Voltage Variation of a Contactless AC Electric Machine Amplifier When an Inductive Load Is Switched On
Abstract Generated abstract
This paper analyzes the initial voltage dip in a contactless AC electromechanical amplifier with phase compounding when an inductive load, representing the starting condition of an asynchronous motor, is switched on. The authors derive transient equations for the amplifier without feedback at the first switching instant, obtain an approximate expression for the output voltage envelope, and simplify it in terms of stator and rotor leakage reactances and load reactance. Experimental comparison with a 300 VA, 200 Hz model gives the same order of voltage drop as the calculated value. The study concludes that the dip is governed mainly by leakage reactances of the rotor and output windings, and proposes a design condition limiting their combined per unit value for acceptable dynamic overload performance.
Full Text
CHANGE IN THE VOLTAGE OF A CONTACTLESS AC ELECTROMECHANICAL AMPLIFIER WHEN AN INDUCTIVE LOAD IS SWITCHED ON
M. L. KOSTYREV, A. I. SKOROSPELUSHKIN
(Recommended by the seminar of the Departments of Electrical Machines and General Electrical Engineering)
A contactless electromechanical alternating-current amplifier (BEMU) [1, 2] can operate in the mode of an amplifier-converter in a high-speed electric drive or in the mode of an autonomous generator. When a supplied asynchronous motor is started, the voltage of the BEMU, like that of any alternating-current generator, may change abruptly, which has a substantial effect both on the starting process and on the operation of other consumers supplied by the BEMU.
The power factor of asynchronous motors during starting is small ($\cos \varphi_k = 0.2—0.35$); therefore, during starting they may be regarded as a purely inductive load.
The voltage at the output of the BEMU is restored by means of a fast-acting phase compounding circuit. Such a circuit gives the maximum voltage dip at the first instant of starting, when the feedback has not yet had time to operate, after which the voltage is rapidly restored to the nominal value (Fig. 1).
In connection with this, the maximum voltage dip at the output of the BEMU when the supplied asynchronous motor is started can be estimated,
Fig. 1. Output voltage and current of the compensating winding of the BEMU when the rated inductive load is switched on.
having considered the transient process of switching on the inductive load of the amplifier without feedback and with constant excitation voltage. The differential equations of the contactless electromechanical amplifier, taking into account the initial conditions, in operator form for complex relative quantities [2], are written as follows:
\[ \left| \begin{array}{ccc} \dot U_{f(p)}+p x_f \dot X_{f0}+p x_{m1}\dot I_{R0} & r_f+x_f(p+j) \\[4pt] p x_{m1}\dot I_{f0}+p l_r \dot I_{R0} & p x_{m1} \\[4pt] p x_{m2}\dot I_{R0} & 0 \end{array} \right| = \left| \begin{array}{cc} r_f+x_f(p+j)\\[4pt] p x_{m1}\\[4pt] 0 \end{array} \right| \to \]
\[ \to \left| \begin{array}{cc} x_{m1}(p+j) & 0\\[4pt] r_r+p x_r & p x_{m2}\\[4pt] x_{m2}[p+j(1-S)] & r_s+x[p+j(1-s)] \end{array} \right| \cdot \left| \begin{array}{c} \dot I_{f(p)}\\[4pt] \dot I_{R(p)}\\[4pt] \dot I^{*}_{s(p)} \end{array} \right|. \]
The inductive resistance of the stator circuit is
\[ x=x_L+x_s, \]
where
\[ x_L \text{ is the load resistance,} \]
\[ I_{f0}\ \text{and}\ I_{R0} \text{ are the values of the excitation and rotor currents at the instant of switching on.} \]
Neglecting the active resistance of the rotor, from the no-load mode of the contactless electromechanical amplifier we obtain:
\[ \dot I_{f0}=\frac{\dot U_f}{r_f}, \qquad \dot I_{R0}=-\frac{\dot U_f}{r_f}\cdot\frac{x_{m1}}{x_r}. \]
Having determined the approximate value of the roots of the corresponding characteristic equation, we write the determinant of the system of equations (1) in the form
