STUDY OF THE DEPENDENCE OF THE CURRENT IN THE WINDING OF AN ELECTRICAL MACHINE EQUIPPED WITH A SEMICONDUCTOR COMMUTATOR ON THE WINDING PARAMETERS AND THE SWITCHING FREQUENCY OF THE SEMICONDUCTOR COMMUTATOR

E. F. OBERTAN, A. I. SKOROSPEshKIN

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

This article presents part of the results of a study of an electric-machine converter with a semiconductor commutator in the excitation circuit, used for the purpose of varying, over wide limits, the frequency of the output voltage while the rotational speed of the machine shaft remains constant.

The nature of the behavior of the current in the winding of an electrical machine equipped with a semiconductor commutator, depending on the number of winding terminals and the winding parameters at various switching frequencies of the commutator cells, is of independent interest.

Therefore, the solution of the problem posed in the present article is carried out without taking into account the influence on the current in the excitation winding of the remaining circuits and operating modes of the electrical machine as a whole.

As the excitation winding of the electrical machine, an armature-type winding uniformly distributed over the stator phases has been chosen. The terminals from the winding sections are connected to the cells of a semiconductor commutator, made according to a ring circuit using controlled semiconductor valves and supplied from a direct-current source.

Figure 1 shows part of a semiconductor commutator connected to two diametrically located winding sections.

Fig. 1. Electrical circuit of two cells of a semiconductor commutator connected to diametrically located winding sections.

The notation in Fig. 1 is as follows: \(W\) is part of the parallel branch of the excitation winding, \(W_s\) is a section or part of the winding between terminals, \(U\) and \(I\) are the supply voltage and the current through the winding, \(B\) are controlled valves (thyristors), \(D\) are diodes, and \(C\) are commutating capacitors.

For convenience in considering the process of current commutation in the winding by a semiconductor commutator, let us divide the cycle of current commutation in one section into two time intervals:

\(t_z\) — the time interval of the closed state of the section, and
\(t_p\) — the time interval after the closed section has been opened.

Before the control pulses are applied to the thyristors, the winding is connected to a direct-current source through the open thyristors \(B_1\) and \(B_2\) of the commutator cells connected to diametrical terminals of the winding.

In this case a current flows in the parallel branch of the winding,

\[ i_0=\frac{I}{2}=\frac{U}{R_0}. \tag{1} \]

Here \(R_0=r_1+r_2\) is the ohmic resistance of the parallel branch of the winding, and \(r_2\) is the resistance of the section \(W_s\).

Time interval \(t_z\)

This time interval, during which thyristors \(B_1\), \(B_2\), \(B_3\), \(B_4\) operate, is reckoned from the moment thyristors \(B_3\) and \(B_4\) are opened upon application of a control pulse to the moment thyristors \(B_1\) and \(B_2\) are blocked.

Since, during this time interval, no external e.m.f. is applied to section \(W_s\), the electromagnetic energy of sections \(W_s\) is dissipated, and the current \(i_0\), which flowed in the section at \(t_z=0\), changes according to the exponential law

\[ i_{W_s}(t)=i_0\cdot e^{-\frac{r_1+r_2}{L_2}\cdot t_z}, \tag{2} \]

where \(L_2\) is the inductance of section \(W_s\), and \(i_{W_s}(t)\) is the current in section \(W_s\) in the time interval \(t_z\).

The current in the remaining part of the parallel branch \(W\) then increases, changing according to the law:

\[ i_W(t)=i_0\left[1+\frac{r_2}{r_1}\left(1-e^{-\frac{r_1}{L_1}\cdot t_z}\right)\right], \tag{3} \]

where \(r_1\) and \(L_1\) are the resistance and inductance of the part of the parallel branch \(W\).

\(i_W(t)\) is the current in the part of the parallel branch without section \(W_s\) in the time interval \(t_z\).

