EFFECT OF OVERLAP ON THE RMS VALUE OF CURRENT IN ELECTRIC CIRCUITS WITH VALVES

V. A. Shubenko

(Presented by the scientific seminar of the Faculty of Power Engineering).

In designing converter installations it is always necessary to determine the rms value of the current in the transformer windings and in the valves. The power rating of the transformer feeding the circuit, the design dimensions of the valves, and so forth are determined by the value of this current.

As is known, commutation of currents through valves, which occurs in multiphase converter circuits, affects both the shape of the curve of these currents and their rms value.

The duration of commutation depends on the magnitude of the inductance in the current circuit, on the magnitude of the current being converted, and also on the firing angle of the valves in the case of a converter controlled by means of grids.

Most often the rms value of the current is determined without taking into account the influence of commutation. However, in view of the appearance of high-power converter installations with transformers having considerable leakage reactances, it is of interest to estimate the magnitude of the error made in determining the rms value of the current without taking account of the phenomenon of commutation.

Let us consider the circuit, most suitable for converting large electric powers, proposed by Professor Larionov (Fig. 1). The forms of the current and voltage curves characterizing the operation of this circuit are shown in Fig. 2.

We introduce the following notation:

\(e_a, e_b, e_c\)—instantaneous values of the sinusoidal emfs of the secondary winding of the transformer;

\[ E_2 \]
—the rms value of the secondary emf;

\[ E_{l.\max} = \sqrt{2}\cdot \sqrt{3}\cdot E_2 \]
—the amplitude of the line voltage;

\[ X_a \]
—the transformer reactance, determined from the short-circuit test;

\[ X_g \]
—the reactance in the dc circuit of the converter;

\[ I_2 \]
—the rms value of the current through the valves;

\[ e_s = -X_a \frac{di}{d\vartheta} \]
—the emf of self-induction;

\[ i_1 \text{ and } i_2 \]
—the currents of valve 6, respectively increasing and decreasing during the overlap;

\[ \alpha \]
—the firing angle of the valves, measured from the point of natural firing in rectification;

\[ \gamma \]
—the commutation (overlap) angle.

Fig. 1.

Fig. 2.

With the chosen origin for measuring angles, \(\vartheta=\omega t\) (Fig. 2).

\[ e_a=\sqrt{2}\cdot E_2\cos\vartheta, \]

\[ e_b=-\sqrt{2}\cdot E_2\cos\left(\vartheta+\frac{\pi}{3}\right), \]

\[ e_c=-\sqrt{2}\cdot E_2\cos\left(\vartheta-\frac{\pi}{3}\right). \]

The rms value of the current without allowance for overlap, according to Fig. 2б, is determined as:

\[ I_2=\sqrt{\frac{1}{2\pi}\int_{\alpha+\frac{\pi}{3}}^{\alpha+\pi} I_g^2\cdot d\vartheta} =\frac{1}{\sqrt{3}}\cdot I_g . \tag{1} \]

In order to determine the rms value of the current with allowance for overlap, it is necessary to have equations characterizing the change of the current through the valve during the overlap interval (Fig. 2в).

Neglecting the active resistance of the transformer windings and valves, and assuming that a sufficiently large choke is connected in the dc circuit of the converter, for the commutation circuit in Fig. 3 we write:

\[ e_a+e_{s3}-e_{s1}-e_b=0. \]

This equation expresses the balance of the emf in the commutation circuit and is valid in the interval of angles

\[ \alpha+\frac{\pi}{3}<\vartheta<\alpha+\frac{\pi}{3}+\gamma. \]

\[ \sqrt{2}\cdot E_2\cdot\cos\vartheta +X_a\frac{di_1}{d\vartheta} -X_a\frac{di_3}{d\vartheta} +\sqrt{2}\cdot E_2\cos\left(\vartheta+\frac{\pi}{3}\right)=0. \]

In the presence of a sufficiently large reactance \(X_g\), during overlap

\[ i_1+i_3=I_g. \]

Thus,

\[ di_1=-\frac{E_{2.\max}}{2X_a}\cos\left(\vartheta+\frac{\pi}{6}\right)d\vartheta, \]

\[ i_1=-\frac{E_{2.\max}}{2X_a}\sin\left(\vartheta+\frac{\pi}{6}\right)+K. \]

Using the condition \(i_1=0\) at \(\vartheta=\alpha+\frac{\pi}{3}\), we obtain:

\[ K=\frac{E_{2.\max}}{2X_a}\cos\alpha \]

and

\[ i_1=\frac{E_{2.\max}}{2X_a} \left[ \cos\alpha-\cos\left(\vartheta-\frac{\pi}{3}\right) \right]. \]

