Eng. I. D. Kutyavin.

IRON CURRENT LIMITERS

Limiting short-circuit currents in low-power branches connected to busbars with powerful power sources is absolutely necessary, since the equipment of such branches, rated for the full short-circuit current, becomes bulky and expensive in comparison with that required for nominal currents.

The question of limiting short-circuit currents is especially acute in powerful low-voltage installations up to 500 V, where these currents may likewise reach extremely high values.

The concrete reactors used up to now for voltages up to 6 kV are expensive, bulky, and unreliable in operation. The operating experience of Lenenergo, Mosenergo, Chelyabres, and the Stalin Thermal Power Station has shown that concrete reactors are unable to withstand the overvoltages arising during accidents in the network. Moreover, these reactors are almost entirely unsuitable for installation in unheated rooms, since the high hygroscopicity of concrete leads, during a short circuit beyond the reactor, to flashover of a phase along the concrete support.

Therefore the question of limiting short-circuit currents by means of iron resistances (resistors) is of exceptionally great interest. Resistors, which have found wide use in Germany, are built for up to 6 kV and have incomparably smaller dimensions and cost than reactors. The operating principle of resistors is based on the increase in the resistance of iron with increasing heating temperature.

The normal temperature during a short circuit is taken to be 800–900° C; in this case an increase in the resistance of iron by a factor of 9–10 can be obtained, depending on the quality of the iron. Of all the impurities in iron, carbon has the most significant influence on the temperature coefficient of resistance; with an increase in its content the temperature coefficient decreases.

The shape and dimensions of the iron cross-section also play an important role. In comparatively thick strips the harmful effect of the skin effect is strongly manifested; therefore it is recommended to use strip iron with a thickness of not more than one millimeter.

In the existing literature on this question, the theory of resistor calculation is covered comparatively fully, but the material is presented in a scattered and insufficiently processed form; therefore

use of this material presents difficulties. Consequently, the task of the present article is to give a complete and simple theory of the thermal calculation of a resistor, taking into account the reactance of the circuit, and a theory for calculating short-circuit currents in circuits with resistors. In addition, the literature has so far contained no indications of a method for calculating the thermal stability, with respect to short-circuit currents, of current-carrying parts in installations with resistors; therefore this question is also developed here rather fully.

1. Thermal calculation of the resistor.

The value of the initial resistance of the resistor is limited by the power losses under rated conditions and is usually expressed as the percentage voltage drop from the rated branch current relative to the phase voltage, i.e.

\[ E_{rn}=\frac{r_n J_n}{E_\phi}\,100, \]

where \(r_n\) is the resistance of the resistor in ohms under rated conditions. The losses in the resistor will be expressed by the same percentage of the total power transmitted by the resistor.

Let us now consider the case when a resistor with percentage resistance \(E_{rn}\) is connected to the station busbars. If the total reactance, in percent, of the generators and transformers feeding the busbars, referred to the rated branch current, is denoted by \(E_x\), then the transient short-circuit current will be expressed by the equality:

\[ J_k=\frac{J_n\,100}{\sqrt{E_x^2+E_{rn}^2(1+\alpha\tau)^2}}\ldots\ldots\ldots\ldots\ldots\ldots\ldots (1) \]

where \(\tau\) is the temperature rise of the resistor due to the short-circuit current, and \(\alpha\) is the temperature coefficient of resistance of the resistor.

Since the duration of the short-circuit current is small, owing to the insignificant time delays in circuits with resistors (up to \(0.5\)—\(1\) sec), it may be assumed that the heating process of the resistor takes place adiabatically; then the energy-balance equation will be written in the form:

\[ J_k^2 r\,dt=g\,c\,d\tau \ldots\ldots\ldots\ldots\ldots\ldots\ldots (2) \]

where \(g\) is the weight of the resistor iron per phase, \(C\) is the mean heat capacity of iron within the range \(50\)—\(800^\circ C\) (it may be taken equal to \(550\,\frac{W}{kg\cdot c}\)).

