STUDY OF THE COMMUTATION PROCESS IN THE CONTROL WINDING OF A VARIABLE-FREQUENCY BRUSHLESS ELECTRIC-MACHINE AMPLIFIER (BEMA–VF)

A. I. Skorospeshkin, B. E. Trofimenko

(Recommended by the seminar of the departments of electrical machines and general electrical engineering)

In connection with the development of alternating-current systems, the problem arises of creating electric-machine amplifiers (commutator and brushless) operating on alternating current.

At the Department of Electrical Machines of the Tomsk Polytechnic Institute, work is being carried out on the development and study of brushless electric-machine amplifiers of regulated frequency (BEMA–VF), which are an electric-machine–semiconductor system [1]. One of the important questions in the study of such a system is the investigation of the operation of the semiconductor commutator in interaction with the control winding.

In the present work the commutation process in the control winding of a BEMA–VF equipped with a thyristor commutator is investigated.

Figure 1 presents the circuit of the CSC with the control winding. The designations in the circuit are:

\(E_y\) — a direct-current source feeding the control winding;

\(B_{1-k};\ D_{1-k}\) — controlled valves and diodes connected in the circuit to the positive terminal of the source \(E_y\);

\(B'_{1-k};\ D'_{1-k}\) — controlled valves and diodes connected in the circuit to the negative terminal of the source \(E_y\);

\(W_s\) — a section of the control winding with inductance \(L_s\) and active resistance \(r_s\);

\(C\) — commutating capacitances.

The CSC circuit makes it possible, with varying frequency, to change the diametrical points of connection of the source \(E_y\) to the winding, thereby obtaining a rotating magnetic flux with the required frequency

\[ f_y=\frac{f_p}{K}, \tag{1} \]

where \(f_p\) is the switching frequency of the winding points;

\(K\) is the number of taps from the CW (always even).

In studying the commutation process we consider the case of a steady-state regime, in which the switching frequency \(f_p\) is such that the current in the CW, during the period of succession of switching pulses, has time to increase to the value determined by its active resistance and \(E_y\). In doing so we assume that the other circuits of the BEMA–VF have no influence on the commutation process in the CW, and we neglect the influence of losses in the steel of the magnetic core and the phenomenon of mutual induction.

It is clear from Fig. 1 that the circuit is symmetrical; therefore it will be sufficient to consider the transient process when switching the supply, for example, from points \(1;\ -\dfrac{K}{2}+1\) to points \(2;\ \dfrac{K}{2}+2\) (Fig. 2).

Fig. 1. Electrical circuit of a UPC with OU.

Fig. 2

Let, up to the instant \(t_0=0\), the thyristors \(B_1, B'_1\) be open, i.e., the CU is connected to \(E_y\) at points \(1,\ \dfrac{K}{2}+1\). Through the control winding there flows a current equal to

\[ I_y=\frac{4E_y}{r_s\cdot K}. \tag{2} \]

The commutating capacitors are charged (the polarity is shown in the circuit); moreover, the capacitors located between the cathodes \(B_k—B_1,\ B_1—B_2,\ B'_k—B'_1,\ B'_1—B'_2\) are charged to the maximum voltage possible for the circuit. Below, the dependence of \(U_{c\max}\) on the parameters of the circuit and its mode of operation will be shown. At the instant \(t_0-0\), the control circuit applies pulses to the control electrodes of the thyristors \(B_2, B'_2\), which open instantaneously, while the thyristors \(B_1\) and \(B'_1\) are also rapidly de-energized by the capacitors \(C_1\) and \(C'_1\), respectively, and the reverse voltage of the capacitors is applied to them. From this instant the commutating capacitors lose their charge along the circuit \(B_2—C_1—D_1—\mathrm{CU}—D'_1—C'_1—B'_2\) until it is completely lost. During this time the diodes \(D_2\) and \(D'_2\) will be closed by the voltage on the capacitors \(C_1\) and \(C'_1\). Simultaneously, \(C_1\) and \(C'_1\) lose charge along the paths \(C_1B_1B_2\) and \(C'_1B'_1B'_2\) during the recovery time of the reverse resistance of the thyristors \(B_1\) and \(B'_1\). The recovery time of the reverse resistance of \(B_1, B'_1\) and the magnitude of the charge lost by the capacitors through them are so small in comparison with the entire commutation time and with the initial charge on the plates \(C_1, C'_1\) that, when considering the commutation process, without appreciably reducing accuracy we take \(\Delta t_v=0,\ \Delta q=0\). From the moment when \(U_c=0\), the diodes \(D_2, D'_2\) begin to conduct. The current in the commutated sections \(W_s\) changes direction to the opposite one. The capacitors \(C_1, C'_1\) are charged to \(-U_{c\max}\) (with polarity opposite to the polarity in the circuit of Fig. 1). With the arrival of control pulses to the next pair of thyristors in the circuit, \(B_3, B'_3\), the process is repeated.

