On the Design of a Contactless AC Electric Machine Amplifier
Abstract Generated abstract
This paper examines the design of a contactless alternating-current electric-machine amplifier in which generator stages are combined in a single magnetic core to reduce size and weight. It derives relationships for distributing electromagnetic loadings between cascades, including constraints on magnetic induction and linear loading, and gives formulas for output power, amplification factor, equivalent time constant, quality factor, and leakage reactance. Numerical calculations for a machine based on a standard motor core show that the loading distribution coefficient strongly affects efficiency, output power, response speed, and dynamic overload capacity. The study concludes that design choices should balance amplification, speed, and overload requirements, and that a somewhat enlarged air gap can improve quality factor at maximum power while reducing noise.
Full Text
ON THE DESIGN OF A CONTACTLESS AC EMA
M. L. KOSTYREV, A. I. SKOROSPEШKIN
(Recommended by the scientific seminar of the Departments of Electrical Machines
and General Electrical Engineering)
A contactless electric-machine amplifier (EMA) of alternating current consists of a synchronous generator and an asynchronous frequency converter, rotated by a driving motor [1, 2]. The EMA provides an increased frequency at the output, has good regulating and dynamic properties, and can be used as an autonomous generator or as a frequency-converter amplifier.
To reduce the dimensions and weight of the amplifier, its stages are combined in one magnetic core. The design of single-machine converters, in which the generator and the driving motor are combined, is the subject of works by V. S. Novokshenov, V. M. Pavlinin, V. S. Nersesyan, and a number of others [3, 4, 5]. In the EMA, the stages of the generator part are combined. Such a combination, as well as allowance for the specific requirements imposed on the amplifier, introduces a number of special features into its design.
In the present article, the main attention is devoted to the selection of the electromagnetic loadings of the EMA.
The distribution of electromagnetic loadings between the combined stages is characterized by the coefficients
\[ C_B = \frac{B_{\delta_2}}{B_\delta}, \qquad C_A = \frac{A_2}{A}. \tag{1} \]
Note. 1. All formulas are given in the SI system of units. The conventional notation is given at the end of the article.
2. Here and below, subscript 1 refers to stage 1, and subscript 2 to stage 2.
In its construction the EMA is analogous to standard asynchronous motors. However, unlike the latter, from the thermal standpoint the rotor windings of the EMA are the most heavily loaded. Therefore, by the total linear loading of the EMA one should understand the linear loading of the rotor windings. It is chosen within the limits corresponding to asynchronous motors of the same power and design.
The magnetic induction in the air gap of the EMA may be selected lower, with allowance for the required voltage forcing. For example, with 1.5-fold forcing, the induction is selected at about \(2/3\) of the induction of the corresponding asynchronous motor.
The coefficient \(C_B\) may be varied in the design. We shall show that in this case the coefficient \(C_A\) cannot be chosen arbitrarily. Indeed, from the condition of equality of the number of phases and currents of the rotor windings, the following relation follows:
\[ \frac{A_1}{A_2}=\frac{W_{R_1}}{W_{R_2}} . \tag{2} \]
On the other hand, the ratio of the emf’s of the rotor windings is
\[ \frac{E_{R_1}}{E_{R_2}}=K_E=\frac{W_{R_1}K_{\omega_1}P_2B_{\delta_2}}{W_{R_2}K_{\omega_2}P_1B_{\delta_2}} . \tag{3} \]
From expressions (1, 2, 3) we obtain
\[ C_A=\frac{1-C_B}{1-C_B(1-K_1)} , \tag{4} \]
where
\[ K_1=\frac{K_EK_{\omega_2}P_1}{K_{\omega_1}P_2}. \]
The last expression (4) coincides with that obtained by V. S. Novokshchenov [3], but the coefficient \(K_1\) has a different content.
Thus, having specified the total linear load of the rotor, the induction in the gap, and the coefficient \(C_B\), we determine from formulas (1, 4) the electromagnetic loads of the cascades of the BEMA.
Let us consider the influence of the coefficient \(C_B\), characteristic of combined machines, on the principal parameters of the BEMA.
Let us determine the amplifier parameters for specified dimensions.