If in equation (1) we substitute \(J_k\) and the value

\[ r=r_n(1+\alpha\tau)=\frac{E_\phi E_{rn}}{100J_n}(1+\alpha\tau), \]

then equality (2) will take the form:

\[ \frac{J_n E_\phi\,100 E_{rn}(1+\alpha \tau)\,dt}{E_x^2+E_{rn}^2(1+\alpha \tau)^2}=gC\,d\tau . \]

Denoting by \(\psi=\dfrac{1000g}{J_nE_\phi}\) the weight of iron per \(1\ KVA\), \(\delta=\dfrac{E_x}{E_{rn}}\), and separating the variables, we integrate the preceding expression within the limits \(0—t\) and \(0—\tau\)

\[ \frac{10^5 t}{\psi C E_{rn}} = \delta^2 \frac{1}{\alpha}\ln(1+\alpha\tau) + \bigl[(1+\alpha\tau)^2-1\bigr]\frac{1}{2\alpha}. \tag{3} \]

From the last equality one can determine \(\psi\), if one puts \(t=t_m\)—the time-delay of the response protection—and \(\tau=\tau_m\)—the normally permissible heating temperature of the resistor iron during a short circuit, namely:

\[ \psi= \frac{2\alpha\,10^5 t_m} {E_{rn}C\left\{\bigl[(1+\alpha\tau_m)^2-1\bigr]+\delta^2\ln(1+\alpha\tau_m)^2\right\}} \tag{4a} \]

for \(\alpha=0.0091/^\circ C\), \(\tau_m=800^\circ C\), and \(C=550\,\dfrac{w}{kg^\circ C}\)

\[ \psi= \frac{0.78\,\dfrac{t_m}{E_{rn}}}{15.8+\delta^2}. \tag{4b} \]

In Fig. 1 the curves \(\psi=f\left(\dfrac{t_m}{E_{rn}}\right)\) are plotted for various values of \(\delta\) according to eq. (4b).

The character of the rise in the heating temperature of the resistor during a short circuit. To determine the character of the rise in the heating temperature of the resistor iron over the time during which the short-circuit current flows, let us substitute in eq. (3) the value of \(\psi\) from (4):

\[ \frac{2\alpha 10^5 t}{C E_{rn}} \frac{C E_{rn}\left\{\bigl[(1+\alpha\tau_m)^2-1\bigr]+\delta^2\ln(1+\alpha\tau_m)^2\right\}} {2\alpha 10^5 t_m} = \delta^2\ln(1+\alpha\tau)^2+\bigl[(1+\alpha\tau_m)^2-1\bigr] \]

or

\[ \delta^2\ln(1+\alpha\tau)^2+\bigl[(1+\alpha\tau)^2-1\bigr] = \frac{t}{t_m} \left\{\bigl[(1+\alpha\tau_m)^2-1\bigr]+\delta^2\ln(1+\alpha\tau_m)^2\right\}. \tag{5} \]

Solving this equation with respect to \(\tau\) does not appear possible; therefore we investigate it graphically. In Fig. 2 the curves of temperature change with time are plotted as a function \(\tau=f\left[\dfrac{t}{t_m}\right]\) for \(\tau_m=800^\circ C\), \(\alpha=0.009\ 1/^\circ C\), and \(\delta=0, 1\) and \(2\).

From a comparison of the curves for the separate values of \(\delta\), it may be concluded that: 1) the magnitude of the limiting time overrun, in the case

• failure of operation of the oil circuit-breaker relay, may be allowed to be \((1.5—2)t_m\), and the gradation of the relay time delays must be subordinated to this value; and 2) with sufficient accuracy for all practical calculations one may take the law of temperature variation with time from Eq. (5) in the form of the following simple expression:

\[ (1+\alpha \tau)^2 = 1+\frac{t}{t_m}\left[(1+\alpha \tau_m)^2-1\right], \ldots \ldots \ldots (6) \]

since the quantity \(\delta\) for low-power feeders is small, and from Fig. 2 it is evident that the terms of Eq. (5) with the factor \(\delta\) have very little effect on the result.

Dimensions of the iron of the resistor. The length and cross-section of the iron of the resistor can be determined from the following relations:

\[ \psi=\frac{Sl\gamma}{E_\phi J_n} \]

\[ E_{rn}=\frac{J_n \rho l}{E_\phi S}\,100, \ldots \ldots \ldots (7) \]

where \(S\) is the cross-section of the strip in \(\mathrm{mm}^2\), \(l\) is its length in m, \(\rho\) is the specific resistance of iron, and \(\gamma\) is the specific weight in \(\mathrm{kg}/\mathrm{dm}^3\).

Solving the last two equalities jointly, we obtain:

\[ S=10J_n\sqrt{\frac{\psi\rho}{E_{rn}\gamma}}\ \mathrm{mm}^2 \]

\[ l=0.1E_\phi\sqrt{\frac{\psi E_{rn}}{\rho\gamma}}\ \mathrm{m}, \ldots \ldots \ldots (8) \]

where \(\psi\) is according to Eq. (4b) or from Fig. 1.

