CALCULATION OF THE EXTERNAL CHARACTERISTIC AND SHORT-CIRCUIT CURRENTS OF A FERROMAGNETIC FREQUENCY TRIPLER

Ya. V. Petrov, A. I. Zaitsev

(Recommended by the scientific seminar of the Electromechanics Faculty)

Most known methods for calculating the external characteristic of a ferromagnetic frequency tripler have substantial shortcomings: the need for laborious calculations when taking into account the nonlinearity of the magnetization curve of the tripler transformers; moreover, as a rule, it is necessary to construct a considerable part of the characteristic in order to find on it the point corresponding to a specified load; a large calculation error if, for the purpose of simplifying the calculations, the output of the tripler is represented by an active linear two-terminal network; and, in a number of cases, a lack of universality.

The most general and accurate analytical method for calculating the external characteristic of a tripler was proposed by L. L. Rozhanskii [1], improved by him through the introduction of a system of relative units [2]. Another original analytical method for refined calculations was developed by S. V. Shapiro [3]. Both methods are based on approximating the magnetization curve of the core by a hyperbolic sine. A certain shortcoming of both methods is the unavoidable need to construct a known part of the external characteristic in order to establish the relationship between a given load current and the voltage corresponding to it.

In the literature known to us, the calculation of short-circuit currents beyond the capacitors of series compensation (SC) is not given; meanwhile, when choosing the operating voltage of the capacitors, such currents must be taken into account.

We neglect harmonics higher than the third, losses in the steel, and the leakage of the secondary winding (the latter will be taken into account only in calculating the short-circuit currents beyond the SC capacitors). The leakage resistance and active resistance of the primary winding are taken into account by a corresponding reduction in the value of the fundamental harmonic of the induction. The magnetization curve of the transformer steel, approximated by a hyperbolic sine, is obtained at alternating current of frequency 50 Hz by the ammeter-voltmeter method; moreover, the amplitude values of the induction and magnetic-field strength are calculated from the rms values of the emf and the magnetizing current.

Between the output quantities of the frequency tripler at frequency \(3f\) and one input phase at frequency \(f\) there exists a one-to-one relationship. Thus, under no-load conditions of the tripler, the magnetic-equilibrium equation for the amplitude values of the quantities may be written as

\[ \sqrt{H_{1M}^{2}+H_{3M}^{2}}=\alpha\,\operatorname{sh}\beta\,\sqrt{B_{1M}^{2}+B_{3M}^{2}}, \tag{1} \]

where:

\[ H_{1M}=\frac{\sqrt{2}\,I_{0}w_{1}}{l_{c}} \]

is the amplitude of the equivalent sinusoid of the magnetic-field intensity at frequency \(f\);

\[ H_{3M}=\frac{\sqrt{2}\,I_{3k}\cdot w_{2}}{l_{c}} \]

is the amplitude of the third harmonic of the magnetic-field intensity;

\(B_{1M}, B_{3M}\) are the amplitudes of the fundamental and third harmonics of the magnetic induction;

\(\alpha, \beta\) are the coefficients of the approximating expression;

\(I_{0}\) is the effective value of the magnetizing current of the single-phase transformer included in the tripler, for a specified induction \(B_{1M}\); it is determined from the magnetization curve or by measurement;

\(I_{3k}\) is the short-circuit current of the tripler (without capacitive compensation), assumed to be purely inductive;

\(w_{1}, w_{2}\) are the numbers of turns of the phase of the primary and secondary windings;

\(l_{c}\) is the length of the mean line of magnetic induction.