\[ \Delta=x_f x_r x T'_r\cdot T'_f\cdot a p(p+b+j)[p+j(1-s)], \]
where
\[ a=\frac{1}{T'_f T_r}+\frac{1}{T_f T'_r}-\frac{1}{T'_f T'_r}, \]
\[ b=\frac{1}{a}\left( \frac{1}{T'_f T_f T_r} +\frac{1}{T'_f T_r T'_r} +\frac{1}{T_f T'_r T_s} \right) = \frac{1}{T}. \]
\[ T'_f=\left(x_f-\frac{x_{m1}^{2}}{x_r}\right)\frac{1}{r_f}, \qquad T'_r=\left(x_r-\frac{x_{m2}^{2}}{x}\right)\frac{1}{r_r}, \qquad T_s=\frac{x}{r_s}. \]
The Carson transform of the excitation voltage is
\[ U_f(t)=U_f\cdot e^{-j(t+\delta_1)} \div \dot U_f\,\frac{p}{p+j}. \]
The voltage at the output of the contactless electromechanical amplifier from equations (1) is
\[ \dot U^{*}_{(p)} = -x_L[p+j(1-s)]\dot I^{*}_{s(p)} = \]
\[ = \frac{\dot U_f\cdot x_{m1}x_{m2}x_L}{x_f x x_r T'_r T'_f a} \cdot \frac{[p+j(1-s)]}{(p+b+j)} \left( \frac{p}{p+j}+pT'_f \right). \tag{2} \]
Passing from the transform to the original and neglecting quantities of the second order of smallness, we obtain
\[ \dot U^{*}(t)=sU_s\cdot A[\sin(t+\delta_1)+j\cos(t+\delta_1)], \]
where
\[ U_s=\frac{U_f x_{m1}x_{m2}x_L\cdot T}{x_f\cdot x_r T'_r T'_f a}, \]
\[ A=\left[1-\left(1-\frac{T'_f}{T}\right)e^{-\frac{t}{T}}\right]-\frac{1}{sT}e^{-\frac{t}{T}}. \]
After the inverse transformations, the phase voltage of the actual machine is
\[ U_a(t)=sU_sA\sin(st+\delta_1+\delta_2). \tag{3} \]
If the influence of the active resistances \(r_s\) and \(r_r\) on the voltage amplitude is neglected, then
\[ U_a(t)\simeq sU_s\left[1-\left(1-\frac{T'_f}{T}\right)e^{-\frac{t}{T}}\right]\sin(st+\delta_1+\delta_2). \]
Let us determine the change in the envelope of the voltage at the output of the CEMA at the instant when an unloaded CEMA is switched onto an inductive load. From the no-load condition [2]
\[ U_0=\frac{sU_f x_{m1}x_{m2}}{r_f\cdot x_r}. \]
When switching at the instant \(t=0\), the voltage envelope is
\[ U_{(t=0)}=\frac{sU_f x_{m1}x_{m2}x_L}{x_f x_r T'_r a}. \]
The change in the voltage envelope is
\[ \Delta U_s=\frac{U_0-U_{(t=0)}}{U_0} =1-\frac{x_Lx_f}{x_f xT'_r a}= \]
\[ =\frac{x\cdot x_r x_s-x_{m2}^{2}x_f-x_{m1}^{2}x_s} {x_f x_r(x_s+x_L)-x_{m2}^{2}x_f-x_{m1}^{2}(x_s+x_L)}. \]
The total impedances of the rotor and stator are
\[ x_r=x_{m1}+x_{m2}+x_{\sigma r}, \]
\[ x_{\sigma r}=x_{\sigma r1}+x_{\sigma r2}, \]
\[ x=x_s+x_L=x_{m2}+x_{\sigma s}+x_L. \]
\[ x_f\simeq x_{m1}. \]
Taking these expressions into account, we obtain
\[ \Delta U_s\simeq \frac{x_{\sigma s}+x_{\sigma r}}{x_{\sigma s}+x_{\sigma r}+x_L}. \tag{4} \]
Formula (4), for calculating the initial voltage dip, was checked experimentally. For the CEMA model (300 VA, 200 Hz)
\[ x_{\sigma r}=0.3\ \text{p.u.}\qquad x_{\sigma s}=0.14\ \text{p.u.} \]
\[ \Delta U=\frac{0.3+0.14}{0.3+0.14+1}=0.305. \]
From the oscillogram in Fig. 1, \(\Delta U\simeq 0.4\).
It is known [3] that, when the rated inductive load of an autonomous generator is switched on, the permissible value is
\[ \Delta U\leq 0.25. \]
Therefore, from the conditions of dynamic overload, the permissible value of the BEAMU leakage reactances in relative units is
\[ (x_{\sigma s} + x_{\sigma r}) \ll 0.3. \tag{5} \]
Conclusions
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The voltage dip at the output of a BEAMU with phase compounding when an inductive load is switched on is determined mainly by the leakage inductive reactance of the rotor windings and of the output winding.
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The condition for dynamic overload of a BEAMU (5) has been obtained, and should be used in design.
References
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E. Mishkin. Poly-field Alternating-Current Induction Machines. U.S. Patent No. 2966623, 27.12.1960.
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M. L. Kostyrev, A. I. Skorospeshkin. Questions in the theory of a contactless AC EMU. In the present collection.
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G. A. Meleshkin. Shipboard Synchronous Generators with Automatic Voltage Regulation. Sudpromgiz, 1962.