Calculation of Characteristics of an AC Electromechanical Amplifier
Abstract Generated abstract
This paper develops methods for calculating the operating characteristics of a contactless phase-sensitive AC electric machine amplifier with increased output frequency, formed by combining an inverted synchronous generator and an asynchronous frequency converter in a single magnetic core. It derives exact and simplified equivalent circuits for steady autonomous operation, gives procedures for external and control characteristics under unsaturated conditions, and then extends the calculation to saturation by accounting for the mutual influence of superposed rotating fields through magnetizing current components and Bessel-function expressions. An iterative computation method is proposed and tested on a model using a Minsk-1 digital computer, giving reported errors of about 10 percent for several characteristic points, and the approach is suggested as applicable to other combined-type brushless converters.
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CALCULATION OF CHARACTERISTICS OF AN AC ELECTRIC MACHINE AMPLIFIER
M. L. KOSTYREV, A. I. SKOROSPLESHKIN
(Presented by the seminar of the Department of Electrical Machines and General Electrical Engineering)
In recent years, new circuits have been proposed for use as power amplifiers, including a contactless “phase-sensitive” electric machine amplifier with an output at an increased unregulated frequency (EMA-PF) [1]. Such an amplifier consists of an inverted synchronous generator and an asynchronous frequency converter, rotated by a drive motor. The control signal is applied to the field winding of the synchronous generator.
For electric machine amplifiers, operation with an unsaturated magnetic circuit is characteristic. Nevertheless, it is necessary to have a method that takes saturation into account for calculating operating modes with voltage forcing at the output.
In order to obtain a compact, economical design, both cascades of the EMA-PF are combined in one magnetic core. Owing to the mutual influence of the superposed fields, it is impossible to take saturation of the steel into account by known methods.
The present article is devoted to an analysis of the operation of the EMA-PF in the established autonomous mode and to methods for calculating its characteristics.
In analyzing the operation of the machine, we proceed from the following basic assumptions:
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We consider only the fundamental harmonics of the emf and mmf, since in a properly designed machine with a symmetrical load, even when the cascades are combined, the share of higher harmonics is insignificant [2].
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We neglect the influence of eddy currents in the steel and of hysteresis on the form of the amplifier characteristics.
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We assume that the rotor of the amplifier is driven by a motor with an absolutely rigid mechanical characteristic.
The equivalent circuit of the EMA-PF is shown in Fig. 1. To increase the amplification, reduce the spread of characteristics, and improve the form of the voltage at the amplifier output, capacitors are connected.
The second cascade of the EMA-PF may be regarded as an asynchronous machine with slip
\[ S_2=\frac{f_2}{f_p}=\frac{p_2 \pm p_1}{p_1}. \tag{1} \]
\(P_1\) and \(P_2\) are the numbers of pole pairs of cascades I and II.
In the mode most favorable from the energy standpoint, the rotor and the field of cascade II rotate in the same direction (the plus sign in (1)). In this mode, energy is transmitted electrically from cascade I to cascade II.
Fig. 1. Exact equivalent circuit of an EMA–FC.
In the equivalent circuit corresponding to this operating mode, the active and inductive load resistances \(r_{\mathrm{н}}'\) and \(x_{\mathrm{н}}'\), the active and inductive leakage resistances of the output winding \(r_2'\) and \(x_2'\), and the capacitance resistance at the output \(x_c'\) are referred to the rotor winding of the second stage. Since the slip \(S_2\) remains constant, it is convenient to determine the output resistances at the output frequency and refer them to the rotor by multiplying by the reduction factor
\[ K = K_{e2}\cdot K_{i2} \tag{2} \]
The current transformation ratio of the second stage is
\[ K_{i2}=\frac{I_2}{I_2'}=\frac{m_p\cdot W_{p2}'}{m_2\cdot W_{c2}'}. \tag{3} \]
The emf transformation ratio of the second stage is
\[ K_{e2}=\frac{E_{c2}'}{E_{c2}}=\frac{W_{p2}'}{W_{c2}\cdot S_2}. \tag{4} \]
The reduction factor of the control current to the rotor for a concentric-type control winding is
\[ K_{i1}=\frac{I_y}{I_y'}\frac{0.9\,m_R\cdot W_{p1}'}{W_y\cdot K_f}. \tag{5} \]
Here \(m_p\) and \(m_2\) are the number of phases in the rotor and at the output;
\(W_{p1}'\), \(W_{p2}'\), \(W_{c2}'\) are the numbers of series turns per phase of the rotor windings of the first and second stages and of the output winding, taking the winding factors into account;
\(W_y\) is the number of series turns of the control winding;
\(K_f\) is the form factor of the control field [3].
