Control Systems for Starting and Braking Currents of a Series-Wound Motor
Abstract Generated abstract
This paper examines relay automatic control systems for regulating starting and braking currents in DC series-wound traction motors, addressing limitations caused by sensor speed and hysteresis in comparison devices. It describes systems using thyristor switching, a saturable multivibrator current sensor, and a frequency pulse selector, and develops analytical representations of electromagnetic processes that account for eddy currents, magnetization nonlinearity, leakage inductance, and armature reaction. For regenerative braking, the authors apply harmonic linearization and root-locus methods to estimate self-oscillation amplitude and frequency in quasi-steady operation. The proposed expressions and methods are reported to agree with experimental results within 8 to 15 percent.
Full Text
Systems for Regulating Starting and Braking Currents of a Series-Wound Motor
L. V. Volynets, R. K. Gachik, A. P. Zaitsev, Yu. F. Mikheev
(Presented by the scientific and technical seminar of the Research Institute of Automation and Electromechanics at TPI)
Pulse-width systems for regulating the current of DC electric-drive motors are widely known. Less widespread and insufficiently studied are analogous systems using a relay principle of regulation. This is explained by the difficulty of applying the relay principle when regulating fast processes, such as the electromagnetic process associated with changes in armature current. The insufficient speed of current sensors and the considerable hysteresis of the characteristics of comparison devices produced in series cause significant pulsations of the regulated quantity and, as a consequence, poor energy performance and commutation of electric motors. This circumstance restrains the development of relay current-regulation systems that are simple in structure, possess high speed of response, and make it possible to achieve large amplification factors without violating the stability of the system with respect to the smooth component of the regulated current.
A convenient object of regulation for a relay automatic current-regulation system is a series-wound motor, which has an increased electromagnetic time constant of the armature circuit, thereby somewhat reducing the requirements imposed on the speed of response of the current sensors in the system. The Research Institute of Automation and Electromechanics at TPI, together with the Riga Electric Machine-Building Plant, has developed and tested relay automatic regulation systems for the starting and braking currents of traction motors, the block diagrams of which are given in Fig. 1. Current regulation is carried out by shunting the field winding or the resistance of the armature circuit with a thyristor switch TK. For electric transport with a low-voltage power source, the thyristor switch may be connected in series into the armature circuit for the full source voltage. In rheostatic-regenerative braking, the traction motor is connected according to an independent-excitation circuit, with a thyristor switch in the field-winding circuit. As the current sensor in the systems, a saturable multivibrator with magnetic coupling is used, the frequency of whose output voltage varies as a function of the regulated current; as the element comparing the actual and prescribed values of the armature current, a frequency pulse selector is used, which has no hysteresis in its output characteristic. This made it possible to reduce the amplitude of pulsations of the armature current during regulation to several percent of the average value.
Fig. 1.
For relay control systems it is characteristic that, owing to the high rates of change of the system coordinates, certain factors—which are neglected, or are taken into account by simplified relations, in continuous systems—have a substantial influence on the nature of the processes.
In analyzing quasi-steady-state and transient regimes in relay automatic current-control systems, these factors may be taken into account as follows.
- The eddy-current circuit is represented as an equivalent short-circuited circuit without leakage fluxes, with the time constant
\[ T_{\mathrm{k}} = T_{\mathrm{k}}' \frac{A w_0}{A w_1}, \]
where
\[ T_{\mathrm{k}}' = \frac{4 l_{\mathrm{c}}}{\pi \rho \delta 10^9} \cdot \frac{a \cdot b}{\dfrac{a}{b} + \dfrac{b}{a}}, \]
\(l_{\mathrm{c}}\) — length of the solid section of the core;
\(\rho\) — electrical resistivity of the material of the solid section;
\(a, b\) are the sides of the rectangular cross section of the core;
\(\sigma\) is the length of the air gap;
\(A w_0\) are the ampere-turns expended in overcoming the air gap;
\(A w_1\) are the total magnetizing ampere-turns.
- The inductance of the magnetizing circuit
\[ L_\mu = 2 p w_{\mathrm{v}} \frac{\Delta \Phi}{\Delta I_\mu}\cdot K_2, \]
where
\(p\) is the number of pairs of motor poles;
\(w_{\mathrm{v}}\) is the number of turns of the field winding;
\(\dfrac{\Delta \Phi}{\Delta I_\mu}\) is the slope coefficient of the load characteristic;
\(K_2\) is a coefficient taking into account the influence of hysteresis.