The maximum value of the current \(i_W\) at \(t_z=\infty\) is equal to

\[ i_W=i_0\cdot\frac{r_1+r_2}{r_1}=i'_0. \tag{4} \]

Time interval \(t_p\)

This time interval, during which thyristors \(B_3\) and \(B_4\) operate, is reckoned from the moment valves \(B_1\) and \(B_2\) are blocked. The blocking process of valves \(B_1\) and \(B_2\) is carried out by the commutating capacitors \(C\) through their discharge along the circuits \(C—B_1—B_3\) and \(C—B_4—B_2\). The presence of diodes \(D_1\), \(D_2\), \(D_3\), and \(D_4\) prevents the discharge of capacitors \(C\) through section \(W_s\).

In determining the law of variation of the current \(i_{\mathrm{pb}}(t)\) in the parallel branch of the winding \(W_{\mathrm{pb}}=W+W_s\) over the time interval \(t_p\), we use the superposition principle, considering the transient process as the result of two processes: the steady-state process in the original circuit, which is the parallel branch of the winding with the section \(W_s\) short-circuited, and the additionally established process in the changed circuit, which is the parallel branch of the winding with the section \(W_s\) open.

Then, assuming that by the instant \(t_p=0\) the current \(i_w(t)\) has assumed its steady-state value, the change in current in the parallel branch of the winding \(i_{\mathrm{pb}}(t)\) over the time interval \(t_p\) can be determined from the equation

\[ i_{\mathrm{pb}}(t)=i_0-\frac{e_{\phi}}{z_1(t)+z_2(t)}, \tag{5} \]

where \(z_1(t)\) and \(z_2(t)\) are the total impedances in the transient mode of, respectively, the part of the parallel branch \(W\) of the winding and the circuit connected with the section \(W_s\);

\[ e_{\phi}=i_0'\cdot z_2(t) \]

is a fictitious e.m.f. equal to the change in voltage at the terminals of the section \(W_s\) in the transition from the interval \(t_z\) to the interval \(t_p\).

Solving (5) gives the following law of variation of the current \(i_{\mathrm{pb}}(t)\):

\[ i_{\mathrm{pb}}(t)=i_0'\cdot\frac{1}{1+f(t)}, \tag{6} \]

where

\[ f(t)= \left[ \frac{r_2}{r_1} \left( 1-e^{-\frac{r_1}{L_1}\cdot t_p} \right) + \frac{L_2}{L_1}\cdot e^{-\frac{r_1}{L_1}\cdot t_p} \right] \times \left[ 1-e^{-\frac{r_2}{2L_2}\cdot t_p}\cdot \cos \omega t_p \right]. \]

The obtained equations (2), (3), and (6) describe the behavior of the current in the winding and in the commutated sections over all time intervals of the commutation period.

For the convenience of further analysis, bearing in mind that in practice the resistance and inductance of the entire winding and the number of terminals of the winding connected to the semiconductor commutator are usually specified, we introduce the additional parameters \(L_0\)—the inductance of the parallel branch of the winding—and \(n\)—the number of pairs of terminals, or the number of terminals in the parallel branch of the winding.

Taking into account that

\[ R_0=r_1+r_2;\qquad n=\frac{r_1+r_2}{r_2}=1+\frac{r_1}{r_2};\qquad \tau_0=\frac{L_0}{R_0}, \]

we express the parameters introduced earlier in terms of the common ones:

\[ r_1=R_0\cdot\frac{n-1}{n};\qquad L_1=L_0\cdot\left(\frac{n-1}{n}\right)^2;\qquad \tau_1=\frac{L_1}{r_1}=\tau_0\cdot\frac{n-1}{n} \]

\[ r_2=R_0\cdot\frac{1}{n};\qquad L_2=L_0\cdot\left(\frac{1}{n}\right)^2;\qquad \tau_2=\frac{L_2}{r_2}=\tau_0\cdot\frac{1}{n} \]

