Phase Compounding of an AC BEMU
Abstract Generated abstract
This paper examines phase compounding circuits for a contactless electromagnetic amplifier with increased-frequency alternating-current output, treated in steady state as analogous to a nonsalient-pole synchronous generator. It argues that parallel compounding with capacitors, especially using a three-winding transformer when adjustment is needed, best satisfies amplifier requirements such as absence of self-excitation, controllability under voltage forcing, high response speed, and overload capability. A calculation procedure is developed for selecting transformer ratios and capacitance from no-load and rated-load conditions while accounting for rectifier factors, compensating-winding resistance, capacitive-current magnetizing effects, and desired time constant. Calculations and tests on a 300 W, 200 Hz model showed satisfactory agreement, improved response at smaller parameter values, external-characteristic stiffness, and gain above 1200 at a 70 ms response time.
Full Text
PHASE COMPOUNDING OF AN ALTERNATING-CURRENT BEMU
M. L. KOSTYREV, A. I. SKOROSPESHKIN
(Recommended by the seminar of the Departments of Electrical Machines and General Electrical Engineering)
Phase compounding is widely used for the static excitation of synchronous machines of various power ratings. A considerable number of works [1, 2, and others] have been devoted to the calculation of such circuits as applied to synchronous machines. The present article considers questions of selecting and calculating phase-compounding circuits with allowance for the requirements imposed on an alternating-current amplifier.
The phenomenon of armature reaction in a contactless electromagnetic amplifier with an increased-frequency output (BEMU-IF) (Fig. 1) does not differ qualitatively from the processes in a synchronous generator [3]. It can be shown that, in steady-state operation, a lossless BEMU-IF is equivalent to a synchronous nonsalient-pole generator with synchronous reactance.
Fig. 1. Circuit of a compounded BEMU-IF.
\[ x_{dэ} = \frac{x_{\mu 2}\left(x_{\mu 1}+x_R\right)} {\left(x_{\mu 1}+x_{\mu 2}+x_R\right)K_{i2}K_{e2}^{\,2}} + x_2, \tag{1} \]
where \(x_{\mu 1}, x_{\mu 2}\) are the resistances of the magnetizing circuits of stages 1 and 2 of the amplifier,
\(x_R\) is the total leakage resistance of the rotor windings,
\(x_2\) is the leakage resistance of the output winding,
\(K_{i2}, K_{e2}\) are the coefficients for referring the parameters of the output winding to the rotor.
It is known that, by means of phase compounding in an ideal synchronous generator, one can ensure complete invariance of the output voltage in the steady-state regime with respect to the load [4]. The conclusion that the invariance conditions can be satisfied is also valid for an ideal BEMU with an infinitely large gain.
The invariance conditions can be fulfilled for different ratios of the circuit parameters. When designing compounding circuits for synchronous generators, additional conditions are introduced: minimum dimensions and weight of the system, reliability of self-excitation, etc.
Special requirements are imposed on the phase-compounding circuit of a BEMU: absence of self-excitation, high speed of response, and a specified overload capability.
Selection of the Circuit
Phase-compounding circuits differ in the method of summing currents proportional to the voltage and to the load current, and in the method of control (correction). Among the large number of phase-compounding circuits, one should note parallel-compounding circuits with capacitors as reactive compounding elements, as best meeting the requirements for a BEMU. If reactors are used, their saturation is possible when the output voltage is forced, with loss of controllability. The requirement of reactor linearity leads to an increase in the dimensions and weight of the system. These considerations also apply to the circuit with a three-winding transformer, whose voltage winding is used as the compounding resistance. In the circuit with capacitors, there is no danger of loss of controllability. In addition, such a circuit promotes a faster course of transient processes. In the circuit with a three-winding transformer and capacitors, it is possible to tune and obtain the required speed of response, which is difficult in a circuit with a two-winding transformer when the capacitors are connected directly to the rectifier.
If the BEMU-PC operates in the mode of a generator with stabilized output voltage, then voltage changes caused by changes in frequency, temperature, and other factors can be eliminated by using deviation control. Since the control power of the amplifier amounts to fractions of a watt, it is not difficult to implement such a regulator.
