection of the capacitors and the load.

As the adjustable inductance in circuits (b) and (c), it is possible to use a small saturable reactor having positive feedback with respect to the load current \(3f\). The current is rectified by semiconductor rectifiers.

Fig. 3.

Let us establish the comparative effectiveness of series and parallel connection of capacitors for the phase of the tripler under a load of active-inductive character. As is known, a static tripler can be represented by an approximate equivalent circuit in the form of a source of e.m.f. \(E_{30}\) with internal resistance \(x_3\) (the active resistance of the windings is neglected). With series connection of the capacitor, we obtain the expression for the external characteristic of the tripler (Fig. 4).

\[ U_n=\sqrt{E_{30}^{2}-I_n^{2}\left[(x_3-x_c)^2+2x_n(x_3-x_c)\right]} . \tag{1} \]

Fig. 4.

The apparent power of the load is

\[ S_n=\frac{U_n^{2}}{Z_n} =\frac{E_{30}^{2}-I_n^{2}\left[(x_3-x_c)^2+2x_n(x_3-x_c)\right]} {\sqrt{r_n^{2}+x_n^{2}}}. \tag{2} \]

Equating the derivative

\[ \frac{dS_H}{dx_c}=0, \]

we obtain the value of the optimal capacitive reactance at which the quantity \(S_H\) reaches a maximum:

\[ x_{c\,\mathrm{opt}}=x_3+x_H . \tag{3} \]

The maximum load power delivered by the tripler is

\[ S_{H\max}=\frac{E_{30}^{2} Z_H}{r_H^{2}} . \tag{4} \]

The required capacitor power is

\[ Q_c=I_c^{2}\cdot x_{c\,\mathrm{opt}}=I_H^{2}\cdot x_{c\,\mathrm{opt}} =\frac{E_{30}^{2}(x_3+x_H)}{r_H^{2}}, \tag{5} \]

since at maximum power

\[ I_H=\frac{E_{30}}{r_H}. \]

The specific power output in the optimum mode is

\[ \frac{S_{H\max}}{Q_c}=\frac{Z_H}{x_3+x_H}\,[\mathrm{kVA}/\mathrm{kvar}] . \tag{6} \]

With parallel connection of the capacitor, the external characteristic of the tripler (Fig. 5) is

\[ U_H= \frac{E_{30}\cdot Z_H} {\sqrt{ Z_H^{2} + x_3^{2}\left[ \left(\frac{x_H}{Z_H}-\frac{Z_H}{x_c}\right)^{2} + \left(\frac{r_H}{Z_H}\right)^{2} \right] + 2Z_Hx_3\left(\frac{x_H}{Z_H}-\frac{Z_H}{x_c}\right) }} . \tag{7} \]

Fig. 5.

Equating the derivative

\[ \frac{dS_H}{dx_c} = \frac{d}{dx_c} \left( \frac{U_H^{2}}{Z_H} \right) =0, \]

we obtain the value of the optimal capacitive reactance at which \(S_H\) reaches a maximum.

\[ x_{c\mathrm{opt}}=\frac{x_3 Z_{\mathrm{n}}^{2}}{x_3 x_{\mathrm{n}}+Z_{\mathrm{n}}^{2}}. \tag{8} \]

Maximum load power

\[ S_{\mathrm{n\,max}}=\frac{E_{30}^{2} Z_{\mathrm{n}}^{3}}{r_{\mathrm{n}}^{2} x_3^{2}}. \tag{9} \]

Required capacitor power

\[ Q_c=\frac{U_c^{2}}{x_{c\mathrm{opt}}} =\frac{U_{\mathrm{n}}^{2}}{x_{c\mathrm{opt}}} =\frac{E_{30}^{2} Z_{\mathrm{n}}^{2}\left(x_3 x_{\mathrm{n}}+Z_{\mathrm{n}}^{2}\right)}{r_{\mathrm{n}}^{2}x_3^{3}}, \tag{10} \]

since at maximum power

\[ U_{\mathrm{n}}=\frac{E_{30}Z_{\mathrm{n}}^{2}}{r_{\mathrm{n}}x_3}. \]