After simple transformations:

\[ \sqrt{I_{0}^{2}+I_{3k}^{2}\left(\frac{w_{2}}{w_{1}}\right)^{2}} = \frac{\alpha l_{c}}{\sqrt{2}w_{1}}\, \operatorname{sh}\beta\,\sqrt{B_{1M}^{2}+B_{3M}^{2}}. \tag{2} \]

When the tripler is loaded by a current \(I_{3}\), equality (2) undergoes the following changes:

the reactive component of the load current \(I_{3p}=I_{3}\cdot\sin\varphi_{3}\) is algebraically added to the current \(I_{3k}\) (the plus sign is taken for a capacitive current), in connection with which, for a specified induction \(B_{1M}\), the induction \(B_{3M}\) changes accordingly;

the reaction of the active component of the load current \(I_{3a}=I_{3}\cdot\cos\varphi_{3}\) may also be expressed by a magnetic-equilibrium equation of type (2), if it is assumed that this component of the current of frequency \(3f\) is compensated by an increase in the magnetizing current of frequency \(f\) of the single-phase transformer to

\[ I_{0H}=\sqrt{I_{0}^{2}+I_{3a}^{2}\left(\frac{w_{2}}{w_{1}}\right)^{2}}, \tag{3} \]

to which there corresponds another induction \(B_{1MH}\), determined from the magnetization curve of the transformer at

\[ H_{1MH}=\frac{\sqrt{2}\,I_{0H}\cdot w_{1}}{l_{c}} \tag{4} \]

or, approximately:

\[ H_{1MH}\cdot l_{c}=\sqrt{2}\,I_{0H}\cdot w_{1}=\alpha l_{c}\cdot\operatorname{sh}\beta B_{1MH}, \]

whence

\[ B_{1\mathrm{mH}} \approx \frac{1}{\beta}\ln \frac{2\sqrt{2}\,I_{\mathrm{H}}\cdot w_1}{a\cdot l_c} = \frac{1}{\beta}\ln \frac{ 2\sqrt{2}\,w_1 \sqrt{I_0^2+I_3^2(1-\sin^2\varphi_3)\left(\frac{w_2}{w_1}\right)^2} }{ a\cdot l_c }. \tag{5} \]

Then, instead of (2), we obtain:

\[ \begin{aligned} \sqrt{ I_0^2+\left(I_{3\mathrm{k}}\pm I_3\cdot\sin\varphi_3\right)^2 \left(\frac{w_2}{w_1}\right)^2 } &= \frac{a\cdot l_c}{\sqrt{2}\,w_1}\, \operatorname{sh}\beta \sqrt{B_{1\mathrm{mH}}^2+B_{3\mathrm{mH}}^2} \\ &= \frac{a l_c}{\sqrt{2}\,w_1}\, \operatorname{sh}\beta \sqrt{ \left[ \frac{1}{\beta}\ln \frac{ 2\sqrt{2}\,w_1 \sqrt{ I_0^2+I_3^2(1-\sin^2\varphi_3) \left(\frac{w_2}{w_1}\right)^2 } }{ a\cdot l_c } \right]^2 +B_{3\mathrm{mH}}^2 } \\ &\approx \frac{a\cdot l_c}{2\sqrt{2}\,w_1}\, e^\beta \sqrt{ \left[ \frac{1}{\beta}\ln \frac{ 2\sqrt{2}\,w_1 \sqrt{ I_0^2+I_3^2(1-\sin^2\varphi_3) \left(\frac{w_2}{w_1}\right)^2 } }{ a\cdot l_c } \right]^2 +B_{3\mathrm{mH}}^2 }. \end{aligned} \tag{6} \]

Solving (6) with respect to \(B_{3\mathrm{mH}}\),

\[ \begin{aligned} (\beta B_{3\mathrm{mH}})^2 &= \ln^2 \left[ \frac{2\sqrt{2}\,w_1}{a\cdot l_c} \sqrt{ I_0^2+ \left(I_{3\mathrm{k}}\pm I_3\cdot\sin\varphi_3\right)^2 \left(\frac{w_2}{w_1}\right)^2 } \right] \\ &\quad -\ln^2 \left[ \frac{2\sqrt{2}\,w_1}{a\cdot l_c} \sqrt{ I_0^2+I_3^2(1-\sin^2\varphi_3) \left(\frac{w_2}{w_1}\right)^2 } \right]. \end{aligned} \tag{7} \]

