Determination of the Harmonic Composition of the Current and Voltage of a Ferromagnetic Voltage Converter under Balanced Operating Conditions
Abstract Generated abstract
The paper examines the harmonic composition of current and voltage in a ferromagnetic voltage converter used to transform a balanced three-phase voltage system into a single-phase output for relay-protection applications. Under assumptions neglecting hysteresis and eddy-current losses, the authors approximate the magnetization curve by a trigonometric polynomial and apply a complex calculation method with Bessel-function expressions for magnetic permeability and induction harmonics. The analysis shows that, under symmetrical no-load conditions, only voltage harmonics that are multiples of three appear at the converter output, and it provides formulas for the primary current and third-harmonic output voltage. Experimental measurements using an electrodynamic wattmeter are compared with calculated curves, indicating that the proposed method gives accuracy sufficient for practical determination of converter harmonics.
Full Text
DETERMINATION OF THE HARMONIC COMPOSITION OF THE CURRENT AND VOLTAGE OF A FERROMAGNETIC VOLTAGE CONVERTER UNDER SYMMETRICAL CONDITIONS
N. V. LISETSKII, V. K. SKRIPKO, A. N. BURNASHEV
(Presented by the scientific seminar of the Department of Electric Power Stations)
A significant simplification of relay-protection circuits can be achieved by converting a three-phase system of currents or voltages into a single-phase system of voltages. In [1] the possibility is considered of using for this purpose a frequency-tripling system (Spinelli system). Devices intended for converting a three-phase voltage system into a single-phase one are called by us ferromagnetic voltage converters (FVC).
The primary windings of the FVC are connected in star and are supplied from measuring voltage transformers (Fig. 1). In the normal operating mode, during three-phase short circuits and line swings, the line voltages form a symmetrical three-phase system.
Fig. 1
The analytical determination of the current and voltage harmonics of the FVC in the no-load mode is carried out by the complex method [2] under the following assumptions:
- We neglect hysteresis losses and eddy currents.
- The remagnetization of the core occurs not along the hysteresis loop, but along the basic magnetization curve obtained by the ballistic method.
We approximate the basic magnetization curve by a trigonometric polynomial [3]:
\[ B = \sum_{n=1}^{n} g_{2n-1}\sin(2n-1)\frac{H}{H_0}, \tag{1} \]
where
\(H_0\) is the value of the field intensity corresponding to one radian; in this case \(H\) may vary from \(-\dfrac{\pi H_0}{2}\) to \(+\dfrac{\pi H_0}{2}\),
\(n\) is the index of expansion of the main induction curve into a Fourier series.
The coefficients \(b_{2n-1}\) are determined by the method of trigonometric interpolation [4].
In view of the absence of a neutral conductor, current harmonics that are multiples of three will not flow in the primary circuit. For this reason, the currents in it may approximately be regarded as sinusoidal. Therefore, in the approximating expression (1) it is sufficient to take into account only the first harmonic of the field intensity. Owing to the adopted idealization, the calculation as a whole is greatly simplified. Then equation (1) is written as
\[ B=\sum_{n=1}^{n} b_{2n-1}\sin \frac{(2n-1)H_{m1}\sin\omega t}{H_0}. \tag{2} \]
The complex coefficient relating the quantities \(B\) and \(H\) in complex calculation is the complex magnetic permeability of the \(\nu\)-th order
\[ \bar{\mu}_{\nu}=\frac{K_{\nu}(B)}{K_{\nu}(H)}. \]
In [5] an expression was obtained for the complex magnetic permeability
\[ \bar{\mu}_{\nu}=\frac{2}{\bar{H}_{\nu}}\sum_{n=1}^{n} b_{2n-1}\cdot J_{\nu}\frac{(2n-1)H_{m1}}{H_0}, \tag{3} \]
where
\[ J_{\nu}\frac{(2n-1)H_{m1}}{H_0} \]
is the Bessel function of the first kind of order \(\nu\) with real argument.
Applying Kirchhoff’s second law and the law of total current to the Spinelli circuit, we compose the complex equations of the converter in no-load operation:
\[ \left\{ \begin{aligned} &Z_{\nu 1}(\dot{I}_{m\nu a}-\dot{I}_{m\nu b})+j\nu\omega W_1 S(\bar{\mu}_{\nu a}\dot{H}_{m\nu a}-\bar{\mu}_{\nu b}\dot{H}_{m\nu b})=\dot{U}_{m\nu ab},\\ &\dot{I}_{m\nu a}W_1=\dot{H}_{m\nu a}l,\qquad \dot{I}_{m\nu b}W_1=\dot{H}_{m\nu b}l,\\ &\dot{I}_{m\nu b}=a_{\nu}^{2}\dot{I}_{m\nu a},\qquad \dot{H}_{m\nu b}=a_{\nu}^{2}\dot{H}_{m\nu a}, \end{aligned} \right. \tag{4} \]
where
\(\dot{I}_{m\nu a}\) is the complex amplitude of the \(\nu\)-th harmonic of the current in the primary winding of phase “a”,
\(\dot{H}_{m\nu a}\) is the complex amplitude of the \(\nu\)-th harmonic of the field intensity in the core of phase “a”,
\(W_1\) is the number of turns of the primary winding of one transformer,
\(Z_{\nu 1}\) is the complex impedance of the \(\nu\)-th order of the primary winding of one phase, including the active component and the leakage resistance,
\(S\) is the effective cross-sectional area of the core,
\(l\) is the mean length of the magnetic field line in the core,
\[ a_{\nu}=e^{j\frac{2\pi}{3}\nu} \]
is the phase rotation operator for the \(\nu\)-th harmonic.
