STUDY OF THE COMMUTATION PROCESS IN THE WINDING OF AN ELECTRIC MACHINE EQUIPPED WITH A SEMICONDUCTOR COMMUTATOR

E. F. OBERGAN, A. I. SKOROSPELISHKIN

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

One variety of switching devices in electric machines with semiconductor commutation of current in the windings is a semiconductor commutator made with thyristors.

In constructing the circuit of a semiconductor commutator supplied from a dc source, commutating capacitors are used to transfer the current from one thyristor to another; these capacitors determine the specific character of the current-commutation process also in the sections of the winding.

In the present article the dependence of the overvoltage on the commutated section upon the parameters of the winding and the value of the commutating capacitor is considered.

Figure 1 shows one section \(W\) of the winding of an electric machine and the part of the semiconductor commutator participating in the commutation of current in this section at an arbitrarily chosen instant of time.

Fig. 1. Electrical circuit of a cell of a semiconductor commutator connected with a winding section.

Introducing the notation:
\(t_z\) — time during which section \(W_s\) is in the closed state,

\(t_p\) is the time of the open-circuit state of section \(W_s\); let us consider the commutation process successively in each time interval.

At the initial instant \(t_3 = 0\), the winding is connected to the “+” bus of the power supply at point \(a\) through the open thyristor \(B_1\) and diode \(D_1\).

In this case, in the parallel branch of the winding, which includes section \(W_s\) and has the parameters \(R_0, L_0\), a current \(i_0\) flows, and the voltage on the plates of capacitor \(C\) is equal to the voltage drop across section \(W_s\)

\[ U_c = i_0 \cdot r_s, \tag{1} \]

where \(r_s\) is the ohmic resistance of section \(W_s\).

We shall neglect the voltage drop across diode \(D_1\).

When a control pulse is applied to thyristor \(B_2\), the current in section \(W_s\) varies according to the law

\[ i_{W_s}(t) = i_0 \cdot e^{- n^2 \frac{t_3}{\tau_0}}, \tag{2} \]

where \(i_{W_s}(t)\) is the current in section \(W_s\) during the time interval \(t_3\),

\(n\) is the number of pairs of winding terminals,

\[ \tau_0 = \frac{L_0}{R_0} \]

is the time constant of the parallel branch of the winding,

and \(R_0\) and \(L_0\) are the active resistance and inductance of the parallel branch of the winding.

Simultaneously with the decrease of the current in section \(W_s\), the process of current commutation from thyristor \(B_1\) to thyristor \(B_2\) takes place. This process is caused by the discharge of capacitor \(C\) through the circuit \(C—B_1—B_2\).

From the moment thyristor \(B_1\) is blocked, the time interval \(t_p\) begins. After the transient processes have ended, at \(t_p = \infty\), the winding proves to be connected to the “+” bus of the power supply at point \(b\) through the open thyristor \(B_2\) and diode \(D_2\).

Since, in order to ensure reliable blocking of thyristor \(B_1\), the discharge time of capacitor \(C\) must be somewhat greater than the recovery time of the control properties of thyristor \(B_1\), at the instant when the latter is blocked the voltage on the plates of the capacitor will have some value different from zero.

The current \(i_{W_s}(t_3)\) in section \(W_s\), in the general case, may also differ from zero at the instant the thyristors \(B_1\) are blocked, i.e., at the beginning of the time interval \(t_p\).

However, in order to simplify the mathematical derivations, with accuracy sufficient for practical purposes we shall take as the initial conditions of the time interval \(t_p\) the values \(U_c = 0\) and \(i_{W_s} = 0\).

Then the values \(U_c(t_p)\) and \(i_{W_s}(t_p)\) in the time interval \(t_p\) can be determined from the formulas:

\[ U_c(t_p) = i_{W_s}(t_p) \cdot Z_2(t_p), \tag{3} \]

\[ i_{W_s}(t_p) = i'_0 \cdot \frac{Z_1(t_p)}{Z_1(t_p) + Z_2(t_p)} . \tag{4} \]

In these formulas,

\(Z_1(t_p)\) and \(Z_2(t_p)\) are the total impedances in the transient regime of the part of the parallel branch of the winding without the commutated section \(W_s\) and of the circuit associated with section \(W_s\).

\[ i'_0 = \frac{U}{R_0 - r_s} \]

is the steady-state value of the direct current in the part of the parallel branch of the winding without the commutated section \(W_s\).

Upon substituting (4) into (3), we can find the ratio of the voltage on the capacitor \(C\) in the time interval \(t_p\) to the supply voltage of the winding.

\[ \frac{U_c(t_p)}{U} = \frac{1}{R_0-r_s}\cdot \frac{Z_1(t_p)\cdot Z_2(t_p)}{Z_1(t_p)+Z_2(t_p)} . \tag{5} \]

Having determined, from the circuit (Fig. 1), the expressions for the resistances \(Z_1(t_p)\) and \(Z_2(t_p)\), and after a number of transformations of (5), we obtain:

\[ \frac{U_c(t_p)}{U} = \frac{1}{ (n-1)\left[1-e^{-n\frac{t_p}{\tau_0}}\right] + \left(1-e^{-\frac{n}{\,n-1\,}\cdot\frac{t_p}{\tau_0}}\right) + R_0 C\cdot \frac{n-1}{n}\cdot \frac{1}{t_p} }. \tag{6} \]

Taking the derivative of (6) and setting its numerator equal to zero, we find the time after which the ratio \(\dfrac{U_c}{U}\) will be maximal.

