ION FREQUENCY CONVERTER FOR POWERING A BETATRON

V. G. Stepanov, V. A. Kukhtin

(Presented at the Second Interinstitutional Conference on Electron Accelerators)

Experience in operating betatrons designed and built by the Tomsk Polytechnic Institute has convincingly shown that the frequency stability provided by the industrial power grid is quite sufficient for a betatron powered directly from the grid to produce stable radiation [7].

This gives grounds to suppose that, if for the purpose of increasing intensity the industrial frequency is multiplied several times, then such a betatron, with correspondingly greater effect and smaller dimensions, should operate stably.

Alongside experiments on the creation of an electromagnetic frequency multiplier undertaken at the Tomsk Polytechnic Institute [8], a valve frequency converter [4] is proposed for consideration. Calculations and experiments show that the ion frequency converter according to the circuit proposed by the authors has a number of substantial advantages over existing ones. The principal advantages are: high efficiency; the entire circuit is assembled from standard elements; and relative simplicity and stability in operation.

Below is given a description and analysis of circuits that make it possible to obtain frequencies of 150, 300, 450 hertz and higher.

Ion converters that increase frequency consist, as is known, of rectifier and inverter sections and of an explicitly or implicitly expressed direct-current section

Fig. 1

[2, 3]. The presence of an inverter stage accounts for the relatively low operating reliability and the considerable complexity of the circuit of ion frequency converters.

Figure 1 shows a circuit of an ion converter without an inverter stage, making it possible to obtain at the output a single-phase voltage of tripled frequency, \(\dfrac{f_2}{f_1}=3\), line diagrams of the secondary phase voltages of the transformer, and the voltage at the converter output.

The frequency converter consists of two three-phase controlled ion rectifiers I and II, operating alternately at a delay angle \(\alpha = 120^\circ\) electrical [4].

The converter transformer has six secondary windings connected in two inverse three-phase stars, and a primary winding connected in delta, so that there is no forced-magnetization flux in the transformer core.

By the same principle, a frequency converter can be implemented for another frequency ratio \(\dfrac{f_2}{f_1}=k\), where \(k\) is an integer.

Expansion of the transformed-voltage curve into harmonics

Owing to the symmetry of the curve with respect to the axis of abscissas, the expansion contains no even harmonics (\(n\) even) and no constant term (Fig. 1). The coefficients of the Fourier series

\[ a_n = \frac{2}{T}\int_0^T f(x)\cos n\omega x\,dx; \]

\[ b_n = \frac{2}{T}\int_0^T f(x)\sin n\omega x\,dx \]

when the notation for the curve under investigation is introduced: \(n\) is the order of the harmonic, a multiple of three, \(T=2l\) is the period of the third harmonic, where \(l=\dfrac{\pi}{3}\), \(\omega=\dfrac{2\pi}{T}=\dfrac{\pi}{l}\), take the form

\[ a_n = \frac{1}{l}\int_0^{2l} f(x)\cos n\frac{\pi}{l}x\,dx; \]

\[ b_n = \frac{1}{l}\int_0^{2l} f(x)\sin n\frac{\pi}{l}x\,dx. \]

To variation of \(x\) within the limits \(\alpha < x < \pi\) there corresponds the function \(f(x)=\sin x\); for \(\alpha+\pi < x < \dfrac{4\pi}{3}\), \(f(x)=\sin\left(x+\dfrac{2\pi}{3}\right)\); for \(\dfrac{2\pi}{3}<x<\alpha\) and

\[ \pi < x < \frac{\pi}{3}+\alpha \qquad f(x)=0. \]

We take \(U_m = 1\). For the case \(\dfrac{2\pi}{3} < \alpha < \pi\), after carrying out the integrals we obtain:

\[ a_n = \frac{6}{\pi(9n^2 - 1)} \cdot \kappa_a; \qquad b_n = \frac{6}{\pi(9n^2 - 1)} \cdot \kappa_b, \]

where

\[ \kappa_a = [1 - \cos \alpha \cos 3n\alpha - 3n \cdot \sin \alpha \cdot \sin 3n\alpha], \]

\[ \kappa_b = [3n \sin \alpha \cdot \cos 3n\alpha - \cos \alpha \cdot \cos 3n\alpha]. \]

