ON THE SELECTION OF PARAMETERS FOR THE ACTUATING ELEMENT OF A THYRISTOR AC VOLTAGE REGULATOR

V. M. RAZIN, L. A. VOLYNSKAYA

(Presented by the scientific seminar of the Department of Computer Engineering)

As AC voltage regulators, devices have become widespread that consist of a step-up autotransformer, between the end of whose main winding and the common bus connecting the load to the terminal of the AC voltage source there is connected an actuating element consisting of two thyristors \(T_1\), \(T_2\) connected in anti-parallel (Fig. 1) [1, 2].

The distinctive feature of the AC voltage regulators under consideration, for

\[ \frac{W_1}{W_2} > 1, \]

is that in the quasi-steady-state regime the current \(i_o\) flowing through the actuating element proves to be less than the nominal value of the load current \(i_n\). However, during supertransient processes the magnitude of the current \(i_o\) may substantially exceed the nominal value of the load current. This phenomenon is especially pronounced when the regulator operates with a resonant load of high quality factor.

Fig. 1. Circuit of a thyristor AC voltage regulator

In the general case, if losses in the autotransformer are neglected, then, using the operator form of notation, for the load voltage \(U_n\) and the current \(i_o\) we shall have the following expressions:

\[ U_n(p)=U_c(p)\, \frac{Z_n(p)\left[Z_0(p)+pL_2K(1+K)\right]} {Z_0(p)\cdot Z_n(p)+pL_2\left[K^2Z_n(p)+Z_0(p)\right]}, \tag{1} \]

\[ i_o(p)=U_c(p)\, \frac{\left[Z_n(p)+pL_2(1+K)\right]} {Z_0(p)\cdot Z_n(p)+pL_2\left[K^2Z_n(p)+Z_0(p)\right]}. \tag{2} \]

In expressions (1), (2)

\(Z_n(p)\) — operator impedance of the load circuit;
\(Z_0(p)\) — operator impedance of the circuit of the actuating element;

\(L_2\) is the inductance of the series winding of the autotransformer;

\[ K=\frac{W_1}{W_2}; \]

\(W_1\) 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.

If the load is a resonant circuit formed by a parallel-connected resistance \(R_{\mathrm{n}}\), inductance \(L_{\mathrm{n}}\), and capacitor \(C_{\mathrm{n}}\), then

\[ Z_{\mathrm{n}}(p)= \frac{p}{C_{\mathrm{n}}\left(p^2+\dfrac{p}{R_{\mathrm{n}}C_{\mathrm{n}}}+\dfrac{1}{L_{\mathrm{n}}C_{\mathrm{n}}}\right)}. \tag{3} \]

As is seen from expression (1), in the steady-state mode the maximum value of the voltage \(U_{\mathrm{n}}\) will occur at \(Z_0\approx 0\). In this case

\[ U_{\mathrm{nmax}}=U_{mc}\frac{1+K}{K}\sin\omega t \tag{4} \]

The minimum value of the regulator output voltage corresponds to the case \(Z_0=0\) and is equal to

\[ U_{\mathrm{nmin}}=\frac{U_{mc}}{\sqrt{1+m^2}}\sin(\omega t-\operatorname{arctg}m) \tag{5} \]

In expression (5),

\[ m=\frac{\omega L_2}{R_{\mathrm{n}}}. \]

In this case the current flowing through the coil with inductance \(L_{\mathrm{n}}\) will be equal to

\[ i_L=-\frac{U_{mc}\cdot Q}{R_{\mathrm{n}}\cdot\sqrt{1+m^2}}\cdot\cos(\omega t-\operatorname{arctg}m) \tag{6} \]

where

\[ Q=\omega C_{\mathrm{n}}R_{\mathrm{n}}=\frac{R_{\mathrm{n}}}{\omega L_{\mathrm{n}}} \]

is the quality factor of the resonant circuit.

