CONDITIONS FOR SELF-EXCITATION OF AN ELECTRIC-MACHINE AMPLIFIER WITH A TRANSVERSE FIELD

V. P. Lagunov

(Presented by the scientific seminar of the Electromechanical Faculty)

When operating under load conditions, and also when operating at no load, an electric-machine amplifier with a transverse field (EMA TF) under certain conditions may spontaneously self-excite; in this case its controllability is lost.

Fig. 1.
Schematic diagram of an EMA with a transverse field.

$U_1$ — voltage supplied to the control winding.
$U_3$ — voltage at the terminals of the amplifier’s longitudinal circuit.
$r_1$ — resistance of the control winding.
$r_k$ — resistance of the compensating winding.
$r_0$ — shunting resistance.
$r_g$ — resistance of the commutating-pole winding.
$i_1$ — current in the control winding.
$i_2$ — current of the armature transverse circuit.
$i_k$ — current in the compensating winding.
$d - d$ — longitudinal axis of the amplifier.
$q - q$ — transverse axis of the amplifier.
$\alpha$ — displacement angle of the transverse-circuit brushes.

To determine the conditions under which self-excitation of a transverse-field EMA at no load is possible, as well as the speed of this process, it is necessary to investigate the system of differential equations of the amplifier, which may be represented as follows [1]:

\[ \left. \begin{aligned} U_1 &= i_1 r_1 + L_1 \frac{di_1}{dt} + M_{1k}\frac{di_k}{dt} - M_{12}\frac{di_2}{dt},\\[6pt] 0 &= M_{k1}\frac{di_1}{dt} + i_k R_k + L_k\frac{di_k}{dt} - M_{k2}\frac{di_2}{dt},\\[6pt] e_2 &= k_{21} i_1 + k_{2k} i_k - k_{2p} i_2 = i_2 r_2 + L_2\frac{di_2}{dt},\\[6pt] e_3 &= k_{32} i_2, \end{aligned} \right\} \tag{1} \]

where \(U_1\) is the voltage applied to the control winding;

\(e_2\) is the emf at the brushes of the transverse circuit;

\(e_3\) is the emf at the brushes of the longitudinal circuit;

\(r_1\) and \(L_1\) are the active resistance and inductance of the control winding;

\(M_{1k}\) and \(M_{12}\) are the mutual inductances of the control winding with the compensating winding and with the transverse circuit;

\(r_k\) and \(L_k\) are the active resistance and inductance of the compensating winding;

\(R_k = r_k + r_0\) is the total resistance of the circuit—compensating winding—shunting resistance;

\(M_{k1}\) and \(M_{k2}\) are the mutual inductances of the compensating winding with the control winding and with the transverse circuit;

\(k_{21}\) is the slope coefficient of the emf characteristic of the transverse circuit with respect to the control current;

\(k_{2k}\) is the slope coefficient of the emf characteristic of the transverse circuit with respect to the current of the compensating winding.

\[ k_{2p}=k_z+k_c\pm k_{\mathrm{sh}}, \tag{2} \]

where \(k_z\), \(k_c\), \(k_{\mathrm{sh}}\) are coefficients accounting for the change in the emf of the transverse circuit due to the reaction of the commutation and eddy currents, and also due to the displacement of the brushes of the transverse circuit (\(+\) corresponds to displacement of the brushes in the direction of rotation of the amplifier);

\(r_2\) and \(L_2\) are the active resistance and inductance of the transverse circuit;

\(k_{32}\) is the slope coefficient of the emf characteristic of the longitudinal circuit with respect to the current of the transverse circuit.

The system of differential equations (1) has the following characteristic equation of the third degree with respect to \(p\):

\[ \begin{aligned} &(L_1 L_k L_2 - M_{k1} M_{1k} L_2)p^3 + (k_{2p}L_1L_k + r_2L_1L_k + R_kL_1L_2 + r_1L_kL_2 \\ &\quad - k_{21}M_{1k}M_{k2} + k_{2k}M_{k1}M_{12} - k_{21}M_{12}L_k - k_{2k}M_{k2}L_1 - k_{2p}M_{1k}M_{k2} \\ &\quad - r_2M_{k1}M_{1k})p^2 + (k_{2p}R_kL_1 + k_{2p}r_1L_k + r_2R_kL_1 + r_1r_2L_k + r_1R_kL_2 \\ &\quad - k_{21}R_kM_{12} - k_{2k}r_1M_{k2})p + r_1R_k k_{2p} + r_1r_2R_k = 0. \end{aligned} \tag{3} \]

