Stability Boundary Conditions for a Cross-Field Electric Machine Amplifier in No-Load Operation
Abstract Generated abstract
This paper analyzes stability limits for a transverse-field electric machine amplifier operating at no load, with attention to how adjustable and design parameters affect transient behavior in automatic-control systems. Starting from a linearized system of dynamic equations, the authors derive a characteristic equation and use its roots to distinguish oscillatory, aperiodic stable, and self-excitation regimes. Boundary expressions are obtained for the transition from oscillatory to aperiodic response and from stable operation to self-excitation, expressed through brush displacement, time-constant ratio, and amplifier parameters. The resulting controlled-parameter plane is proposed as a practical aid for selecting operating modes and for designing amplifiers with a wider stable operating range.
Full Text
Boundary Conditions for the Stability of a Transverse-Field EMA in No-Load Operation
V. P. Lagunov, V. M. Riikonen
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
Automatic-control systems (ACS) with transverse-field electric machine amplifiers (EMA TF) have found wide practical application.
In modern ACS it is usually required to ensure stability and a small steady-state error, and it is also often required to ensure the proper course of the transient process.
The problems posed in the study of transient processes may be very diverse. One of them may be the problem of selecting the parameters of the system, including those of the amplifier, such that the transient process would satisfy the requirements imposed on it. In such investigations it is very important to know the boundary values of the parameters characterizing the operation of the amplifier at the boundary between two transient processes adjacent in quality.
The indicated boundary values of the parameters of the transverse-field EMA may be obtained from an analysis of the system of equations characterizing the dynamic properties of the amplifier. Under a linear consideration of the parameters, this system has the following form:
\[ \begin{aligned} \Phi_1 &= \lambda_1(w_y i_y + w_k i_k - w_p i_2),\\ \Phi_2 &= \lambda_2 w_{\mathrm{я}}(1 - x)i_2,\\ U_y &= r_y i_y + \sigma_y w_y p\Phi_1,\\ e_2 &= c\Phi_1 = r_2 i_2 + \sigma_{\mathrm{я}} w_{\mathrm{я}} p\Phi_2,\\ e_3 &= c\Phi_2 + (\sigma_{\mathrm{я}} w_3 - \sigma_k w_k)p\Phi_1 - r_k i_k,\\ 0 &= \sigma_k w_k p\Phi_1 + R_k i_k, \end{aligned} \tag{1} \]
where
\[ w_p = \pm xw_{\mathrm{я}} + w_z + w_c;\qquad w_3 = w_{\mathrm{я}} - w_{\mathrm{д}};\qquad R_k = r_k + r_{\mathrm{ш}}; \]
\(\lambda_1, \lambda_2\) are the magnetic permeances along the longitudinal and transverse axes;
\(\sigma_{\mathrm{я}}, w_{\mathrm{я}}\) are the leakage coefficient and the number of turns of the armature winding;
\(w_c, w_z\) are fictitious turns corresponding to the magnetomotive forces due to losses in the armature steel and the reaction of commutation currents in the transverse circuit;
\(\sigma_y, r_y, w_y\) are the leakage coefficient, active resistance, and number of turns of the control winding;
\(\sigma_k, r_k, w_k\) are the leakage coefficient, active resistance, and number of turns of the compensating winding;
\(r_2\) — active resistance of the transverse circuit;
\[ x=\frac{2p\beta}{\pi} \]
— relative displacement from the neutral of the brushes of the transverse circuit.
The system of equations (1) has the characteristic equation:
\[ (1-x)T_1T_2z^2+[T_1+(1-x)T_2]z+k_p+1=0, \tag{2} \]
where
\[ T_1=T_y+T_k \]
\(T_y, T_k, T_2\) — the time constants of the control winding, the compensating winding, and the transverse armature circuit;
\(k_p\) — the transfer coefficient due to the armature reaction along the longitudinal axis of the amplifier, caused by the current of the transverse armature circuit. It is equal to
\[ k_p=\frac{\lambda_1 C}{r_2}(\pm xw_{\text{я}}+w_z+w_c). \tag{3} \]
To analyze the quality of the transient process of the EMA, we examine the characteristic equation (2). The equation has two roots:
\[ z_{1,2}= \frac{-[T_1+(1-x)T_2]\pm\sqrt{[T_1-(1-x)T_2]^2-4(1-x)T_1T_2k_p}} {2(1-x)T_1T_2}. \tag{4} \]
It follows from expression (4) that the roots \(z_{1,2}\) for a particular amplifier are determined by two time constants \(T_1, T_2\) and by the value of the coefficient \(k_p\). In the process of adjusting the amplifier operation, the quantities of the time constant \(T_1\) and of the coefficient \(k_p\) may vary; the first by changing the degree of compensation and the value of the active resistance included in the circuit of the control winding, and the second by shifting the brushes of the transverse circuit from the geometrical neutral. Here \(T_1\) always remains positive, while the quantity \(k_p\) may take either positive or negative values. It may be negative for a certain displacement of the transverse brushes against the direction of rotation of the armature (a minus sign is placed before \(x\) in equation 3).
Depending on the relationship of these parameters, three types of transient regimes are possible:
- Oscillatory regime.
- Aperiodic stable regime.
- Self-excitation regime.
From (4) it is clear that the transition from the oscillatory regime to the aperiodic stable regime is characterized by the value of \(k_p\) equal to
\[ k_p= \frac{[T_1-(1-x)T_2]^2}{4(1-x)T_1T_2} = \frac{[1-(1-x)\gamma]^2}{4(1-x)\gamma}, \tag{5} \]
or, neglecting the decrease of the transverse flux when the brushes are shifted, by the value of the displacement of the transverse brushes \(x\), equal to
\[ x=\frac{w_c-w_z}{w_{\text{я}}} -\frac{r_2}{\lambda_1}\cdot \frac{(1-\gamma)^2}{4w_{\text{я}}\gamma}, \tag{6} \]
where
\[ \gamma=\frac{T_2}{T_1}. \]
From (2 and 4) it follows that the self-excitation regime occurs when \(k_p<-1\), i.e., the transition from the aperiodic stable regime to the self-excitation regime is characterized by the condition \(k_p=-1\). Substituting
Fig. 1. Schematic diagram of a transverse-field EMA.
Substituting into the last condition \(k_p\) according to expression (3), we find the displacement of the transverse brushes \(x\) (in this case the brushes are shifted opposite to the direction of rotation). It is equal to:
\[ x = \frac{w_c + w_z}{w_{\mathrm{a}}} + \frac{r_2}{\lambda_1 C w_{\mathrm{a}}}. \tag{7} \]
In the plane with coordinates \(x, \gamma\), curves have been constructed from equations (6, 7) (Fig. 2), which characterize the transition of the amplifier from one type of transient process to another. The curves are constructed for the parameters of the EMU-5P amplifier.
Fig. 2. Plane of adjustable parameters of the transverse-field EMA.
The reduced plane, which may be called the plane of the controlled parameters of the EMA, helps provide a clear representation of their influence on the quality of the transient process, while expressions (6, 7) reveal the influence of the amplifier’s design parameters on the indicated boundary conditions.
Equation (7) shows that the amplifier’s stability margin increases with an increase in losses in the armature steel, in the reaction of commutation currents, and in the active resistance of the transverse circuit, and also with a decrease in magnetic permeability along the longitudinal axis.
Conclusions
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The boundary conditions obtained can be used in the design of amplifiers in order to expand the range of their stable operation.
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The reduced plane of the controlled parameters of a transverse-field EMA makes it possible to select the optimal operating mode of the amplifier during operation.