The specific power output in the optimum mode is

\[ \frac{S_{\mathrm{n\,max}}}{Q_c} = \frac{x_3 Z_{\mathrm{n}}}{x_3 x_{\mathrm{n}}+Z_{\mathrm{n}}^{2}}. \tag{11} \]

Dividing (11) by (6), we establish the effectiveness of parallel or series connection of capacitors under optimum conditions:

\[ \left(\frac{S_{\mathrm{n\,max}}}{Q_c}\right)_{\mathrm{par}} : \left(\frac{S_{\mathrm{n\,max}}}{Q_c}\right)_{\mathrm{ser}} = \frac{x_3^{2}+x_3 x_{\mathrm{n}}}{x_3 x_{\mathrm{n}}+Z_{\mathrm{n}}^{2}}. \tag{12} \]

The dependence

\[ \left(\frac{S_{\mathrm{n\,max}}}{Q_c}\right)_{\mathrm{par}} : \left(\frac{S_{\mathrm{n\,max}}}{Q_c}\right)_{\mathrm{ser}} = f\!\left(\frac{Z_{\mathrm{n}}}{x_3}\right) \]

is represented by a hyperbolic curve; moreover, when \(\frac{Z_{\mathrm{n}}}{x_3}=1.0\), the ratio of the specific powers is equal to 1.0, that is, in this case both methods of connecting the capacitors prove equivalent (Fig. 6). Series connection of capacitors is more effective for a tripler with \(x_3<Z_{\mathrm{n}}\); parallel connection, on the contrary, is more effective for a tripler with \(x_3>Z_{\mathrm{n}}\). This circumstance should be taken into account when designing a tripler for a specified load \(Z_{\mathrm{n}}\) corresponding to the maximum-power mode.

Fig. 6.

Let us compare the two capacitor-connection circuits when stabilizing the voltage, assuming \(U_{\mathrm{n}}=E_{30}\) for various loads.

From expression (1), for \(U_{\mathrm{n}}=E_{30}\) we obtain two values of the stabilizing capacitive reactance of a series-connected capacitor.

\[ x_{ccT\cdot 1}=x_3+2x_h; \]

\[ x_{ccT\cdot 2}=x_3. \tag{13} \]

For a variable motor load, the stabilizing reactance \(x_{ccT\cdot 2}=x_3\) is of great practical interest; it does not depend on the load and makes it possible automatically to maintain a constant load voltage independently of its magnitude and character (see Fig. 4b).

With stabilized voltage, the load power is

\[ S_{h\,cT}=\frac{U_h^2}{Z_h}=\frac{E_{30}^2}{Z_h}. \tag{14} \]

It can be shown that, in the absence of a capacitor in the tripler circuit, the maximum load power is

\[ S_{h\cdot \max}=\frac{E_{30}^2}{2X_3}. \]

Thus, by using a capacitor, it is possible not only to maintain an almost constant voltage on the varying load, but also to deliver to it a power that exceeds the maximum power in the absence of a capacitor by

\[ \frac{S_{h\cdot cT}}{S_{h\cdot \max}}=\frac{2X_3}{Z_h} \]

times. The utilization of the tripler is thereby considerably increased.

For a series capacitor

\[ Q_c=I_c^2 x_{ccT}=I_h^2 x_{ccT}=\frac{E_{30}^2\cdot x_3}{Z_h^2}, \tag{15} \]

since \(x_{ccT}=x_3\) and \(I_h=\dfrac{E_{30}}{Z_h}\).