The resistances of the magnetizing circuits of the first and second stages, \(x_{\mu1}\) and \(x_{\mu2}\), are determined from the known formulas [3] at the rotor frequency for windings \(W_{p1}\) and \(W_{p2}\). The saturation coefficients when operating on the linear portion of the no-load characteristic may be taken as
\[ K_{\mu1}\approx K_{\mu2}=1.1\div 1.3. \]
The exact equivalent circuit reduces to the simplified one in Fig. 2 if the magnetizing circuit of the second stage is brought out to the output terminals and iron losses are neglected. The parameters of the simplified circuit are
\[ \begin{aligned} x_{\mu2}' &= x_{\mu2}\cdot \sigma,\\ x_p' &= (x_{p1}+x_{p2})\cdot \sigma,\\ r_p' &= (r_{p1}+r_{p2})\cdot \sigma. \end{aligned} \tag{6} \]
Correction coefficient
\[ \sigma \simeq 1+\frac{x_{p1}+x_{p2}}{x_{\mu 2}} . \tag{7} \]
Fig. 2. Simplified equivalent circuit of the EMU-FC.
The resistances \(x_2''\), \(r_2''\), \(x_c''\), \(x_H''\), \(r_H''\) are obtained by multiplying \(x_2'\), \(r_2'\), \(x_c'\), \(x_H'\), \(r_H'\) by \(\sigma^2\).
When calculating the external characteristics for \(I_y=\mathrm{const}\), \(x_c=\mathrm{const}\), we specify the values of \(r_H\), \(x_H\) and determine the current and voltage at the output
\[ U_H=U_H''\frac{1}{\sigma\cdot K_{e2}}, \tag{8} \]
\[ I_H=I_H''\cdot K_{i2}\cdot \sigma, \]
where \(I_H''\) and \(U_H''\) are determined from the calculation of the circuit in Fig. 2.
The calculation of the control characteristic reduces to determining \(I_y\) from the circuit of Fig. 2 at \(U_H=\mathrm{const}\) and varying \(x_H\), \(r_H\).
The proposed method makes it possible to calculate the external and control characteristics of the EMU-FC with satisfactory accuracy (10–20%) for an unsaturated magnetic circuit, when the induction in any steel section does not exceed \(1.4\div -1.5\) T.
A sufficiently good approximation is also given by the following formula for calculating the external characteristic:
\[ U_H=K_U\cdot U_y\cdot \cos\theta' - X_d I_H\cdot \sin\varphi_H, \tag{9} \]
where \(\theta'\) is the angle between the vectors of the emf of the control winding and the resultant emf of the first stage at load current \(I_H\) and load power factor \(\cos\varphi_H\).
\[ \sin\theta'=\frac{K_{i1}}{K_{i2}}\cdot\frac{I_H}{I_y}\cdot \cos\varphi_H \tag{10} \]
\(K_U\) is the voltage amplification factor of the two stages at no load,
\(U_y\) is the voltage applied to the control winding.
The resistance \(x_d\) is determined from the no-load and short-circuit tests at the output, at the nominal control current.
The characteristics in the saturated mode should be calculated with allowance for the mutual influence of the combined fields. The first harmonics of the m.m.f. on the steel sections of the magnetic circuit depend only on the maxim—
…of the inductions of the combined rotating fields and do not depend on the initial angles and the mutual velocity of displacement of the fields [2].
The magnetizing current may be written as the sum of the component of the current for the air gap and for the steel sections of the magnetic circuit:
\[ I_{o1}=I_{\delta 1}+I_{\mathrm{st}1} =\frac{E_{p1}}{x_{\delta 1}}+f_1(E_{p1},E_{c2}'), \tag{11} \]
\[ I_{o2}=I_{\delta 2}+I_{\mathrm{st}2} =\frac{E_{c2}'}{x_{\delta 2}}+f_2(E_{p1},E_{c2}'), \]
\(x_{\delta 1}\) and \(x_{\delta 2}\) are equal to the resistances of the magnetizing circuits, \(x_{\mu 1}\) and \(x_{\mu 2}\), when the saturation coefficients \(K_{\mu 1}=K_{\mu 2}=1\).
