Some Results of Studying the Switching Process Taking into Account Mutual Inductance

A. I. Skorospeshkin

(Presented at the scientific and technical conference of the EMF)

Analytical study of the switching process is, as a rule, carried out without taking into account the mutual inductance between the sections being switched.

It is known that the latter has a substantial effect on the process of current change in a section when it passes from one parallel branch to another.

In real machines the brush width \((b_{\mathrm{щ}})\) is usually greater than the width of a commutator segment \((b_k)\), and mutual-inductive coupling between the sections being switched is always present.

In connection with this, a study was carried out of the switching process taking mutual inductance into account.

For the analytical solution, one of the special cases of the dependence of the transient resistance of the brush contact on current density is used—a hyperbolic dependence. Such a dependence ensures constancy of the transient voltage drop. Thereby the character of the differential equations is simplified and their solution is facilitated.

The consideration is carried out for two cases:

\[ 1)\ b_k < b_{\mathrm{щ}} < 2b_k \quad \text{and} \quad 2)\ b_{\mathrm{щ}} = 2b_k. \]

1. \(b_k < b_{\mathrm{щ}} < 2b_k\)

As applied to the circuit shown in Fig. 1, the differential equations describing the change of current in switching circuits 1 and 2 are expressed in the following form:

\[ L_c \frac{di_1}{dt} + i_1 r_c + M \frac{di_5}{dt} = 0, \tag{1} \]

\[ L_c \frac{di_5}{dt} + i_5 r_c + M \frac{di_4}{dt} = 0. \tag{2} \]

Fig. 1.

In Fig. 1 and in equations (1), (2) the following notation is used:

\(t_1\)—current time;

\(i_a\)—current of the parallel branch;

\(i_4, i_5\)—currents in the commutated sections 1, 2, corresponding to the time \(t_1\);

\(i_1=i_a-i_4,\quad i_2=i_4-i_5,\quad i_3=i_a-i_5\)—currents in the connecting wires, corresponding to the position of the brush shown in Fig. 1, during the time \(t_1\);

\(r_c\)—resistance of a section;

\(L_c\)—inductance of a section;

\(M\)—mutual inductance between sections.

The time \(t_1\) is actually from 0 to \((T-T_k)\), where

\(T\)—the commutation period corresponding to the full width of the brush;

\(T_k\)—the commutation time determined by the width of the commutator segment.

Point \(O\) (Fig. 1) is taken as the initial position of the brush \((t_1=0)\), and time is reckoned from it.

In compiling equations (1), (2) it is assumed that the commutating emf is absent \((e_k=0)\) and that the resistance of the connecting wires is small in comparison with the resistance of a section.

After a number of transformations, the system of equations (1) and (2) with respect to the current \(i_5\) for the 2nd commutated section can be represented by the equation

\[ -\frac{d^2 i_5}{dt_1^2}\left(\frac{M}{L_c}-\frac{L_c}{M}\right) -\frac{di_5}{dt_1}\cdot\frac{2r_c}{M} -i_5\frac{r_c^2}{L_cM}=0. \tag{3} \]

The solution of this equation is written in the form

\[ i_5=C_1 e^{\kappa_1 t_1}+C_2 e^{\kappa_2 t_1}, \tag{4} \]

where

\(\kappa_1\) and \(\kappa_2\)—roots of the characteristic equation;

\(C_1\) and \(C_2\)—constants of integration.

From equation (3)

\[ \kappa_1=\frac{r_cL_c+r_cM}{M^2-L_c^2} \quad\text{and} \tag{5} \]

\[ \kappa_2=\frac{r_cL_c-r_cM}{M^2-L_c^2}. \tag{6} \]

The constants of integration are determined from the initial conditions. For \(t_1=0\)

\[ i_5=i_a, \]

\[ \frac{di_5}{dt_1}=0. \tag{7} \]

As a result, we find

\[ C_1=-\frac{\kappa_2 i_a}{\kappa_2-\kappa_1} \tag{8} \]

and

\[ C_2=\frac{\kappa_1 i_a}{\kappa_2-\kappa_1}. \tag{9} \]

After substituting (8), (9) into (4), we obtain

\[ i_5=i_a\left(\frac{M-L_c}{2M}e^{\kappa_1 t_1} +\frac{M+L_c}{2M}e^{\kappa_2 t_1}\right). \tag{10} \]

Thus, during the time \(t_1\), sections 1 and 2 are commutated simultaneously.

With the brush position shown in Fig. 2, only one section, (2), will be commutated during the time \(t_2\). The time \(t_2\) is valid from \((T-T_k)\) to \(T_k\).

The process of current variation in the section in this case is described by the following differential equation:

\[ L_c\frac{di_6}{dt_2}+i_6 r_c=0. \tag{11} \]

Fig. 2.

In Fig. 2 and in equation (11) the following notation is used:

\(t_2\)—the current time;

\(i_6\)—the current in section (2), corresponding to time \(t_2\);

\(i_7=i_a+i_6,\ i_8=i_a-i_6\)—the currents in the connecting wires corresponding to the brush position shown in Fig. 2 during the time \(t_2\).

The solution of equation (11) is the expression

\[ i_6=C_3 e^{\kappa_3 t_2}, \tag{12} \]

where \(\kappa_3\) is the root of the characteristic equation;

\(C_3\) is the integration constant.