The numerator of the derivative is

\[ (n-1)\left[\frac{1}{\tau_0}\cdot n\cdot e^{-n\frac{t_p}{\tau_0}}\right] + \frac{n}{n-1}\cdot \frac{1}{\tau_0}\cdot e^{-\frac{n}{\,n-1\,}\cdot\frac{t_p}{\tau_0}} - R_0 C\cdot \frac{n-1}{n}\cdot \frac{1}{t_p^2} =0. \]

Analysis of equation (6) shows that the time \(t_{pм}\), after which the ratio \(\dfrac{U_c}{U}\) reaches a maximum, is of the order of \(10^{-7}\)--\(10^{-4}\) sec.

For such a value of \(t_{pм}\),

\[ e^{-n\cdot\frac{t_{pм}}{\tau_0}} \approx e^{-\frac{n}{\,n-1\,}\cdot\frac{t_{pм}}{\tau_0}} \approx 1, \tag{7} \]

then

\[ (n-1)\frac{n}{\tau_0} + \frac{n}{n-1}\cdot \frac{1}{\tau_0} - R_0\frac{n-1}{n}\cdot C\frac{1}{t_{pм}} =0, \]

whence

\[ t_{pм} = \frac{n-1}{n}\cdot \sqrt{L_0 C}\cdot \sqrt{\frac{1}{n^2-2(n-1)}} . \tag{8} \]

Substituting the value \(t_{pм}\) from (9) into (6) and taking condition (7) into account, we obtain

\[ \left(\frac{U_c}{U}\right)_{\max} = \frac{1}{R_0}\cdot \sqrt{\frac{L_0}{C}\cdot \frac{1}{n^2-2(n-1)}} . \tag{9} \]

The dependence \(\left(\dfrac{U_c}{U}\right)_{\max}=f(C)\) for various \(n\) and arbitrarily selected \(L_0=0.2\ \text{H}\), \(R_0=2\ \Omega\) is shown in Fig. 2.

Using equation (9), for any parameters and any number of winding leads one can determine the value of the capacitance of the capacitor that ensures overvoltages, permissible for the selected elements of the semiconductor commutator and the winding design, on the commutated section.

At the same time, it should be remembered that, since this same capacitor is the commutating capacitor for the thyristors, the value of its capacitance,

Fig. 2. Plot of the dependence of the overvoltage on the commutating capacitor \(C\) on the value of its capacitance.

Vertical axis: \(\left(\dfrac{u_C}{U}\right)_{\max}\)

Horizontal axis: \(C\), \(\mu\mathrm{F}\)

Curves: \(n=6\); \(n=18\).

calculated from (9), must also satisfy the condition

\[ t_{\text{dis}} > t_{\text{r}}, \tag{10} \]

where \(t_{\text{dis}}\) is the discharge time of capacitor \(C\) through the circuit

\[ C - B_1 - B_2, \]

and \(t_{\text{r}}\) is the recovery time of the control properties of the thyristor, which is a nameplate parameter of the thyristor.

Fulfillment of this condition is checked by calculation; for this purpose it is first necessary to determine from (6) the initial voltage on the capacitor \(U_{c n}\) at the instant \(t_3 = 0\). In this case the value of the time \(t_p\) substituted into (6) is equal to

\[ t_p = \frac{1}{f} - t_{\text{r}}, \]

where \(f\) is the frequency of the control pulses supplied to the semiconductor commutator.

Specifying the ratio \(\dfrac{U_c}{U_{c n}}\) of the voltage on the capacitor at the instant of blocking of thyristor \(B_1\) to the initial voltage, and knowing the forward and reverse resistances \(r_{\text{fwd}}\) and \(r_{\text{rev}}\) of thyristors \(B_2\) and \(B_1\) in the open state, one can determine the value of the capacitance of capacitor \(C\) that ensures satisfaction of condition (10):

\[ t_{\text{dis}} = - (r_{\text{fwd}} + r_{\text{rev}})\, C \ln \frac{U_c}{U_{c n}} > t_{\text{r}}, \]

whence

\[ C > - \frac{t_{\text{r}}}{\ln \dfrac{U_c}{U_{c n}} \cdot (r_{\text{fwd}} + r_{\text{rev}})} . \tag{11} \]

If, for example, by the end of the time \(t_{\text{r}}\), i.e., by the instant of blocking of thyristor \(B_1\), the reverse voltage applied to it must be not less than 1% of \(U_{c n}\), then

\[ \ln \frac{U_c}{U_{c n}} = \ln 0.01 = -2.3 . \]

In this case

\[ C > \frac{t_{\text{r}}}{2.3(r_{\text{fwd}} + r_{\text{rev}})} . \]

In each particular case, for the selected winding circuit and the type of elements of the semiconductor commutator, the capacitance values of capacitor \(C\) are determined from equation (9) and inequality (11), and the larger value is chosen.

Conclusions

The analytical dependences presented in this article, which characterize the process of current commutation in the winding of an electric machine equipped with a semiconductor commutator, show the dependence of the overvoltage on the commutated winding sections on the parameters of the winding and of the elements of the semiconductor commutator.

The calculation formulas obtained make it possible, when designing electric machines with semiconductor current commutation in the wind-

as to ensure optimal commutation conditions by matching the parameters of the electric machine with the parameters of the elements of the semiconductor commutator.

REFERENCES

1. Sh. I. Lutidze. Semiconductor commutation of electric machines. In: Operating Modes of Electric Power Systems and Regulation of Synchronous Machines, Nauka Publishing House, 1964.

2. Sh. I. Lutidze. Control and circuits of electric machines with a controlled semiconductor commutator. Proceedings of the Academy of Sciences of the USSR, No. 6, 1964.