RESONANT THREE-PHASE FREQUENCY TRIPLER

A. I. ZAITSEV, Ya. V. PETROV

(Presented by Doctor of Technical Sciences I. D. Kutyavin)

For supplying power to hand tools, as well as induction heating, frequency converters are required. An electrified hand tool operating at an increased frequency is considerably lighter than the same tool operating at industrial frequency. A lightweight tool is of especially great importance for the coal and timber industries. Sources of increased frequency may be electric-machine frequency converters, synchronous generators of increased frequency, and static converters.

Rotating converters, in comparison with static ones, have advantages that consist mainly in the good waveform of the voltage curve and the relatively simple regulation of the increased-frequency voltage. However, rotating converters also have a serious drawback, namely the presence of moving parts, which significantly reduces reliability.

Despite a number of valuable merits of static electromagnetic frequency multipliers—simplicity of construction, low cost, high reliability, etc.—they have not yet found practical use because of their low power factor (0.2–0.3), soft external characteristic, and the difficulty of regulating the increased-frequency voltage. Still more complicated is the problem of obtaining three-phase currents of increased frequency. Thus, to obtain three-phase current of tripled frequency according to the classical circuit, nine single-phase transformers are necessary.

The installation in this case becomes bulky. At present there are a number of circuits by means of which tripled frequency can be obtained with a smaller number of transformers [1], by using four- and five-limb transformers, etc.

Tripled frequency can be obtained if, for this purpose, the phenomenon in a ferroresonant circuit is used. The circuit of the proposed tripler is shown in Fig. 1. With an appropriate selection of the circuit parameters, compensation of the fundamental harmonic current in the external circuit can be achieved, and through transformer 1 there will flow mainly a current of triple frequency. The instantaneous value of the current in the external circuit will be equal to the difference of the instantaneous values of the currents flowing through capacitor 2 and saturation reactor 3. The volt-ampere characteristics of such a circuit are given in Fig. 2. To obtain a larger output of triple-

frequencies, it is necessary that the operating voltage be greater than the resonance voltage. The form and phase of the change in the external current depend on the magnitude of the voltage applied to the circuit. When a voltage equal to the resonance voltage is reached, the instantaneous value of the current in the external circuit is zero. With a further increase in voltage, the phase of the external current changes, and its amplitude will be determined by the amplitude value of the applied voltage.

[Diagram labels: \(f\), \(3f\).]

Fig. 1.

Thus, owing to the summation of the currents in the capacitance and the choke, the external current will vary at triple frequency. Let us consider certain relations characterizing the operation of such a converter. The principal indicator of the operation of the tripler is the current of triple frequency in the external circuit.

[Diagram labels: \(U\), \(i_c\), \(i_c - i_{\mathrm{dr}}\), \(i_{\mathrm{dr}}\), \(I_c\), \(I_{\mathrm{dr}}\).]

Fig. 2.

The instantaneous value of the current in the external circuit is determined by the geometric difference of the currents of the capacitor \(i_c\) and the choke \(i_{\mathrm{dr}}\)

\[ i_{\mathrm{vn}} = i_c - i_{\mathrm{dr}}. \]

Let us determine the expression for the current in the saturation choke. In view of the fact that the magnetization curve is nonlinear, it is necessary to represent it in the form of some analytic dependence. One of the known equations of the magnetization curve is an equation of the form

\[ AW = \alpha \cdot \operatorname{Sh} \beta B. \]

The quantities \(\alpha\) and \(\beta\) are determined from the condition that the selected analytical curve intersect the actual magnetization curve at two points with coordinates \(B_1, AW_1\), and \(B_2, AW_2\) [3, 4] (Fig. 3).

To determine the coefficients entering into the approximate equation of the magnetization curve, we have two equations

\[ AW_1=\alpha\,\operatorname{Sh}\beta B_1, \]

\[ AW_2=\alpha\,\operatorname{Sh}\beta B_2. \]

Fig. 3.

