CALCULATION OF PERIODIC PROCESSES IN A FERROMAGNETIC CURRENT CONVERTER BY THE COMPLEX METHOD WITH CURRENT TRANSFORMERS CONNECTED IN STAR

V. K. Skripko

(Presented by the scientific seminar of the Department of Electrical Stations)

A ferromagnetic current converter (FCC), serving to convert a three-phase system of currents into a single-phase system of voltages, consists of three single-phase transformers with saturated cores, whose windings are connected analogously to the arrangement of the frequency-tripling system of Spinelli [1]. The conversion of frequency and number of phases in the FCC occurs owing to the saturation effect of the steel of the cores of the single-phase transformers. Two equivalent variants of connecting current converters in relay-protection circuits are possible.

1. The secondary windings of the current transformers (CT) are connected in delta, and the primary windings of the FCC are connected in star.

2. The secondary windings of the current transformers are connected in star, and the primary windings of the converter—in an equal-arm zigzag.

In the present paper a method is set forth for calculating periodic processes in a current converter under three-phase and two-phase short circuits in the supply network, using variant 2 as an example. The analytical determination of the voltage harmonics at the output of the FCC in no-load operation is carried out by the complex method [2] under the following assumptions:

1. Leakage fluxes, hysteresis losses, and eddy currents are neglected.

2. Remagnetization of the core occurs not along the hysteresis loop, but along the principal magnetization curve.

We approximate the principal magnetization curve by the trigonometric polynomial [3]

\[ B = \sum_{n=1}^{n} b_{2n-1} \sin (2n - 1)\frac{H}{H_0}, \tag{1} \]

where \(n\) is the index of expansion of the principal magnetization curve into a Fourier series. \(H_0\) is an arbitrarily chosen value of the field strength corresponding to one radian; in this case \(H\) may vary from \(-\dfrac{\pi H_0}{2}\) to \(+\dfrac{\pi H_0}{2}\).

The coefficients \(b_{2n-1}\) are determined by the method of trigonometric interpolation [4].

The form of the currents in the primary circuit of the converter remains sinusoidal over a wide range of their variation; therefore, in the calculation we proceed from the condition of sinusoidality of the magnetizing force:

\[ B=\sum_{n=1}^{n} b_{2n-1}\sin \frac{(2n-1)H_m\sin(\omega t+\alpha)}{H_0}, \tag{2} \]

where \(\alpha\) is the initial phase of the current.

Expanding this expression in a series in Bessel functions of increasing order [5], we obtain

\[ B=\sum_{n=1}^{n}\sum_{k=1}^{\infty}2b_{2n-1}J_{2k-1}\frac{(2n-1)H_m}{H_0}\sin(2k-1)(\omega t+\alpha), \tag{3} \]

where \(J_{2k-1}\dfrac{(2n-1)H_m}{H_0}\) is the Bessel function of the first kind of order \((2k-1)\) with a real argument.

Passing from (3) to the complex representation of the induction, we obtain:

\[ K_\nu[B]=\dot{B}_\nu=\frac{j2}{T}\int_{0}^{T}e^{-j\nu\omega t} \left[ \sum_{n=1}^{n}\sum_{k=1}^{\infty}2b_{2n-1}J_{2k-1}\frac{(2n-1)H_m}{H_0}\times \right. \]

\[ \left. {}\times \sin(2k-1)(\omega t+\alpha) \right]dt, \tag{4} \]

where \(\nu\) indicates the ordinal number of the harmonic.

Changing in (4) the order of integration and summation and using the property of the definite integral, we arrive at the following relation

\[ \dot{B}_\nu= \sum_{n=1}^{n}\sum_{k=1}^{\infty} 2b_{2n-1}J_{2k-1}\frac{(2n-1)H_m}{H_0} \int_{0}^{T}\frac{j2}{T}e^{-j\nu\omega t}\times \]

\[ {}\times \sin(2k-1)(\omega t+\alpha)\,dt. \tag{5} \]

The integral in expression (5) is the complex representation of the trigonometric function \(\sin(2k-1)(\omega t+\alpha)\).

In complex calculation it is proved that

\[ K_\nu[\sin(2k-1)(\omega t+\alpha)] = \begin{cases} 0, & \text{if } \nu\ne 2k-1,\\ e^{j\nu\alpha}, & \text{if } \nu=2k-1. \end{cases} \]

Consequently, the expression for the complex magnetic induction will have the following form

\[ \dot{B}_\nu= 2e^{j\nu\alpha}\sum_{n=1}^{n}b_{2n-1}J_\nu\frac{(2n-1)H_m}{H_0}. \tag{6} \]

The complex amplitude of the \(\nu\)-th harmonic of the voltage on the secondary winding of one FCT transformer will be equal to

\[ \dot{U}_{m\nu} = j2\nu\omega W_{11}Se^{j\nu\alpha} \sum_{n=1}^{n}b_{2n-1}J_\nu\frac{(2n-1)H_m}{H_0}, \tag{7} \]

where \(S\) is the active cross-sectional area of the core;

\(W_{11}\) is the number of turns of the secondary winding of one transformer.

In Fig. 1, a shows the current distribution in the primary circuit of the current converter under symmetrical conditions.

Fig. 1

The currents in the secondary windings of the current transformers will be equal in magnitude and shifted relative to one another by \(120^\circ\):

\[ \left. \begin{aligned} i_a &= \sqrt{2}\,I \sin(\omega t+\alpha),\\ i_b &= \sqrt{2}\,I \sin\left(\omega t+\alpha-\frac{2\pi}{3}\right),\\ i_c &= \sqrt{2}\,I \sin\left(\omega t+\alpha+\frac{2\pi}{3}\right). \end{aligned} \right\} \tag{8} \]

where \(I\) is the rms value of the current in the secondary winding of the CT,
\(\alpha\) is the initial phase of the current.

