ON THE ISSUE OF SUPPRESSING THE MAGNETIC FIELD OF A SYNCHRONOUS GENERATOR USING A VARISTOR RESISTANCE

I. V. Pavlov

The Ideal Method of Field Suppression

The most widespread circuit for field suppression (f.s.) on generators in the Soviet Union is the circuit of the Kharkov Electromechanical Plant (KhEMZ). A certain acceleration of f.s. under this circuit can be achieved by increasing the suppressing resistance—\(R_g\). The factor limiting the speed of f.s. is the magnitude of the overvoltage on the rotor rings, equal to the voltage drop in the suppressing resistance:

\[ U_k = iR_g . \]

If the active resistance of the excitation winding is neglected, then the value of \(U_k\) will be equal to

\[ U_k = -W\frac{d\Phi}{dt}; \]

i.e., the magnitude of the voltage on the rings will be proportional to the rate of decrease of the field, \(-\frac{d\Phi}{dt}\). As is known, in the KhEMZ circuit the voltage on the rings reaches a maximum only at the first instant of f.s., and then decreases according to the law:

\[ U_k = U_{k.\max} e^{-\frac{t}{T}}, \]

i.e., the speed of f.s. during the entire process is below the permissible value.

If in some way the voltage on the rings is maintained at the maximum permissible limit, then f.s. will also proceed at the greatest possible speed. Such an f.s. process may be called ideal. Under ideal f.s., the magnetic flux and the current decay according to a straight-line law \(\left(\frac{di}{dt} = \text{const},\right.\) if there is no saturation of the steel).

To carry out the ideal f.s. process, it is necessary to increase the suppressing resistance according to a hyperbolic law:

\[ R_g = \frac{U_{\max.\,\mathrm{dop}}}{i} = \frac{i_0 R_0}{i_0 - kt}, \]

where \(i_0, R_0\) are the initial values of the current and of the suppressing resistance; \(t\) is time; \(k\) is the coefficient determining the slope of the straight line of current decay.

The graph of the variation of the quantities \(i, R_g, U_k\) during the ideal f.s. process is given in Fig. 1. The straight line of current decrease under ideal f.s. coincides with the tangent to the current curve \(i = i_0 e^{-\frac{t}{T}}\) of the KhEMZ circuit at the instant of the beginning of suppression.

\(t = 0\). Hence it follows that the field-suppression time by the ideal method will be 4–5 times shorter than the field-suppression time of the HEMZ circuit.

In practice, it is very difficult to implement a suppressing resistance that varies according to a hyperbolic law. The introduction of additional contacts into the field-suppression circuit is undesirable. One of the simplest methods

Fig. 1

of accelerating field suppression may be the use of a suppressing resistance made of nonlinear elements having a volt–ampere characteristic as close as possible to the ideal one:

\[ U=\mathrm{const}. \]

Volt–ampere characteristic of vilite and its use for field suppression

The vilite disks that have hitherto been used in arresters have a \(V\)–\(A\) characteristic of the form:

\[ U=ci^a, \]

Fig. 2

where, for standard disks, one may take on average:

\[ C=700,\qquad a=0.2. \]

The dependence \(U=700i^{0.2}\) is shown in Fig. 2. This curve is quite close to the ideal straight line \(U=\mathrm{const}\) over the portion lying to the right of the bend.

Let us investigate the question—which part of the \(V-A\) characteristic is most suitable for f. s.

A vilite damping resistance may be designed for a higher or lower current density, i.e., the number of disks connected in parallel can be varied. At the same time, in order to obtain the required voltage on the rings, it will be necessary to take disks of different heights or connect them in series.

When the current density is increased, the point at which the f. s. process begins on the \(V-A\) characteristic (Fig. 2) shifts to the right. The curve \(U = ci^{\alpha}\), as \(i\) increases, approaches more and more closely a horizontal straight line. It would seem that the f. s. process would approach the ideal. It turns out, however, that the speed of f. s. does not depend on the current density in the vilite disks. This is confirmed by the following considerations:

1. The degree of approximation of the actual curve to the ideal characteristic \(U=\mathrm{const}\) can be estimated by the ratio of the areas enclosed under these characteristics, starting from the origin of coordinates and up to the current \(i_{\max}\) (Fig. 2). Let us calculate this ratio:

\[ z=\frac{S_1}{S_2} = \frac{\displaystyle\int_0^{i_{\max}} U_1\,di}{U_2 i_{\max}} = \frac{\displaystyle\int_0^{i_{\max}} ci^{\alpha}\,di}{U_2 i_{\max}} = \frac{1}{1+\alpha}, \]

where

\[ c=\frac{U_{\max.\ \mathrm{adm}}}{i_{\max}^{\alpha}} = \frac{U_2}{i_{\max}^{\alpha}}. \]

Thus, the ratio of the areas does not depend on the current density.