Whence we obtain that

\[ \frac{AW_2}{AW_1}=\frac{\operatorname{Sh}\beta B_2}{\operatorname{Sh}\beta B_1}. \]

In the region of interest to us it may be assumed that

\[ \frac{\operatorname{Sh}\beta B_2}{\operatorname{Sh}\beta B_1}\simeq e^{\beta(B_2-B_1)}, \]

whence

\[ \beta=\frac{\ln\left(\dfrac{AW_2}{AW_1}\right)}{B_2-B_1}. \]

Next we have

\[ AW=AW_1\frac{\operatorname{Sh}\beta B}{\operatorname{Sh}\beta B_1}= \]

\[ =AW_1 \frac{ e^{\frac{B}{B_2-B_1}\ln\left(\frac{AW_2}{AW_1}\right)} - e^{-\frac{B}{B_2-B_1}\ln\left(\frac{AW_2}{AW_1}\right)} }{ e^{\frac{B_1}{B_2-B_1}\ln\left(\frac{AW_2}{AW_1}\right)} - e^{-\frac{B_1}{B_2-B_1}\ln\left(\frac{AW_2}{AW_1}\right)} }. \]

Let us denote

\[ \frac{B_2}{B_1}=\kappa \quad\text{and}\quad \frac{AW_2}{AW_1}=e^s \quad\text{or}\quad s=\ln\frac{AW_2}{AW_1}. \]

Then

\[ AW=AW_1 \frac{ \operatorname{Sh}\left[\dfrac{s\cdot B}{B_1(\kappa-1)}\right] }{ \operatorname{Sh}\left(\dfrac{s}{\kappa-1}\right) }. \]

In order to simplify the solution, we neglect active losses in the steel, residual magnetism, and leakage fluxes. Let us write the equations of electrical and magnetic equilibrium.

The equation of electrical equilibrium is

\[ W_a\frac{d\Phi}{dt}=U_{\text{max}}\sin(\omega t+\psi). \tag{1} \]

Magnetic-equilibrium equation

\[ iW_a=-\frac{AW_1}{\operatorname{Sh}\left(\dfrac{s}{k-1}\right)}\,l_{cp}\, \operatorname{Sh}\left[\frac{s\cdot \Phi}{(k-1)g_c\cdot B_1}\right], \tag{2} \]

where \(W_a\) is the number of turns in the alternating-current winding;
\(g_c\) is the useful cross section of the magnetic core;
\(l_{cp}\) is the mean length of the magnetic core.

Integrating equation (1),

\[ \int_{0}^{t} d\Phi = \int_{0}^{t}\frac{U_{\max}}{W\cdot \omega}\sin(\omega t+\psi)\,d(\omega t), \]

\[ \Phi_t=\Phi_0-\frac{U_{\max}\cdot \cos(\omega t+\psi)}{W\cdot \omega} +\frac{U_{\max}}{W\cdot \omega}\cos\psi . \]

For the steady-state regime the constant term is equal to zero. This corresponds to the value \(\psi=\dfrac{\pi}{2}\). We have agreed to neglect the value \(\Phi_0\).

Under the condition \(\psi=\dfrac{\pi}{2}\),

\[ \Phi = -\frac{U_{\max}\cdot \cos\left(\omega t+\dfrac{\pi}{2}\right)}{W\cdot \omega} = \frac{U_{\max}\cdot \sin\omega t}{W\omega}. \]

Substitute the value of \(\Phi\) into equation (2):

\[ iW= \frac{AW_1}{\operatorname{Sh}\left(\dfrac{s}{k-1}\right)}\,l_{cp}\, \operatorname{Sh}\left[ \frac{S\cdot U_{\max}\cdot \sin\omega t\cdot 10^8} {(k-1)\,g_c\cdot B_1\cdot W\cdot \omega} \right]. \tag{3} \]

Denote

\[ Q=\frac{S\cdot U_{\max}\cdot 10^8}{(k-1)g_cB_1\cdot W\cdot \omega}. \]

We rewrite the last equation:

\[ iW= \frac{AW_1}{\operatorname{Sh}\left(\dfrac{s}{k-1}\right)}\,l_{cp}\, \operatorname{Sh}\left[Q\cdot \sin\omega t\right]. \]

It is known that

\[ \operatorname{Sh}\left[Q\sin\omega t\right] = \frac{e^{Q\sin\omega t}-e^{-Q\sin\omega t}}{2}. \]

Both exponential terms are periodic functions and can be represented by Fourier series.

\[ e^{Q\sin\omega t}=\sum_{-\infty}^{+\infty} I_n(jQ)e^{-jn\omega t}= \tag{4} \]

\[ =\sum_{-\infty}^{+\infty}\left[I_n(jQ)\cos n\omega t-jI_n(Q)\sin n\omega t\right], \]

\[ e^{-Q\sin\omega t}=\sum_{-\infty}^{+\infty} I_n(jQ)e^{jn\omega t}= \tag{5} \]

\[ =\sum_{-\infty}^{+\infty}\left[I_n(jQ)\cos n\omega t+jI_n(Q)\sin n\omega t\right]. \]