The m.m.f. for the steel sections of the magnetic circuit in a machine with combined fields is determined through Bessel functions from the maximum inductions of the fields [2].
\[ \begin{aligned} I_{\mathrm{st}1} &=f_1(E_{p1},E_{c2}') =\frac{p_1}{0.9m_p\cdot W_{p1}'}\, \bigl[ 2L_{zp}\cdot \alpha \cdot I_0(\beta B_{zp2})\cdot I_1(\beta B_{zp1})\\ &\quad +2L_{zc}\cdot \alpha \cdot I_0(\beta B_{zc2})\cdot I_1(\beta B_{zc1})\\ &\quad +\frac{4}{\pi}\,L_{ap1}\cdot \alpha \cdot I_0(\beta B_{ap2})\cdot I_1(\beta B_{ap1}) +\frac{4}{\pi}\,L_{ac1}\cdot \alpha \cdot I_0(\beta B_{ac2})\cdot I_1(\beta B_{ac1}) \bigr]. \tag{12} \end{aligned} \]
\[ \begin{aligned} I_{\mathrm{st}2} &=f_2(E_{p1},E_{c2}') =\frac{p_2}{0.9m_p\cdot W_{p2}'}\, \bigl[ 2L_{zp}\cdot \alpha \cdot I_0(\beta B_{zp1})\cdot I_1(\beta B_{zp2})\\ &\quad +2L_{zc}\cdot \alpha \cdot I_0(\beta\cdot B_{zc1})\cdot I_1(\beta\cdot B_{zc2}) +\frac{4}{\pi}L_{ap2}\cdot \alpha \cdot I_0(\beta B_{ap1})I_1(\beta_{ap2})\\ &\quad +\frac{4}{\pi}L_{ac2}\cdot \alpha \cdot I_0(\beta B_{ac1})\cdot I_1(\beta_{ac2}) \bigr], \end{aligned} \]
\(I_0(\beta B)\) and \(I_1(\beta B)\) are Bessel functions of a purely imaginary argument of the first kind, of zero and first order,
\(L_{zp}, L_{zc}, L_{ap1}, L_{ap2}, L_{ac1}, L_{ac2}\) are the lengths of sections of the path of the magnetic fluxes of stages I and II in the teeth and yokes of the rotor and stator.
The amplitudes of the inductions in the teeth and yokes of the rotor and stator from the fields of stages I and II, \(B_{zp1}, B_{zp2}, B_{zc1}, B_{zc2}, B_{ap1}, B_{ap2}, B_{ac1}, B_{ac2}\), are determined through the e.m.f. \(E_{p1}, E_{c2}'\) and the dimensions of the machine,
\(\alpha\) and \(\beta\) are coefficients depending on the grade of steel and on the section of the magnetization curve [2].
The saturation coefficients of stages I and II are determined through the e.m.f.
\[ K_{\mu 1}=1+\frac{x_{\delta 1}}{E_{p1}}\cdot f_1(E_{p1},E_{c2}'), \]
\[ K_{\mu 2}=1+\frac{x_{\delta 2}}{E_{p2}}\cdot f_2(E_{p1},E_{c2}'). \tag{13} \]
It is proposed to calculate the characteristics of the machine, taking into account the mutual influence of the combined fields, by the method of iterations (successive approximations).
Having first specified the saturation coefficients, we calculate \(E_{p1}\) and \(E_{c2}'\) from the exact equivalent circuit, determine \(K_{\mu 1}\) and \(K_{\mu 2}\) by formula (13), and repeat the calculation until the required accuracy is attained.
By the proposed method, the characteristics of an EMU-PCh model were calculated on the “Minsk-1” electronic digital computer. The Bessel functions were represented
... by power series [4]. The machine time for calculating 5–6 points of the characteristic does not exceed 5 min, with an error of no more than 10 percent.
The proposed method is applicable for calculating the characteristics of asynchronous brushless converters and other machines of a combined type.
REFERENCES
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E. Mishkin, Poly-field Alternating-Current Induction Machines, United States Patent Office, Dec. 27, 1960, No. 2966623.
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V. S. Novokshchenov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk, 1960.
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M. P. Kostenko, L. M. Piotrovskii. Electrical Machines, Part II, Energiya, 1965.
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E. Gray and G. B. Mathews. Bessel Functions and Their Application to Physics and Mechanics, IL, 1949.