Determination of Current and Voltage Harmonics of a Ferromagnetic Converter by the Point Method
Abstract Generated abstract
This paper addresses the calculation of current and voltage harmonics in ferromagnetic voltage converters used in relay protection, where nonsinusoidal currents and voltages arise especially under asymmetrical network faults. It applies the point transformation method to the nonlinear differential equations of a three-phase input, single-phase output converter, representing instantaneous values by vectors and using matrix operations to obtain currents, phase voltages, output voltage, and harmonic spectra. The method is formulated for both symmetrical and asymmetrical operating modes and is suitable for digital computation. Comparison with experimental data for a converter tested at Tomsk Polytechnic Institute showed agreement within about 5 to 7 percent, which the authors regard as adequate for practical calculations.
Full Text
Determination of Current and Voltage Harmonics of a Ferromagnetic Converter by the Point Method
A. N. Burnashev, N. V. Lisetskii
(Presented by the scientific seminar of the Department of Electric Power Stations and Electrical Systems and Networks)
In [4, 5], methods are proposed for the analytical determination of the harmonic composition of the output voltage of a ferromagnetic current converter (FCT). For the implementation of certain types of relay protection (directional, distance, etc.), in addition to the indicated converters, ferromagnetic voltage converters (FVC) are also required. Frequency triplers with a three-phase input and a single-phase output (the Spinelli circuit), which make use of the phenomenon of saturation of steel cores and the properties of multiphase systems [3], are used as FVCs (Fig. 1).
Fig. 1. Circuit of a ferromagnetic voltage converter
The operating conditions of an FVC and an FCT differ substantially. In an FCT the primary currents are sinusoidal. An FVC operates under conditions in which both currents and voltages are nonsinusoidal. In the case where a symmetrical system of voltages is applied to the primary windings of the FVC (normal operation or a three-phase short circuit), the output voltage contains only harmonics that are multiples of three [1, 7]. Thus, when supplied by a symmetrical voltage system, the FVC operates as a frequency tripler.
Under asymmetrical faults in the protected network, the triangle of line voltages is distorted; therefore the cores of the FPN transformers become saturated unequally, and, in addition to the third harmonics, other odd harmonics also appear in the output-voltage curve \((1, 5, 7,\) etc.).
The approximate method for determining the third harmonic of the output voltage described in [1] is suitable only for the symmetrical mode and, moreover, gives a large error, especially when the supply voltage varies over wide limits.
For harmonic analysis of the output quantities of ferromagnetic voltage converters used for single-system relay protections, a more universal method is required, one equally applicable both to symmetrical and to any asymmetrical modes, as well as to cases in which both the currents and the voltages are nonsinusoidal. In such cases the most effective methods are those that make it possible to determine the instantaneous values of the quantities sought (currents or voltages). Knowing the instantaneous values of the nonsinusoidal quantity sought within the period of its variation, it is easy to compute approximate values of the amplitudes of the harmonics being taken into account, as well as their initial phases.
An approximate method for determining the instantaneous values of a nonsinusoidal quantity in nonlinear circuits was first proposed by Professor V. Yu. Lomonosov [6]. In [2] the method was further developed as applied to the calculation of circuits with rectifiers and was called the method of point transformations (point method).
Briefly, the essence of the method may be set forth as follows: by means of the so-called direct point transformation, a point representation of the original nonlinear differential equation (or system of such equations) is written. As a result of such a transformation, we obtain a matrix algebraic equation for the quantity sought. Solving this equation by one of the known methods, we find the sought quantity in point form. By means of the inverse point transformation, from the point representation of the sought quantity we pass to the “original,” that is, to the expression of the sought quantity as a function of time.