The amplitudes of the \(n\)-th harmonic are

\[ U_{nm} = \frac{6}{\pi(9n^2 - 1)} \cdot \kappa_\lambda, \tag{1} \]

where

\[ \kappa_\lambda = \sqrt{\kappa_a^2 + \kappa_b^2} = \]

\[ = \sqrt{1 - 2(\cos \alpha \cdot \cos 3n\alpha + 3n \sin \alpha \cdot \sin 3n\alpha) + \cos^2 \alpha + 9n^2 \cdot \sin^2 \alpha}. \]

For the case \(\dfrac{\pi}{3} < \alpha < \dfrac{2\pi}{3}\) we obtain

\[ U_{nm} = \sqrt{a_n^2 + b_n^2}, \]

where

\[ a_n = \frac{3}{\pi(1 - 9n^2)} \cdot \left[ 2\sqrt{3}\cos 3n\alpha \cdot \sin(60 - \alpha) + 9n \cdot \sin \alpha \cdot \sin 3n\alpha \right], \]

\[ b_n = \frac{3}{\pi(1 - 9n^2)} \cdot \left[ 2\sqrt{3}\sin 3n\alpha \cdot \sin(60 - \alpha) - 6\sqrt{3}\cos(60 - \alpha)\cos 3n\alpha \right]. \tag{2} \]

Fig. 2

As a result of calculations using formulas (1, 2), the curve

\[ \frac{U_{nm}}{U_m}=f(\alpha) \]

(Fig. 2) was constructed for \(\alpha=\dfrac{\pi}{3}\div\pi\).

For the case \(\alpha=\dfrac{2\pi}{3}\), the effective value at the converter output is

\[ U_{1n}=\frac{0.716\cdot U_m}{\sqrt{2}}=0.505\cdot U_m, \]

and the effective value of the voltage at the converter output is

\[ U_n=\sqrt{\frac{3}{\pi}\int_{\frac{2\pi}{3}}^{\pi} U_m^2\sin^2\omega t\,d\omega t}=0.542\cdot U_m, \]

where \(U_m\) is the maximum voltage of the transformer secondary phase.

Experimental investigation

After checking the operation of the circuit of Fig. 1, assembled in the laboratory using thyratrons, the circuit was investigated under factory conditions.

A TMF-560/35 transformer of 441 kva was used, with a \(\mathrm{Y}/\mathrm{Y}\) 6600/6\(\times\)60 V connection scheme and brought-out neutrals of the secondaries; rectifiers of the RMNV-1000\(\times\)6 type, 825 V rectified voltage, manufactured by the “Uralelektroapparat” plant, consisting of six ion valves (excitron type, two-grid design), mounted on one frame and insulated from one another.

Changes were made in the grid-control, ignition, and excitation circuits of the ShRV-41 cabinet in order to insulate the valve cathodes of the even and odd stars.

The supply of the control grids of one star was disconnected from the ShRV-41 cabinet and connected to a separate grid-control cabinet. Phase regulators were included in the primary circuits of both grid transformers, making it possible to set the required delay angle \(\alpha\). The load is a resistance rheostat.

Figure 3 gives oscillograms taken at

\[ \alpha=\frac{2\pi}{3} \]

of the transformer secondary phase voltage \(u_{2\phi}\), the voltage at the converter output \(u_n\), the converter current \(i_n\), and the current in the common cathode bus of the odd star \(i_t\).

Fig. 3

At no load, the burning of valve No. 5 corresponds to a commutation current greater than in the other valves, owing to inaccurate adjustment of the voltage peaks on the control grids of the valves.

With a load on active resistance, the waveform of the current \(i_n\) (especially for valves 1, 2, 3) is close to a sinusoid.