1. Let us consider the simplest case of f. s. of an inductive coil with an unsaturated core (Fig. 3).

Fig. 3

In the damping resistance there are \(n\) parallel branches, and each consists of \(m\) disks connected in series.

Let us write the equation of electrical equilibrium:

\[ -L\frac{dI}{dt}=c_1 i_1^{\alpha}, \]

where \(I=ni_1\), \(c=mc_1\), \(c_1\) and \(\alpha\) are the parameters of one disk.

Integrating, we obtain:

\[ i_1= \left[ i_{10}^{\,1-\alpha} - \frac{1-\alpha}{nL}mc_1 t \right]^{\frac{1}{1-\alpha}}. \]

At a current equal to zero, we find the field-suppression time:

\[ t_{\text{f.s.}}=\frac{i_0^{\,1-\alpha}Ln}{(1-\alpha)mc_1}. \]

Since

\[ mc_1=\frac{U_{\max.\,\text{allow}}}{i_0^\alpha}. \]

then

\[ t_{\text{f.s.}}=\frac{i_0L}{(1-\alpha)U_{\max.\,\text{allow}}}. \]

that is, the field-suppression time does not depend on the dimensions of the suppressing resistance \(n\) and \(m\), and consequently not on the current density. This proposition was verified experimentally by the author of the article.

It is obvious that one can approach the ideal method of f.s. only by reducing the nonlinearity coefficient \(\alpha\).

Thus, from the standpoint of the speed of field suppression, the current density is of no importance. However, the current density has a very strong effect on the volume of the suppressing resistance and, consequently, on its cost, dimensions, etc. Thus, if the volume of the suppressing resistance at a maximum current density of \(20\,a\) per disk is taken as unity, then at a current density of \(1\,a\) per disk the volume of the resistance increases 23-fold. The character of the change in the volume of the suppressing resistance as a function of the maximum current density is shown in Fig. 4.

Fig. 4

Obviously, it is necessary to adopt the maximum permissible current density in order to obtain the minimum volume of resistance.

Calculation of the Vilite Suppressing Resistance

The purpose of the calculation is to determine the volume of the suppressing resistance, or the total number of standard disks \(N\), as well as to determine the numbers \(n\) and \(m\) (see above).

Vilite disks can carry very large currents, but only for a very short time. If the time during which current is passed is increased, the disk over-

...it overheats, and thermal breakdown occurs. The current-carrying capacity of a disk can be determined by the amount of energy—\(Q_{\text{disk}}\)—that the disk can absorb before breakdown during a single passage of current. The value \(Q_{\text{disk}}\) is not constant and depends on the time \(\Delta t\) during which the current is passed. If, in addition, the current changes over the interval \(\Delta t\), then \(Q_{\text{disk}}\), obviously, also depends on the current waveform. Data on the current-carrying capacity of a varistor for times \(\Delta t > 1\) sec are not yet available in the literature. Still less has the current-carrying capacity been investigated for currents varying according to an exponential law.

Thus, an experimental investigation is needed of the current-carrying capacity of the varistor for the indicated cases.

If the value \(Q_{\text{disk}}\) is determined for a standard disk, then the value of the damping element will be equal to:

\[ N = k \frac{Q_{\text{full}}}{Q_{\text{max}}}\quad (\text{pieces}), \]

where \(Q_{\text{full}}\) is the greatest amount of energy that can be released in the damping resistance in the worst case of field suppression, i.e., a two-pole short circuit of the stator,

\(k\) is the strength safety factor.

Knowing the number of disks \(N\) and the maximum value of the excitation current \(I_{\text{max}}\) during field suppression, one can determine \(n\) and \(m\) from the following conditions:

\[ \text{a)}\qquad n \cdot m = N, \]

\[ \text{b)}\qquad m c_1 \left(\frac{I_{\text{max}}}{n}\right)^a = U_{\text{max. allow.}}; \]

hence

\[ n = \left[ \frac{N c_1 I_{\text{max}}^a}{U_{\text{max. allow.}}} \right]^{\frac{1}{1-a}}, \]

\[ m = \frac{N}{n}. \]

For rough preliminary calculations one may use the relation:

\[ n = \frac{I_{\text{max}}}{10}, \qquad m = 1.5, \qquad N = 0.15 I_{\text{max}}, \]

where \(I_{\text{max}}\) should be taken as the maximum value of the excitation current for a three-pole short circuit of the stator. The maximum voltage on the rings will then be about 1700 V.

Conclusion

For the same magnitude of overvoltage on the rotor rings, field suppression using a varistor resistance proceeds 2.5–3 times faster than with the KHEMZ circuit. These figures were obtained by the author of the article both by calculation and experimentally (a salient-pole generator of 15 kVA was used). When a varistor is used, the field-suppression circuit breaker need not have a normally open contact, since at the exciter voltage the current flowing through the damping resistance is very small. Taking these advantages of the new method of field suppression into account, the gaps in the calculation method noted above should be filled, and the new circuit should be tested more widely under production conditions.