Using the phase operator, one can write
\[ (1-a_\nu^2)\left(Z_{\nu 1}\frac{l}{W_1}+j\nu\omega W_1S\bar{\mu}_\nu\right)\dot{H}_{m\nu a}=\dot{U}_{mab}, \tag{5} \]
where
\[ \bar{\mu}_\nu=\bar{\mu}_{\nu a}=\bar{\mu}_{\nu b}=\bar{\mu}_{\nu c} \]
by virtue of the symmetry of the circuit. Setting \(\nu=1\) in (5), we obtain:
\[ \dot{H}_{m1a}=H_{m1a}e^{j\alpha}= \frac{\dot{U}_{mab}\cdot e^{-j\frac{\pi}{6}}} {\sqrt{3}\left(\dfrac{lZ_{11}}{W_1} +j\dfrac{2\omega W_1S}{H_{m1a}} \sum_{n=1}^{n}\varepsilon_{2n-1}\cdot J_{\frac{(2n-1)H_{m1a}}{H_0}}\right)}. \tag{6} \]
This equation is solved by the iteration method.
The rms value of the current in the primary circuit of the converter is determined on the basis of the total-current law
\[ \dot{I}_{1a}=\frac{l\dot{H}_{m1a}}{\sqrt{2}W_1}. \tag{7} \]
The complex amplitude of the \(\nu\)-th harmonic of the voltage at the output of the FVC in no-load operation will be equal to:
\[ \dot{U}_{m\nu 11}=j\nu\omega W_{11}S(\dot{B}_{m\nu a}+\dot{B}_{m\nu b}+\dot{B}_{m\nu c}). \tag{8} \]
Or, taking the phase operator into account, one can write:
\[ \dot{U}_{m\nu 11}=j\nu\omega W_{11}S\dot{B}_{m\nu a}(1+a_\nu^2+a_\nu). \tag{9} \]
Analogously to the complex coupling coefficient (3), one can write the complex representation of the amplitude for the \(\nu\)-th harmonic of the induction [5]
\[ \dot{B}_{m\nu a}=2e^{j\nu\alpha}\sum_{n=1}^{n}\varepsilon_{2n-1}\cdot J_\nu\frac{(2n-1)H_{m1}}{H_0}. \tag{10} \]
And since
\[ \Pi_\nu=\frac{1}{3}(1+a_\nu+a_\nu^2)= \begin{cases} 0, & \text{if } \nu\ne 3p,\\ 1, & \text{if } \nu=3p \end{cases} \quad (p\text{ is an integer}), \]
only harmonics that are multiples of three will be present at the output of the converter.
Thus, the rms value of the third-harmonic voltage at the output of the FVC will be equal to
\[ \dot{U}_{3II}=j\frac{18}{\sqrt{2}}\omega W_{II}Se^{j3\alpha} \sum_{n=1}^{n}\varepsilon_{2n-1}\cdot J_3\frac{(2n-1)H_{m1a}}{H_0}. \tag{11} \]
The experimental determination of the harmonic composition of the current and voltage of the FVC was carried out with the aid of an electrodynamic wattmeter of type D568 (Figs. 2, a and 2, b). The industrial-frequency voltage was supplied to the converter through a three-phase autotransformer.
When measuring current harmonics, the current winding of the wattmeter was connected in series with the winding of one of the converter phases. The parallel winding of the wattmeter was supplied from an audio-frequency generator (A.F.G.), the output voltage of which was monitored by voltmeter \(V_2\). With a smooth change in the frequency in the parallel winding of the wattmeter, the maximum deflection of its moving system corresponded to the absence of a phase shift between the current and the voltage of one fre-
Fig. 2.
Labels in the diagram:
a, b, c; LATR; FVC; \(V_1\); \(V_2\); \(W\); from SG; CT; \(A\).
Subfigures: a., b.
Fig. 3.
Subfigure a: vertical axis \((\mathrm{mA})\, J_\nu\); horizontal axis \(U_1\) \((\mathrm{V})\). Curves labeled \(1\), \(5\), \(7\).
Subfigure b: vertical axis \((\mathrm{V})\, U_\nu\); horizontal axis \(U_1\) \((\mathrm{V})\). Curves labeled \(U_{3\mu}\), \(1\), \(3\), \(9\).
frequency. In this case the magnitude of each current harmonic is equal to the quotient of the power measured by the wattmeter divided by the voltage.
Figure 3,a shows the curves of variation of the 1st, 5th, and 7th current harmonics, obtained experimentally as functions of the line voltage applied to the FVC. The dashed line shows the dependence of the first current harmonic on voltage, calculated by formula (7).
The voltage harmonics were measured according to the circuit shown in Fig. 2,b. The series winding of the wattmeter was fed through an intermediate transformer (I.T.) from an audio-frequency generator (A.F.G.). One of the secondary windings of the converter was connected to the parallel winding of the wattmeter [6].
Figure 3,b presents curves of variation of the voltage harmonics (up to and including the 9th) on the secondary winding of one transformer, as well as the dependence of the third voltage harmonic at the output of the FVC. The dashed line shows the curve of variation of the third voltage harmonic at the output of the FVC, calculated by formula (10).
In the calculations, only the first six induction harmonics were taken into account in the approximating expression (1).
Conclusions
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To simplify relay-protection circuits by means of ferromagnetic voltage converters, a three-phase voltage system can be converted into a single-phase system.
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The calculation method set forth in the present article makes it possible, with accuracy sufficient for practice, to determine the current and voltage harmonics of the converter under symmetrical conditions.
References
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V. K. Skripko, N. V. Lisetskii. Calculation of a Spinelli system frequency tripler by the complex method. In: Electrification of Metallurgical Enterprises of Siberia, Tomsk, 1971.
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