Then equations (2), (3), and (6) can be written as follows:

\[ i_{w_s}(t)=i_0\cdot e^{-n^2\cdot\frac{t_z}{\tau_0}}, \tag{7} \]

\[ i_w(t)=i_0\left[1+\frac{1}{n-1}\left(1-e^{-\frac{n}{n-1}\cdot\frac{t_z}{\tau_0}}\right)\right], \tag{8} \]

\[ i_{\mathrm{pv}}(t)= \frac{i_0\cdot \dfrac{n}{n-1}} {1+\left[\dfrac{1}{n-1}\left(1-e^{-\frac{n}{n-1}\cdot\frac{t_p}{\tau_0}}\right)+\dfrac{1}{(n-1)^2}\cdot e^{-\frac{n}{n-1}\cdot\frac{t_p}{\tau_0}}\right]} \times \frac{1} {\left[1-e^{-\frac{n}{2}\cdot\frac{t_p}{\tau_0}}\cdot\cos \omega t_p\right]} . \tag{9} \]

The difference between the established values of the currents in the time intervals \(t_z\) and \(t_p\) is

\[ \Delta i_0=i_{w\,t_z=\infty}-i_{\mathrm{pv}\,t_p=\infty} =i_0'-i_0=i_0\cdot\frac{1}{n-1}, \]

whence

\[ i_0'=i_0+\Delta i_0=i_0\frac{n}{n-1}. \]

For \(t_z\ne\infty\) and \(t_p\ne\infty\), a current will flow through the winding whose value in the parallel branch will be equal to:

\[ i_{\mathrm{cp}}=i_0+\Delta i_{\mathrm{cp}}. \tag{11} \]

In this equality, \(\Delta i_{\mathrm{cp}}\)—the mean value of the current increment during operation of the semiconductor commutator—is a constant quantity and depends on the winding parameters, the number of terminals \(n\), and the durations of the time intervals \(t_z\) and \(t_p\).

Taking into account that

\[ t_z+t_p=T=\frac{1}{f}, \]

where \(T\) is the commutation period,
\(f\) is the switching frequency of the semiconductor commutator,
the determination of \(\Delta i_{\mathrm{cp}}=f(n,f,\tau_0)\) will be carried out by a graphical-analytical method.

Having assigned arbitrary values of \(L_0\), \(R_0\), and \(n\), one can, from (8) and (9), construct the curves of the variation with time of the currents \(i_w(t)\) and \(i_{\mathrm{pv}}(t)\). Analysis of these exponential curves shows their almost complete identity, except for a certain discrepancy in the initial portions.

This circumstance makes it possible to simplify considerably the derivation of formulas for determining \(\Delta i_{\mathrm{cp}}\).

In Fig. 2 the nature of the change of current in the winding at different commutator switching frequencies is shown graphically: \(f_1=10\) hertz and \(f_2=42\) hertz, the number of winding terminals \(n=10\), and the arbitrarily chosen \(t_z=0.02\) sec. The value \(t_z=0.02\) sec is determined by the recovery time of the control properties of the thyristors and depends only on the selected type of thyristors.

The abscissa axis in Fig. 2 is chosen at the level of the current \(i_0\) at \(t_z=0\), i.e., before the beginning of commutator switching.

From Fig. 2 it is seen that the current increment in the winding \(\Delta i_{\mathrm{cp}}\), for any ratios of \(t_z\) and \(t_p\), reaches a steady-state value after a certain time \(t_y\).

To determine the dependence \(\Delta i_{\mathrm{cp}}=f(n,f,\tau_0)\), we shall take the time \(t_y\) as the origin of coordinates and consider in Fig. 3 the current curve over one period of the switching frequency of the semiconductor commutator.

Fig. 2. Graphical determination of the average current increment in the winding \(\Delta i_{\mathrm{avg}}\) when the semiconductor commutator is switched at frequencies of 10 hertz and 42 hertz.

Fig. 3. Curve of the change in the current increment \(\Delta i_0\) in the winding over one switching period of the semiconductor commutator.