At the instant of time characterized by the angle

\[ \vartheta=\alpha+\frac{\pi}{3}+\gamma;\quad i_1=I_g \]

and

\[ \cos\alpha-\cos(\alpha+\gamma)=\frac{2X_a I_g}{E_{2.\max}}. \tag{2} \]

Thus, for the current increasing during overlap, we obtain

\[ i_1 = I_g \cdot \frac{\cos \alpha - \cos\left(\vartheta - \frac{\pi}{3}\right)} {\cos \alpha - \cos(\alpha+\gamma)} . \tag{3} \]

Reasoning in an analogous manner, for the interval \(\alpha+\pi<\vartheta<\alpha+\pi+\gamma\) we obtain

\[ i_2 = I_g \left[ 1 - \frac{\cos \alpha + \cos \vartheta} {\cos \alpha - \cos(\alpha+\gamma)} \right] . \tag{4} \]

The rms value of the current, taking overlap into account, is determined as

\[ I_2 = \sqrt{ \frac{1}{2\pi} \left[ \int_{\alpha+\frac{\pi}{3}}^{\alpha+\frac{\pi}{3}+\gamma} i_1^2 \, d\vartheta + \int_{\alpha+\frac{\pi}{3}+\gamma}^{\alpha+\pi} I_g^2 \, d\vartheta + \int_{\alpha+\pi}^{\alpha+\pi+\gamma} i_2^2 \, d\vartheta \right] }. \tag{5} \]

Fig. 3.

Fig. 4.

Substituting equations (3) and (4) into (5) and integrating, we obtain:

\[ I_2 = I_g \sqrt{ \frac{1}{2\pi} \left\{ \frac{ 2\gamma - 2\sin\gamma + \cos(2\alpha+\gamma)(\gamma\cos\gamma-\sin\gamma) } {\left[\cos\alpha-\cos(\alpha+\gamma)\right]^2} + \frac{2\pi}{3} - \gamma \right\} }. \tag{6} \]

It appears possible to obtain a simpler expression for determining \(I_2\), if the influence of overlap on the shape of the current curve is taken into account in a simplified manner. According to Fig. 4, the equation for the variation of the current increasing during overlap is

\[ i_1 = I_g \frac{\vartheta}{\gamma} \tag{7} \]

and the equation for the variation of the current decreasing during overlap, after

\[ \frac{2\pi}{3} \]

is

\[ i_2 = -\frac{I_g}{\gamma} \left[ \left(\frac{2\pi}{3}+\gamma\right)-\vartheta \right]. \tag{8} \]

Substitution of equations (7) and (8) into (5), followed by integration, gives

\[ I_2 = I_g \sqrt{\frac{1}{3} - \frac{\gamma}{6\pi}} . \tag{9} \]

The angle \(\gamma\) entering expressions (6) and (9) can be determined from condition (2).

However, for greater convenience in determining \(\gamma\), condition (2) can be transformed by introducing, for this purpose, a system of relative units.

If the following are chosen as base quantities:
\(I_{gн}\)—the rated current in the direct-current circuit, and
\(E_1\)—the primary phase voltage, then it can be shown that

\[ \frac{2X_a I_g}{E_{\lambda m}} = X_{a*} I_{g*}. \]

Thus, a convenient equation is obtained for determining the angle \(\gamma\):

\[ \cos \alpha - \cos(\alpha + \gamma) = X_{a*} I_{g*}. \]

All other conditions being equal, as the firing angle \(\alpha\) increases, the shape of the current curve approaches the ideal one shown in Fig. 2б. Larger angles \(\alpha\) correspond to a shorter commutation duration, since as \(\alpha\) increases, the emf determining the commutation rate increases. The larger the angle \(\gamma\), the stronger the influence of overlap on the value of \(I_2\). This is also indicated by formulas (6) and (9). Consequently, when posing the problem of estimating the greatest error in determining \(I_2\) by the simplified formulas (1) and (9), the comparison must be made for the case \(\alpha = 0\).

This also corresponds to the practically recommended operating mode of high-power rectifiers, which, in order to reduce the consumption of reactive power from the supply system, should operate with firing angles close to or equal to zero.

Comparative calculations carried out for identical conditions showed that the error in determining \(I_2\) by the simplified formula (9), over the range of reactances \(2\text{–}14\%\), does not exceed \(1\%\).

Determination of \(I_2\) from formula (1), for transformer reactances of \(10\text{–}14\%\), gives an error of no more than \(3\%\).