From equation 11 we find

\[ \kappa_3=-\frac{r_c}{L_c}. \tag{13} \]

For \(t_2=0\),

\[ i_6=i_5(T-T_k) \tag{14} \]

and, consequently, the integration constant is

\[ C_3=i_5(T-T_k). \tag{15} \]

Thus, the current of the section is

\[ i_6=i_5(T-T_k)e^{\kappa_3 t_2}. \tag{16} \]

In the next time interval \((t_3)\), actual from \(T_k\) to \(T\), two sections (2 and 3) will again be commutated. Figure 3 presents the circuit corresponding to this time interval.

Fig. 3.

The differential equations describing the change of current in the commutated circuits (2 and 3) are expressed in the following form:

\[ L_c \frac{di_9}{dt_3} + i_9 r_c + M \frac{di_{10}}{dt_3} = 0. \tag{17} \]

\[ L_c \frac{di_{10}}{dt_3} + i_{10} r_c + M \frac{di_9}{dt_3} = 0. \tag{18} \]

In Fig. 3 and in equations (17), (18), the following notation is used:

\(t_3\) is the current time;

\(i_9, i_{10}\) are the currents in sections (2 and 3), corresponding to time \(t_3\);

\(i_{11} = i_a + i_{10}\), \(i_{12} = i_9 - i_{10}\), \(i_{13} = i_a - i_9\) are the currents in the connecting conductors, corresponding to the brush position shown in Fig. 3, during time \(t_3\).

After a number of transformations, the system of equations (17), (18) with respect to the current \(i_9\) is determined by the equation

\[ \frac{d^2 i_9}{dt_3^2} \left( \frac{M}{L_c} - \frac{L_c}{M} \right) - \frac{di_9}{dt_3}\frac{2r_c}{M} - i_9 \frac{r_c^2}{L_c M} = 0. \tag{19} \]

Its solution is written as the expression

\[ i_9 = C_4 e^{\kappa_4 t_3} + C_5 e^{\kappa_5 t_3}, \tag{20} \]

where \(\kappa_4\) and \(\kappa_5\) are the roots of the characteristic equation;

\(C_4\) and \(C_5\) are integration constants.

It is necessary to take into account that

\[ \kappa_4 = \kappa_1, \]

\[ \kappa_5 = \kappa_2. \tag{21} \]

To find the integration constants, the following conditions are available:

\[ \begin{aligned} \text{at } t_3 &= 0 \qquad && i_9 = i_{6(T_k)},\\ t_3 &= T \qquad && i_9 = -i_a, \end{aligned} \tag{22} \]

where \(i_{6(T_k)}\) is the value of the current \(i_6\) at the instant of time corresponding to \(T_k\).

Taking the known conditions into account, we find

\[ C_4 = \frac{ i_a + i_{6(T_k)} e^{\kappa_5 T} }{ e^{\kappa_5 T} - e^{\kappa_4 T} }, \tag{23} \]

\[ C_5 = \frac{ -i_a - i_{6(T_k)} e^{\kappa_4 T} }{ e^{\kappa_5 T} - e^{\kappa_4 T} }. \tag{24} \]

After substituting (23), (24) into (20), we obtain

\[ i_g = \frac{i_a - i_{i(T_k)} e^{k_3 T}}{e^{k_3 T} - e^{k_4 T}}\, e^{k_4 T_3} - \frac{-i_a - i_{i(T_k)} e^{k_4 T}}{e^{k_3 T} - e^{k_4 T}}\, e^{k_3 T_3} = 0. \tag{25} \]

Thus, for the case when \(b_k < b_{\mathrm{sh}} < 2b_k\), the process of current variation in the section being commutated is determined by expressions (10), (16), (25). Moreover, each of them is valid for a definite time interval.

Fig. 4.

Fig. 5.

2. \(b_{\mathrm{sh}} = 2b_k\).

For the circuit shown in Fig. 4, the same system of equations as (1), (2) can be set up. The only difference is that the time \(t_1\) is in fact from 0 to \(T_k\).

The expression for the current \(i_3\) is found in the same way as in the first case.

For the circuit shown in Fig. 5, where the notation is the same as in Fig. 3, the same system of equations as (17) and (18) can be set up.

The expression for the current \(i_1\) is found in the same way as in the first case.

Thus, when \(b_{\mathrm{sh}} = 2b_k\), two sections take part in the commutation process at all times. Moreover, from 0 to \(T_k\), sections 1 and 2 are commutated simultaneously, and from \(T_k\) to \(T\), sections 2 and 3.

Using the equations obtained for the case when \(b_k < b_{\mathrm{sh}} < 2b_k\), a calculation was carried out for the following parameters:

\[ i_a = 1.5\,a, \qquad T = 7.4 \cdot 10^{-4}\ \text{sec}, \]

\[ L_c = 1.48 \cdot 10^{-4}\ \text{henry}, \qquad T_k = 4.62 \cdot 10^{-4}\ \text{sec}, \]

\[ r_c = 0.1\ \text{ohm}, \qquad M = 0.83\,c. \]

As a result, curve 1 was constructed (Fig. 6).

For the same parameters, an experimental check was carried out on an electric-machine amplifier. Brushes of grade MG-4 were installed. It was established experimentally that they provide a hyperbolic dependence of the transient resistance of the brush contact on the current density.

Oscillographic recording of the current curve of the commutated section was carried out with the collector in a steady thermal state.

Curve (2), shown in Fig. 6, is experimental. The deviations present in curves (1) and (2) are explained by the fact that the calculation does not take into account the armature reaction that occurs in the electric-machine amplifier.

Fig. 6.

Thus, the method set forth makes it possible to investigate the commutation process while taking into account the mutual inductance between the commutated sections. It can also be extended to the case when \(b_{\text{щ}} > 2b_{\text{к}}\).