Having briefly considered the physical picture of the commutation process, we proceed to its mathematical investigation, for which we shall make a number of assumptions:

1) the thyristors and diodes in the UPC circuit have an ideal characteristic;

2) the power source \(E_y\) has an internal resistance equal to zero;

3) only two capacitors \(C_1,\ C'_1\) participate in the commutation process; the influence of the remaining capacitances of the ring is negligible;

4) the intrinsic capacitance and inductance of the wiring are neglected.

For the analytical investigation of the process, we divide the total commutation time into two stages: the first stage, \(t_0—t_1\), is determined by the discharge time of the commutating capacitors onto the CU; the second stage, \(t_1—t_2\), is the time of establishment of the reverse current in the section \(W_s\) and of charging the capacitor to \(-U_{c\max}\).

In Fig. 3 a circuit is presented for investigating the stage \(t_0—t_1\), where

\(L_y\) — inductance of the entire control winding,

\(r_y\) — active resistance of the CU.

The equation for determining the current has the following form:

\[ E_y=L_y\cdot\frac{di_y}{dt}+i_y\cdot r_y-\frac{2}{C}\int_{t_0}^{t_1} i_y\cdot dt. \tag{3} \]

Fig. 3. Circuit for investigating the first stage of commutation \((t_0 — t_1)\).

From equation (3) we find the current in the control winding

\[ i_y=\left(\frac{E_y+4U_{c\max}}{2L_y\omega_y}\sin\omega_y\cdot t+I_y\cdot\cos\omega_y\cdot t\right)\cdot e^{-\alpha_y\cdot t}, \tag{4} \]

where

\[ \omega_y=\sqrt{\frac{2}{L_y\cdot C}-\alpha_y^2}\simeq \sqrt{\frac{2}{L_y\cdot C}}, \]

\[ \alpha_y=\frac{r_y}{2L_y}, \quad \text{with } \alpha_y \ll \frac{2}{L_y\cdot C}, \]

\[ i_y(t)=i_c(t). \tag{5} \]

In equation (4), which determines the current through the CW, the factor \(e^{-\alpha_y}\) may be neglected, since \(e^{-\alpha_y t}\simeq 1\).

From the equation for the voltage across the capacitor we find the analytical expression for the time of the first stage:

\[ U_c=\frac{1}{C}\int_0^1 i_c(t)\cdot dt=0. \tag{6} \]

Solving (4, 5, 6) jointly, we obtain

\[ t_1=\frac{1}{\omega_y}\arcsin \frac{\alpha_y\cdot\omega_y\cdot(B^2-A^2)+AB\cdot(\omega_y^2-\alpha_y^2)} {(\alpha_y^2+\omega_y^2)\cdot(A^2+B^2)}, \tag{7} \]

where

\[ A=\frac{E_y+4U_{c\max}}{2L_y\omega_y}, \quad B=I_y. \]

The value of the current in the CW at the end of the first stage is determined by substituting \(t_1\) from (7) into equation (4). We shall denote the magnitude of the current at this instant by \(I_y^1\).

Thus, the beginning of the second stage is characterized by the following: the voltage on the commutating capacitors is zero, a current \(I_y^1\) flows in the CW, and the diodes \(D_2, D'_2\) begin to conduct.