The coefficients of the series \(I_n(jQ)\) are Bessel functions. For even \(n\) they will be real quantities, and for odd \(n\)—purely imaginary ones. The difference of the series (4) and (5) depends on time in an odd manner, and therefore the cosine terms cancel, and then

\[ Sh\,(Q\sin\omega t)=-j\sum_{-\infty}^{+\infty} I_n(jQ)\sin n\omega t. \]

Since the left-hand side is a real quantity, the terms with even index in the right-hand side cancel [4]. If it is taken into account that

\[ I_{-n}(Q)=(-1)^n I_n(Q), \]

then we shall have

\[ Sh\,(Q\sin\omega t)=-2j\sum_{0}^{\infty} I_{2n+1}(jQ)\sin[(2n+1)\omega t]. \]

We substitute the equation obtained into equation (3)

\[ iW=2\frac{AW_1}{Sh\left(\dfrac{s}{\kappa-1}\right)}\,l_{cp} \sum_{0}^{\infty}-jI_{2n}(jQ)\sin[(2n+1)\omega t]. \tag{6} \]

From the properties of Bessel functions we have that the products

\[ -jI_1(jQ);\qquad -jI_5(jQ);\qquad -jI_9(jQ) \quad \text{are positive,} \]

\[ -jI_3(jQ);\qquad -jI_7(jQ) \quad \text{are negative.} \]

Substituting all values into equation (6), we obtain the magnitude of the magnetizing current of the saturation reactor.

\[ i_{gp}=2\,\frac{A W_1 l_{cp}}{WSh\left[\dfrac{\ln\dfrac{A W_2}{A W_1}}{\dfrac{B_2}{B_1}-1}\right]}\times \]

\[ \times \sum_{0}^{\infty}\left\{-j I_{2n+1}\left[ j\,\frac{\ln\dfrac{A W_2}{A W_1}\,U_{\max}\cdot 10^8} {\left(\dfrac{B_2}{B_1}-1\right)g_c\omega B_1 W} \right]\sin(2n+1)\omega t\right\}. \]

From the last equation we can determine all harmonic components of the current in the choke.

The first harmonic of the choke current is

\[ i_{1gp}=2\,\frac{A W_1 l_{cp}}{WSh\left[\dfrac{\ln\dfrac{A W_2}{A W_1}}{\dfrac{B_2}{B_1}-1}\right]}\, j I_1\left[ j\,\frac{\ln\dfrac{A W_2}{A W_1}\,U_{\max}\cdot 10^8} {\left(\dfrac{B_2}{B_1}-1\right)g_c\cdot\omega B_1 W} \right]\sin\omega t. \]

The third harmonic of the choke current is

\[ i_{3gp}=-2\,\frac{A W_1 l_c}{WSh\left[\dfrac{\ln\dfrac{A W_2}{A W_1}}{\dfrac{B_2}{B_1}-1}\right]}\, j I_3\left[ j\,\frac{\ln\dfrac{A W_2}{A W_1}\,U_{\max}\cdot 10^8} {\left(\dfrac{B_2}{B_1}-1\right)g_c\omega B_1 W} \right]\sin 3\omega t. \]

The fifth harmonic of the choke current is

\[ i_{5gp}=2\,\frac{A W_1 l_{cp}}{WSh\left[\dfrac{\ln\dfrac{A W_2}{A W_1}}{\dfrac{B_2}{B_1}-1}\right]}\, j I_5\left[ j\,\frac{\ln\dfrac{A W_2}{A W_1}\,U_{\max}\cdot 10^8} {\left(\dfrac{B_2}{B_1}-1\right)g_c\cdot\omega B_1 W} \right]\sin 5\omega t. \]

etc.

The phase shifts of the first, fifth, ninth, etc., harmonics relative to the induction are equal to zero, while the phase shifts of the third, seventh, etc., harmonics relative to the induction are equal to \(180^\circ\) (a lagging phase shift). If hysteresis is neglected, the variation of the current in the choke can be constructed graphically. Such a graph is shown in Fig. 4.

In view of the fact that odd harmonic e.m.f.’s are induced in the saturating choke, while the applied voltage of the source contains no higher harmonics, one can write the second Kirchhoff equation on the basis of the equivalent circuit shown in Figs. 5a and 5b. As is known, the inductive reactance increases with increasing frequency, and

the reactance of the capacitor decreases by the same factor by which the frequency increases. Owing to the presence of higher-harmonic emf components in the reactor, the current in the capacitor will not be sinusoidal. The higher harmonics of the current, flowing through the capacitor, distort the waveform of the current.