Calculation of the Phase-Compounding Circuit
The compounding circuit must provide the nominal voltage in the no-load mode and at nominal load. If the invariance conditions are satisfied at these limiting points, then, owing to the linearity of the characteristics, they will also be satisfied at intermediate load values. From the calculation of the amplifier the following are known: the nominal phase voltage \(U\), the load current \(I\) and \(\cos\varphi\) in the nominal mode, the excitation current in the no-load mode \(I_{f0}\) and at nominal load \(I_{fн}\), and also the resistance of the compensation winding at the operating temperature \(r_k\). The calculated value \(r_k\) must take into account the resistance of the valves and the protective resistance connected in parallel with the winding. The calculated value of the excitation current \(I'_{fн}\), taking into account losses in the compounding circuit and in the compensation winding, can be determined from the formula:
\[ I'_{fн}=I_{fн}\left(1+\frac{I_{fн}^{2} r_k}{\eta \cdot m \cdot U \cdot I}\right), \tag{2} \]
where \(m\) is the number of phases at the amplifier output,
\(\eta\) is the transformer efficiency.
It is required to determine the transformation ratios
\[ K_{\mathrm{n}}=\frac{W_1}{W_2}, \qquad K_{\mathrm{t}}=\frac{W_2}{W_3} \]
and the capacitance \(C\).
Since the system is symmetrical, for a symmetrical load it is possible to use a single-phase equivalent circuit (Fig. 2). All parameters of the circuit are referred to winding \(W_1\).
Fig. 2. Equivalent circuit of the compounding circuit.
According to the equivalent circuit, the current in the secondary winding of the transformer is
\[ i_2=\frac{U+j I x_c}{R\left(1-\frac{x_c}{x_{\mu t}}\right)-i x_c}. \tag{3} \]
The plus sign before \(I\) is taken with allowance for the fact that, for correct geometrical addition of the components of the compounding currents, windings \(W_1\) and \(W_3\) are connected in opposition. In analyzing the circuit we assume that
\[ \frac{x_c}{x_{\mu t}}\ll 1. \]
The required current of winding \(W_k\) in the no-load mode can be provided at different values of \(x_c\) and \(K_{\mathrm{n}}\). To evaluate the variants it is convenient to use the parameter proposed by V. M. Yusim [1],
\[ a=\frac{R}{\sqrt{x_c^2+R^2}}. \tag{4} \]
This parameter varies from 0 to 1.
The referred resistance is
\[ R=r_k\frac{k_{\mathrm{n}}^2}{\beta_u\cdot \beta_i}, \tag{5} \]
where \(\beta_i,\ \beta_u\) are the transfer coefficients with respect to current and voltage of the rectifier,
\[ \beta_u=\frac{U_{\mathrm{cp}}}{U_{\mathrm{eff}}}, \qquad \beta_i=\frac{I_{\mathrm{eff}}}{I_{\mathrm{cp}}}. \]
In synchronous generators, the parameter \(a\) is chosen from the conditions of the minimum rated power of the excitation-system elements, which gives a substantial saving in the design of high-power machines. These conditions correspond to the optimal value \(a=0.6\text{–}0.8\).
In a BEMU–FC it is important to ensure high speed of response with maximum circuit simplicity. The m.m.f. of the control winding is small; therefore it can be made with a small time constant. Then the speed of response of the amplifier will be determined primarily by the time constant of the compensating winding. The inclusion of a reactance on the alternating-current side leads to its reduction without reducing the gain. The equivalent time constant of the compensating winding [1] is
\[ T_{k\mathrm{e}} \simeq \frac{L_k}{R_k+x_c/K_{\mathrm{n}}^2} = T_k \frac{a}{a+\sqrt{1-a^2\beta_u\beta_i}}, \tag{6} \]
where \(T_k\) is the intrinsic time constant of the compensating winding with the rotor locked.
It follows from expression (6) that, in order to reduce \(T_{k\mathrm{e}}\), it is necessary to reduce \(a\), choosing \(a=0.2\) or less. But at small \(a\), especially in low-power machines, the current through the capacitors becomes considerable, and its magnetizing effect must be taken into account. Then, from equations (3), (4), the transformation ratio is
\[ K_{\mathrm{n}}= \frac{ aU\beta_u+\sqrt{(aU\beta_u)^2+4I'_{f0}\cdot\gamma\cdot a\cdot U\beta_u r_k} }{ 2I'_{f0}r_k }. \tag{7} \]
With the aid of the coefficient \(\gamma\), the capacitive current is referred to the compensating winding.
\[ \gamma=\frac{0.m\, W'_{c2}}{W'_k}\,\beta_i\sqrt{1-a^2}, \tag{8} \]
\(W'_{c2}, W'_k\) are the number of turns in the phase of the output winding and the number of turns of the compensating winding, taking into account their winding factors.
The calculated excitation current in the no-load mode is
\[ I'_{f0}=I_{f0}-I'_y, \]
\[ I'_y=(0.1\div0.3)I_{f0} \]
is the magnitude of the control current referred to the compensating winding.