It was indicated at the beginning that, in order to avoid the harmful influence of the skin effect, the thickness of the iron strip should be no more than \(1\ \mathrm{mm}\), but the dimensions of the iron cross-section are also limited by the maximum permissible thermal load on the surface of the strip from the rated current. The cooling surface of the strip may be represented in the form:

\[ Q=2\frac{Sl}{1000\,b}\ \mathrm{m}^2, \]

where \(b\) is the thickness of the strip in mm, and the power losses in the resistor per phase are

\[ P=\frac{E_{rn}}{100}J_nE_\phi\ \mathrm{watts}. \]

Then the surface thermal load will be

\[ p=\frac{P}{Q} = \frac{10\,b\,E_{rn}E_\phi J_n}{2Sl}\ w/\mathrm{m}^2 \ldots \ldots \ldots (9) \]

and must not exceed the maximum permissible values, since the heating temperature of the resistor iron at the rated current would be high and the range of temperature fluctuation during a short circuit would be narrowed.

Substituting in expression (9) for \(S\) and \(l\) their values from (8), we then obtain that

\[ p = \frac{100 \varepsilon E_{rn} E_{\phi} J_n} {2.10 J_n 0.1 E_{\phi} \sqrt{\frac{\psi^2 \rho E_{rn}}{\gamma^2 \rho E_{rn}}}} \]

or

\[ p=\frac{5\varepsilon E_{rn}\gamma}{\psi}\frac{w}{m^2}\ . . . . . . . . . . (10) \]

On the basis of the last equality one can determine the thickness of the strip \(b\), if for freely radiating surfaces one takes

\[ p_{\mathrm{adm}}=500—550\ \frac{\text{watt}}{m^2} \]

(which corresponds to a temperature rise above the ambient medium of \(20—30^\circ C\)), i.e.,

\[ b=\frac{p_{\mathrm{adm}}\psi}{5E_{rn}\gamma}\ \text{mm}, . . . . . . . . . . . (11) \]

and, knowing the thickness and the cross section of the strip, one can also determine its width.

However, the strip must not be excessively thin, since then bringing in the current becomes more difficult, the mechanical strength is weakened, and its durability is reduced as a result of oxidation.

In practice the thickness of the strip is taken within the range \(0.5—1\ \text{mm}\). The temperature rise of the resistor above the ambient medium, under normal operating conditions and for known strip dimensions, can be determined from expression (10), if one sets \(p_{\mathrm{adm}}=\tau_n q_{\mathrm{adm}}\), where \(\tau_n\) is the temperature rise, and \(q_{\mathrm{adm}}=20\ \frac{w}{m^2{}^\circ C}\) (on the average for \(0—50^\circ C\)), i.e.,

\[ \tau_n=\frac{5\varepsilon E_{rn}\gamma}{q_{\mathrm{adm}}\psi}^\circ C . . . . . . . . . . (12) \]

Cooling time of the resistor. The cooling time of the resistor after a short circuit down to normal temperature \((50^\circ C)\) plays an essential role in operation. The oil circuit breaker, after it has been disconnected by the action of the relay, may be switched on only when the resistor has cooled to normal temperature; otherwise, re-energizing into a short circuit will lead to burnout of the resistor iron. If it is assumed that the cooling process exactly follows an exponential law, then

\[ \tau_n=t'_m e^{-\frac{t'}{T}} . . . . . . . . . . (13) \]

where \(\tau_n\) is the initial temperature under normal operating conditions,

\(\tau_m'\)—the final temperature under short-circuit conditions,

\(t'\)—the cooling time of the resistor, and

\[ T=\frac{gC}{Qq_{\mathrm{dop}}} \]

is the time constant.

The final heating temperature of the resistor will be \(\tau_m'=\tau_n+\tau\), where \(\tau\) is found from Eq. (6), i.e.

\[ \tau= \frac{ \sqrt{1+\frac{t}{t_m}\left[(1+\alpha\tau_m)^2-1\right]}-1 }{\alpha}. \]

Next, for the time constant \(T\), we substitute the values

\[ g=\frac{Sl\gamma}{1000}\ \mathrm{kg} \]

and

\[ Q=\frac{2Sl}{1000\,\varepsilon}\ \mathrm{m}^2; \]

then

\[ T=\frac{\varepsilon\gamma C}{2q_{\mathrm{dop}}}. \tag{14} \]

We now solve Eq. (13) with respect to \(t'\) and, substituting into it the values of \(\tau_m'\) and \(T\), obtain:

\[ t'=T(\ln \tau_m'-\ln \tau_n)=T\ln\frac{\tau_m'}{\tau_n} \]

or

\[ t'=\frac{\varepsilon\gamma C}{2q_{\mathrm{dop}}} \ln \frac{ \sqrt{1+\frac{t}{t_m}\left[(1+\alpha\tau_m)^2-1\right]}-1 }{\alpha\tau_n} +1. \tag{15} \]