Figure 4 shows the circuit for investigating the commutation process in the second period \(t_1 — t_2\), where \(W_0\) is the parallel branch of the winding without one commutated section, \(L_0\) is its inductance, and \(r_0\) is its active resistance.

Fig. 4. Circuit for studying the second stage of commutation \((t_1—t_2)\).

The initial equations for determining the currents of the second stage are as follows:

\[ E_y=\frac{1}{C}\int_{t_1}^{t_2} i_c\cdot dt+r_0\cdot i_0+L_0\cdot \frac{di_0}{dt}; \tag{8} \]

\[ 0=i_s\cdot r_s+L_s\cdot \frac{di_s}{dt}-\frac{1}{C}\int_{t_1}^{t_2} i_c\cdot dt. \]

Solving equations (8) with respect to the currents \(i_s\), \(i_0\), \(i_c\) by the operational method, we obtain a third-degree equation. When finding the original from the image, the first cannot practically be expressed in a general form through the coefficients of the third-degree equation. Therefore, taking into account the fact that \(L_0 \gg L_s\), we regard the current through \(L_0\) during the second stage as constant, equal to \(i_{y1}=\frac{1}{2}I_y^1\). This assumption satisfies the accuracy required for a practical calculation of the circuit. In reality, the current in \(L_s\) and \(L_0\) by the end of the second stage has the value \(i_{y2}\ll i_{y1}\). The quantity \(\Delta i_y=i_{y1}-i_{y2}\) will be determined below.

The initial equations for determining the currents \(i_s\) and \(i_c\) under the condition adopted above are:

\[ L_s\cdot \frac{di_s}{dt}+r_s\cdot i_s=U_c; \]

\[ i_{y1}=i_s+C\cdot \frac{dU_c}{dt}. \tag{9} \]

From them we find

\[ i_s=i_{y1}\cdot (2e^{-a_s\cdot t}\cdot \cos \omega_s\cdot t-1); \tag{10} \]

\[ i_c=I_y^1\cdot e^{-a_s\cdot t}\cdot \cos\omega_s\cdot t, \tag{11} \]

where

\[ a_s=\frac{r_s}{2L_s},\qquad \omega_s=\sqrt{\frac{1}{L_s\cdot C}-a_s};\qquad \frac{1}{L_s\cdot C}\gg a_s. \]

From equation (10) it is seen that the current \(i_s\) will become equal to \(-i_{y1}\) when the argument of the cosine is equal to \(\frac{\pi}{2}\), i.e. \(\omega_s\cdot t=\frac{\pi}{2}\),

whence

\[ t_2=\frac{\pi}{2\omega_s}\simeq \frac{\pi}{2}\sqrt{L_s\cdot C}. \tag{12} \]

The voltage to which the capacitor is charged during the time \(t_1—t_2\) is found from the equation:

\[ U_{c\max}=\frac{1}{C}\int_{t_1}^{t_2} I_y^1\cdot e^{-\alpha_s t}\cos\omega_s t\cdot dt, \tag{13} \]

whence

\[ U_{c\max}=\frac{I_y^1}{C\cdot \omega_s}\simeq I_y^1\cdot \sqrt{\frac{L_s}{C}}. \tag{14} \]

Summing the time intervals of the first and second stages, we determine the total commutation time.

\[ t_k=\frac{1}{\omega_y}\arcsin \frac{\alpha_y\cdot \omega_y\cdot (B^2-A^2)+AB\cdot(\omega_y^2-\alpha_y^2)} {(\alpha_y^2+\omega_y^2)\cdot(A^2+B^2)} +\frac{\pi}{2\omega_s}. \tag{15} \]

Taking into account the fact that, in the steady-state regime, \(4U_{c\max}\gg E_y\) and, for the given circuit, \(\alpha_y\ll \sqrt{\dfrac{2}{L_y\cdot C}}\), the time of the first stage of commutation is written as:

\[ t_1\simeq L_y\cdot \frac{\sqrt{L_s\cdot C}}{2L_s+L_y}, \tag{16} \]

whence

\[ t_k\simeq L_y\frac{\sqrt{L_s\cdot C}}{2L_s+L_y} +\frac{\pi}{2}\sqrt{L_s\cdot C}\,[\text{sec}]. \tag{17} \]

Having determined the time and current of commutation in the section \(W_s\), from the condition of constancy of the current in \(W_0\), we can now calculate with sufficient accuracy the change of current in the part of the winding \(W_0\) over the period \(t_1—t_2\).