Commutation current occurs at \(\alpha<150^\circ\) el. owing to the positive value of the sum of the instantaneous phase e.m.f.’s \(e_1+e_2\).

in the circuit, valve 2—the current of valve 1 and is equal to the current of the preceding phase from the moment at which the valve of the following phase begins to operate (Fig. 1).

The commutation current overloads the valves and the transformer, and reduces the voltage and efficiency of the converter.

If the burning of the preceding valve is interrupted before the ignition of the following valve, there will be no commutation current.

The necessary duration of the interval is determined by the time during which the valve’s blocking properties are restored. The interval should preferably be minimal, because the output voltage of the converter decreases as the interruption in valve conduction is increased.

In ion valves of the ignitron type, which do not have a continuously burning excitation arc, when the anode load disappears the cathode spot disappears in a time of \(\simeq 10^{-5}\) sec, or \(0.18^\circ\) electrical angle at industrial frequency [6].

We shall take an interval of \(10^\circ\) electrical angle as quite sufficient for stable operation of the converter under a certain load with valves of the excitron type. With an active converter load \(P_a\), a capacitor bank of power

\[ P_c = P_a \cdot \operatorname{tg} 10^\circ = 0.18 \cdot P_a \]

will be required to create a current voltage lead by an angle of \(10^\circ\) electrical angle.

Calculation of the Typical Transformer Power

Let us take the transformation ratio \(k_T = 1\). In the circuit of Fig. 1, during a period \(T\), one half-sinusoid of current passes in a phase of the secondary winding of the transformer, and two half-sinusoids of current pass in a phase of the primary winding. Suppose that the duration of a current half-sinusoid is equal to \(\dfrac{T_2}{2}\).

Let us determine the effective value of the current in the secondary phase of the transformer:

\[ i_{2\phi} = I_m \cdot \sin 3\omega t \]

\[ I = \sqrt{\frac{1}{2\pi}\int_{\frac{2\pi}{3}}^{\pi} I_m^2 \cdot \sin^2 3\omega t \cdot dt} = I_m \cdot \]

\[ \cdot \sqrt{\frac{1}{2\pi}\int_{\frac{2\pi}{3}}^{\pi} \left[\frac{\omega t}{2} - \frac{\sin 6\omega t}{12}\right]} = \frac{i}{\sqrt{12}} \]

or

\[ I_{2\phi} = \frac{I_n}{\sqrt{6}}, \]

where \(I_n\) is the effective value of the current at the output of the converter.

The effective value of the first harmonic of the voltage at the converter output is

\[ U_n = \frac{U_{m_1}}{\sqrt{2}}. \]

The effective value of the current in the primary winding of the transformer

\[ I_{1\phi}=\sqrt{2}\cdot I_{2\phi}=\frac{I_n}{\sqrt{3}}, \]

having

\[ U_{m_1}=0{,}716\cdot U_m,\qquad U_\phi=\frac{U_m}{\sqrt{2}}, \]

we obtain

\[ U_\phi=\frac{1}{0{,}716}\cdot U_n. \]

The apparent power at the converter output is

\[ P_n=I\cdot U_n=\frac{P_a}{\cos\varphi_n}, \tag{3} \]

where \(P_a\) is the active power passing through the converter.

The apparent powers of the windings are

\[ P_2=m_2\cdot U_\phi\cdot I_{2\phi}, \tag{4} \]

\[ P_2=6\cdot\frac{1}{0{,}716}\cdot U_n\cdot\frac{1}{\sqrt{6}}\cdot I_n =3{,}43\cdot P_n; \]

\[ P_1=m_1\cdot U_\phi\cdot I_{1\phi}, \tag{5} \]

\[ P_1=3\cdot\frac{1}{0{,}716}\cdot U_n\cdot\frac{1}{\sqrt{3}}\cdot I_n =2{,}43\cdot P_n. \]

The transformer rating is

\[ P_{\mathrm{T}}=\frac{P_1+P_2}{2}, \tag{6} \]

\[ P_{\mathrm{T}}=\frac{3{,}43+2{,}43}{2}\cdot P_n =\frac{2{,}93}{\cos\varphi_n}\cdot P_a. \]

In order to reduce the required transformer rating, the circuit of Fig. 4 may be used.