In Fig. 3 a curve of the dependence of the form

\[ \frac{t'}{\varepsilon}=f\!\left(\frac{t}{t_m}\right) \]

is plotted for

\[ \tau_n=50^\circ C,\quad \tau_m=800^\circ C,\quad \alpha=0.009\ \frac{1}{^\circ C},\quad \gamma=7.9\ \frac{kg}{dm^3},\quad C=550\ \frac{w}{kg^\circ C} \]

and

\[ q_{\mathrm{dop}}=20\ \frac{w}{M^\circ C}. \]

When using the curve, depending on \(\frac{t}{t_m}\), one can determine the ratio \(\frac{t'}{\varepsilon}=m\), and then \(t'=m\varepsilon\).

2. The nature of the change in short-circuit current over time.

Of very great interest is the law of variation of the short-circuit current with time. If the law of current decay during a short circuit is known, then one can judge the magnitude of the currents being interrupted and the nature of the heating of current-carrying parts. It is also interesting to trace how the magnitude of the percentage reactance affects these changes.

In the event of a sudden short circuit of the station busbars through a resistor, the current will decrease, first, owing to the increased demagnetizing action of the armature flux and, second, owing to the increase in the resistance of the resistor. Let us first consider the influence of the armature reaction on the magnitude of the short-circuit current. As is known, at the first instant of the short circuit the armature reaction cannot demagnetize the machine, because of the damping action of the rotor system; subsequently, however, the influence of the demagnetizing armature flux will gradually begin to make itself felt; the resultant flux will weaken, and with it the induced EMF, and consequently the short-circuit current, will decrease.

To clarify the character of the decay of the current in a short circuit through a resistor as a result of armature reaction, we shall make use of Professor Rüdenberg’s solution for the sudden short circuit of synchronous three-phase machines.

As is known, in addition to the steady currents in the windings, after a sudden short circuit there arise free currents obeying the following relations:

\[ L_1 \frac{di_1}{dt} + R_1 i_1 + M \frac{di_2}{dt'} = 0 \]

\[ L_2 \frac{di_2}{dt} + R_2 i_2 + M \frac{di_1}{dt'} = 0; \tag{16} \]

where \(L_2\) and \(L_1\) are the self-inductances of the rotor and stator, \(R_1 = R' + r\) is the sum of the resistances of the stator winding and the resistor,

\(R_2\)—the resistance of the rotor system,

\(M\)—the coefficient of mutual induction.

Prof. Rüdenberg solves this system of differential equations by assuming a harmonic law for the variation of the currents in the stator with frequency \(\alpha\) and in the rotor with frequency \(\beta\):

\[ i_1 = J_1 e^{i\alpha t} \]

\[ i_2 = J_2 e^{i\beta t}, \tag{17} \]

where the frequencies \(\alpha\) and \(\beta\) are connected by the relation \(\alpha = \omega + \beta\).

The last terms of equations (16) express the induction between the moving rotor and the stationary stator, and the action of the rotor frequency \(\beta\) on the stator must be increased by the amount \(\omega\), while the action of the stator frequency \(\alpha\) on the rotor must be decreased by the amount \(\omega\), in order correctly to take into account the action of the rotor’s rotation.

In a sudden short circuit in circuits without a resistor the frequencies \(\alpha\) and \(\beta\) are constant and depend only on the ohmic and inductive resistances of the stator windings and of the rotor system. Let us now suppose that, when the station busbars are short-circuited onto a resistor,

\[ \alpha = f_1(t) \quad \text{and} \]

\[ \beta = f_2(t); \]

then, from equations (16), upon substituting the values (17), we obtain the relation between the amplitudes and frequencies of the currents:

\[ jL_1J_1\frac{d(\alpha t)}{dt}+R_1J_1+jMJ_2\frac{d(\alpha t)}{dt}=0 \]

\[ jL_2J_2\frac{d(\beta t)}{dt}+R_2J_2+jMJ_1\frac{d(\beta t)}{dt}=0. \tag{18} \]

To obtain an expression for the unknown frequencies, we eliminate the current amplitudes:

\[ \frac{J_1}{J_2} = \frac{-jM\frac{d(\alpha t)}{dt}} {R_1+jL_1\frac{d(\alpha t)}{dt}} = \frac{R_2+jL_2\frac{d(\beta t)}{dt}} {-jM\frac{d(\beta t)}{dt}}. \]

