Calculation of a Heating Resistance for Automatic Field Suppression

I. V. Pavlov

The greater the magnitude of the suppressing resistance in automatic field-suppression circuits (a.f.s.), the faster the process of de-excitation of a synchronous machine proceeds. The magnitude of the suppressing resistance is limited by a certain value, which depends on the strength of the insulation of the field winding and on the initial value of the excitation current. The maximum voltage on the slip rings is expressed by the formula:

\[ U_m = i_0 R . \tag{1} \]

Since, in the process of field suppression, the excitation current decreases, the resistance can be increased so that the product \(iR\) does not exceed the maximum permissible voltage on the slip rings.

For automatic increase of the resistance during the suppression process, its weight can be calculated in such a way that it is strongly overheated by the excitation current. By making the resistance from a material with a high temperature coefficient, an increase in \(R\) during the field-suppression process by several times can be achieved.

Initial system of equations

To investigate the transient process in an \(L, R\) circuit with a heating resistance, we have compiled a system of equations for the electrical equilibrium of the circuit, the thermal balance of the resistance, and the law of variation of the resistance with temperature:

\[ - L \frac{di}{dt} = ir, \tag{2} \]

\[ i^2 r\,dt = GC\,d\tau, \tag{3} \]

\[ r = r_0(1+\alpha \tau), \tag{4} \]

where \(L\) is the inductance of the field winding, \(r\) is the magnitude of the resistance, \(\tau\) is the temperature of the resistance, \(i\) is the excitation current, \(G\) is the weight of the resistance, \(C\) is the mean heat capacity of the resistance, \(r_0\) is the initial value of the resistance, \(\alpha\) is the temperature coefficient, and \(t\) is time.

The solution is obtained in the following form:

\[ i = i_0 \sqrt{\frac{K+1}{e^{\frac{t}{T}}+K}}, \tag{5} \]

\[ r = r_0 \frac{K+1}{K e^{-\frac{t}{T}}+1}. \tag{6} \]

where

\[ K=\frac{i_0^2 L\alpha}{2GC} \]

is a constant coefficient, which we have called the overheating coefficient,

\[ T=\frac{L}{2r_0(1+K)} \tag{6a} \]

is a coefficient equal to one half of the circuit time constant at the end of the process.

The overheating coefficient \(K\) is so called because it shows the ratio of the increase in resistance at the end of the process (at \(t=\infty\)):

\[ r_{\max}=r_0(1+K), \tag{7} \]

and, therefore, the magnitude of the overheating of the resistance.

Determination of the Most Advantageous Values of

\(K\) and \(r_0\)

The voltage curve at the rings

\[ U=ri=U_0\,\frac{K+1}{K e^{-\frac{t}{T}}+1}\sqrt{\frac{K+1}{e^{\frac{t}{T}}+K}} \tag{8} \]

has a maximum in the middle of the process.

The graph of function (8) is shown in Fig. 1.

Fig. 1

The relative magnitude of the overvoltage peak as a function of \(K\) is expressed by the graph in Fig. 2.

To determine the most advantageous overheating coefficient, we construct the curves \(i=f(t)\) for various \(K\) and under the condition \(U_m=\text{const}\). In this case, the value \(r_0\) must be selected according to the relation:

\[ \frac{U_m}{U_0}=f(K). \tag{9} \]

Since \(U_0=i_0 r_0\), the initial value of the resistance is equal to:

\[ r_0=\frac{U_m}{i_0\cdot f(K)}. \tag{10} \]

The curves of the decay of the excitation current, and consequently also of the magnetic field of excitation, are shown in Fig. 3.

Fig. 2

It follows from Fig. 3 that the most advantageous overheating coefficient is equal to 5.

Fig. 3

Method for calculating the resistance

Since the value of the energy \(Q\) released in the field-suppression resistance is known\(^{1)}\), then, for a given \(K\), it is easy to find the weight of the resistance from

\(^{1)}\)
\[ Q=\frac{i_0^2 L}{2} \]
is the energy of the magnetic field of excitation released during field suppression with the windings on the stator open.

\[ Q_3=\frac{Q_0}{\sigma}=\frac{Q_0 X'_d}{X_d} \]
is the energy released during field suppression in the case of a three-phase short circuit of the stator.

formula (see 6a):

\[ K=\frac{Q^{a}}{GC}. \tag{11} \]

Thus, the weight is equal to:

\[ G=\frac{Q^{a}}{CK}. \tag{12} \]

The value of the resistance in the cold state is determined by formula 10. Knowing the weight and the value of the resistance, we find the length and cross-section of the conductor:

\[ l=\sqrt{\frac{r_{0}G}{\rho d}}, \]

\[ S=\frac{G}{l\cdot d}, \]

where
\(\rho\) — specific resistance,
\(d\) — specific weight,
\(l\) — length,
\(S\) — cross-section.