In the primary and secondary windings of the transformer, over the period \(T_1\), two half-waves of current pass; consequently,

\[ P_{\mathrm{T}}=\frac{P_1+P_2}{2}=P_1=2{,}43\cdot P_n. \]

Circuits using both half-sinusoids of the phase voltage of the secondary winding of the transformer, analogous to the circuit of Fig. 4, may be used for converters with frequency ratios

\[ \frac{f_2}{f_1}=3,\ 5,\ 7,\ 9,\ \text{etc.}, \]

which becomes evident upon consideration

linear diagrams of the phase voltages of the converter transformers for these frequencies.

As a result of the increase in the amplitude of the first harmonic of the converted voltage as \(\alpha\) decreases, the required rated power of the transformer decreases.

Fig. 4

The rated power of the transformer for the circuit of Fig. 4 was calculated by formulas (4), (5), and (6). The result of the calculation is summarized in Table 1.

Table 1

For \(\alpha < 90\text{–}95^\circ\) el., \(P_T\) begins to increase noticeably as a result of the increase in the capacitive current at the converter output.

A further increase in the capacitance connected at the converter output will cause an increase in the required transformer power and may be justified by the desire to improve the converter \(\cos\varphi\).

An analogous calculation for the circuit of Fig. 5, which has at the output a frequency \(f_2 = 6 f_1\), and for circuits giving \(f_2 = 9 f_1\), was performed for two values of \(\alpha\).

The results of the calculation are entered in Table 2.

Fig. 5

Power Factor of the Converter

For \(\alpha = 120^\circ\), the electrical angle of displacement of the voltage and current curves in the primary phase of the transformer, neglecting the magnetizing current, is

\[ \varphi_1 = 60^\circ \text{ el.}, \quad \text{and} \quad \cos \varphi_1 = 0.5. \]

The power factor is \(\chi = \nu \cos \varphi_1\), where \(\nu\) is the distortion factor, equal to

\[ \nu = \frac{1}{\sqrt{1 + \varepsilon \left(\frac{I_{nn}}{I_{1m}}\right)^2}}. \]

The amplitude values of the harmonic components of the primary current are determined by expanding the primary-current curve into a trigonometric series. As a result, taking into account the seventeenth harmonic, we obtain the distortion factor

\[ \nu = \frac{1}{\sqrt{1 + 0.2977}} = 0.878 \]

and the power factor

\[ \chi = \nu \cos \varphi_1 = 0.878 \cdot 0.5 = 0.439. \]

A converter with an inverter stage in a three-phase circuit

Table 2

rectification has the best \(\chi\), namely: \(\nu = 0.827,\ \cos \varphi_1 = 0.966,\ \chi = 0.799\), but requires the use of a capacitor bank of power \(P_c = 0.57\,P_a\) [2, 3].

If a capacitance \(P_c = 0.57\,P_a\) is connected to the terminals of the primary winding of the transformer in the circuit of Fig. 1, then \(\cos \varphi_1\) will change to 0.7, and we obtain \(\chi = 0.7 \cdot 0.878 = 0.615\).

The power factor of the circuit in Fig. 1 can be substantially improved if a capacitor bank is connected at the output of the converter, thereby changing the instant of extinction of the valves, and at the same time the angle \(\alpha\) is reduced by turning the rotors of the phase regulators. By connecting at the converter output \(P_c = 0.57\,P_a\) and setting \(\alpha = 90^\circ\) el., we obtain \(\varphi_n = 30^\circ\) el., \(\varphi_1 = 30^\circ\) el., \(\cos \varphi_1 = 0.866\), and \(\chi = 0.866^2 = 0.76\).

To obtain \(\chi = 0.799\), it is necessary to connect an additional capacitance \(P_c^1 = 0.15\,P_A\) at the converter output.