- The leakage inductance
\[ L_s = 2 p w_{\mathrm{v}} \frac{\Phi_{\mathrm{n}}(\sigma_{\mathrm{n}} - 1)} {I_{\mathrm{vn}}(2 - \sigma_{\mathrm{n}})}, \]
where
\(\Phi_{\mathrm{n}}\) is the rated useful flux,
\(\sigma_{\mathrm{n}}\) is the rated leakage coefficient,
\(I_{\mathrm{vn}}\) is the rated excitation current.
- The demagnetizing effect of the armature reaction is conveniently taken into account as a separate element in the structural diagram with transfer coefficient
\[ K_{\mathrm{r}} = \frac{\Delta \Phi_{\mathrm{r}}}{\Delta I_{\mathrm{ya}}} \quad \text{for } I_{\mathrm{v}} = \mathrm{const}. \]
With the above allowance for eddy currents, the nonlinearity of the magnetization curve, leakage inductance, and armature reaction, the electromagnetic processes in the circuits of Fig. 1 for each state of the TC are described by second-order differential equations, which can be solved by known methods, for example, by the phase-plane method. The generalized transfer function of such automatic control systems over intervals of constant parameters has the form
\[ W(p) = W_f(p)\cdot F(p) + W_\delta(p), \]
where
\(F(p)\) is the image of the control action,
\(W_i(p)\) is the transfer function with respect to the main input,
\(W_\delta(p)\) is the transfer function with respect to the \(\delta\)-input.
The transfer function with respect to the main input can be represented in the form
\[ W_i(p) = \frac{T_k p + 1} {T_s T_k p^2 + (T_s + T_k + T_\mu)p + (k_e + 1)}, \]
\(T_s\) is the time constant of leakage of the main poles;
\(T_\mu\) is the time constant of the magnetizing circuit;
\(K_e\) is a coefficient taking into account the counter-emf of the motor.
The transfer function with respect to the \(\delta\)-input takes into account the initial conditions of motion in the system:
\[ W_\delta(p) = \frac{ T_s T_k i'_0 + \left[T_s T_k p + (T_s + T_k + T_\mu)\right] i_0 - T_k U_{c0} } { T_s T_k p^2 + (T_s + T_k + T_\mu)p + (k_e + 1) }, \]
where
\(i_0\), \(i'_0\), and \(U_{c0}\) are, respectively, the initial values of the current, its first derivative, and the voltage of the supply network.
The electromagnetic processes in the circuit of Fig. 2 are described by third-order differential equations, and investigation of the quasi-steady processes in the relay-type automatic control system of Fig. 2 by the phase-plane method is difficult. Below is an investigation of quasi-steady processes in the optimal regenerative-braking system described in [1] and constructed according to the circuit of Fig. 2. In the investigation the following assumptions were adopted:
Fig. 2
1) the motor speed of rotation is constant,
2) the line voltage does not change,
3) the thyristor switch is ideal,
4) the linear part of the system is a good low-frequency filter.
Fig. 3
The block diagram of the system under investigation is shown in Fig. 3, where
\[
W_1=\frac{\Delta U_{\mathrm{v}}(p)}{\Delta I_{\mathrm{v}}(p)}
=
\frac{1/r_{\mathrm{v}}\,(T_k p+1)}
{T_s T_k p^2+(T_s+T_k+T_{\mu})p+1}
\]
— is the transfer function of the field winding;
\[
W_2=\frac{\Delta \Phi(p)}{\Delta I_{\mathrm{v}}(p)}
=
\frac{1/r_{\mathrm{ya}}}{T_k p+1}
\]
— is the transfer function of the magnetic circuit of the motor;
\[
k_2=C_e n
\]
— is the motor transfer coefficient between e.m.f. and flux;
\[
W_3=\frac{\Delta I_{\mathrm{ya}}(p)}{\Delta U_{\mathrm{ya}}(p)}
=
\frac{1/r_{\mathrm{ya}}}{T_{\mathrm{ya}}p+1}
\]
— is the transfer function of the armature circuit of the motor;
\[
k_3
\]
— is the transfer coefficient of the control unit (armature-current sensor, computing device);
\[ K(A)=\frac{4B}{\pi A}e^{-i\varphi(A)},\quad \varphi(A)=\arcsin\frac{B}{A} \]
is the complex transfer coefficient of the relay element with an ambiguity zone \(2\varepsilon\) and output signal \(B\);
\[ T_{\mathrm{a}}=\frac{L_{\mathrm{a}}}{r_{\mathrm{a}}} \]
is the time constant of the armature circuit.
