Investigation of the Transient Process During a Short Circuit in the Circuit of a Compensated Ferromagnetic Frequency Tripler Using an Analog Machine

Ya. V. Petrov, M. S. Komaltilov, S. N. Lysenko

(Presented by the scientific-technical seminar of the Research Institute of AEM and the Department of EPA, TPI)

Short circuits (s.c.) beyond the series-compensation capacitors (SC) of a ferromagnetic frequency tripler can cause dangerous overvoltages of the capacitors and converter windings. To select the insulation level of the windings, the operating voltage of the capacitors, and means of protection against overvoltages, it is necessary to estimate in advance the possible limiting short-circuit currents in the installation. A short circuit of a compensated tripler is a complex electromagnetic oscillatory process in a nonlinear circuit, both components of the magnetic characteristic \(B(H)\) of whose transformers are nonsinusoidal. Therefore, calculation of short-circuit currents, especially in the transient mode, is extremely difficult and can be performed, under a number of assumptions, only approximately.

Let us obtain a general expression for the currents of the transient and steady-state short-circuit modes, as well as for the voltages across the SC capacitors, applying the method of N. A. Galochkin, developed by him for analyzing the steady-state processes of a tripler with transverse compensation and active load [1]. We shall perform the calculation for one output phase; we shall neglect losses in the steel, leakage of the secondary windings, and harmonics above the third; we shall approximately take into account the resistances of the primary windings of the transformer group, assuming \(E_{1l}/U_{1l} < 1\).

The m.m.f. equations of the transformer cores of the group are

\[ \begin{aligned} i_a w_1 + i_2 w_2 &= l_c H_a;\\ i_b w_1 + i_2 w_2 &= l_c H_b;\\ i_c w_1 + i_2 w_2 &= l_c H_c, \end{aligned} \tag{1} \]

where \(i_a, i_b, i_c\) are the primary currents;
\(i_2\) is the secondary current;
\(w_1, w_2\) are the numbers of turns of the input and output phases (for a T-connected output circuit, take \(w_2 = 1.5 w_{2\gamma}\), where \(w_{2\gamma}\) is the number of turns of the phase of the equivalent star);
\(l_c\) is the length of the mean line of magnetic induction;
\(H\) is the instantaneous value of the magnetic-field intensity of the corresponding phase.

Adding the equations and taking into account that in a three-phase three-wire system

\[ i_a + i_b + i_c = 0, \]

we obtain

\[ i_2 = \frac{l_c}{3W_2}\sum_{i=a,b,c} H_i . \tag{2} \]

The voltage equation of the secondary circuit, consisting of a series-connected nonlinear inductance, capacitance, and active resistance—

… of the tripler winding

\[ \frac{d\psi_3}{dt}+\frac{1}{C}\int i_2\,dt+i_2 r_2=0, \tag{3} \]

where \(\psi_3\) is the flux linkage of the third harmonic for three phases.

Since

\[ \psi_3=3w_2\Phi_3=3w_2B_3F_c=3w_2B_{1\mathrm{m}}F_c\frac{B_3}{B_{1\mathrm{m}}}=3w_2B_{1\mathrm{m}}F_c x, \tag{4} \]

where \(F_c\) is the active cross section of the core, then (3) is rewritten as

\[ 3w_2B_{1\mathrm{m}}F_c\frac{dx}{dt}+\frac{1}{C}\int i_2\,dt+i_2r_2=0. \tag{5} \]

Differentiating (5) with respect to time,

\[ 3w_2B_{1\mathrm{m}}F_c\frac{d^2x}{dt^2}+\frac{i_2}{C}+r_2\frac{di_2}{dt}=0. \tag{6} \]

We approximate the main magnetization curve of the core by a third-degree polynomial

\[ H=a_1B+a_3B^3. \tag{7} \]

