Methodology for Calculating a Thyristor AC Voltage Stabilizer
Abstract Generated abstract
This paper develops a calculation method for a thyristor AC autotransformer voltage stabilizer, focusing on the influence of magnetic saturation on the voltage drop across the regulating element. Under simplifying assumptions that neglect core losses, winding losses, and leakage flux, the authors formulate nonlinear circuit equations using an odd polynomial approximation of the flux linkage characteristic and solve them by a reverse method. The analysis yields approximate expressions for load current, including fundamental, third, and fifth harmonic components, as well as formulas for the current and voltage of the regulating element. These relations are intended to support selection of parameters for the stabilizer’s regulating devices.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV, HOLDER OF THE ORDER OF THE OCTOBER REVOLUTION
AND THE ORDER OF THE RED BANNER OF LABOR
Volume 277 1977
METHODOLOGY FOR CALCULATING A THYRISTOR AC VOLTAGE STABILIZER
L. A. VOLYNSKAYA, V. M. RAZIN
(Presented by the scientific seminar of the Department of Computing Technology)
Most high-power compensation stabilizers are characterized by the presence of a transformer rated for the full output power of the stabilizer, owing to which the weight of active materials per unit of stabilized power can reach 20–30 kg/kVA [1]. The use of autotransformer stabilizers, in which the regulating element is connected in series with the main winding of the autotransformer, makes it possible to reduce the weight of the active materials of the stabilizer to 7–10 kg/kVA when magnetic amplifiers are used [2], and to 4.5–5 kg/kVA when the regulating element is used in the form of two antiparallel-connected thyristors [3].
A distinctive feature of the AC autotransformer voltage stabilizers under consideration is that the amplitude value of the voltage drop across the regulating element depends substantially on saturation of the magnetic system of the autotransformer; however, this fact has not been considered sufficiently fully in the literature [4, 5].
The circuit of the autotransformer stabilizer is shown in Fig. 1, where \(W_1\) is the number of turns of the main winding of the autotransformer, \(W_2\) is the number of turns of the series winding of the autotransformer, \(Z\) is the load resistance, and \(Z_0\) is the effective value of the resistance of the regulating element.
Fig. 1. Circuit of an AC voltage stabilizer.
Let us analyze the autotransformer circuit under the following basic assumptions:
a) the dependence of flux linkage on current is determined by an odd function
\[ \Psi=L\left(i-C_1 i^3+C_2 i^5\right), \tag{1} \]
where
\(L\) is the winding inductance in the absence of saturation;
b) at \(Z_0=0\) there is no saturation of the magnetic system;
c) magnetic losses in the core and electrical losses in the windings are absent;
d) leakage fluxes are absent.
Taking assumptions (a--d) into account, for the circuit in Fig. 1 we have the following equations:
\[ U=i_0\left(Z_0+k^2L_2p\right)-k^2C_1L_2p(i_0)^3+k^2L_2C_2p(i_0)^5 -kL_2pi_n+kC_1L_2p(i_n)^3-kC_2L_2p(i_n)^5, \tag{2} \]
\[ U(p)=i_n\left(Z_n+L_2p\right)-C_1L_2p(i_n)^3+C_2L_2p(i_n)^5 -kL_2pi_0+kC_1L_2p(i_0)^3-kC_2L_2p(i_0)^5, \tag{3} \]
where \(L_2\) is the inductance of the series winding of the autotransformer in the absence of saturation,
\[ k=\frac{W_1}{W_2}. \]
