Optimization of a Contactless Electric Machine Amplifier of Increased Frequency
Abstract Generated abstract
This paper formulates an optimization problem for a brushless high-frequency electric-machine amplifier intended as a regulated source for automated systems operating at 100 to 400 Hz. It proposes weight and cost as principal optimality criteria, defines constraints on winding temperature, gain, response time, leakage reactance, magnetic induction, and technological dimensions, and describes a digital-computer calculation procedure using selected electromagnetic and geometric variables with a Gauss-Seidel search. Calculations for 0.5 to 5 kW enclosed ventilated machines identify practical optimal ranges for magnetic loading, linear loading, current density, induction distribution, and diameter-to-length ratio. Comparisons with standard high-frequency generators and converter units suggest that the optimized amplifiers can offer lower weight and cost, good efficiency, and favorable dynamic and regulating properties, especially for variable motor-type loads.
Full Text
Optimization of a Brushless High-Frequency Electric Machine Amplifier
M. L. Kostyrev, A. I. Skorospeleshkin
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
For the introduction of automated systems at an increased frequency of 100–400 Hz, regulated sources are required; a brushless electric-machine amplifier (BEMA) may be used as such a source [1]. The amplifier consists of a synchronous generator and an asynchronous frequency converter, combined in a single magnetic circuit and rotated by a drive motor. To obtain rigid external characteristics and high dynamic properties, phase compounding has been applied in the amplifier [2].
As a result of investigations carried out at the Department of Electrical Machines of TPI, a theory of the operation of the BEMA and a method for its design have been developed [2, 3, 7]. In the present article the problem of optimizing the BEMA is posed.
As is known, the problem of finding an optimum belongs to the field of applied mathematics—mathematical (or optimal) programming—and consists in determining the values of independent variables for which the optimality criterion has the minimum (or maximum) possible value, provided that the variables take only positive values and the constraint conditions are satisfied.
If the number of independent variables does not exceed 3–4, then the search for the optimum can be carried out by scanning, in a definite order, the nodes of a multidimensional grid in the space of independent variables. With a larger number of variables, the method of random trials, gradient methods, and other methods are used [4, 5].
Of the known optimality criteria, the following are most appropriate as applied to the BEMA:
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Minimum cost of manufacture and operation (minimum cost).
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Minimum dimensions and weight.
The second criterion is more important if the BEMA is intended for operation in mobile installations.
Taking into account the features of the BEMA as an element of an automated system, the following constraint conditions are recommended:
- The temperature rise of the stator and rotor windings must not exceed the permissible value
\[ \theta_{\mathrm{CuS}} \leqslant \theta_{\mathrm{dop}}, \qquad \theta_{\mathrm{CuR}} \leqslant \theta_{\mathrm{dop}} . \]
- The power gain coefficient must be no less than the specified value
\[ K_{\mathrm{y}} \geq K_{\mathrm{y\,dop}} . \]
- The response speed of the amplifier must be no worse than the specified value
\[ T \leq T_{\mathrm{dop}} . \]
- The total leakage reactance of the windings of the brushless electric-machine amplifier must not exceed that permissible from the conditions of dynamic overload capacity [2, 3, 7]:
\[ \sum X_s < X_{s\,\mathrm{dop}} . \]
- The inductions in the steel sections of the magnetic circuit must not exceed those specified by the conditions of voltage forcing at the output
\[ B_{a1}+B_{a2}\leq B_{a\,\mathrm{dop}}, \]
\[ B_{z1}+B_{z2}\geq B_{z\,\mathrm{dop}} . \]
- Technological conditions. For example, the tooth width and the yoke height must not be less than the permissible values. Taking into account the specific conditions of manufacture and operation, additional constraints may be introduced (for example, a constraint on the flywheel moment when operating from a diesel engine).
Previously conducted studies of the design features of a combined-type brushless electric-machine amplifier [3], as well as the use of known experience in designing asynchronous motors on a digital computer [6], made it possible to develop a calculation procedure for searching for optimal variants of the brushless electric-machine amplifier. The procedure is recommended for closed, ventilated machines with a power of 0.5–5 kW and a frequency of 100–400 Hz.
According to the procedure [7], the following are adopted as independent variables: the total electromagnetic loads \(A\), \(B\), \(\delta\); the coefficient of distribution of the induction between the stages \(C_B\); the current densities in the stator and rotor windings \(\Delta_S\), \(\Delta_R\); the stator bore diameter \(D\); and the slot heights of the stator and rotor \(h_{ns}\), \(h_{nR}\).
Preliminarily, for the specified frequency and rotor speed, taking account of the combination, the number of pole pairs of the stages, the number of rotor phases, the winding connections, their pitches and winding factors are selected, as well as the numbers and shape of the stator and rotor slots and the air gap. The outer diameter of the stator is selected from the conditions of standardization or from a series of normalized diameters, and the shaft diameter under the rotor core from the conditions of mechanical strength.
The basic parameters of the amplifier are calculated on the basis of the formulas proposed in [3]. The losses and heating of the windings are calculated by the known procedure [8]. When calculating the cost of the brushless electric-machine amplifier, the capital costs \(K\) are reduced to operating costs \(C_1\) according to the known formula
\[ C=C_1+E_nK, \]
where
\[ E_n=0.15\text{–}0.2 \]
is the coefficient of economic efficiency.
Transportation, installation, and maintenance costs are not taken into account in the calculations, since they are relatively small and depend almost not at all on the use of materials, i.e., they do not affect the position of the optimum.
Using the developed procedure, optimized variants of the brushless electric-machine amplifier were calculated on the M-20 digital computer. The machine time for calculating one variant was about 2 sec.