According to Fig. 4, we write the equation:

\[ E_y-L_s\cdot \frac{di_s}{dt}-L_0\cdot \frac{di_0}{dt}=0, \tag{18} \]

whence we obtain

\[ \Delta i_y=\int_{t_1}^{t_2}\frac{E_y}{L_0}\,dt -\frac{L_s}{L_0}\int_{i_{y1}}^{-i_{y1}}di_s. \]

Finally

\[ \Delta i_y=\frac{E_y\cdot \pi}{L_0\cdot 2\omega_s} -\frac{L_s}{L_0}\,2i_{y1}. \tag{19} \]

By the end of the second stage, a current will be established in the control winding equal to

\[ I_{y2}=I_y^1-2\cdot\left(\frac{L_s}{L_0}I_y^1-\frac{E_y\cdot \pi}{2L_0\cdot \omega_s}\right), \tag{20} \]

after which it will settle to the initial value \(I_y\) according to the law

\[ i_y=I_{y2}+\left(\frac{E_y}{r_y}-I_{y2}\right)\cdot \left(1-e^{-\frac{r_y\cdot t}{L_y}}\right). \tag{21} \]

In practice it was observed that, when the circuit was initially switched on at low switching frequencies, the collector did not enter the steady-state operating mode. A short-circuit mode was observed, due to the fact that the thyristor which opened first in the circuit did not close when the current was commutated to the next thyristor.

Let us examine the process of switching on the SPC to the winding. Let the supply voltage be applied and the thyristors be locked. At some instant we switch on the control circuit; the instant \(t_0\) is fixed by the application of pulses to the thyristors, for example, \(B_1\), \(B'_1\) (Fig. 1), which open practically instantaneously.

The time constant of the circuit for charging the capacitances \(C\) is much smaller than the time constant of the control winding,

\[ \Sigma r \cdot C^1 \ll \frac{L_y}{r_y+\Sigma r}, \tag{22} \]

where

\[ C^1=\frac{4\cdot C}{K}. \]

\(\Sigma r\) is the sum of the forward resistances of two diodes and two open thyristors \(D_1, D_{\frac{k}{2}+1}, B_1, B'_1\); moreover, \(\Sigma r \ll r_y\).

Therefore we assume that the capacitors are charged practically instantaneously (relative to the time of current establishment in the CW) to the value

\[ U_c=\frac{E_y\cdot 2}{K}. \tag{23} \]

In the control winding the current is established according to the law

\[ i_y=\frac{E_y}{r_y}\cdot\left(1-e^{-\frac{r_y}{L_y}\cdot t}\right). \tag{24} \]

It was indicated above that the time of the first commutation stage \(t_1\)—that is, the time during which a reverse voltage, equal to the voltage on the commutating capacitor, is applied to the preceding thyristor in the SPC ring. The voltage \(U_c\) during \(t_0\)–\(t_1\) falls to zero, i.e.,

\[ t_1 \geq t_v \tag{25} \]

will be the condition for stable starting of the collector.

\(t_v\) is the recovery time of the control properties of the thyristor in the forward direction.

From equation (7) it is seen that \(t_1\) depends on the parameters of the winding and the SPC, \(U_{c\max}, E_y, I_y\). For \(E_y=\mathrm{const}\), \(U_c=\frac{2\cdot E_y}{K}=\mathrm{const}\), the parameters of the CW remain constant; only \(i_y\) can change.

From equation (24) it is seen that \(i_y\) depends on \(t\). During the time

\[ t=T_p=\frac{1}{f_p} \]

the current in the winding increases to \(i_{yp}\).