Usually, in the theoretical consideration of transient processes of a transverse-field EMA, the coupling coefficient between the control and compensating windings is taken equal to unity [1]. If this is assumed—

… and divide the characteristic equation obtained by \(r_1\), \(r_2\), and \(R_k\), then it will have the following form (4):

\[ a_0 p^2 + a_1 p + a_2 = 0, \tag{4} \]

where the coefficients \(a_0\), \(a_1\), and \(a_2\) are equal to:

\[ a_0 = T_2 (T_1+T_k) + \frac{k_{21}}{r_2}(T_{1k}T_{k2}-T_kT_{12}) + \frac{k_{2k}}{r_2}(T_{12}T_{k1}-T_{k2}T_1), \tag{5} \]

\[ a_1=\left(\frac{k_{2p}+r_2}{r_2}\right)(T_1+T_k)+T_2-\frac{1}{r_2}(k_{21}T_{12}+k_{2k}T_{k2}), \tag{6} \]

\[ a_2=-\frac{k_{2p}+r_2}{r_2}, \tag{7} \]

where

\[ T_1=\frac{L_1}{r_1} \]

is the time constant of the control-winding circuit;

\[ T_k=\frac{L_k}{R_k} \]

is the time constant of the compensating-winding circuit;

\[ T_2=\frac{L_2}{r_2} \]

is the time constant of the transverse circuit;

\[ T_{12}=\frac{M_{12}}{r_1} \]

is the time constant determined by the mutual inductance of the control winding with the transverse circuit;

\[ T_{k2}=\frac{M_{k2}}{R_k} \]

is the time constant determined by the mutual inductance of the compensating winding with the transverse circuit;

\[ T_{k1}=\frac{M_{k1}}{R_k} \]

is the time constant determined by the mutual inductance of the compensating winding with the control winding;

\[ T_{1k}=\frac{M_{1k}}{r_1} \]

is the time constant determined by the mutual inductance of the control winding with the compensating winding.

The characteristic equation (4) has two roots

\[ p_{1,2}=\frac{-a_1\pm\sqrt{a_1^2-4a_0a_2}}{2a_0}. \]

The nature of the transient process of the increase of the emf at the longitudinal brushes of the transverse-field electric-machine amplifier as a whole is determined by the roots \(p_{1,2}\). In order to determine the conditions under which self-excitation of the amplifier is possible, and also to establish how various factors affect the rate of this process, let us investigate the roots of the characteristic equation (4).

Self-excitation of the amplifier is possible only if one of the roots takes the value of a real positive number. Such a value of the roots is possible in two cases, namely:

1. For \(a_1>0\) and \(a_2<0\).

In this case

\[ p_{1,2}=\frac{-a_1\pm\gamma a_1}{2a_0}, \]

where \(\gamma>1\).

Thus, in this case we shall have at least one real positive root.

1. For \(a_1<0\) and \(a_2<0\).

Moreover, it is necessary to note that the coefficient \(a_1\) becomes less than zero only in the case when \(a_2<0\). This follows from an analysis of the structure of the coefficients of the characteristic equation.

Case 2 is not characteristic of the self-excitation mode of the amplifier for the following reason. The equivalent time constant of the transverse-field electric machine amplifier has the following expression [2]

\[ T_{\text{eq}}=\frac{2a_0}{a_1}. \tag{8} \]

Considering the expression for the equivalent time constant of the amplifier (8), we see that, if \(a_1\) assumes a value equal to zero or less than zero, then self-excitation of the amplifier becomes impossible.

Thus, for the self-excitation mode of the amplifier the first case is characteristic, i.e., the case when \(a_1>0\) and \(a_2<0\). Let us determine under what conditions the coefficient \(a_2\) is negative.