If, for simplification, we introduce the leakage coefficient

\[ \sigma=\frac{L_1L_2-M^2}{L_1L_2} \]

and, moreover, replace \(\frac{d(\alpha t)}{dt}=u\), and

\[ \frac{d(\beta t)}{dt} = \frac{d[(\alpha-\omega)t]}{dt} = u-\omega, \]

then we obtain:

\[ u^2-u\left[\omega+j\left(\frac{R_1}{\sigma L_1}+\frac{R_2}{\sigma L_2}\right)\right] = \sigma\frac{R_1}{\sigma L_1}\frac{R_2}{\sigma L_2} - j\omega\frac{R_1}{\sigma L_1}. \]

The solution of this quadratic equation gives approximate roots of the form:

\[ \frac{d(\alpha_1t)}{dt} = u_1 = \frac{d(\gamma't)}{dt} + j\frac{d(\rho't)}{dt} \]

\[ \frac{d(\alpha_2t)}{dt} = u_2 = \frac{d(\gamma''t)}{dt} + j\frac{d(\rho''t)}{dt}, \]

and integration of these roots gives the values for the frequencies of the equalizing currents

\[ \begin{aligned} \alpha_1t&=\gamma't+j\rho't\\ \alpha_2t&=\gamma''t+j\rho''t \end{aligned} \tag{19} \]

We see that, both for the frequency of the transient current and for its damping coefficient, two values are obtained; consequently, the change of the current in time, upon substituting the values (19) into (17), is expressed by:

\[ i_1=i'_1+i''_1 = J_1\left(e^{-\rho't}\cos\gamma't+e^{-\rho''t}\cos\gamma''t\right). \tag{20} \]

The first component of the transient current \(i'_1\) has an extremely

a low frequency and is, as it were, an instantaneous direct current, while the component \(i''_1\) varies with the frequency

\[ \frac{\rho''_t}{t}=\omega . \]

Since we are interested in the damping of free currents, we shall consider \(\rho'_t\) and \(\rho''_t\), whose values have the form:

\[ \begin{aligned} \frac{d(\rho'_t)}{dt} &= \frac{R_1}{\sigma L_1} + \frac{R_2}{\sigma L_2}\, \frac{\left(\dfrac{R_1}{\sigma L_1}\right)^2(1-\sigma)} {\omega^2+\left(\dfrac{R_1}{\sigma L_1}\right)^2}, \\[6pt] \frac{d(\rho''_t)}{dt} &= \frac{R_2}{\sigma L_2} - \frac{R_2}{\sigma L_2}\, \frac{\left(\dfrac{R_1}{\sigma L_1}\right)^2(1-\sigma)} {\omega^2+\left(\dfrac{R_1}{\sigma L_1}\right)^2}. \end{aligned} \tag{21} \]

Let us make the following substitution in the preceding expressions, neglecting the ohmic resistances of the stator in comparison with the resistance of the resistor,

\[ \frac{R_1}{\sigma L_1}=\frac{\omega r}{x_1} \quad\text{and}\quad \frac{R_2}{\sigma L_2}=\frac{\omega R_2}{x_2}, \]

where \(x_1\) and \(x_2\) are the inductive reactances of the stator and rotor windings; moreover, on the basis of equation (6)

\[ r=r_n(1+\alpha\tau) = r_n\sqrt{\,1+\frac{t}{t_m}\left[(1+\alpha\tau_m)^2-1\right]\,} = r_n\sqrt{\,1+A\frac{t}{t_m}\,}; \]

therefore

\[ \frac{R_1}{\sigma L_1} \simeq \frac{\omega r_n\sqrt{\,1+A\dfrac{t}{t_m}\,}}{x_1} = \frac{\omega}{\delta}\sqrt{\,1+A\frac{t}{t_m}\,}, \quad \text{where } \delta=\frac{x_1}{r_n}=\frac{E_x}{E_m}. \]

Then equations (21) will take the form:

\[ \begin{aligned} \frac{d\rho'_t}{dt} &= \omega \left\{ \frac{\sqrt{\,1+A\dfrac{t_m}{t}\,}}{\delta} + \frac{R_2}{X_2} \left[ \frac{(1-\sigma)\left(1+A\dfrac{t}{t_m}\right)} {\delta^2+\left(1+A\dfrac{t}{t_m}\right)} \right] \right\}, \\[6pt] \frac{d\rho''_t}{dt} &= \omega\frac{R_2}{X_2} \left[ \frac{\delta^2+\sigma\left(1+A\dfrac{t}{t_m}\right)} {\delta^2+\left(1+A\dfrac{t}{t_m}\right)} \right]. \end{aligned} \tag{22} \]

