On Estimating the Variation of Shunts for High Direct Currents
Abstract Generated abstract
This paper examines variation in stationary shunts for high direct currents caused by nonuniform current supply, a problem that becomes more significant as terminal height increases. Using a simplified symmetrical shunt model, the authors derive expressions for the potential field by conformal mapping and obtain formulas for estimating voltage variation as a function of shunt geometry and potential clamp position. Calculations for a normalized shunt length are compared with experiments using an electrically conductive paper analogue, and the measured points are reported to agree sufficiently with the analytical dependence. The results indicate that variation may be positive or negative, has a positive maximum near a specified terminal height range, becomes zero near another height, and can be reduced markedly by increasing the distance between the potential clamp and the current-carrying bus connection.
Full Text
ON ESTIMATING THE VARIATION OF SHUNTS FOR HIGH DIRECT CURRENTS
L. G. Fuks, V. V. Ivanov
(Presented by the Department of Boiler Construction and Boiler Installations)
The standard for stationary shunts, GOST 8042-61, provides that the maximum variation of a shunt arising from nonuniformity of current supply must lie within one half of the nominal error of the corresponding accuracy class. The danger of large variations increases as the height of the tip increases, i.e., in the transition to shunts for high currents. The greatest nonuniformity of current distribution occurs with symmetrical connection of the current-carrying buses on both sides of the shunt, as shown in Fig. 1-b.
If, in a shunt connected into the circuit according to the scheme (Fig. 1-a), a voltage drop \(V_1\) arises, while for any other connection—for example, as in Fig. 1-b—the voltage drop at the same current is \(V_2\), then the magnitude of the variation is determined as
Fig. 1. \(a\)—uniform current supply, \(b\)—nonuniform current supply.
1—plate, 2—terminal, 3—current-carrying bus, 4—potential clamp.
Fig. 2.
\[ \Delta V=\frac{V_2-V_1}{V_1}\,100\%. \]
For a theoretical study of the shunt variation we shall use a simplified scheme (Fig. 2-a), which was obtained on the basis of the following assumptions:
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At the point where the current-carrying bus is connected \((x=0,\;0<y<h)\), the potential will be considered constant and equal to \(t_0\).
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In view of the fact that the shunt and the current leads are symmetrical with respect to the transverse axis of the shunt, it is possible to consider only one half and to assume that at the middle of the shunt \((x=l)\) the potential is the same and equal to \(t_c\).
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The potential terminal is shown in Fig. 2-a by the coordinates \((a,\;1/2)\).
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The height of the shunt is taken equal to unity.
Thus, the problem is reduced to finding the potential in the region shown in Fig. 2-a.
To solve the problem we shall apply the method of conformal mappings. The function
\[ W=\frac{2}{\pi}\operatorname{arth}\left\{\cos\frac{\pi h}{2}\sqrt{\operatorname{th}^{2}\frac{\pi z}{2}+\operatorname{tg}^{2}\frac{\pi h}{2}}\right\} \tag{1} \]
maps the region under consideration of the \(z\)-plane onto the infinite strip \(0\leq v\leq 1\) of the \(W\)-plane (Fig. 2-b) [1].
The distribution of the potential in this strip is given by the formula
\[ t=t_0-\frac{t_0-t_c}{l^*}\,u. \tag{2} \]
The length of the shunt \(l\) becomes \(l^*\), the distance from the potential terminal to the point where the buses are connected \(a\) becomes \(a^*\), and the points \(A, B, C,\) and \(D\) are transformed into the corresponding points: \(A'\), \(B'\), \(C'\), \(D'\).
Taking into account that, for determining the variation, the constant-current mode is characteristic, we obtain the value of the variation in passing from \(h=0\) to a certain value \(h\) (Fig. 2)
\[ \Delta V=\frac{(l^*-l)-(a^*-a)}{l}. \tag{3} \]
The length \(l^*\) is found from (1)
\[ l^*=\frac{2}{\pi}\operatorname{arth}\left\{\cos\frac{\pi h}{2}\sqrt{\operatorname{th}^{2}\frac{\pi l}{2}+\operatorname{tg}^{2}\frac{\pi h}{2}}\right\}. \tag{4} \]
Since for large values of \(l\) it is difficult to find the quantity \(l^*\), a simplified method is proposed for determining the difference \(l^*-l\). For this purpose we transform formula (4) to the form
\[ \operatorname{th}^{2}\frac{\pi l^*}{2} = \cos^{2}\frac{\pi h}{2}\operatorname{th}^{2}\frac{\pi l}{2} + \sin^{2}\frac{\pi h}{2}. \tag{5} \]
Replacing the hyperbolic tangent, for large values of \(l\), by its approximate value
\[ \operatorname{th}^{2}\pi l \simeq 1 - 4e^{-\pi l}, \]
we obtain
\[ l^{*}-l=-\frac{1}{\pi}\ln \cos^{2}\frac{\pi h}{2}. \tag{6} \]
To determine the new abscissa of the potential terminal, we use function (1) and find its real part \(Re(W)\) for \(x=a,\ y=1/2\). This quantity is found by means of a sequence of operations convenient for calculation:
\[ a^{*}=\frac{1}{\pi}\ln \frac{\sqrt{1+P^{2}-2R}} {1-\sqrt{2(P+R)+P}}, \tag{7} \]
where
\[ P=\sqrt{R^{2}+S^{2}}, \tag{8} \]
\[ R=\sin^{2}\frac{\pi h}{2}+\cos^{2}\frac{\pi h}{2}\,M, \tag{9} \]
Fig. 3.
\[ S=\cos^2 \frac{\pi h}{2}\,N . \tag{10} \]
\[ M=\frac{\operatorname{sh}^2 \pi a-1}{\operatorname{sh}^2 \pi a+1}, \tag{11} \]
\[ N=\frac{2\operatorname{sh}\pi a}{\operatorname{sh}^2 \pi a+1}. \tag{12} \]
Using formulas (3)–(12), the variations of a shunt of total length 100 units were calculated for various values of \(a\) and \(h\). The results of the calculation are shown in Fig. 3 by solid lines.
To check the dependences obtained above, experiments were carried out based on the electrical analogy of potential fields. The shunt model was cut out of electrically conductive paper having a resistance of about 300 ohm/m. A strip of paper 200 mm wide and 1214 mm long was clamped between two massive copper current leads, one of which could be moved along the width of the strip. By means of this displacement, various values of \(h\) were simulated. During the experiment a constant value of the current was maintained. The variation was measured with the aid of a probe placed at various distances \(a\) from the movable current lead.
The results of the experiments, calculated for a shunt length equal to 100 units, are plotted in Fig. 3 as points. As can be seen from the figure, the experimental values confirm the analytical dependence (3) with sufficient accuracy.
Considering Fig. 3, one may note:
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Both positive and negative values of the variation may arise in the shunt.
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There is a maximum of the positive variation, lying within the limits \(h=0.4 \div 0.45\).
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Near \(h=0.6\) the variation passes through zero.
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Negative variations of sufficiently large magnitude occur for \(h>0.75\).
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The maximum variation decreases sharply as the distance \(a\) increases, i.e., the distance from the potential clamp to the place where the current-carrying buses are connected.
Conclusions
It is possible to estimate the magnitudes of the shunt variation and to give recommendations for reducing these magnitudes to normal values that do not exceed the limits of the standard requirements.
Literature
- Lavrent’ev M. A., Shabat B. V. Methods of the Theory of Functions of a Complex Variable, Gostekhizdat, 1951.