In [1] it is shown that, with such an analytic representation of the magnetic characteristic \(H(B)\) and sinusoidal linear e.m.f.’s, we have

\[ \sum_{i=a,b,c} H_i=(3a_1B_{1\mathrm{m}}+4.5a_3B_{1\mathrm{m}}^3)x+3a_3B_{1\mathrm{m}}^3x^3+0.75a_3B_{1\mathrm{m}}^3\cos 3\omega_1t. \tag{8} \]

We substitute (8) into (2), then differentiate the resulting expression with respect to time and, finally, enter the values of the current \(i_2\) and its derivative \(di_2/dt\) into (6); after simple transformations we obtain

\[ \frac{d^2x}{dt^2} +\frac{r_2l_c}{w_2^2F_c} \left(\frac{a_1}{3}+0.5a_3B_{1\mathrm{m}}^2+a_3B_{1\mathrm{m}}^2x^2\right) \frac{dx}{dt} +\frac{l_c}{w_2^2CF_c} \left(\frac{a_1}{3}+0.5a_3B_{1\mathrm{m}}^2\right)x + \]

\[ +\frac{l_ca_3B_1^2x}{3w_2^2CF_c} = \frac{l_ca_3B_{1\mathrm{m}}^2}{12w_2^2F_c} \left(3r_2\omega_1\sin 3\omega_1t-\frac{1}{C}\cos 3\omega_1t\right). \tag{9} \]

We introduce the dimensionless time \(\tau=\omega_1t\) and the following substitutions:

\[ \frac{r_2l_c}{\omega_1w_2^2F_c} \left(\frac{a_1}{3}+0.5a_3B_{1\mathrm{m}}^2\right)=2\delta; \]

\[ \frac{r_2l_ca_3B_{1\mathrm{m}}^2}{\omega_1w_2^2F_c}=\delta_1; \]

\[ \frac{l_c}{\omega_1^2w_2^2CF_c} \left(\frac{a_1}{3}+0.5a_3B_{1\mathrm{m}}^2\right)=\gamma; \]

\[ \frac{l_ca_3B_{1\mathrm{m}}^2}{3\omega_1^2w_2^2CF_c}=\lambda; \]

\[ \frac{r_2l_ca_3B_{1\mathrm{m}}^2}{4\omega_1w_2^2F_c}=k_1; \]

\[ \frac{l_ca_3B_{1\mathrm{m}}^2}{12\omega_1^2w_2^2CF_c}=k_2. \tag{10} \]

Then (9) will have the following form:

\[ \frac{d^2x}{d\tau^2} +\left(2\delta+\delta_1x^2\right)\frac{dx}{d\tau} +\gamma x+\lambda x^3 = k_1\sin 3\tau-k_2\cos 3\tau. \tag{11} \]

A nonlinear differential equation of the second order with respect to the third harmonic of induction has been obtained in dimensionless form. Without the second term of the coefficient at the first derivative, it is known as the extended (with a damping term) Duffing equation.

Solving equation (11) then makes it possible to find the short-circuit current according to (2) and (8)

\[ i_2=\frac{l_c}{\omega_2}\left[(a_1B_{1\mathrm{m}}+1.5a_3B_{1\mathrm{m}}^3)x+a_3B_{1\mathrm{m}}^3x^3+0.25a_3B_{1\mathrm{m}}^3\cos 3\tau\right] \tag{12} \]

and the capacitor voltage

\[ u_c=\frac{1}{\omega_1C}\int i_2\,d\tau =\frac{l_c}{\omega_1\omega_2C}\int\left[(a_1B_{1\mathrm{m}}+1.5a_3B_{1\mathrm{m}}^3)x+a_3B_{1\mathrm{m}}^3x^3+\right. \]

\[ \left.+0.25a_3B_{1\mathrm{m}}^3\cos 3\tau\right]\,d\tau. \]

Using equations (11) … (13), the process of three- and two-phase short circuits in the circuit of a tripler with a rated power of 300 VA was calculated by means of an MN-14 analog computer.