The total power of the BEMA is
\[ P_{\mathrm{out}}^{1}=\frac{\pi^2K_{\omega s}}{\sqrt{2}K_{Es}}D^2lA_sB_{\delta_2}n_s . \tag{5} \]
From the vector diagram (Fig. 1), neglecting the angle \(\gamma\), we determine
\[ F_s=\sqrt{F_{R_2}^{\,2}-(F_2\cos\varphi)^2}-F_2\sin\varphi . \]
Hence the linear load of the output winding is
\[ A_s=A_2\frac{K_{\omega_2}F_s}{K_{\omega s}F_{R_2}} = \frac{K_{\omega_2}}{K_{\omega s}} \left[ \sqrt{ A_2^2-\left(\frac{K_2B_{\delta_2}\cos\varphi}{D}\right)^2 } -\frac{K_2B_{\delta_2}\sin\varphi}{D} \right], \]
where
\[ K_2=\frac{\sqrt{2}\,\delta'K_{\mu_2}P_2}{\mu_0K_{\omega_2}} . \]
The amplification factor of the BEMA without feedback is
\[ K_y=\frac{P_{\mathrm{out}}}{i_f^{\,2}R_f}. \]
Expressing the current and resistance of the excitation winding in terms of the specific loads and the dimensions of the machine, we obtain
\[ K_y=\frac{P_{\mathrm{out}}}{\pi(e+\tau_1)D j A_f\rho}. \tag{6} \]
From the vector diagram of the amplifier,
\[ A_f=\frac{2\sqrt{2}K_{\omega_1}}{\pi K_f}A_1 \sqrt{ 1+\left(\frac{K_3B_{\delta_1}}{A_1D}\right)^2 +\frac{2K_3B_{\delta_1}\cos\varphi_R}{A_1D} }, \]
where
Fig. 1. Vector diagram of the CEMA
\[ \cos \varphi_R = \sqrt{1-\left(\frac{K_{fs} A_s}{K_{\omega_2} A_2}\right)^2}, \]
\[ K_3=\frac{\sqrt{2}\,\delta' K_{\mu_1} P_1}{\mu_0 K_{\omega_1}} . \]
The response speed of the amplifier is characterized by the equivalent time constant [6]
\[ T= \frac{\pi^2 \mu_0 K_f^2 A_f D^2 e} {32\delta' K_{\mu_1} p_1^2 \left[1+\left(\frac{C_B K_E}{1-C_B}\right)^2\right](e+\tau_1) j \rho}. \tag{7} \]
The quality factor of the amplifier without feedback is
\[ D=\frac{K_y}{T}. \tag{8} \]
The losses and efficiency of the CEMA are calculated by known methods [3].
The dynamic overload capacity of the amplifier is determined by the leakage reactance of its windings [7]. In relative units,
\[ x_\sigma=x_{\sigma s}+x_{\sigma R_1}+x_{\sigma R_2}= \]
\[ = \frac{4\pi\mu_0 p_2 K_{Es} A_s}{V^2 K_{\omega s} B_{\delta 2}} \left[ \frac{\sum \lambda_s}{z_s} + \frac{\sum \lambda_1}{z_R} \left(\frac{A_1 K_{\omega s}}{A_2 K_{\omega 2}}\right)^2 + \frac{\sum \lambda_2}{z_R} \left(\frac{K_{\omega s}}{K_{\omega 2}}\right)^2 \right]. \tag{9} \]
From expressions (5–9) it is evident that the principal parameters of a contactless electric-machine amplifier substantially depend on the distribution of electromagnetic loadings between the stages.
On the “Promin” digital computer, the parameters of a contactless electric-machine amplifier were calculated in the iron of a series motor A02-42 at \(n=3000\) rpm, \(\cos\varphi=1\), \(A=210\) A/cm, \(B_\delta=0.5\) T, \(2p_1=2\), \(2p_2=6\), \(f_{\text{out}}=200\) Hz. As can be seen from Fig. 2, the maximum efficiency coincides with the maximum output power, which corresponds to the known proposition concerning the coincidence of the optimum in volume and losses for combined machines [3]. With an increase in \(C_B\), the amplification factor decreases and the speed of response improves. The dynamic overload capacity also increases, since the relative leakage reactance decreases.