Let us consider the calculation of the harmonic composition of the FPN output voltage in the no-load mode by the point method. On the basis of Kirchhoff’s first and second laws, one can write a system of differential equations for the unknown primary currents. In our case there are two such unknown currents (the third current can be determined from Kirchhoff’s first law). Neglecting the active resistance of the windings, we have:
\[ U_{\mathrm{AB}}=(L_{\mathrm{A}}+L_{S})\frac{di_{\mathrm{A}}}{dt} -(L_{\mathrm{B}}+L_{S})\frac{di_{\mathrm{B}}}{dt}, \]
\[ U_{\mathrm{BC}}=(L_{\mathrm{C}}+L_{S})\frac{di_{\mathrm{A}}}{dt} +(L_{\mathrm{C}}+L_{\mathrm{B}}+2L_{S})\frac{di_{\mathrm{B}}}{dt}. \tag{1} \]
Let us write the same equations in point form:
\[ \vec{U}_{\mathrm{AB}}=\omega\left[ J(\check{L}_{\mathrm{A}}+\check{L}_{S})\vec{i}_{\mathrm{A}} - J(\check{L}_{\mathrm{B}}+\check{L}_{S})\vec{i}_{\mathrm{B}} \right], \]
\[ \vec{U}_{\mathrm{BC}}=\omega\left[ J(\check{L}_{\mathrm{C}}+\check{L}_{S})\vec{i}_{\mathrm{A}} + J(\check{L}_{\mathrm{B}}+\check{L}_{\mathrm{C}}+2\check{L}_{S})\vec{i}_{\mathrm{B}} \right], \tag{2} \]
where
\[
\vec{U}_{\mathrm{AB}},\ \vec{U}_{\mathrm{BC}},\ \vec{U}_{\mathrm{CA}}
\]
— point representations of the line voltages applied to the primary windings of the FPN;
\[
\vec{i}_{\mathrm{A}},\ \vec{i}_{\mathrm{B}},\ \vec{i}_{\mathrm{C}}
\]
— point representations of the primary currents;
\[
\check L_A,\ \check L_B,\ \check L_C
\]
— point representations of the nonlinear differential inductances of the FPN transformers;
\[
\check L_S
\]
— point representation of the leakage inductance;
\[
J
\]
— matrix differentiation operator.
By point representations of currents and voltages are meant vectors whose components are their instantaneous values within the time interval under consideration.
The point representation of a differential inductance is a diagonal matrix whose components are the values of the inductance at the corresponding instantaneous values of the current [2].
The form of the differentiation matrix is determined by the nature of the approximating expression for the current and by the method of choosing the approximation nodes.
In the symmetrical mode, the primary currents contain no harmonics whose numbers are multiples of three, and the approximating expression has the form
\[ i(\theta)=I'_1\sin\theta+I'_5\sin5\theta+I'_7\sin7\theta +I''_1\cos\theta+I''_5\cos5\theta+I''_7\cos7\theta . \tag{3} \]
Higher-order harmonics are neglected because of their smallness. When the triangle of line voltages is distorted, third harmonics appear in the magnetizing current. In this case the approximating trigonometric polynomial takes the form
\[ i(\theta)=I'_1\sin\theta+I'_3\sin3\theta+I'_5\sin5\theta +I''_1\cos\theta+I''_3\cos3\theta+I''_5\cos5\theta . \tag{4} \]
If the hysteresis phenomenon is disregarded, the magnetizing-current curve is bisymmetric, and therefore it is possible to restrict the consideration to only one quarter of the period of the fundamental frequency. We choose the nodes according to the expression
\[ \theta_k=\frac{\pi}{2n}K, \tag{5} \]
where
\[
n
\]
— the number of nodes \((n \ge 2p,\ p\) is the number of harmonics taken into account).
