UDC 621.316.722

STATIC CHARACTERISTICS OF AN AUTOTRANSFORMER ELECTROMECHANICAL AC VOLTAGE REGULATOR

Yu. B. Volynskii

(Presented by the scientific seminar of the Faculty of Physics and Technology)

In the general case, the circuit of an autotransformer AC voltage regulator consists (Fig. 1) of a step-up autotransformer 1, a bridge rectifier circuit 2, and an electromechanical amplifier 3 controlled by a feedback circuit.

Fig. 1. Autotransformer electromechanical AC voltage regulator

If the inductance of the series winding of autotransformer 1 with the number of turns \(W_2\) is denoted by \(L_2\), then, neglecting losses in the windings and in the magnetic core and neglecting leakage fluxes, one may write

\[ U_c = i_0 R_0 + K^2 L_2 \frac{d i_0}{d t} - K L_2 \frac{d i_{\mathrm{н}}}{d t}, \tag{1} \]

\[ U_c = L_2 \frac{d i_{\mathrm{н}}}{d t} + i_{\mathrm{н}} R_{\mathrm{н}} - K L_2 \frac{d i_0}{d t}, \tag{2} \]

where \(U_c\) is the mains voltage;
\(i_0\) is the current of the actuator;
\(i_{\mathrm{н}}\) is the load current;
\(R_0\) is the equivalent resistance of the actuator;
\(R_{\mathrm{н}}\) is the load resistance.

Using the operational form of notation [1], from (1, 2) we have

\[ U_{\mathrm{н}}(p) = \frac{ U_c(p) R_{\mathrm{н}} \left[ R_0 + p L_2 K(1+K) \right] }{ R_0 R_{\mathrm{н}} + p L_2 \left( K^2 R_{\mathrm{н}} + R_0 \right) }, \tag{3} \]

\[ U_0(p) = \frac{ U_c(p) R_0 \left[ R_{\mathrm{н}} + p L_2 (1+K) \right] }{ R_0 R_{\mathrm{н}} + p L_2 \left( K^2 R_{\mathrm{н}} + R_0 \right) }, \tag{4} \]

where \(U_{\mathrm{н}}\) is the voltage across the load;
\(U_0\) is the voltage across the actuator.

The operating principle of the electromechanical actuator is based on the fact that, when the rectifier operates into a load having counter-

counter-electromotive force, current will exist only in the case when the voltage applied to the anode of the valve has a positive sign relative to its cathode, i.e., operation of the rectifier takes place with current cutoff [2]. The current cutoff angle is determined by the magnitude of the counter-electromotive force; therefore, by changing the magnitude of the counter-electromotive force \(E_{\mathrm{EMU}}\), it is possible to vary the effective value of the current \(i_0\), which corresponds to varying the magnitude of a certain equivalent resistance \(R_0\).

For \(E_{\mathrm{EMU}}=0\), according to (3, 4), we have

\[ U_{\mathrm{n}}=U_{mc}\frac{1+K}{K}\sin \omega t, \tag{5} \]

\[ i_0= U_{mc} \frac{\sqrt{1+b^2(1+K^2)}}{R_{\mathrm{n}}\,bK^2} \sin\left[ \omega t+\operatorname{arctg}\frac{1}{b(1+K)} \right], \tag{6} \]

where

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

If, at the instant when the line voltage passes through zero, the counter-electromotive force \(E_{\mathrm{EMU}}\) is switched on, then for \(\omega t>0\) the rectifier bridge operates with current cutoff, and the current \(i_0\) will exist only under the condition \(U_0-E_{\mathrm{EMU}}>0\). For \(E_{\mathrm{EMU}}>U_0\),

\[ U_{\mathrm{n}}= U_{mc}\left[ \frac{b}{1+b^2}e^{-\frac{\omega t}{b}} + \frac{1}{\sqrt{1+b^2}}\sin(\omega t-\operatorname{arctg} b) \right], \tag{7} \]

\[ U_0= U_{mc}\left\{ \sqrt{\frac{1+b^2(1+K)^2}{1+b^2}} \sin\left[ \omega t+\operatorname{arctg} \frac{Kb}{1+b^2(1+K)} \right] - \frac{Kb}{1+b^2}e^{-\frac{\omega t}{b}} \right\}. \tag{8} \]