Each current will correspond to its own magnetic-field intensity in the core; moreover, the resultant field intensity in the core of the phase “\(a\)” transformer will be equal to

\[ H_a=\frac{\sqrt{2}\sqrt{3}\,W_1}{l}\,I\sin\left(\omega t+\alpha+\frac{\pi}{6}\right), \tag{9} \]

where \(W_1\) is the number of turns of the primary winding of the transformer,
\(l\) is the mean length of the magnetic line of force in the core.

According to equation (7), the complex of the rms value of the \(\nu\)-th harmonic of the voltage on the secondary winding of the phase “\(a\)” transformer will be equal to

\[ U_{\nu a} = j\,\frac{2}{\sqrt{2}}\,\nu\omega W_{\mathrm{II}}S e^{j\nu\left(\alpha+\frac{\pi}{6}\right)} \sum_{n=1}^{\infty} b_{2n-1}J_\nu \frac{2.45(2n-1)W_1}{H_0l}\,I. \tag{10} \]

Under symmetrical conditions, the periodic processes in the phases will be shifted relative to one another by \(1/3\) of the period of the fundamental frequency. These phase shifts can be taken into account in the calculation by means of the phase rotation operator for the \(\nu\)-th harmonic

\[ a_\nu = e^{j\frac{2\pi}{3}\nu}. \]

The complex rms value of the voltage at the output of the current converter for the \(\nu\)-th harmonic is equal to the geometric sum of the \(\nu\)-th harmonics of the phase voltages

\[ \dot U_{\nu \mathrm{II}} = j\frac{2}{\sqrt{2}}\,\nu\omega W_{\mathrm{II}}S e^{j\nu\left(\alpha+\frac{\pi}{6}\right)} \sum_{n=1}^{n} b_{2n-1}\cdot J_\nu \frac{2.45(2n-1)W_{\mathrm{I}}I}{H_0 l} \left(1+a_\nu^2+a_\nu\right). \tag{11} \]

And since

\[ \Pi_\nu=\frac{1}{3}\left(1+a_\nu^2+a_\nu\right) = \begin{cases} 0, & \text{if } \nu \ne 3p,\\ 1, & \text{if } \nu = 3p, \end{cases} \quad (p\text{ is an integer}), \]

only voltage harmonics that are multiples of three will be present at the converter output.

The complex rms value of the third-harmonic voltage at the output of the FPT will be

\[ \dot U_{3\mathrm{II}} = j\frac{18}{\sqrt{2}}\,\omega W_{\mathrm{II}}S e^{j3\left(\alpha+\frac{\pi}{6}\right)} \sum_{n=1}^{n} b_{2n-1}\cdot J_3 \frac{2.45(2n-1)W_{\mathrm{I}}I}{H_0 l}. \tag{12} \]

Figure 1, b shows the current distribution in the primary circuit of the current converter for a two-phase short circuit between phases B and C, without taking the load currents into account.

With the adopted positive directions of the currents \(I_b=-I\), the instantaneous values of the field intensities in the cores of the converter phases will respectively be equal to:

\[ \left. \begin{aligned} H_a &= -\frac{\sqrt{2}W_{\mathrm{I}}}{l}\,I\sin(\omega t+\alpha),\\[4pt] H_b &= \frac{2\sqrt{2}W_{\mathrm{I}}}{l}\,I\sin(\omega t+\alpha),\\[4pt] H_c &= -\frac{\sqrt{2}W_{\mathrm{I}}}{l}\,I\sin(\omega t+\alpha). \end{aligned} \right\} \tag{13} \]

Then the complex rms value of the voltage at the converter output for the \(\nu\)-th harmonic in a two-phase short circuit will be equal to the geometric sum of the \(\nu\)-th harmonics of the phase voltages

\[ \dot U_{\nu \mathrm{II}} = j\frac{2}{\sqrt{2}}\,\nu\omega W_{\mathrm{II}}S e^{j\nu\alpha} \left( \sum_{n=1}^{n} b_{2n-1}\cdot J_\nu \frac{2\sqrt{2}(2n-1)W_{\mathrm{I}}I}{H_0 l} - 2\sum_{n=1}^{n} b_{2n-1}\cdot J_\nu \frac{\sqrt{2}(2n-1)W_{\mathrm{I}}I}{H_0 l} \right). \tag{14} \]

When the first six harmonics are taken into account in the approximating expression, the calculation error does not exceed 10%.

Conclusions

1. The voltage at the output of the ferromagnetic current converter has a harmonic composition that depends on the type of fault.

2. The calculation method presented in the article makes it possible, with accuracy sufficient for practice, to determine the voltage harmonics at the converter output in various modes.

References

1. N. V. Lisetskii, V. I. Khuduguev, Yu. A. Kulaga, V. K. Skripko, A. N. Burnashev. Single-system relay-protection circuits. — In: Electrification of Metallurgical Enterprises of Siberia. Tomsk State University Press, Tomsk, 1971.

2. G. E. Pukhov. Complex Calculus and Its Application. Publishing House of the Academy of Sciences of the USSR, Kiev, 1961.

3. L. S. Goldfarb, G. R. Gerts(en)berg. Determination of current and voltage harmonics in electrical circuits containing iron. — Elektrichestvo, 1939, No. 1.

4. E. Whittaker, G. Robinson. Mathematical Processing of Observation Results. ONTI, 1935.

5. E. Filippov. Nonlinear Electrical Engineering. Moscow, “Energiya,” 1968.