Under the assumptions made, the parameters of self-oscillations in a quasi-steady process can be determined by carrying out a harmonic linearization of the braking system. In doing so, we assume that only the first harmonic of the oscillations circulates in the system. The relay element is replaced by the complex transfer coefficient \(K(A)\), and it is assumed that the amplitude and frequency of the first harmonic of the self-oscillations of the relay system are, in the first approximation, equal to the amplitude and frequency of the self-oscillations of the regulated quantity of the harmonically linearized system located at the stability boundary. With this approach, the transfer function of the open-loop linearized system is written as
\[ W_{\mathrm{p}}(p)= \frac{K_{\mathrm{l}}\cdot K(A)} {\left[T_{\mathrm{s}}T_{\mathrm{k}}p^{2}+\left(T_{\mathrm{s}}+T_{\mathrm{k}}+T_{\mu}\right)p+1\right]\left(T_{\mathrm{a}}p+1\right)}, \tag{1} \]
where
\(K_{\mathrm{l}}\) is the transfer coefficient of the linear part of the system.
Further investigation of the self-oscillations may be carried out by any method of the linear theory of automatic control, but since \(K(A)\) is a complex quantity, it is simplest to conduct the subsequent investigation by the root-locus method.
The zero approximation of the self-oscillation frequency is determined by
\[ \omega_{0}=\sqrt{p_{1}\cdot p_{1}}, \]
where
\(p_{1}\) and \(p_{2}\) are the poles of the transfer function of the open-loop system that are nearest to the origin and are not compensated by nearby zeros.
Assigning frequency values close to \(\omega_{\mathrm{n}}^{0}\), we write the phase equation:
\[ -\varphi(A)-\sum_{i=1}^{3}\operatorname{arc\,tg}\frac{\omega_{0}}{p_{i}}=+180^{\circ}. \tag{2} \]
From (2) we determine the value \(\varphi(A)\), and from the expression for \(K(A)\) and \(\varphi(A)\) we determine the oscillation amplitude at the input of the relay element \(A\) and the modulus \(K(A)\). The values of the total transfer coefficient of the linearized system at which the system will be at the stability boundary are found from the modulus equation:
\[ K_{\mathrm{gr}}=K_{\mathrm{l}}\cdot |K(A)|= a\prod_{i=1}^{3}\sqrt{\omega_{0}^{2}+p_{i}^{2}}. \tag{3} \]
From (3) we determine the value of \(K_{\mathrm{l}}\) at which self-oscillations with frequency \(\omega_{0}\) will occur in the system. Having carried out such transformations for several values of \(\omega_{0}\), we construct the dependences \(\omega_{0}=f(K_{\mathrm{l}})\) and \(A=f(K_{\mathrm{l}})\), from which, for a known \(K_{\mathrm{l}}\), the parameters of the self-oscillations can be determined, or, for given \(\omega_{0}\) and \(A\), the necessary \(K_{\mathrm{l}}\) can be determined.
The methods and expressions presented above made it possible to investigate electromagnetic processes in relay control systems according to the circuits in Figs. 1 and 2. The results of calculations and experiments agree with an accuracy of 8–15 percent.
References
-
A. P. Zaitsev, R. K. Gachik. Optimal control system for electric braking of an electric train. Reports to the Fifth Scientific and Technical Conference on Problems of Production Automation. Vol. 1, 1967, Tomsk.
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R. K. Gachik, A. P. Zaitsev, L. M. Trakhtman, L. Yu. Veitsman et al. Method of smooth automatic rheostatic-regenerative braking of electric rolling stock. Inventor’s Certificate No. 244373, Bulletin, No. 18, 1969.
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E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. “Energiya,” 1967.