Connecting the points A, B, and C by straight lines, we obtain the triangle ABC. We shall take the mean value of the current in the winding, \(\Delta i_{\mathrm{avg}}\), to be equal to

\[ \Delta i_{\mathrm{avg}}=\frac{1}{2}\left(\Delta i_A+\Delta i_B\right) \tag{12} \]

Taking into account the assumption adopted above concerning the identity of the exponential current curves \(i_w(t)\) and \(i_{\mathrm{pv}}(t)\), we obtain expressions for the current increments at points A and B:

\[ \Delta i_B=\Delta i_0 \frac{1-e^{-\frac{t_3}{\tau_1}}} {1-e^{-\frac{t_3+t_p}{\tau_1}}}, \tag{13} \]

\[ \Delta i_A=\Delta i_0 \frac{1-e^{-\frac{t_3}{\tau_1}}} {1-e^{-\frac{t_3+t_p}{\tau_1}}} \cdot e^{-\frac{t_p}{\tau_1}} . \tag{14} \]

Here \(\tau_1=\tau_0\frac{n-1}{n}\) is the adopted common time constant of the transient current processes \(i_w(t)\) and \(i_{\mathrm{pv}}(t)\). Substituting \(\Delta i_A\) and \(\Delta i_B\) from (14) and (13) into (12), we obtain

\[ \Delta i_{\mathrm{avg}}=\Delta i_0\cdot t_3\cdot f\cdot \left(1-\frac{t_p}{2\cdot\tau_1}\right). \tag{15} \]

Introducing into (15) the values of \(\tau_1\) and \(\Delta i_0\) from (10), we obtain

\[ i_0\cdot\frac{1}{n-1} \left[ \left(1+\frac{t_3\cdot n}{2\cdot\tau_0(n-1)}\right) \cdot t_3\cdot f - \frac{t_3\cdot n}{2\cdot\tau_0(n-1)} \right]. \tag{16} \]

Equation (16) is valid for values of the switching frequencies \(f\) of the semiconductor commutator lying within the limits

\[ \frac{1}{2\cdot\tau_0\cdot\frac{n-1}{n}+t_3}<f<\frac{1}{t_3}. \tag{17} \]

The upper limit is determined by the operating principle of the proposed semiconductor-commutator circuit, namely, at the frequency

\[ f>\frac{1}{t_3}\qquad \frac{t_3}{T}<1. \]

Such an operating mode cannot be permitted, since in this case two, three, or more winding sections may be closed simultaneously, and the operation of the commutator is disrupted. The lower limit of the frequency values in (17) is due to the fact that the value of \(\Delta i_{\mathrm{avg}}\) at low frequencies, when the value of \(\Delta i_B\) does not depend on \(T\), can be determined only by an integral method.

Conclusions

In the winding of an electrical machine equipped with a semiconductor commutator and supplied from a direct-current source, at any switching frequency of the commutator there flows a pulsating direct current, the mean value of which is determined by the switching frequency-

the switching of the cells of the semiconductor commutator, increasing as the indicated frequency increases.

At the same time, the minimum value of the direct current in the winding is determined by the supply voltage applied to it through the semiconductor commutator and by the ohmic resistance of two parallel branches of the winding.

The magnitude of the maximum increment of the mean value of the direct current, at the maximum commutator switching frequency for the given circuit, is equal to the ratio of the ohmic resistance of the commutated section to the ohmic resistance of the parallel branch of the winding.

With a sufficiently large number of winding terminals, in calculations of electrical machines it is possible to assume that the current in the winding is independent of the switching frequency of the semiconductor commutator.

REFERENCES

1. I. L. Kaganov. Electronic and Ionic Converters, Part III. Gosenergoizdat, 1956.

2. Sh. I. Lutidze. Electrical machines with a controlled semiconductor commutator. Academy of Sciences of the USSR, Elektroenergetika, issue 5, 1962.