Let us now integrate these equations over the limits \(0\)—\(t\); we obtain:

\[ \begin{aligned} \rho'_t &=\omega\left\{ \frac{2t_m}{3A\delta} \left[ \left(1+A\frac{t}{t_m}\right)^{3/2}-1 \right] +\frac{(1-\sigma)t_m\dfrac{R_2}{x_2}}{A} \left[ A\frac{t}{t_m} -\delta^2 \ln \left( \frac{\delta^2+1+A\dfrac{t}{t_m}}{\delta^2+1} \right) \right] \right\},\\[6pt] \rho''_t &=\omega\frac{R_2}{x_2} \left[ \frac{t_m\delta^2}{A}(1-\sigma)\ln \left( \frac{\delta^2+1+A\dfrac{t}{t_m}}{\delta^2+1} \right) +\sigma t_m\frac{t}{t_m} \right]. \end{aligned} \tag{23} \]

In Fig. 4, curves are plotted for the relative damping of the free currents according to equations (23) as a function of \(\dfrac{t}{t_m}\)

\[ \frac{i'_1}{J_1}=e^{-\rho'_t}\text{—curve }1' \]

\[ \frac{i''_1}{J_1}=e^{-\rho''_t}\text{—curves }1,\ 2,\ 3,\ 4,\ 5\text{ and }6. \]

In constructing them it was assumed that

\[ \frac{R_2}{X_2}=0.01,\qquad \tau_m=800^\circ C,\qquad \alpha=0.009\ \frac{1}{^\circ C} \]

and the leakage coefficient \(\sigma=0.1\).

As is seen from Fig. 4 (curve 1), the constant component of the transient current by the instant of the short-circuit current jump \((t=0.01\ \mathrm{sec.})\), for \(\delta=1\) and \(t_m=1\), decreases to \(6\)—\(7\%\) of its initial value. For values \(\delta<1\), \(t_m<1\), the damping curves of the constant component lie between the ordinate axis and the curve shown, so that the periodic current will be almost unaffected in amplitude during the first half-wave.

In Fig. 5, curves are plotted for the relative damping of the constant component as a function of \(\dfrac{t}{t_m}\) for various values of

\[ \frac{t_m}{\delta}, \]

where the second term of equation (23) for \(\rho''_t\) has been neglected because of the smallness of the factors \(\dfrac{R_2}{X_2}\) and \(\delta\). We see that with an increase in the relative reactance \(\delta\), the damping time of the constant component increases; it also increases with increasing \(t_m\).

The damping of the alternating component of the transient current with time, and likewise of the generator emf, is insignificant (Fig. 4: curves 4, 5, and 6 for \(\delta=0\) and 1, 2, and 3 for \(\delta=1\)); it increases with increasing relative inductance \(\delta\) and the calculated time delay \(t_m\). This damping is due exclusively to the demagnetizing action of the longitudinal and transverse armature fields; moreover, the transverse field, unlike the short circuit in circuits without a resistor, plays the predominant role.

Therefore, in calculating short-circuit currents, taking into account the influence of the armature reaction on the magnitude of the current is hardly meaningful. Moreover, all modern electric stations are usually equipped with fast-acting automatic voltage regulators; therefore, a slight decrease in the emf during a short circuit will be compensated by their action (if current limiters are absent).

In fact, at the moment of a sudden short circuit, owing to the released part of the generator’s magnetic flux, there occurs a step-like increase in the excitation current, whose decay should obey the law of decay of the generator emf. The decay of the emf, as well as of the generator field, proceeds gradually, whereas the terminal voltage, as a result of the increased voltage drop in the stator windings at the instant of the initial jump of the short-circuit current, will fall sharply and will actuate the voltage regulator. The increase of current in the generator excitation circuit under ordinary conditions proceeds according to an exponential law, with time constant \(\sigma L_2/R_2\), but this law of current rise will not be valid during a short circuit, since in this case the task of the voltage regulator will be not to increase, but to maintain the already increased current in the generator excitation circuit.

Calculation of short-circuit currents. On the basis of the preceding considerations, it may be assumed that the emf of the generators feeding the short circuit remains constant throughout the entire duration of the short circuit; then all calculations are greatly simplified. With constant emf of the generators feeding the short circuit, the change of the short-circuit current with time will be determined only by the change in the heating temperature of the resistor iron.