The value \(I'_y\) is determined by the manufacturing tolerances of the amplifier, tolerances on capacitors and transformers, permissible changes in rotor speed, etc. In the case of a three-winding transformer, a smaller value of \(I'_y\) may be chosen, since adjustment of the amplifier is possible.
Having determined the coefficient \(K_{\mathrm{n}}\), we determine the capacitive reactance
\[ x_c=\frac{U\sqrt{1-a^2}\,K_{\mathrm{n}}}{I_{k0}\beta_i}, \tag{9} \]
where the compensating-winding current in the no-load mode is
\[ I_{k0}=\frac{I'_{f0}\cdot K_{\mathrm{n}}}{K_{\mathrm{n}}+\gamma}. \tag{10} \]
The capacitance of the selected capacitor must not exceed the value corresponding to the calculated reactance \(x_c\).
The transformation ratio \(K_t\) is determined from the rated condition:
\[ K_t= \frac{ I\sqrt{1-a^2} }{ I_{k\mathrm{n}}\beta_i\left(-K\sin\varphi+\sqrt{1-K^2\cos^2\varphi}\right) }. \tag{11} \]
Current of the compensating winding at nominal load
\[ I_{\mathrm{kn}}=I'_{\mathrm{fn}}-I'_{\mathrm{y}}-\gamma\sqrt{1-\alpha^{2}}\cdot \frac{I_{\mathrm{ko}}}{K_{\mathrm{n}}}, \tag{12} \]
\[ K=\frac{I_{\mathrm{ko}}}{I_{\mathrm{kn}}}. \tag{13} \]
When using expression (11), \(\sin\varphi>0\) must be assumed in the case of an active-inductive load.
From the maximum voltage, the amplifier overload current, and the transformation ratios determined from (6), (11), the transformer is calculated.
In the case where \(K_{\mathrm{n}}\) is close to or less than unity, it is advisable to switch to a two-winding transformer. Taking \(K_{\mathrm{n}}=1\), we determine
\[ \alpha=\frac{I_{\mathrm{ko}}\cdot r_{\mathrm{k}}}{U\cdot \beta_{\mathrm{i}}}. \tag{14} \]
Under the condition \(\alpha^{2}\ll 1\), it may be assumed that the current \(I_{\mathrm{ko}}\) does not depend on \(\alpha\) and is equal to
\[ I_{\mathrm{ko}}=\frac{I'_{\mathrm{fo}}}{1+\gamma}, \tag{15} \]
where
\[ \gamma \simeq \frac{0.9m\,W'_{\mathrm{c2}}\beta_{\mathrm{i}}}{W'_{\mathrm{k}}}. \tag{16} \]
Using the proposed method, a number of variants of the compounding circuit were calculated for a model BEMU-PC with a power of 300 W and a frequency of 200 Hz at various \(\alpha\). Experimental studies showed good agreement with the calculation and confirmed that, with decreasing \(\alpha\) and an unchanged gain, the equivalent time constant of the rise of the amplifier voltage when the control current is switched on decreases (Table 1).
Table 1
| \(\alpha\) | 0.4 | 0.11 | 0.023 | Note |
|---|---|---|---|---|
| \(T\), ms | 80 | 70 | 45 | \(T_{\mathrm{y}}=25\) ms |
| 30 | 20 | 15 | \(T_{\mathrm{y}}<3\) ms |
The external characteristics of the model (Fig. 3) at \(\alpha=0.11\) and \(K_{\mathrm{n}}=1\) (two-winding transformer) have satisfactory stiffness and linearity. The gain at a response speed of 70 ms was more than 1200.
Thus, the proposed method makes it possible to calculate the amplifier compounding circuit so as to obtain the required response speed at high gain and satisfactory external characteristics.
Fig. 3. External characteristics of a compounded BEMU-FC.
— — — \(\cos\varphi = 1\),
—— \(\cos\varphi = 0.8\) (ind.).
Visible graph labels: ordinate \(U/U_n\); abscissa \(I/I_n\); \(\rho_y = 0.45\ \mathrm{W}\), \(\rho_y = 0.17\ \mathrm{W}\), \(\rho_y = 0.07\ \mathrm{W}\).
REFERENCES
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V. M. Yusin. Selection of optimal parameters of static excitation systems. Transactions of VEI, issue 73, 1966.
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D. V. Vilesov, I. A. Rabinovich. Selection of the main parameters of self-excited synchronous-generator excitation circuits. Elektrichestvo, 1960, No. 3.
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M. L. Kostyrev, A. I. Skorospeshkin. Calculation of the characteristics of an alternating-current EMU. Proceedings of the Tomsk Polytechnic Institute, vol. 160, 1966.
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D. V. Vilesov. Automation of Electric-Power Systems, Part 1, VMOLA, Leningrad, 1961.