From equation (23) we find the switching frequency

\[ f_p=\frac{1}{\tau_y\cdot \ln \frac{E_y}{E_y-i_{yp}\cdot r_y}}. \tag{26} \]

From (7) we obtain

\[ \sin \omega_y t= \frac{4E_y\cdot L_y\cdot \omega_y\cdot K\,(K+8)\cdot i_{yp}} {\left[E_y\cdot (K+8)\right]^2+\left(4L_y\cdot \omega_y\cdot K\cdot i_{yp}\right)^2}, \]

Solving for \(i_{\mathrm{up}}\), we find

\[ i_{\mathrm{up}}=\frac{2\cdot E_y\cdot (K+8)}{8\cdot L_y\cdot \omega_y\cdot K\cdot \sin\omega_y\cdot t}. \tag{27} \]

Substituting (27) into (26), we find the frequency for stable starting of the collector

\[ f_n \geq \frac{1}{\tau_y \cdot \ln \dfrac{S}{S-r_y\cdot (K+8)}} , \tag{28} \]

where

\[ S=2L_y\cdot \omega_y\cdot K\cdot \sin\omega_y\cdot t_{\mathrm{v}}. \]

On the basis of the theoretical analysis and the analytical expressions obtained above, an experimental specimen of a thyristor collector with a control winding for the BEMU—RF was designed, manufactured, and tested. The TCC is made according to the circuit of Fig. 1.

The control winding is a ring winding of the armature type with \(K=12\), inductance \(L_y=52.5\cdot 10^{-3}\) H, and active resistance \(r_y=0.36\ \Omega\).

The total inductance of a section is \(L_s=2.6\cdot 10^{-3}\) H.
The active resistance of a section is \(r_s=0.12\ \Omega\).
The inductance of a part of the winding is \(L_o=36.0\cdot 10^{-3}\) H.

Table 1 gives the theoretical and experimental data for the times of the first and second stages of commutation for a CW with the above parameters. As can be seen from the table, the calculated data agree quite accurately with the experimental data.

Table 1

In Fig. 5, curves are plotted for the change of current in the commutated section for \(I_y=5\) A. Each of the three curves corresponds to a definite value of the commutated capacitance: a) \(C=10\ \mu\text{F}\); b) \(C=5\ \mu\text{F}\); c) \(C=1\ \mu\text{F}\).

Figure 6a gives an oscillogram of the current in the parallel branch of the CW, taking commutation into account, and Fig. 6b the current of the commutated section.

An experimental check of the limiting frequency of stable starting of the collector fully confirmed the theoretical data obtained.

Thus, the results of the work carried out may be formulated as follows:

1. A circuit of a thyristor collector has been developed, an experimental specimen of the TCC has been manufactured, taking into account the operating features of the control wind-

Fig. 5. Commutation-current curves in the section.
a) \(C = 10\ \mu\mathrm{F}\), b) \(C = 5\ \mu\mathrm{F}\), c) \(C = 1\ \mu\mathrm{F}\).
Points 1 on the curves correspond to the end of the first stage.

a

Fig. 6a. Section current with allowance for commutation.

b

Fig. 6b. Commutation current in the section.

... in the circuit of a brushless electric-machine amplifier of adjustable frequency.

1. The physical picture of the commutation process has been examined, and equivalent circuits have been constructed on its basis.

2. Equations have been obtained for the current in the control winding and the commutated section, and for the overvoltage on the commutated section and on the capacitor; these are valid for any operating modes of the UPK and the OU.

3. The commutation time and the frequency of stable starting of the UPK have been determined.

4. Experimental verification has confirmed the correctness of the theoretical propositions.

The experimental prototype of the UPK operates stably under various conditions.

REFERENCES

1. E. F. Obergans, A. I. Skorospeshkin. A synchronous generator with frequency regulation by means of a semiconductor collector, Izvestiya TPI, vol. 145, 1966.

2. M. I. Kontorovich. Operational Calculus and Nonstationary Phenomena in Electrical Circuits, State Publishing House of Technical-Theoretical Literature, Moscow, 1953.