For a known form of the approximating expression and known approximation nodes, the differentiation matrix can be calculated from the formula
\[ J=W'\cdot[W]^{-1}, \tag{6} \]
where
\[
W
\]
— a matrix whose components are the instantaneous values of the trigonometric functions, a linear combination of which is chosen for approximating the unknown nonsinusoidal quantity. Thus, for the unsymmetrical mode the matrix will have the following form:
\[ W= \left| \begin{array}{cccccc} \sin\theta_1 & \sin3\theta_1 & \sin5\theta_1 & \cos\theta_1 & \cos3\theta_1 & \cos5\theta_1 \\ \sin\theta_2 & \sin3\theta_2 & \sin5\theta_2 & \cos\theta_2 & \cos3\theta_2 & \cos5\theta_2 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \sin\theta_n & \sin3\theta_n & \sin5\theta_n & \cos\theta_n & \cos3\theta_n & \cos5\theta_n \end{array} \right| \tag{7} \]
The components of the matrix \(W'\) are the corresponding values of the derivatives of the same functions.
Substituting (6) into (2), we obtain a matrix equation with respect to the point images of the unknown currents, whose order is equal to \(2n\). Solving this equation, say, by the method of iterations, we find the currents \(i_A\) and \(i_B\) in point form. The primary phase voltages are found by means of expressions of the form:
\[ \begin{aligned} \overrightarrow{U}_A &= J\omega L_A^{\vee}\overrightarrow{i}_A,\\ \overrightarrow{U}_B &= J\omega L_B^{\vee}\overrightarrow{i}_B,\\ \overrightarrow{U}_C &= J\omega L_C^{\vee}\overrightarrow{i}_C . \end{aligned} \tag{8} \]
Then the secondary phase voltages are calculated and, finally, the point image of the voltage at the converter output is determined:
\[ \overrightarrow{U}_{\text{out}}=(\overrightarrow{U}_A+\overrightarrow{U}_B+\overrightarrow{U}_C)N \tag{9} \]
or, taking (8) into account:
\[ \overrightarrow{U}_{\text{out}}=J\omega N\left(L_A^{\vee}\overrightarrow{i}_A+L_B^{\vee}\overrightarrow{i}_B+L_C^{\vee}\overrightarrow{i}_C\right), \tag{10} \]
where
\(N\) is the transformation ratio of the FPN transformers.
With the aid of the inverse point transformation we find the vector of the amplitude spectrum of the output-voltage harmonics
\[ \overrightarrow{S}=W^{-1}\overrightarrow{U}_{\text{out}}. \tag{11} \]
The matrix \(J\) is constant and can be calculated in advance. The entire process of calculating the harmonics of the FPN output voltage can be carried out on an electronic digital computer, especially since standard subroutines are available for all the matrix operations considered. A comparison of the results of a calculation performed on the BESM-4 computer with the data of an experimental investigation of an FPN manufactured at the Department of Electric Power Stations of the Tomsk Polytechnic Institute showed their fairly close agreement (the deviation does not exceed 5–7%). Such accuracy is quite satisfactory for practical purposes.
Parameters of the FPN: \(W_1=760\) turns, \(W_2=880\) turns, \(S_{\text{st}}=1.75\cdot10^{-4}\ \text{m}^2\), \(l_{\text{av}}=0.11\ \text{m}\).
Conclusions
-
It has been shown that, for calculating the current and voltage harmonics of an FPN in various operating conditions, the point method can be used.
-
The error of calculation by means of this method does not exceed 7%.
Literature
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A. M. Bambas, V. A. Kulinich, S. V. Shapiro. Static Electromagnetic Converters of Frequency and Number of Phases. GEI, 1961.
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K. A. Krug. Fundamentals of Electrical Engineering. ONTI, 1936.
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Yu. A. Kulaga. Calculation of the output quantities of a ferromagnetic converter. Tomsk TsNTI, information sheet No. 6, 1970.
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N. V. Lisetskii, V. I. Khuduguev, Yu. A. Kulaga. Determination of the harmonic composition of a functional current converter. Proceedings of the East Siberian Technological Institute, issue III, vol. 4, 1968.
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V. Yu. Lomonosov. Periodic processes in nonlinear circuits. “Elektrichestvo,” No. 7, 1952.
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L. L. Rozhanskii. Static electromagnetic frequency converters. GEI, 1959.