The emf of the secondary circuit under load and with longitudinal capacitive compensation is

\[ \begin{aligned} E_3 &= 3\cdot 4.44\cdot 3 f w_2 B_{3\mathrm{mH}}\cdot Q_c\cdot 10^{-8} = A w_2 B_{3\mathrm{mH}}\cdot Q_c = I_3 Z_3 \\ &= I_3\frac{x_{\mathrm{H}}-x_c}{\sin\varphi_3} = I_3\frac{Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c}{\sin\varphi_3}, \end{aligned} \tag{8} \]

where

\[ A=3\cdot 4.44\cdot 3 f\cdot 10^{-8}; \]

\[ \sin\varphi_3 = \frac{x_{\mathrm{H}}-x_c}{ \sqrt{(r_3+r_{\mathrm{H}})^2+(x_{\mathrm{H}}-x_c)^2} } = \frac{Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c}{ \sqrt{ (r_3+Z_{\mathrm{H}}\cdot\cos\varphi_{\mathrm{H}})^2+ (Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c)^2 } }. \tag{9} \]

(here \(\cos\varphi_{\mathrm{H}}, x_{\mathrm{H}}, r_{\mathrm{H}}\) are, respectively, the power factor and the load resistances).

Comparing (7) and (8) and substituting \(\sin\varphi_3\) from (9), we finally obtain, for any type of load and longitudinal capacitive compensation:

\[ \begin{aligned} &\ln^2 \left\{ \frac{2\sqrt{2}\,w_1}{a\cdot l_c} \sqrt{ I_0^2+ \left[ I_{3\mathrm{k}}\pm I_3 \frac{Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c}{ \sqrt{ (r_3+Z_{\mathrm{H}}\cdot\cos\varphi_{\mathrm{H}})^2+ (Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c)^2 } } \right]^2 \left(\frac{w_2}{w_1}\right)^2 } \right\} \\ &\quad -\ln^2 \left\{ \frac{2\sqrt{2}\,w_1}{a\cdot l_c} \sqrt{ I_0^2+ I_3^2 \left[ 1- \frac{(Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c)^2}{ (r_3+Z_{\mathrm{H}}\cdot\cos\varphi_{\mathrm{H}})^2+ (Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c)^2 } \right] \left(\frac{w_2}{w_1}\right)^2 } \right\} \\ &= \left(\frac{\beta I_3}{A w_2 Q_c}\right)^2 \left[ (r_3+Z_{\mathrm{H}}\cdot\cos\varphi_{\mathrm{H}})^2+ (Z_{\mathrm{H}}\cdot\sin\varphi_{\mathrm{H}}-x_c)^2 \right]. \end{aligned} \tag{10} \]

Thus, to calculate the voltage across the load, it is necessary to have specified the current and power factor of the load, the value of the longitudinal-compensation capacitance, the design data and the magnetization curve of the transformer of the tripler, and the short-circuit current of the latter. Solving the transcendental equation (10) by one of the known approximate methods, we find the value of the load resistance \(Z_{\mathrm{H}}\), and then also the voltage

\[ U_{\mathrm{H}}=I_3 Z_{\mathrm{H}}. \]

With a purely active load and longitudinal compensation, equation (10) is somewhat simplified:

\[ \ln^2\!\left[ \frac{2\sqrt{2}\,w_1}{\alpha l_c} \sqrt{I_0^2+\left(I_{3k}\pm I_3\sin\varphi_3\right)^2 \left(\frac{w_3}{w_1}\right)^2} \right] - \ln^2\!\left[ \frac{2\sqrt{2}\,w_1}{\alpha l_c} \sqrt{I_0^2+I_3^2\left(1-\sin^2\varphi_3\right) \left(\frac{w_2}{w_1}\right)^2} \right] = \left( \frac{\beta I_3 x_c}{A w_2 Q_c \sin\varphi_3} \right)^2, \tag{11} \]

that is, in this case first \(\sin S_3\) is determined, then the load resistance

\[ r_n=x_c\operatorname{ctg}\varphi_3-r_3 \]

and, finally, the voltage across the load \(U_n\).