To determine the nature of the time variation of the voltage on the load \(U_{\mathrm{n}}\) and the current of the actuating element \(i_0\) under an instantaneous change in the resistance of the actuating element from the maximum value \((Z_0\simeq\infty)\) to the minimum \((Z_0=r_0)\), it is necessary to solve, under the initial conditions,

\[ U_{\mathrm{n}}(0)=-\frac{U_{mc}\cdot m}{1+m^2}\cdot\cos\alpha_{\mathrm{k}} \left(1-\frac{1}{m}\operatorname{tg}\alpha_{\mathrm{k}}\right) \tag{7} \]

\[ i_L(0)=-\frac{U_{mc}\cdot Q}{R_{\mathrm{n}}(1+m^2)}\cdot\cos\alpha_{\mathrm{k}} \left(1+m\,\operatorname{tg}\alpha_{\mathrm{k}}\right) \tag{8} \]

Equations (1), (2), and equation (9)

\[ i_L(p)=U_c(p)\, \frac{Z_0(p)+pL_2K(1+K)} {Z_0(p)\,pL_{\mathrm{n}}\left(1+\dfrac{pL_2}{R_{\mathrm{n}}}\right) +pL_2\left[K^2pL_{\mathrm{n}}+Z_0(p)\left(1+p^2L_{\mathrm{n}}C_{\mathrm{n}}\right)\right]}. \tag{9} \]

Here \(\alpha_{\mathrm{k}}\) is the commutation instant of the thyristor of the actuating element;
\(r_0\) is the minimum resistance of the thyristor actuating element.

After the instantaneous operation of the thyristor actuating element at the instant of time \(\omega t_k=\alpha_k\), the voltage at the regulator output varies in accordance with the expression

\[ \begin{aligned} U_{\mathrm{н}} ={}& U_{mc} \sqrt{ \frac{ 1+\dfrac{m^2}{a_0^2}K^2(1+K)^2 }{ 1+\dfrac{m^2}{a_0^2}(K^2+a_0^2) } } \sin\left[ \omega t+\alpha_k+ \operatorname{arctg} \frac{m a_0(1-a_0)} {a_0^2+m^2K^2(K^2+a_0^2)(1+K)} \right] \\[0.6em] &+ \frac{ U_{mc}\,e^{-\frac{\omega t}{2Qa_0}(a_0+K^2)} }{\sqrt{A}} \times \\[0.6em] &\times \left\{ \frac{\cos\alpha_k(1+m\,\operatorname{tg}\alpha_k)} {1+m^2} \sqrt{1+\frac{1}{mQ}} \sqrt{ A+ \left[ 1+m\frac{2Q\cdot a_0}{a_0+K^2} \times \right. } \right. \\[0.6em] &\left. \left. \times \frac{ 1-\dfrac{1}{m}\operatorname{tg}\alpha_k }{ 1+m\,\operatorname{tg}\alpha_k } \right]^2 } \times \sin\left[ \frac{\omega t}{2Qa_0}(a_0+K^2)\sqrt{A} -\operatorname{arctg} \right. \right. \\[0.6em] &\left. \left. \times \frac{ \sqrt{A} }{ 1+\dfrac{2Qa_0}{m(a_0+K^2)} \left(1+\frac{1}{Qm}\right) \dfrac{1+m\,\operatorname{tg}\alpha_k} {1-\dfrac{1}{m}\operatorname{tg}\alpha_k} } \right] \right. \\[0.6em] &\left. + \frac{ \sin\alpha_k \left[ 1+\dfrac{m}{a_0}K(1+K)\operatorname{ctg}\alpha_k \right] }{ \sqrt{ 1+\dfrac{m^2}{a_0^2}(a_0^2+K^2)^2 } } \times \right. \\[0.6em] &\left. \times \sqrt{ A+ \left[ 1- \frac{a_0}{mK(1+K)} \cdot \frac{ 1-\dfrac{m}{a_0}K(1+K)\operatorname{tg}\alpha_k }{ 1+\dfrac{a_0}{mK}\cdot\dfrac{\operatorname{tg}\alpha_k}{1+K} } \right]^2 } \times \right. \\[0.6em] &\left. \times \sin\left[ \frac{\omega t}{2Qa_0}(a_0+K^2)\cdot\sqrt{A} + \operatorname{arctg} \times \right. \right. \\[0.6em] &\left. \left. \times \frac{ \sqrt{A} }{ \dfrac{2Qa_0}{(a_0+K^2)} \cdot \frac{ \left[ 1-\dfrac{m}{a_0}K(1+K)\cdot\operatorname{tg}\alpha_k \right] }{ mK(1+K) \left[ 1+\dfrac{a_0}{mK}(1+K)\operatorname{tg}\alpha_k \right] } -1 } \right. \right. \\[0.6em] &\left. \left. + \operatorname{arctg} \frac{ \sqrt{A} }{ 1-\dfrac{2Qa_0}{(a_0+K^2)^2m} } \right] \right\}. \tag{10} \end{aligned} \]