Fig. 4.

Fig. 5.

Taking this circumstance into account, one can construct a graph of the variation of the current in the capacitor. Similarly to the reactor, we shall have the phase of the current of the fifth and ninth harmonics equal to zero with respect to the fundamental harmonic, while the phase shifts of the third and seventh rela-

relative to the fundamental harmonic, equal to \(180^\circ\) (lagging phase shift). The resulting current waveform in the capacitor is shown in Fig. 6 and in the oscillogram of Fig. 7.

Fig. 6.

Fig. 7.

The magnitude of the current of the higher harmonic component flowing through the capacitor can be determined from the equation, when the e.m.f. is known,

\[ i_{c(2n+1)} = E_{(2n+1)} (2n+1)\,\omega c, \]

and when the e.m.f. cannot be determined,

\[ i_{c(2n+1)} = i_{\partial p(2n+1)} \frac{(2n+1)\,\omega L_{\Sigma}} {(2n+1)\omega L_{\Sigma} + \dfrac{1}{(2n+1)\omega c}}, \]

where \(L_{\Sigma}\) is the total inductance in the external circuit,
\(i_{\partial p(2n+1)}\) is the current of the higher harmonic component in the choke.

As the frequency increases, the current through the capacitor increases. Therefore it may be assumed that mainly a current of triple frequency will flow through the external circuit. This current will be equal to the difference between the currents in the saturating choke and in the capacitor.

When this current flows through the tripled-frequency transformer, an e.m.f. of triple frequency will be induced in the secondary winding, as shown in the oscillogram of Fig. 8.

Maintaining a constant e.m.f. can be accomplished by varying the current flowing through the higher-frequency transformer. This is achieved either by changing the applied voltage

to the circuit, or by changing the inductance of the saturating choke by premagnetizing this choke with direct current. In both cases the e.m.f. of tripled frequency in the choke increases. The presence of higher harmonics leads to distortion of the e.m.f. of triple frequency, which is a drawback.

To obtain a three-phase e.m.f. of triple frequency, special measures must be taken. For this it is sufficient to obtain a two-phase system of voltages with a displacement of \(90^\circ\), and then, by means of the well-known Scott connection, obtain a three-phase system. A \(90^\circ\) displacement of the tripled-frequency system can be obtained if one circuit is connected to a line voltage and the other to the free phase and neutral. The conversion of the two-phase system into a three-phase system can be combined with the transformers of tripled frequency, if the secondary winding of one of the transformers is made with a tap at the midpoint and the secondary windings are connected according to the scheme shown in Fig. 9.

Fig. 8.

In order to equalize the load in all phases, it is desirable to divide the secondary winding of that transformer which is connected with the circuit to the line voltage. To obtain a symmetrical system of voltages of tripled frequency, it is necessary that the voltage \(U_3\) in Fig. 9 be greater than the voltages \(U_1\) and \(U_2\) by a factor of \(\sqrt{3}\), as is seen from the vector diagram of Fig. 10:

Fig. 9.

Fig. 10.

\[ U_3 = \sqrt{3}\,U_1 = \sqrt{3}\,U_2 . \]

Compared with existing frequency-converter circuits, the proposed circuit has a number of advantages, which are as follows.

1. The power sources are completely relieved of the flow through them of large magnetizing currents of the reactors, as occurs in known circuits. Only current of the increased frequency will flow through the power sources.

2. The power factor at the fundamental frequency of such a converter approaches unity, as compared with 0.2–0.3 for existing transformer converters.

3. When ordinary saturating reactors are replaced by reactors with premagnetization, the voltage of the increased frequency can be regulated comparatively simply.

4. The production of three-phase current of increased frequency is achieved with only two circuits and two transformers of increased frequency.

5. Owing to the absence of current of the fundamental frequency in the external circuit (the power-source circuit), the efficiency of such a converter will be higher than that of ordinary transformer converters.

LITERATURE

1. Kotaria A. A. Magnetically saturated transformers as frequency multipliers. Author’s abstract of dissertation. Tbilisi, 1951.

2. Osovets S. M. Design and calculation of saturating reactors. Elektrichestvo, No. 5, 1948.

3. Bessonov L. A. Electric circuits with steel. GEI, 1948.

4. Whittaker and Watson. A Course of Modern Analysis, Part II, GTTI, 1934.