Fig. 1

The block diagram of the connections of the solving elements is shown in Fig. 1. The solutions were performed with the following initial data: \(B_{1\mathrm{m}}=1.84\ \mathrm{T}\) (steel E31—0.35; average phase capacitance \(C=365\ \mu\mathrm{F}\); zero initial conditions \(\left[x(0)=0,\ \frac{dx(0)}{d\tau}=0,\ u_c(0)=0\right]\) and nonzero initial conditions \([x(0)=0.0943,\) which corresponds to a residual induction of the steel \(B_{1r}=1.18\ \mathrm{T},\ \frac{dx(0)}{d\tau}=0.592,\ u_c(0)=U_{c\mathrm{m}}=25.4\ \mathrm{V}\)—the amplitude of the voltage on the capacitors at the rated load of the converter]; the coefficients of the polynomial approximating the magnetization curve

\[ a_1=-17.8\cdot10^2\ \frac{\mathrm{m}}{\Omega\cdot\mathrm{s}}, \qquad a_3=21.6\cdot10^2\ \frac{\mathrm{m}^5}{\Omega\cdot\mathrm{V}^2\cdot\mathrm{s}^3} \]

As an example, Figs. 2 and 3 show the solutions—oscillograms of the current and of the voltage on the capacitors during a three-phase short circuit and with nonzero initial conditions, obtained from the screen of the machine’s cathode-ray indicator. Analysis of the solutions and comparison of them with experimental oscillograms of the processes of the same installation show the following.

1. The currents of a three-phase short circuit exceed the currents of a two-phase short circuit by a factor of 1.1 … 1.5.

1. The qualitative agreement between the calculated quantities and the measured ones is satisfactory, and in the established regime the accuracy of the solutions is also acceptable (the errors do not exceed ±10 … 15%); the simulated transient process has sharper extrema, which can be explained by the assumptions adopted and by the approximation error; apparently, these same causes underlie the shorter duration of the experimental transient processes (3 … 4 periods of the supply-network current) in comparison with the calculated ones (about 4 … 5 of the same periods).

Fig. 2.

Fig. 3.

1. The short-circuit surge current and the greatest amplitude of the capacitor voltage occur after 2.25 or 2.5 periods (of the load current) from the beginning of the short circuit and exceed the amplitudes of the corresponding steady-state values by no more than a factor of 1.2 … 2.

1. The initial conditions have an insignificant effect on the transient process (for nonzero initial conditions, the largest amplitudes of the current and voltage in the non-steady-state regime increase by 5–10%).

2. The curves \(i_2(t)\) and \(u_c(t)\) in the transient regime have the character of beats and, with a known approximation, may be regarded as sinusoidal functions of time whose angular frequency is close to the angular frequency of the forced oscillations of the output \(\omega_2 = 3\omega_1\), while the amplitude varies harmonically with angular frequency \(\dfrac{\omega_2 - \omega'}{2}\), where \(\omega'\) is the angular frequency of the decaying free component of the short-circuit current, which has an oscillatory character, with \(\omega' \to \omega_2\) [2]. In other words, the short-circuit transient process of the compensated tripler has much in common with the transient process of switching a linear \(r_2—L_{2\mathrm{k}}—C\) circuit onto a sinusoidal voltage, when

\[ \frac{r_2}{2L_{2\mathrm{k}}} \ll \omega' = \sqrt{\frac{1}{L_{2\mathrm{k}}C} - \left(\frac{r_2}{2L_{2\mathrm{k}}}\right)^2}, \]

where \(L_{2\mathrm{k}}\) is the output inductance of the tripler with allowance for the load reaction; that is, the short-circuit regime can be calculated by invoking the well-developed theory of oscillations in linear circuits.

REFERENCES

1. N. A. Galochkin. Investigation of certain properties and operating modes of ferromagnetic frequency triplers with capacitive compensation. Dissertation, 1960.

2. L. R. Neiman and P. L. Kalantarov. Theoretical Foundations of Electrical Engineering, Part II, GEI, 1959.