The specific power output in the stabilization mode is

\[ \frac{S_{h\cdot cT}}{Q_c}=\frac{Z_h}{x_3}. \tag{16} \]

The values of the stabilizing capacitive reactances with a capacitor connected in parallel are obtained from (7), taking \(U_h=E_{30}\):

\[ x_{ccT\,1,2}= \frac{x_3x_h+Z_h^2\pm\sqrt{Z_h^4-x_3^2r_h^2}} {x_3+2x_h}. \tag{17} \]

We take the plus sign before the radical, since this ensures a smaller capacitor power. Here the magnitude of the capacitive reactance depends on the load resistance; therefore, when the latter fluctuates, it is necessary to change correspondingly the inductance of the choke connected in series with the capacitor. This is a significant shortcoming of this method of stabilization, incre—

— increasing, moreover, owing to additional power losses in the adjustable choke. The specific power output in the stabilization mode with parallel connection of the capacitor is

\[ \frac{S_{\mathrm{nst}}}{Q_c} = \frac{x_3 x_n + Z_n^2 + \sqrt{Z_n^4 - x_3^2 r_n^2}} {Z_n (x_3 + 2x_n)} . \tag{18} \]

Dividing (18) by (16), we obtain for the voltage-stabilization modes

\[ \left( \frac{S_{\mathrm{nst}}}{Q_c} \right)_{\mathrm{par}} : \left( \frac{S_{\mathrm{nst}}}{Q_c} \right)_{\mathrm{ser}} = \frac{x_3 \left( x_3 x_n + Z_n^2 + \sqrt{Z_n^4 - x_3^2 r_n^2} \right)} {Z_n^2 (x_3 + 2x_n)} . \tag{19} \]

The last dependence is described by a hyperbolic curve, similar to Fig. 6; moreover, both methods of voltage stabilization are equivalent with respect to the specific power consumption of the capacitors when \(\frac{Z_n}{x_3} \leq 1.0\) for \(\cos \varphi_n = 0 \div 1.0\). Series connection of the capacitor is more economical beginning with values of \(\frac{Z_n}{x_3}\) somewhat less than 1.0 or greater than 1.0 (depending on \(\cos \varphi_n\)). Thus, the range of loads for which voltage stabilization by a capacitor connected in parallel is more economical proves to be narrower than under optimal conditions.

In practice it is possible to make a tripler with such inductive resistance that series connection of the capacitor at a given maximum load (minimum resistance \(Z_n\)) is more effective in terms of the power consumption of the stabilizing capacitor per unit of load power. The choice of the values \(x_3\) should be made taking expression (16) into account. Voltage stabilization by a capacitor connected in series does not require the use of auxiliary adjustable resistances, which appreciably complicate the overall circuit of the tripler and cause additional power losses, thereby lowering the operating indices of the installation as a whole. Thus we arrive at the conclusion that the method of stabilizing the load voltage by a capacitor connected in series is the simplest and most economical. In the event of a short circuit of phases between the motor and the tripler, considerable overvoltages of the capacitors and of the tripler windings may arise because of voltage resonance at frequency \(3f\). This shortcoming of series connection of capacitors when \(x_{ccT} = x_3\) must be taken into account, and special overvoltage protection must be provided.

Conclusions

1. The most acceptable circuit for a three-phase tripler should be considered the variant of six single-phase transformers with subsequent conversion of two-phase current into three-phase current according to the Scott connection.

2. To reduce the weight and dimensions of the tripler and to increase the efficiency of the installation, it is recommended that the conversion of two-phase current into three-phase current be carried out without the use of supplementary transformers—on the windings of the tripler.

1. Stabilization of the voltage of tripled frequency, with a simultaneous increase in the utilization of the tripler, is rationally accomplished by using capacitors connected in series with the load.

REFERENCES

1. Ivanov E. A. A three-phase power frequency converter. Proceedings of the Leningrad Order of Lenin Institute of Railway Engineers, issue 138, 1948.

2. Kotaria A. A. A static electromagnetic frequency tripler in a single transformer unit. Proceedings of the Georgian Polytechnic Institute, No. 29, 1953.

3. Dahlgren F. and Littke—Persson A. K. Static Frequency Transformers for Small Electric Motors. Tekniska Högskolans Handlingar, No. 47, 1951, Göteborg, Sweden.

4. Kratirov A. D. Frequency converter. Description of invention No. 73764, class 21d², 51.

5. Neiman L. R. and Kalantarov P. L. Theoretical Foundations of Electrical Engineering, part II, Gosenergoizdat, 1954.