The instant at which the current \(i_0\) arises is determined by the condition

\[ \omega t_{k_1} = \arcsin\left\{ \frac{E_{\mathrm{EMU}}}{U_{mc}} \sqrt{\frac{1+b^2}{1+b^2(1+K)^2}} + \frac{Kb\,e^{-\frac{\omega t_{k_1}}{b}}} {\sqrt{(1+b^2)\left[1+b^2(1+K)^2\right]}} \right\} - \operatorname{arctg} \frac{Kb}{1+b^2(1+K)}. \tag{9} \]

For \(\omega t>\omega t_{k_1}\), neglecting losses in the armature circuit of the EMU, we have

\[ U_{\mathrm{n}} = U_{\mathrm{n}}(\omega t_{k_1}) - \frac{U_{mc}(1+K)}{K}\sin \omega t_{k_1} + U_{mc}\frac{(1+K)}{K}\sin \omega t, \tag{10} \]

where \(U_{\mathrm{n}}(\omega t_{k_1})\) is the load voltage at the time \(\omega t_{k_1}\), determined according to expression (7).

The current \(i_0\) lags in phase behind the voltage \(U_{\mathrm{n}}\) by the angle

\[ \varphi_0=\operatorname{arctg}\frac{1}{b(1+K)} \]

and becomes equal to zero at the time

\[ \omega t_{k_2} = \pi - \arcsin\left[ \sin\omega t_{k_1} - \frac{U_{\mathrm{n}}(\omega t_{k_1})}{U_{mc}}\cdot\frac{K}{1+K} \right] + \operatorname{arctg}\frac{1}{b(1+K)}. \tag{11} \]

In the interval \(\omega t > \omega t_{k_2}\),

\[ U_{\mathrm{n}} = \left[ U_{\mathrm{n}}(\omega t_{k_2}) - \frac{U_{mc}}{\sqrt{1+b^2}}\sin(\omega t_{k_2}-\arctg b) \right] l^{-\frac{\omega(t-t_{k_2})}{b}} + \frac{U_{mc}}{\sqrt{1+b^2}}\sin(\omega t-\arctg b), \tag{12} \]

\[ \begin{aligned} U_0 &= U_{mc}\sqrt{\frac{1+b^2(1+K)^2}{1+b^2}}\, \sin\left[ \omega t+\arctg\frac{Kb}{1+b^2(1+K)} \right] \\ &\quad -K\left[ U_{\mathrm{n}}(\omega t_{k_2})- \frac{U_{mc}}{\sqrt{1+b^2}}\sin(\omega t_{k_2}-\arctg b) \right] l^{-\frac{\omega(t-t_{k_2})}{b}} . \end{aligned} \tag{13} \]

The next instant at which the current \(i_0\) appears \((\omega t=\omega t_{k_3})\) occurs at \(U_0=-E_{\mathrm{emf}}\). For \(\omega t>\omega t_{k_3}\) the process proceeds in the same way as in the first case for the positive half-period of the supply voltage. A quasi-steady-state regime occurs when \(\omega t_{kn+2}=\omega t_{k_n}+\pi\), where \(n\) is an integer. For the case \(b=2\text{--}5\), the duration of the transient process does not exceed 3–6 periods of the supply voltage.

At the instant when the current \(i_0\) passes through zero, the voltage \(U_{\mathrm{n}}\) in most cases is much less than the maximum instantaneous value of the load voltage \(\dfrac{1+K}{K}U_{mc}\); therefore one may assume that, when \(i_0=0\), \(U_{\mathrm{n}}\approx 0\). In this case, for the device under consideration we have relations analogous to (3), (4),

\[ U_c(p)=i_0(p)pL_2K^2-i_{\mathrm{n}}(p)pL_2K+E_{\mathrm{emf}}, \tag{14} \]

\[ U_c(p)=i_{\mathrm{n}}(p)R_{\mathrm{n}}+i_{\mathrm{n}}(p)pL_2-i_0(p)pL_2K, \tag{15} \]

therefore, in the quasi-steady-state regime, the voltage across the load will exist only in the time interval \(\omega t_{k_1}<\omega t<\pi-\omega t_k\) and will be determined by the expression

\[ U_{\mathrm{n}}=\frac{1+K}{K}U_{mc}\sin\omega t-\frac{E_{\mathrm{emf}}}{K}. \tag{16} \]

The values of \(\omega t_{k_1}\) are determined for the various cases from (16) at \(U_{\mathrm{n}}=0\).