Comparison of frequency-conversion circuits

Figure 6 shows the commonly used circuit of an ion converter with a three-phase rectifier section, in which the same number of valves is used as in the circuits of Figs. 1 and 4. The rated power of the transformers in the circuit of Fig. 6 is \(2.9\,P_\alpha\), and the power of the choke is \(0.4\,P_\alpha\) [1, 2, 3], where \(P_\alpha\) is the transmitted active power of the converter, without losses, determined on the rectified-current side.

Fig. 6

The total rated power of the transformers is \(3.3\,P_\alpha\). Obviously, \(P_\alpha = P_a\).

The external characteristic of this converter is represented by a curve that rises as the converter load decreases [2, 6].

To obtain a sufficiently flat external characteristic, practically the same as for the circuits of Figs. 1, 4, and 5, magnetization of the magnetic core of the inverter transformer by a direct current depending on the inverter load may be used [5]. In this case the rated power of the transformers of the circuit increases. If the ratio of the weights of copper and steel of a normal single-phase transfor-

of the inverter transformer equal to \(1:4\), the standard power rating of the transformers in the circuit will be \(\approx 4.23\,P_n\).

In the circuits of Figs. 1, 4, and 5 the output characteristic is sufficiently rigid, since the voltage losses in the converter—if the voltage drop in the arc of the ion valves is not taken into account, it being practically unchanged over wide limits of load variation—are determined by the voltage losses in the transformer, usually amounting to 5–7% when the load varies from 0 to nominal.

For the circuit of Fig. 4 an ordinary mains transformer with a \(\Delta/Y\) connection circuit and with the neutral point of the star brought out may be used; in this case it will be necessary to increase the cross section of the neutral tap and bring it out to the transformer cover. To obtain higher frequencies, transformers of special design are required, for example, with phase-shifting windings on the primary or secondary sides, made in order to create the required angle of displacement between the voltages of the secondary phases of the transformers.

The required transformer power \(P_T\) increases with increasing ratio of \(f_2\) and \(f_1\), and at

\[ \frac{f_2}{f_1}=9 \]

it becomes approximately equal to \(P_T\) for the circuit of Fig. 6 with magnetization of the inverter transformer (Table 2). The displacement factor decreases with growth, but can be increased considerably by connecting an additional capacitance at the converter output. The efficiency of the described converters with an inverter link is higher because of the absence of a choke and an inverter transformer.

A disadvantage of the converter is the need to have a comparatively large number of ion valves, especially for the higher frequencies.

A feature of the circuits is the constant frequency ratio

\[ \frac{f_2}{f_1}=\text{const}. \]

In the inverter this ratio can be changed when the frequencies of the supply network and of the control grid circuit of the inverter fluctuate.

Conclusions

The investigation carried out gives grounds to conclude that the proposed principle of frequency conversion makes it possible to create converters that have advantages over known converters of increased frequencies with an inverter link:

1) lower cost, especially at a frequency \(f_3=3f_1\), since the expenditures on the additional capacitance are several times less than the saving obtained from reducing the standard power rating of the transformers and the conversion losses;

2) higher efficiency;

3) great simplicity and reliability in operation.

REFERENCES

1. Kaganov I. L. Electronic and Ion Converters, Part I. Gosenergoizdat, 1950.

2. Kaganov I. L. Electronic and Ion Converters, Part III. Gosenergoizdat, 1956.

3. Shilling V. Circuits of Rectifiers, Inverters, and Frequency Converters. Gosenergoizdat, 1950.

4. Stepanov V. G. Ion Frequency Converter. Scientific Department of UPI, 1952.

5. Shevchenko G. I. Elektrichestvo, No. 5, 57, 1953.

6. Kostenko M. P. Electrical Machines. Gosenergoizdat, 1949.

7. Vorob’ev A. A. Charged-Particle Accelerators. Gosenergoizdat, 1952.

8. Klyukin A. F., Titov B. N. Static frequency tripler for betatron supply. Report at the Interuniversity Scientific Conference on the problem “Electron Accelerators,” 1958.

9. Izv. TPI, Vol. 96, 1.