If the transformation ratio of the autotransformer is not higher than 1.5 and \(i_0\leq 0.5 i_n\), then the simultaneous solution of equations (2) and (3) under the conditions \(Z_n=R_n\) and \(Z_0=R_0\) gives the following equation:
\[ U=i_nR_n\frac{d(p+b)}{b(p+d)} -i_n^3\frac{R_0C_1}{k(1+k)(p+d)} +i_n^5\frac{R_0C_2}{k(1+k)(p+d)}, \tag{4} \]
where
\[ b=\frac{R_0}{L_2\left(k^2+\dfrac{R_0}{R_n}\right)}, \]
\[ d=\frac{R_0}{kL_2(1+k)}. \]
Nonlinear differential equations of the form (4) can be solved by the reverse method [6], with the solution proposed in the form
\[ i_n(t)=i_{n1}(t)+i_{n3}(t)+i_{n5}(t)\ldots \tag{5} \]
In the first approximation we have
\[ i_{n1}(p)=\frac{U(p)\cdot b(p+d)}{R_nd(p+b)}, \tag{6} \]
which gives the expression
\[ i_{n1}(t)=\frac{U_m}{R_n}M\sin\left(\omega t+\arctan\frac{\omega}{d}-\arctan\frac{\omega}{b}\right), \tag{7} \]
describing the current flowing through the load in the absence of saturation of the magnetic system of the autotransformer. In expression (7)
\[ M=\sqrt{\frac{1+\dfrac{\omega^2}{d^2}}{1+\dfrac{\omega^2}{b^2}}}. \tag{8} \]
The second term of expression (5) can be obtained by solving the equation
\[ i_{H3}(p)= \frac{R_0 b C_1}{R_H \cdot d \cdot k(1+k)(p+b)}\,i_{H1}(p). \tag{9} \]
For the third term we have
\[ i_{H5}(p)=3\frac{R_0}{R_H}\, \frac{b C_1}{R_H \cdot d \cdot k(1+k)(p+b)} \,[i_{H1}(p)]^2 i_{H3}(p) -\frac{R_0}{R_H}\, \frac{b C_2}{d k(1+k)(p+b)} \,[i_{H1}(p)]^5 . \tag{10} \]
Solving equations (9) and (10), and taking into account that \(\dfrac{\omega^2}{b^2}\gg 1\), we obtain an approximate expression for the load current including the 3rd and 5th harmonics:
\[ \begin{aligned} i_H(t)=& \left[ I_H \cdot M+ \frac{A C_1}{k^2+A}\cdot \frac{3}{4} I_H^3 M^3 +\frac{5}{8} I_H^5 M^5 \left( \frac{3A^2 C_1^2}{(k^2+A)^2} -\frac{A C_2}{k^2+A} \right) \right] \sin\left(\omega t-\arctg\frac{\omega}{d}-\arctg\frac{\omega}{b}\right) \\ &- \left[ \frac{A C_1}{k^2+A}\frac{I_H^3 M^3}{4} +\frac{5}{16} \left( 3\frac{A^2 C_1^2}{(k^2+A)^2} -\frac{A C_2}{k^2+A} \right) I_H^5 M^5 \right] \\ &\times \sin\left(3\omega t+3\arctg\frac{\omega}{d}-3\arctg\frac{\omega}{b}\right) \\ &+ \frac{1}{16} I_H^5 M^5 \left( \frac{3A^2 C_1^2}{(k^2+A)^2} -\frac{A C_2}{k^2+A} \right) \sin\left(5\omega t+5\arctg\frac{\omega}{d}-5\arctg\frac{\omega}{b}\right), \end{aligned} \tag{11} \]
where
\[ A=\frac{R_0}{R_H},\qquad I_H=\frac{U_m}{R_H}. \]
For the current \(i_0\) and the voltage \(U_0\), we shall have
\[ i_0(t)=\frac{1+k}{R_0}U_m \sin\omega t-\frac{R_H}{R_0}k i_H(t), \tag{12} \]
\[ U_0(t)=(1+k)U_m \sin\omega t-kR_H i_H(t). \tag{13} \]
The obtained relations can be used in selecting the parameters of the elements of the regulating devices.
References
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A. M. Ilyukovich, B. R. Shulman. Stabilizers and Stabilized AC Power Supplies. Moscow, “Energiya,” 1965.
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A. Ya. Yakushev. Application of thyristors for voltage stabilization in the auxiliary circuits of ER-9-type AC electric trains. Problems in the automation of electric-traction devices. Collected works of LIIZhT, issue 277, 1968.
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S. G. Obukhov, L. V. Panteleimonova. A precision AC voltage stabilizer. Industrial Electronics. Proceedings of MEI, issue 55, 1965, p. 113.
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V. I. Nekrasov, A. Ya. Yakushev. Features of calculating a ferrothyristor voltage stabilizer. Proceedings of LIIZhT, issue 293, 1969.
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