Here

\[ A=\frac{4Q^2a_0^2}{(a_0^2+K^2)^2}\left(1-\frac{1}{mQ}\right), \qquad a_0=\frac{r_0}{R_{\mathrm{н}}}. \]

For the supertransient process, for the current \(i_0\) we have

\[ \begin{aligned} i_0={}&\frac{U_{mc}}{r_0} \sqrt{ \frac{1+m^2(1+K)^2} {1+\dfrac{m^2}{a_0^2}(K^2+a_0)^2} } \sin\left[ \omega t+\alpha_k+ \right. \\ &\left. +\operatorname{arctg} \frac{mK(a_0+K)} {a_0+m^2(1+K)(K^2+a_0)} \right] +\frac{U_{mc}e^{-\frac{\omega t}{2Qa_0}(a_0+K^2)}}{\sqrt{A}} \times \\ &\times \left\{ \frac{\sin[\alpha_k+\operatorname{arctg}m(1+K)]}{r_0} \cdot \sqrt{ \frac{1+m^2(1+K)^2} {1+\dfrac{m^2}{a_0^2}(K^2+a_0)^2} } \times \right. \\ &\left. \times \sqrt{ A+\left\{ 1-\frac{2Qa_0}{a_0+K^2}\operatorname{ctg}[\alpha_k+\operatorname{arctg}m(1+K)] \right\}^2 } \times \right. \\ &\left. \times \sin\left[ \frac{\omega t}{2Qa_0}(a_0+K^2)\sqrt{A} +\operatorname{arctg} \frac{\sqrt{A}} {\dfrac{2Qa_0}{a_0+K^2}\operatorname{ctg}[\alpha_k+\operatorname{arctg}m(1+K)]-1} +\right. \right. \\ &\left. \left. +\operatorname{arctg} \frac{\sqrt{A}} {1-\dfrac{2Qa_0^2}{m(a_0+K^2)^2}} \right] -\frac{(a_0+K^2)K^2}{R_n\cdot 2Qa_0(1+m^2)} \cos\alpha_k\cdot(1+m\operatorname{tg}\alpha_k) \times \right. \\ &\left. \times \sqrt{ A+ \left[ 1+\frac{2Qa_0m}{a_0+K^2}\cdot \frac{1-\dfrac{1}{m}\operatorname{tg}\alpha_k} {1+\dfrac{1}{m}\operatorname{tg}\alpha_k} \right]^2 } \times \right. \\ &\left. \times \sqrt{ A+ \left[ 1-\frac{2Qa_0^2}{m(a_0+K^2)^2} \right]^2 } \times \cos\left[ \frac{\omega t}{2Qa_0}(a_0+K^2)\sqrt{A} -\operatorname{arctg} \right. \right. \\ &\left. \left. \times \frac{ 1+\dfrac{2Qa_0m}{a_0+K^2}\cdot \dfrac{1-\dfrac{1}{m}\operatorname{tg}\alpha_k} {1+m\operatorname{tg}\alpha_k} } {\sqrt{A}} -\operatorname{arctg} \frac{\sqrt{A}} {1-\dfrac{2Qa_0^2}{m(a_0+K^2)^2}} \right] \right\}. \end{aligned} \tag{11} \]

As is seen from expression (11), for \(Q\gg 1\) and \(a_0\ll 1\), immediately after the commutation instant the current through the actuating element is determined mainly by the free component of the current \(i_{0\mathrm{sv1}}\), caused by the energy stored in the resonant circuit up to the commutation instant,

\[ i_{0\mathrm{sv1}}\simeq -\frac{ U_{mc}K^2\cos\alpha_k(1+m\operatorname{tg}\alpha_k) }{ a_0R_n(1+m^2)\sqrt{\dfrac{4Q^2a_0^2}{K^4}-1} } \times \]

\[ \times \sqrt{ \frac{4Q^2a_0^2}{K^4} -1+ \left[ 1+\frac{2Qa_0(m-\operatorname{tg}\alpha_k)} {K^2(1-m\operatorname{tg}\alpha_k)} \right]^2 } \cdot e^{-\frac{\omega K^2 t}{2Qa_0}} \times \]