The expression for the coefficient \(a_2\), according to (7), is equal to:

\[ a_2=-\frac{k_{2p}}{r_2}+1. \tag{9} \]

From equation (9) it follows that the coefficient \(a_2\) can become negative in the case when

\[ k_{2p}<-r_2. \tag{10} \]

Turning to expression (2), we can establish that inequality (10) exists in the case of displacement of the brushes of the transverse circuit against the direction of rotation. In other words, self-excitation of the amplifier becomes possible for a certain displacement of the brushes of the transverse circuit against the direction of rotation of the amplifier.

Taking into account that the coefficient \(k_{\text{щ}}\) is proportional to the angle of displacement of the brushes of the transverse circuit, we can determine condition (11), under which self-excitation of the amplifier occurs.

Condition (11) can be determined as follows. Substituting in expression (10) the value of \(k_{2p}\) according to (2), we obtain the following expression:

\[ k_z+k_c-k_{\text{щ}}<-r_2 \]

or

\[ k_{\text{щ}}>k_z+k_c+r_2. \tag{11} \]

As the angle of displacement of the brushes of the transverse circuit against the direction of rotation increases, the coefficient \(a_1\) (6) decreases, approaching zero. At \(a_1=0\), as already noted earlier, self-excitation ceases. From the last expression one can determine the maximum angle of rotation at which self-excitation still exists:

\[ a_1= \frac{k_{2p}}{r_2}T_1+ \frac{k_{2p}}{r_2}T_k+ T_1+T_k+T_2- \frac{k_{21}}{r_2}T_{12}- \frac{k_{2k}}{r_2}T_{k2}=0 \]

or

\[ k_{\text{щ}}=k_c+k_z+r_2\left(1+\frac{T_2}{T_1+T_k}\right) -\frac{k_{21}T_{12}+k_{2k}T_{k2}}{T_1+T_k}. \tag{12} \]

Thus, the displacement angle of the brushes of the transverse circuit at which self-excitation of the transverse-field EMA is possible can be determined from the coefficient \(k_{\mathrm{shch}}\), whose value in this case lies within the following limits:

\[ k_c-k_z-r_2\left(1+\frac{T_2}{T_1+T_k}\right) -\frac{k_{21}T_1+k_{ek}T_k}{T_1+T_k} \geq k_{\mathrm{shch}} \geq k_z+k_c+r_2 . \tag{13} \]

The rate of rise of the emf of the longitudinal circuit during self-excitation is determined by the equivalent time constant (8), which is inversely proportional to the coefficient \(a_1\). Analyzing the expression for the coefficients \(a_1\) and \(a_0\), we may conclude that the rate of rise of the emf of the longitudinal circuit during self-excitation depends on a number of parameters that are constant for a given amplifier and, in addition, depends on the displacement angle of the brushes of the transverse circuit and on the magnitude of the shunting resistance \(r_0\), which may vary when the amplifier is adjusted. Consideration of the equivalent time constant of the amplifier shows that increasing the displacement angle of the brushes of the transverse circuit against the direction of rotation and decreasing the shunting resistance \(r_0\) leads to an increase in the equivalent time constant of the amplifier.

Conclusions

1. Self-excitation of the transverse-field EMA at no load occurs when \(k_{\mathrm{shch}}<-r_2\), which corresponds to shifting the brushes of the transverse circuit by a certain angle against the direction of rotation. The minimum value of the angle at which self-excitation of the amplifier can occur is determined by expression (11).

The maximum angle of rotation of the brushes of the transverse circuit at which the transverse-field EMA is still in the self-excitation regime can be determined from (12).

1. The rate of rise of the emf at the longitudinal brushes depends on the magnitude of the displacement angle of the brushes of the transverse circuit; moreover, when the displacement angle of these brushes against the direction of rotation is increased, the self-excitation process proceeds more slowly.

The value of the equivalent time constant is affected by the circuit of the compensating winding.

When the shunting resistance \(r_{ek}\) of the amplifier is increased, it decreases; in other words, the self-excitation process proceeds more rapidly.

References

1. V. V. Rudakov, “Characteristics of electric-machine amplifiers with a transverse field,” Vestnik elektropromyshlennosti, No. 10, 1952.

2. V. V. Rudakov, “Influence of the displacement of the brushes of the transverse circuit on the transient process of an electric-machine amplifier,” Elektrichestvo, No. 11, 1952.