Analysis of Statistical Characteristics of an AC Voltage Regulator with a Transistor Actuator

L. A. Volynskaya, I. G. Smyshlyaeva

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

An AC voltage regulator with a transistor actuator is a bridge circuit, two adjacent arms of which are formed by the windings \(W_1\) and \(W_2\) of a step-up autotransformer, while the other two are formed by the load and a rectifier bridge circuit, in whose DC diagonal a transistor controlled by a feedback circuit is connected [1] (Fig. 1).

Let us analyze the operation of the regulator for the case of an active load, assuming that losses in the autotransformer are absent, the diodes of the rectifier bridge circuit have an ideal characteristic, and the current-voltage characteristics of the transistor in the common-emitter circuit are specified in the form of a family of broken lines with a single break point, for which, in the initial segment up to the break point, the slope does not depend on the base current [2]. In the active region, as the base current increases, the slope of the current-voltage characteristic increases.

Fig. 1. Circuit of an AC voltage regulator with a transistor actuator

If an alternating sinusoidal voltage is applied to the bridge circuit, then after the voltage passes through zero the load voltage \(U_{\mathrm{n}}\) and the voltage \(U_0\) will be determined by expressions (1), (2):

\[ U_{\mathrm{n}}=U_{mc} \sqrt{ \frac{a_0^2+m^2 a_0\left(K^2(1+K)^2\right)} {a_0^2+m^2\left(K^2+a_0\right)^2} } \sin\left[ \omega t+\operatorname{arctg} \frac{m a_0(K-a_0)} {a_0^2+m^2K(K^2+a_0)(1+K)} \right] \tag{1} \]

\[ U_0=U_{mc} \sqrt{ \frac{a_0^2+m^2a_0^2(1+K)^2} {a_0^2+m^2(K^2+a_0)^2} } \sin\left[ \omega t+\operatorname{arctg} \frac{mK(a_0-K)} {a_0+m^2(1+K)(K^2+a_0)} \right] \tag{2} \]

In expressions (1), (2)

\[ a_0=\frac{\Delta U_{ce}}{\Delta I_c}\cdot\frac{1}{R_{\mathrm{n}}}; \]

\[ m=\frac{\omega L_2}{R_{\mathrm{n}}};\qquad K=\frac{W_1}{W_2}. \]

For power transistors it may be assumed that, in the initial segment of the current-voltage characteristic,

\[ \frac{\Delta U_{ce}}{\Delta I_k} \simeq 10^{-1} \div 10^{-2}\ \Omega, \]

therefore

\[ U_0=\frac{a_0(K+1)\cdot U_{mc}}{K^2}\sin\left[\omega t-\operatorname{arctg}\frac{1}{m(1+K)}\right] \tag{3} \]

and the voltage across the load is equal to the input voltage multiplied by the transformation coefficient of the autotransformer.

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

As is seen from expression (3), the voltage \(U_0\), which is the emitter-collector junction voltage \(U_{ce}\) for the transistor actuator, is much smaller than the input voltage. As long as the operating point is in the initial segment of the current-voltage characteristic, the voltage across the load changes according to expression (4).

With an increase in the instantaneous value of the input voltage, upon transition to the active segment of the current-voltage characteristic, an instantaneous increase in the coefficient \(a\) occurs, since in this segment, for power transistors, depending on the magnitude of the base current,

\[ a=\frac{\Delta U_{ce}}{\Delta I_k}\cdot\frac{1}{R_n}=(1\div 10^2)\ \Omega\cdot\frac{1}{R_n}. \]

The instant of transition to the active segment of the characteristic is determined by the expression

\[ \omega t_{k1}=\arcsin\frac{U_{0k}}{U_{mc}}\cdot\frac{K^2}{a_0(1+K)}+\operatorname{arctg}\frac{1}{m(1+K)}, \tag{5} \]

where \(U_{0k}\) is the voltage \(U_{ce}\) corresponding to the knee of the current-voltage characteristic at the specified base current \(i_b\).

