On the Error of a Discrete Voltage Control System for a Synchronous Generator with Implicitly Defined PWM
Abstract Generated abstract
The paper examines the steady-state error of a discrete voltage regulation system for a low-power synchronous generator using a relay amplifier and an implicitly defined pulse-width modulator. Instead of replacing the relay characteristic by a vibrationally linearized model, it derives switching conditions from the actual relay thresholds, sawtooth signal, excitation winding dynamics, and first-order elements of the continuous part of the system. The resulting relations determine pulse duration, mean generator voltage, voltage ripple, and regulation error as functions of load current and system time constants. Comparison with experimental data indicates that the exact calculation better reflects observed behavior, and that variation of the excitation circuit time constant with current can reduce the static regulation error.
Full Text
ON THE ERROR OF A DISCRETE VOLTAGE CONTROL SYSTEM FOR A SYNCHRONOUS GENERATOR WITH IMPLICITLY EXPRESSED PWMs
E. G. Orlov, N. V. Borodovitsyn, B. V. Uskov
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
For regulating the voltage of low-power synchronous generators (SG), used as power sources, transistor regulators operating in a switching mode are widely employed [1].
The switching mode of the transistors is ensured by applying to the input of an amplifier with a relay or near-relay characteristic a periodic sawtooth signal contained in the control action or supplied separately. The indicated regulators belong to the class of discrete regulators with an “implicitly expressed pulse-width modulator” (PWM). In the literature, when such systems are studied, the relay characteristic of the amplifier is replaced by a linearized one in accordance with the theory of vibrational linearization, and the entire system is considered as linear [1], [2], which often leads to substantial inaccuracies in determining the regulation error.
In this work, an analysis is given of the error in the voltage control system of an SG with an excitation winding (EW) and with an implicitly expressed PWM, by an exact method taking into account the real characteristics of the elements.
The control system under consideration, for small deviations from the established operating mode, can be described by the following equations:
\[ U_{\mathrm{g}} = E - k_1 i_{\mathrm{n}} - k_2 i_{\mathrm{H}} \]
\[ i_{\mathrm{p}} = \frac{U_{\mathrm{p}}}{R_{\mathrm{op}}} \cdot \frac{1}{T_1 p + 1} \tag{1} \]
\[ U_1 = \frac{k_3}{T_2 p + 1} U_{\mathrm{g}} \]
\[ U_{\mathrm{ch}} = k_4 (U_1 - U_0). \]
Here the notation is as follows:
\(U_{\mathrm{g}}\) — SG voltage;
\(E\) — generator emf;
\(i_{\mathrm{H}}\) — load current;
\(i_{\mathrm{p}}\) — EW current;
\(U_{\mathrm{p}}\) — voltage at the amplifier output;
\(R_{\mathrm{op}}\) — EW resistance;
\(U_{\mathrm{ch}}\) — voltage at the output of the sensing element (SE);
\(U_o\) — reference voltage;
\(K_1, K_2, K_3, K_4\) — constant coefficients;
\(T_1, T_2\) — time constants of the OP and the ChE filter, respectively.
The value of the OP time constant \(T_1\) depends on the magnitude of the current \(i_{\mathrm{a}}\) according to a hyperbolic law because of saturation of the armature back (Fig. 2).
Fig. 1. Block diagrams of the SG voltage-control system
Fig. 2. Graphs of the dependences \(T_1=\psi(\gamma)\) and \(\beta_1=\psi(\gamma)\)
The amplifier has a relay characteristic without delay, with pickup voltage \(\sigma_1\), release voltage \(\sigma_2\), and output voltage \(U_p\). As the external periodic signal we shall consider a sawtooth signal \(f(t)\), supplied from a special shaper (1), with amplitude \(A\), trailing-edge duration \(\mu_2\), leading-edge duration \(\mu_1\), and constant component \(\frac{1}{2}A\).
The structural diagram of the control system is shown in Fig. 1, a. To simplify the presentation, it is expedient to transform the structural diagram to the form of Fig. 1, b, where the disturbing and reference actions are applied at the relay input.
For the steady-state operating mode and \(E=\mathrm{const}\),
\[ q=\kappa_3\kappa_4\left(E-\kappa_2 i_{\mathrm{n}}\right)-\kappa_4 U_0 . \tag{2} \]
In accordance with Fig. 1, b, the input quantity of the relay will be
\[ q+f(t)-x(t). \tag{3} \]
The transfer function of the continuous part of the system can be decomposed in a known manner into elementary first-order transfer functions.