The dependences \(\omega t_{k_1}=f\left(\dfrac{E_{\mathrm{emf}}}{U_{mc}}\right)\) for various values of \(K\) are given in Fig. 2 as dashed lines. In the same figure, solid lines show curves representing the dependence of the relative value of the load voltage on the ratio \(\dfrac{U_{\mathrm{dn}}}{U_{mc}}=f\left(\dfrac{E_{\mathrm{emf}}}{U_{mc}}\right)\).

The rms value of the load voltage is equal to

\[ U_{\mathrm{dn}}= \frac{U_{mc}(1+K)}{K\sqrt{\pi}} \sqrt{ \frac{\pi}{2} -\omega t_{k_1} -\frac{3}{2}\sin 2\omega t_{k_1} +(\pi-2\omega t_{k_1})\sin^2\omega t_{k_1} }. \tag{17} \]

The periodic odd function of the form (16) satisfies the Dirichlet conditions and can be expanded into an incomplete Fourier sine series over arcs that are multiples of two. The amplitudes of the harmonics of the Fourier series are equal to

\[ B_{nm}=\frac{1}{\pi}U_{mc}\frac{1+K}{K}\left\{ \frac{\sin\left[(n-1)(\pi-\omega t_{k1})\right]}{n-1} -\frac{\sin\left[(n+1)(\pi-\omega t_{k1})\right]}{n+1} +\frac{\sin(n+1)\omega t_{k1}}{n+1} -\frac{\sin(n-1)\omega t_{k1}}{n-1} \right\}. \tag{18} \]

From (18), the amplitude of the first harmonic is

\[ B_{1m}=\frac{(1+K)}{K\pi}U_{mc}\left[\pi-2\omega t_{k1}+\sin 2\omega t_{k1}\right], \tag{19} \]

the amplitudes of the even harmonics are equal to zero, and the amplitudes of the odd higher harmonics are determined by the expression

Fig. 2. Dependences
\[ \omega t_{k1}=f\!\left(\frac{E_{\mathrm{emu}}}{U_{mc}}\right) \]
and
\[ \frac{U_{\mathrm{dn}}}{U_{mc}}=f\!\left(\frac{E_{\mathrm{emu}}}{U_{mc}}\right) \]
for various values of the coefficient \(K\)

Fig. 3. Dependence of the harmonic coefficient \(K_f\) on
\[ \frac{E_{\mathrm{emu}}}{U_{mc}} \]
for various values of the coefficient \(K\)

\[ B_{nm}=\frac{2}{\pi}U_{mc}\frac{1+K}{K}\times \left[ \frac{\sin(n+1)\omega t_{k1}}{n+1} -\frac{\sin(n-1)\omega t_{k1}}{n-1} \right], \tag{20} \]

\[ n=3,\;5,\;7\ldots \]

The dependences of the harmonic coefficient \(K_f\) of the output voltage on the ratio

\[ \frac{E_{\mathrm{emu}}}{U_{mc}} \]

for various values of the coefficient \(K\) are presented in Fig. 3.

Conclusions

1. The use of an autotransformer AC voltage regulator with an electromechanical actuating element makes it possible to vary the rms value of the load voltage by a factor of 2.5–3.

2. When the regulator output voltage is varied over a range of 2–2.5 times, the harmonic coefficient of the output voltage changes by no more than 15–20%.

References

1. M. I. Kontorovich. Operational Calculus and Processes in Electrical Circuits. Moscow, “Nauka,” 1964.

2. V. N. Aksenov. Rectifiers and Transformer Substations. Moscow, Svyazizdat, 1961.