Fig. 2. Influence of \(C_B\) on the parameters of a contactless electric-machine amplifier
When the motor-generator part is combined, the value of \(C_B\) that is optimal from the point of view of volume and losses varies within narrow limits, \(C_{B\text{opt}}=0.33\text{–}0.5\) [3]. As can be seen from Fig. 2, for a contactless electric-machine amplifier \(C_{B\text{opt}}=0.4\text{–}0.6\), and in order to increase the quality factor and overload capacity, an even larger value of \(C_B\) must be chosen.
The influence of the air gap on the power and quality factor of a contactless electric-machine amplifier is shown in Fig. 3. The basic data of the amplifier are the same as for Fig. 2.
When the gap is varied over wide limits, the maximum power and quality factor change only slightly. The maximum of the quality factor shifts toward smaller values of \(C_B\) as the gap increases, approaching the power maximum. In this connection, in a contactless electric-machine amplifier it is advisable to increase the air gap somewhat in comparison with co-
Fig. 3. Effect of the air gap on the parameters of the CEMA
corresponding asynchronous motors. This gives an increase in the quality factor at maximum power in a specified volume, and also leads to a reduction in the noise of the machine.
Conclusions
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Formulas are proposed for taking into account the amplifying and dynamic properties of the CEMA in design.
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It is shown that the choice of electromagnetic loadings of the CEMA cascades is substantially influenced by the requirements imposed on the amplification factor, speed of response, and overload capacity of the amplifier.
Symbols
\(A\) — total linear loading of the rotor,
\(A_1, A_2, A_s, A_f\) — linear loadings of the CEMA windings,
\(B_\delta\) — total magnetic induction in the air gap,
\(B_{\delta 1}, B_{\delta 2}\) — amplitudes of the first harmonic inductions of the CEMA cascades,
\(W_{R1}, W_{R2}\) — numbers of series turns in a phase of the rotor windings,
\(K_{w1}, K_{w2}, K_{ws}\) — winding factors of the rotor and output windings,
\(p_1, p_2\) — number of pole pairs,
\(K_{Es}\) — coefficient equal to the ratio of the emf to the voltage of the output winding,
\(\cos \varphi\) — load power factor,
\(F_{R1}, F_{R2}, F_s, F_f\) — mmfs of the CEMA windings,
\(F_1, F_2\) — resultant mmfs,
\(D, e\) — bore diameter and calculated length of the stator stack,
\(n_s\) — speed of rotation of the field of the second stage relative to the stator,
\(\delta^1\) — reduced air gap,
\(K_{M1}, K_{M2}\) — saturation coefficient of the steel of the magnetic circuit,
\[ M_0 = 4\pi \cdot 10^{-7}\ \frac{\mathrm{H}}{\mathrm{m}} \]
— magnetic permeability of air,
\(\rho\) — specific resistance of copper,
\(\tau_1\) — pole pitch of the first stage,
\(j\) — current density,
\(K_f\) — field-form coefficient of the excitation winding,
\(Z_s, Z_R\) — numbers of stator and rotor slots.
\(\sum \lambda_s, \sum \lambda_1, \sum \lambda_2\) — leakage magnetic permeances per unit length of the machine,
\(*\ \eta\) — efficiency of the amplifier (without the driving motor).
LITERATURE
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U.S. Patent, No. 2966623, 27.12.1960.
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M. L. Kostyrev, A. I. Skorospeshkin. Phase compounding of an AC contactless electric-machine amplifier. Izvestiya TPI, vol. 172, 1967.
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V. S. Novokshonov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk, 1960.
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V. M. Pavlinin, N. S. Syunov. Properties and effectiveness of the OPC. IVUZ — Elektromekhanika, No. 1, 1960.
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V. S. Nersesyan. Elements of the calculation of single-machine contactless frequency converters. Proceedings of the Gorky Polytechnic Institute, vol. 19, issue 3, 1963.
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M. L. Kostyrev, A. I. Skorospeshkin. Questions in the theory of the contactless AC electric-machine amplifier. In the present collection.
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M. L. Kostyrev, A. I. Skorospeshkin. Variation of the voltage of a contactless AC electric-machine amplifier when an inductive load is switched on. In the present collection.