The relative value of the transient current as a function of the temperature of the resistor iron may be written from (1), taking (6) into account:

\[ \frac{J_k}{J_n} = \frac{100}{\sqrt{E_x^{2}+E_{rn}^{2}(1+\alpha\tau)^2}} = \frac{100}{E_{rn}\sqrt{\delta^{2}+1+\frac{t}{t_m}\left[(1+\alpha\tau_m)^2-1\right]}} \]

\[ \text{or}\quad \frac{J_k}{J_n}E_{rn}=K= \frac{100}{\sqrt{\delta^{2}+1+\frac{t}{t_m}\left[(1+\alpha\tau_m)^2-1\right]}} \qquad\ldots\ldots (24a) \]

For \(\alpha=0.009\ 1/^\circ\mathrm{C}\) and \(\tau_m=800^\circ\mathrm{C}\), the preceding equation takes the form:

\[ \frac{J_k}{J_n}E_{rn}=K= \frac{100}{\sqrt{\delta^{2}+1+66\,\frac{t}{t_m}}} \qquad\ldots\ldots (24в) \]

In Fig. 6, curves have been constructed from equation (24*b) for \(\delta=0,1,2\) as functions

\[ k=f\left(\frac{t}{t_m}\right). \]

From equation (24 a), for known \(\delta,\ \tau_m,\ \alpha\), and \(t_m\), all values of the transient current can be determined by substituting the corresponding values of the time \(t\). Thus, for example, the effective value of the first half-wave of the transient current can be determined from (24 a) at \(t=0.01\) sec., i.e.

\[ J_{uv}= \frac{J_n 100} {E_m\sqrt{\delta^2+1+\frac{0.01}{t_m}\left[(1+\alpha\tau_m)^2-1\right]}} ,\ . . . . . (25) \]

and the surge current, or the amplitude of the first half-wave, is \(J_s=\sqrt{2}\,J_{uv}\), since, on the basis of what was said above, the constant component almost completely dies out during \(t=0.01\) sec.

The breaking current for selecting oil circuit breakers at \(t=0.25\) sec. will be:

\[ J_e= \frac{J_n 100} {E_m\sqrt{\delta^2+1+\frac{0.25}{t_m}\left[(1+\alpha\tau_m)^2-1\right]}} .\ . . . . (26) \]

And the steady-state current at \(t=t_m\)

\[ J_{ky}= \frac{J_n 100} {\sqrt{\delta^2+(1+\alpha\tau_m)^2}} .\ . . . . . . (27) \]

Here \(J_n\) is the rated current of the feeder.

3. Calculation of the Current-Carrying Parts.

If it is assumed that all the heat released by the short-circuit current in the current-carrying part goes toward raising its temperature, then the energy balance will be expressed by the relation:

\[ 0.24\,J_k^2 R\,dt = g_1 c_1\,d\tau, \ . . . . . . . . . (28) \]

where \(g_1\) is the weight of the current-carrying part in grams, and

\[ c_1 \]

is the mean heat capacity of the current-carrying part in

\[ \frac{\text{cal}}{\text{g}^{\circ}\mathrm{C}}. \]

If \(J_k\) is substituted from equation (24a),

\[ R=R_0(1+\alpha_1\tau)=\frac{\rho_1 l}{q}(1+\alpha_1\tau) \]

is the resistance of the current-carrying part, and \(g_1=lq\gamma_1\) is its weight, then relation (28) will take the form:

\[ \frac{ 0.24\cdot 10^4 J_n^2 \rho_1 \frac{l}{q}(1+\alpha_1\tau)\,dt }{ E_m^2\left\{\delta^2+1+\frac{t}{t_m}\left[(1+\alpha\tau_m)^2-1\right]\right\} } = lq\gamma_1 c_1\,d\tau \]

or, separating the variables, we obtain:

$$ \frac{J_n^2}{q^2}\cdot \frac{0.24\cdot 10^4\rho_1}{\gamma_1 C_1 E_{rn}^2}\cdot \frac{dt}{\left(\delta^2+1+A\frac{t}{t_m}\right)} = \frac{d\tau}{1+\alpha_1\tau}, $$

where \(A=[(1+\alpha\tau_m)^2-1]\).

Integrating the preceding expression over the limits \(0—t\) and \(0—\tau\), we shall have:

$$ \frac{J_n^2}{q^2}\cdot \frac{t_m}{A E_{rn}^2}\cdot \frac{0.24\cdot 10^4\alpha_1\rho_1}{\gamma_1 C_1} \ln\left[ \frac{\delta^2+1+A\frac{t}{t_m}}{\delta^2+1} \right] = \ln(1+\alpha_1\tau). \tag{29} $$

Next we introduce the following notation:

$$ \frac{J_n^2 t_m}{q^2 E_{rn}^2}=\Delta \quad \text{and} \quad \frac{0.24\cdot 10^4\alpha_1\rho_1}{A\gamma_1 C_1}=B, \tag{30} $$

where \(B\) is a constant depending on the conductor material; then

$$ \Delta= \frac{\ln(1+\alpha_1\tau)} {B\ln\left[ \frac{\delta^2+1+A\frac{t}{t_m}}{\delta^2+1} \right]} \tag{31} $$

According to equation (31), Fig. 7 gives a nomogram for copper, and Fig. 8 for aluminum.