\[ \times \cos \left[ \frac{\omega t K^{2}}{2 Q a_{0}} \sqrt{\frac{4 Q^{2} a_{0}^{2}}{K^{4}}-1} -\arctg \frac{ 1+\dfrac{2 Q a_{0}}{K^{2}}\cdot \dfrac{m-\tg \alpha_{k}}{1+m \tg \alpha_{k}} }{ \sqrt{\dfrac{4 Q^{2} a_{0}^{2}}{K^{4}}-1} } -\arctg \sqrt{\frac{4 Q^{2} a_{0}^{2}}{K^{4}}-1} \right]. \tag{12} \]

According to expression (12), for \(\dfrac{2 Q a_{0}}{K^{2}}>1\) the transient process for the current \(i_{\mathrm{ocв}1}\) is oscillatory and damped. For \(\dfrac{2 Q a_{0}}{K^{2}}=1\),

\[ i_{\mathrm{ocв}1} = -\frac{U_{mc} K^{2}}{a_{0} R_{\mathrm{н}}(1+m^{2})} \cdot \left|m\cos \alpha_{k}-\sin \alpha_{k}\right| \cdot e^{-\frac{\omega K^{2}}{2 Q a_{0}}\cdot t}. \tag{13} \]

For the case \(\dfrac{2 Q a_{0}}{K^{2}}<1\), the damping process of the current \(i_{\mathrm{ocв}1}\) is aperiodic and proceeds without overshoots.

For a quality factor of the resonant circuit of \(60 \div 80\), we usually have \(\dfrac{2 Q a_{0}}{K^{2}} \gg 1\), and the duration of the supertransient process for the current \(i_{\mathrm{ocв}1}\), at a supply-voltage frequency of \(50\) Hz, does not exceed several milliseconds; i.e., by the next instant of thyristor firing the supertransient process has ended.

Fig. 2. Dependence of the amplitude of the current \(i_{\mathrm{ocв}1}\) on the thyristor firing angle

Analysis of expression (12) shows that the amplitude value of the free component of the current \(i_{\mathrm{ocв}1}\) depends on the value of the coefficient \(a_{0}\) and on the firing angle \(\alpha_{k}\) of the thyristor of the actuating element. The dependences

\[ \frac{I_{m\mathrm{ocв}1}}{I_{m\mathrm{н}}}=f(\alpha_{k}) \]

for various values of \(a_{0}\), \(m\), and \(K=1.4\) are shown in Fig. 2. The amplitude value of the regulator load current \(I_{m\mathrm{н}}\) is defined as the ratio

\[ \frac{U_{mc}}{R_{\mathrm{н}}}\cdot \frac{1+K}{K}. \]

Curve 1 is calculated for the case \(m=2,\ a_{0}=0.01\); curve 2, for \(m=5,\ a_{0}=0.01\); curve 3, \(m=2,\ a_{0}=0.05\); curve 4, \(m=5,\ a_{0}=0.05\). At \(\alpha_{k}=\arctg m\), the free component of the current \(i_{\mathrm{ocв}1}\) is equal to zero. As the value of the coefficient \(a_{0}\) decreases, the current \(i_{\mathrm{ocв}1}\) increases.

Although for \(a_{0}=0.01\) the amplitude value of the current \(i_{\mathrm{ocв}1}\) at the instant of commutation exceeds the forced component of the current \(i_{\mathrm{ocв}}\) by tens of times, nevertheless, for \(Q=60 \div 80\) the instantaneous value of the current \(i_{\mathrm{ocв}}\) decreases by a factor of ten already after \(0.002 \div 0.003\) sec.

The obtained expression (12) makes it possible to estimate the short-term overloads that arise during the initial firing of the actuating element of a regulator operating into a resonant load, and

select the type of thyristors for the actuating element capable of withstanding these overloads.

REFERENCES

1. A. Ya. Yakushev. Application of thyristors for stabilizing the voltage of auxiliary circuits of AC electric locomotives of the ER-9 type. Proceedings of LIIZhT, issue 277, 45, L., 1968.

2. V. I. Nekrasov, A. Ya. Yakushev. Features of calculating a ferro-thyristor voltage stabilizer. Proceedings of LIIZhT, issue 293, 85, L.