Equation (10) or (11) can be used in other variants: to find points of the external characteristic of a tripler without compensation \((x=0)\) and with transverse compensation (the capacitance is included in the load); to calculate the value of the capacitance for longitudinal or transverse compensation for specified

\[ I_3,\ U_n,\ \cos\varphi_n \quad \text{and} \quad Z_n=\frac{U_n}{I_3}; \]

to determine the steady-state short-circuit currents beyond the capacitors of the longitudinal-compensation circuit.

For low-power installations with a tripler, the greatest current in the circuit will occur during a short circuit directly beyond the capacitors of the longitudinal-compensation circuit, since the active resistance of the conductor is at least an order of magnitude higher than the inductive resistance, and as the short-circuit point moves away from the capacitors the currents decrease. Consequently, it is possible to confine the calculation to the short-circuit current directly beyond the capacitors of the longitudinal-compensation circuit; for this it is convenient to use expression (11), substituting into it the known value

\[ \sin\varphi_3= \frac{x_{3s}-x_c} {\sqrt{(r_3)^2+(x_{3s}-x_c)^2}}, \]

where \(x_{3s}\) is the leakage reactance of the secondary winding of the tripler (calculated as for ordinary reactors with a steel core).

Since under short-circuit conditions \(\sin\varphi_3 \to 1.0\), it is permissible to take \(\sin\varphi_3=1.0\); in this case expression (11) is noticeably simplified.

If the three-phase output of the tripler is made according to a T-connection, the calculation of the external characteristic and of the short-circuit currents contains no fundamentally new elements—it is only necessary to carry out all calculations for the phase of the equivalent \(Y\).

Although the fundamental harmonic of the induction does not enter directly into the final expressions (10) and (11), the currents \(I_0\) and \(I_{3k}\), especially \(I_0\), depend to a considerable degree on its value, since the tripler operates in the region of strong saturation. The quality of the calculations is largely determined by the reliability of the values \(I_0\) and \(I_{3k}\); therefore, the value of the fundamental harmonic of the induction should preferably be established as accurately as possible, that is, taking into account the voltage drop in the resistances of the primary winding of the tripler. For an experimental model of a 300-va tripler (steel E31, 0.35 mm; \(B_{1m}=15700\text{–}20300\) G) good results were obtained with a ratio of emf to input voltage \(E_1=(0.96\div0.98)U_1\).

The accuracy of the calculations is also increased if the coefficients \(\alpha\) and \(\beta\) are found not by the method of selected points, but by the method of least squares or of averages.

The error in calculating the external characteristics of the indicated model of the tripler \(\left(B_{1\mathrm{m}} = 15700—20300\ \text{G},\ \dfrac{I_3}{I_{3\mathrm{k}}} = 0 \div 3.3\right)\), with SC and active or active-inductive load, did not exceed \(\pm 6\%\); the steady-state currents of three- and two-phase short circuits beyond the capacitors were determined with errors of no more than \(\pm 10\%\).

Conclusions

1. Expressions have been proposed for the analytical or graphical-analytical calculation of the external characteristic of a ferromagnetic frequency tripler and of the steady-state short-circuit currents beyond the capacitors of the longitudinal compensation.

2. The expressions are suitable over a wide range of inductions for loads of different character and magnitude and for various grades of electrical steels; i.e., they possess a certain universality.

3. The principal drawback of the proposed method is the laboriousness of solving the transcendental equation.

Literature

1. L. L. Rozhanskii. Static frequency multipliers in telecontrol and telesignaling circuits. Dissertation, 1947.

2. L. L. Rozhanskii. Static electromagnetic frequency converters. Gosenergoizdat, 1959.

3. S. V. Shapiro. Analytical calculation of the external characteristic of a frequency tripler with longitudinal capacitive compensation. Proceedings of the A. A. Zhdanov GPI, vol. XVI, issue 5, 1960.