In constructing it, it was assumed: 1) for the resistor, \(\tau_m=800^\circ C\) and \(\alpha=0.009\ \frac{1}{^\circ C}\), i.e. \(A=66\); 2) for copper, \(\gamma_1=8.9\ \mathrm{g/cm^3}\), \(C_1=0.093\ \frac{\mathrm{cal}}{\mathrm{g}\,^\circ C}\), \(\rho_1=0.0178\ \frac{\Omega\,\mathrm{mm^2}}{\mathrm{m}}\), and \(\alpha_1=0.00428\ 1/^\circ C\); 3) for aluminum, \(\gamma_1=2.7\ \mathrm{g/cm^3}\), \(C_1=0.219\ \frac{\mathrm{cal}}{\mathrm{g}\,^\circ C}\), \(\rho_1=0.03\ \frac{\Omega\,\mathrm{mm^2}}{\mathrm{m}}\), and \(\alpha_1=0.0039\ 1/^\circ C\).

Use of the nomogram is very simple. When checking a selected cross section of the current-carrying part for heating by short-circuit currents, we determine from (30) the value of \(\Delta\), and then from the values \(\Delta\), \(\delta=\frac{E_x}{E_{rn}}\), and \(\frac{t}{t_m}\) on the nomogram we determine the heating temperature of the current-carrying part. When choosing the cross section of the current-carrying part from the conditions of permissible heating by short-circuit currents, we determine \(\Delta\) from the nomogram for specified \(\frac{t}{t_m}\), \(\delta\), and \(\tau_{dop}\); then the cross section is determined from (30), i.e.

$$ q_{min}=\frac{J_n}{E_{rn}}\sqrt{\frac{t_m}{\Delta}}. \tag{32} $$

List of References Used:

1. Butkevich and Kulakova. Iron resistances for limiting short-circuit currents. Elektrichestvo, 1933, No. 2, p. 31.
2. Butkevich, Mikhailov and Rogauz.—Reactors. Moscow, 1933.
3. Poyarkov. Central Electric Power Stations. Moscow, 1933.
4. Rüdenberg. Short-circuit currents in the practical operation of large power stations. Translated from the German by I. B. Mandelstam. Leningrad, 1930.

SUMMARY.

The principal aim of the present article is to give the most complete and simple theory for the calculation of iron resistors, having the function to limit the short circuit currents in the schemes of power stations.

In the first part of this article the author gives a resolution of principal equations for determining the parameters of resistors which depend on the normal working regime of the network as well as on that of breakage taking the reactance of the scheme into consideration. The law of time changes of a resistor heating temperatures, during the short circuit as well as the time of cooling of the resistor after the short circuit are also defined.

The question of the time change of a short circuit current depending on the influence of the reaction of the generator armature and on the increase of resistance in the resistor during the short circuiting is also narrowly examined.

In order to determine the influence of the generator armature reaction during the passage of the short circuit through the resistor on the character of damping of the transient current, the author made use of Prof. Rüdenberg’s method of resolution for the sudden short circuit of synchronous three phase machines, it being ascertained that the reaction of the armature during the short circuiting in circuits with resistors influences but very little the value of the short circuit current.

The constant component of the transient current toward the moment of the short circuit skip (10,01 sec.) damps to several per cent from its initial value. The damping of the variable component of the transient current is negligible, therefore, in equations recommended by the author, for the calculation of short circuit currents in circuits with resistors only the influence of the resistance of the resistor and the reactance of the scheme on the value of the transient current is taken into consideration.

These equations are very simple and convenient for use.

At the end of the article a theory for the calculation of the current-conducting parts is given with consideration of conditions of their heating by the short circuit current in circuits with resistors. To simplify the calculations, nomograms for aluminium and copper are given.

Fig. 1.

Vertical axis: \(\psi\)

Horizontal axis: \(\dfrac{t_m}{\xi z_m}\)

Curve labels: \(\delta = 0\), \(\delta = 1\), \(\delta = 2\), \(\delta = 3\).

Fig. 2.

Vertical axis: \(\tau\)

Horizontal axis: \(\dfrac{t}{t_m}\)

Curve labels: \(\delta = 2\), \(\delta = 1\), \(\delta = 0\).