Beginning from the instant \(\omega t_{k1}\), the voltage across the load can be determined by solving the differential equation

\[ U_n(p)=U_c(p)\, \frac{a+p\frac{mK}{\omega}(1+K)} {a+p\frac{m}{\omega}(a+K^2)} \tag{6} \]

under the initial condition

\[ U_n(0)= \frac{U_{0k}Km(1+K)} {a_0\sqrt{1+m^2(1+K)^2}} \left[ 1+\frac{U_{mc}a_0}{K^2mU_{0k}} \sqrt{ 1-\frac{U_{0k}^2K^4}{U_{mc}^2a_0^2(1+K)^2} } \right]. \tag{7} \]

The solution of equation (6) has the following form:

\[ U_n=U_{mc}\left\{ \frac{U_{0k}}{a_0U_{mc}}\cdot \frac{Km(1+K)}{\sqrt{1+m^2(1+K)^2}} \left[ 1+\frac{U_{mc}a_0}{U_{0k}K^2m} \times \right.\right. \]

\[ \left.\left. \times \sqrt{ 1-\frac{U_{0k}^2K^4}{U_{mc}^2a_0^2(1+K)^2} } \right] - \sqrt{ \frac{a^2+m^2K^2(1+K)^2} {a^2+m^2(K^2+a)^2} } \times \right. \]

\[ \begin{aligned} &\times \sin \left[\omega t_{k_1}+\operatorname{arctg} \frac{ma(K-a)}{a^2+m^2K(K^2+a)(1+K)}\right]\times e^{-\frac{at'}{m(K^2+a)}}+\\ &+\sqrt{\frac{a^2+m^2K^2(1+K)^2}{a^2+m^2(K^2+a)^2}}\, \sin \left[\omega t+\operatorname{arctg} \frac{ma(K-a)}{a^2+m^2K(K^2+a)(1+K)}\right] \end{aligned} \tag{8} \]

The voltage \(U_0\) is then determined by the expression

\[ \begin{aligned} U_0=U_{mc}\Bigg\{& \left[ \frac{U_{0k}}{U_{mc}}- \sqrt{\frac{a^2+a^2m^2(1+K)^2}{a^2+m^2(K^2+a)^2}}\, \sin\left(\omega t_{k_1}+\right.\right.\\ &\left.\left. +\operatorname{arctg} \frac{mK(a-K)}{a+m^2(1+K)(K^2+a)} \right)\right]\cdot e^{-\frac{at'}{m(K^2+a)}}+\\ &+\sqrt{\frac{a^2+a^2m^2(1+K)^2}{a^2+m^2(K^2+a)^2}}\, \sin\left[ \omega t+\operatorname{arctg} \frac{mK(a-K)}{a+m^2(1+K)(K^2+a)} \right]\Bigg\} \end{aligned} \tag{9} \]

In expressions (8), (9), \(t'=t-t_k\). At the time instant \(\omega t=\omega t_{k2}\), the voltage again becomes equal to \(U_{0k}\). Making the assumption that

\[ e^{-\frac{at_{k_2}}{m(K^2+a)}} \simeq 1, \]

we have \(\omega t_{k2}=180^\circ-\omega t_{k1}\).

Further, until the end of the half-period, the load voltage is determined by expression (4).

As indicated above, the voltage \(U_{0k}\) and the coefficient \(a\) are functions of the current \(i_b\); moreover, as \(i_b\) increases, \(U_{0k}\) increases, while \(a\) decreases. Therefore, according to expression (8), by varying the current \(i_b\), it is possible, for a specified value of \(U_{mc}\), to vary the voltage \(U_n\) over wide limits.

The maximum ratio of variation of the rms value of the regulator output voltage \(U_n\), at constant input voltage, is determined by the expression

\[ \frac{U_{n\max}}{U_{n\min}} = \frac{1+K}{K}\cdot \frac{a^2+m^2(K^2+a)^2}{a^2+m^2K(1+K)(K^2+a)}. \tag{10} \]

REFERENCES

1. V. M. Razin, L. A. Volynskaya. AC voltage stabilizer, Author’s Certificate No. 401981. Bulletin of Inventions, No. 41, 1973.

2. G. I. Atabekov, A. B. Timofeev, S. S. Khukhrikov. Nonlinear Circuits. Moscow, “Energiya,” 1970.