For the \(i\)-th component in the steady-state operating mode, the change in the output coordinate will be determined by the following relations:
For a voltage pulse at the amplifier output,
\[ X_{2i}=C_i k_p \left[ 1+ \frac{e^{-\frac{T_{\mathrm{p}}}{T_i}}-1} {1-e^{-\frac{T}{T_i}}} e^{-\frac{t}{T_i}} \right], \tag{4} \]
during the pause,
\[ X_{1i}=C_i k_p \frac{1-e^{-\frac{T_{\mathrm{i}}}{T_i}}} {1-e^{-\frac{T}{T_i}}} e^{-\frac{t}{T_i}}, \tag{5} \]
where \(i=1,2\);
\(C_i\) is the expansion coefficient;
\(T_{\mathrm{p}}, T_{\mathrm{i}}\) are the durations of the pause and the pulse, respectively;
\[ k_p=\frac{U_p}{R_{\mathrm{op}}}. \]
The origin of the time count in (4), (5) coincides with the relay switching instants,
\[ x(t)=\sum_{i=1}^{2} x_i(t). \tag{6} \]
The relay switching instants will be determined by the conditions:
on operation,
\[ \sigma=\sigma_1,\quad \dot{\sigma}>0, \tag{7} \]
on release,
\[ \sigma=\sigma_2,\quad \dot{\sigma}<0. \tag{8} \]
Denoting by \(\varphi\) the shift of the relay operation instant relative to the instant corresponding to the maximum value of the signal \(f(t)\), we write the equations for the rising and falling edges of the signal \(f(t)\), respectively:
\[ f_1(t)=\frac{A}{\mu_1}\left(T_{\mathrm{i}}+t-\varphi-\mu_2\right), \tag{9} \]
\[ f_2(t)=\frac{A}{\mu_2}\left(\mu_2+\varphi-t\right). \tag{10} \]
Introduce the relative values of the parameters
\[ \frac{T_{\mathrm{i}}}{T}=\gamma T;\qquad \frac{T}{T_i}=\beta_i, \]
where \(T\) is the pulse repetition period.
From (7), (8), taking into account (4)–(6), (9), (10), we obtain the conditions for the relay switching instants:
\[ q+\frac{A}{\mu_2}\left(\mu_2+\varphi-T_{\mathrm{i}}\right)-\sum_{i=1}^{2} c_i k_p \lambda_{2i}=\sigma_2, \]
\[ q+\frac{A}{\mu_1}\left(T-\mu_2-\varphi\right)-\sum_{i=1}^{2} c_i k_p \lambda_{3i}=\sigma_1, \tag{11} \]
where
\[ \lambda_{2i}=\frac{1-e^{-\gamma\beta_i}}{1-e^{-\beta_i}};\qquad \lambda_{3i}=\frac{e^{-(1-\gamma)\beta_i}-e^{-\beta_i}}{1-e^{-\beta_i}}. \]
Eliminating the quantity \(\varphi\) from these relations, we obtain:
\[ q=\frac{1}{T}\left[\mu_2\sigma_2+\mu_1\sigma_1+\mu_2\sum_{i=1}^{2} c_i k_p \lambda_{2i}+\mu_1\sum_{i=1}^{2} c_i k_p \lambda_{3i}-AT(1-\gamma)\right], \tag{12} \]
This relation determines the dependence of the relative duration \(\gamma\) on the input \(q\) when a forced-oscillation mode with period \(T\) exists in the system.
Usually, in operation of the systems under consideration, one is interested in the mean value of the controlled quantity, which, in accordance with Fig. 1, б, is determined for the steady-state mode as
\[ U_{\mathrm{gcp}}=E-\kappa_2 i_{\mathrm{n}}-\kappa_1 k_p \gamma . \]
In this case the swing of the voltage pulsations \(U_g\), in accordance with (4), (5), will be
\[ \delta U_g=\kappa_1 k_p\left(\lambda_{21}-\lambda_{31}\right). \]
Figure 3 gives the dependences, calculated from relations (2), (12), (13), of the voltage regulation error
\[ \Delta U_{\mathrm{gcp}}=U_{\mathrm{gcp}}-\frac{U_0}{k_4} \]
on the load current for \(\mu=0\); \(A=1.5\ \mathrm{V}\); \(\sigma=0.1\ \mathrm{V}\); \(\kappa_1=5\ \mathrm{V/A}\); \(\kappa_2=0.4\ \mathrm{V/A}\); \(k_3=0.6\); \(k_p=1.5\ \mathrm{A}\); \(k_4=10\); \(\beta_1=\mathrm{const}\); \(\beta_2=0.5\). \(U\beta_1=\mathrm{var}\)—variation of the time constant of the OP as a function of the value \(i_{\mathrm{n}}\) (the duty ratio \(\gamma\)). The experimental results are also given there.
Analysis of relations (2), (12), (13) and Fig. 3 shows that the magnitude of the regulation error in the mean value depends substantially on the values of the time constants of the system elements and differs from the value calculated by the method of vibrational linearization. Variation of the time constant of the OP as a function of the current \(i_{\mathrm{n}}\) has a favorable effect on the magnitude of the static error, reducing its value.
The method presented for calculating the error of the voltage regulation system of an SG makes it possible to calculate exact system characteristics, which agree well with the experimental results.
Fig. 3. Plots of the dependences
\[
\Delta U_{\mathrm{g.avg}}=\psi(I_{\mathrm{n}})
\]
for \(\beta_2=\mathrm{const}\) and \(\mu=0\).
Legend in the figure: solid line—at \(\beta_1=\mathrm{const}\); dashed line—taking into account the dependence \(\beta_1=\psi(\alpha)\); circles—experiment.
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V. P. Bunakov, R. G. Gasparov. Semiconductor Voltage and Frequency Regulators for Electric Machines. “Energiya,” 1966.
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A. A. Krasovskii and G. S. Pospelov. Fundamentals of Automation and Engineering Cybernetics. Gosenergoizdat, 1962.