The optimum was determined by the Gauss–Seidel method [5]. Both minimum weight and minimum cost were used as the criterion. In cost optimization, constraints were introduced on temperature, relative leakage, and maximum inductions in the steel sections of the magnetic circuit.
The calculation results (without the drive motor) for cost optimization are given in Table 1. The following initial data were used for the calculations: \(n=3000\) rpm, \(\cos\varphi_n=0.8\), \(D_n=208\) mm, \(\delta=0.4\) mm, \(d_B=0.48\) mm.
Table 1
| Parameter | Symbol | Unit | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| Power | $P_{\text{out}}$ | kVA | 3 | 3 | 3 | 3 |
| Frequency | $t_2$ | Hz | 200 | 300 | 400 | 400 |
| Number of poles | $2p_1$ | — | 2 | 2 | 2 | 4 |
| Number of poles | $2p_2$ | — | 6 | 10 | 14 | 12 |
| Number of slots | $z_s$ | — | 36 | 30 | 42 | 36 |
| Number of slots | $z_R$ | — | 24 | 20 | 28 | 24 |
| Coefficient | $C_B$ | — | 0.47 | 0.57 | 0.54 | 0.45 |
| Induction | $B\delta$ | T | 0.5 | 0.55 | 0.5 | 0.44 |
| Linear load | $A$ | A/cm | 197 | 183 | 200 | 165 |
| Bore diameter | $D$ | mm | 140 | 145 | 150 | 150 |
| Stack length | $l$ | mm | 144 | 144 | 139 | 224 |
| Slot height | $h_{ns}$ | mm | 16 | 17 | 16 | 18 |
| Slot height | $h_{nR}$ | mm | 27 | 28 | 26 | 26 |
| Current density | $\Delta_s$ | A/mm² | 4.1 | 3.8 | 4.45 | 3.3 |
| Current density | $\Delta_R$ | A/mm² | 3.96 | 3.7 | 4 | 3.3 |
| Tooth width | $b_{zs}$ | mm | 3.7 | 5 | 3.9 | 4.8 |
| Tooth width | $b_{zR}$ | mm | 5.5 | 7.4 | 5.2 | 6 |
| Copper weight | $G_{\text{cu}}$ | kg | 10.7 | 10.1 | 8.2 | 11.7 |
| Weight of BEMU | $G$ | kg | 66 | 66 | 65 | 100 |
| Efficiency | $\eta$ | — | 0.85 | 0.84 | 0.84 | 0.82 |
| Winding overheating | $\Theta_{\text{cu}R}$ | °C | 67 | 67 | 67 | 59 |
| Winding leakage | $X_{\sigma}$ | — | 0.49 | 0.49 | 0.5 | 0.5 |
| Wholesale price | $K_1$ | rub. | 101 | 100 | 92 | 130 |
Steel E42, 0.35 mm; winding wire PEV-2; insulation class E.
In calculating the cost it was assumed that the BEMU operating time under load over a service life of 5–7 years, at a power load factor $K_{\text{l}} = 0.7$, is $t = 10000$ h.
Optimization calculations showed that, for the indicated powers, frequencies, and designs, the optimal loads of the BEMU lie within the ranges $B\delta = 0.4$–$0.6$ T, $A = 150$–$200$ A/cm, $\Delta = 3$–$5$ A/mm². The optimal values are
$C_B = 0.35$–$0.55$, and the ratio of diameter to length is $\dfrac{D}{l} = 0.6$–$1.1$. The parameters
$C_B \dfrac{D}{l}$ increase with growth of the pole ratio $\dfrac{p_2}{p_1}$.
The excitation losses in the optimized machines amounted to 2–3% of the output power; the efficiency is sufficiently high, $\eta = 0.77$–$0.85$.
Optimization calculations showed that the basic parameters of the BEMU, including weight and cost, improve as the ratio $\dfrac{p_2}{p_1}$ increases.
In Table 2, the BEMAs optimized with respect to cost are compared with standard high-frequency generators. As can be seen, in the frequency range up to 400 Hz the developed BEMAs have a higher efficiency, are 30% lighter, and are 1.5–1.6 times cheaper in comparison with standard contactless high-frequency generators. In addition, BEMAs possess high dynamic and regulating properties, which gives them additional advantages for use in automated systems.
Table 2
| 12GIS-2 | BEMA | 16GIS2 | BEMA | GPCh12/400 | ||
|---|---|---|---|---|---|---|
| Power | kVA | 3,75 | 3 | 3,75 | 3 | 15 |
| $\cos\varphi_{\mathrm n}$ | — | 0,8 | 0,8 | 0,8 | 0,8 | 0,8 |
| Speed | rpm | 3000 | 3000 | 3000 | 3000 | 1500 |
| Frequency | Hz | 300 | 300 | 400 | 400 | 400 |
| Efficiency | — | 0,83 | 0,84 | 0,82 | 0,84 | |
| Weight | kg | 110 | 66 | 110 | 65 | 280 |
| kg/kVA | 29,3 | 22 | 29,3 | 21,7 | 18,5 | |
| Price | rubles | 192 | 100 | 192 | 92 | 1770 |
| rubles/kVA | 51,2 | 33,3 | 51,2 | 30,7 | 118 | |
| Design | — | enclosed | enclosed | contactless | contactless | contact-protected |
| $\Delta U$ | % | — | 10 | — | 10 | 2 |
| $K_y$ | — | — | 1000 | — | 1000 | — |
| T | s | — | 0,05 | — | 0,05 | — |
In Tables 3 and 4, BEMAs in aggregate design are compared with known electric-machine and static converters. The AOЛ2-31-2 motor is taken as the BEMA drive motor.
Table 3
| BEMA | VPCh-150 | MPG-1 | S-572 | OPCh-3 | I-165 | ||
|---|---|---|---|---|---|---|---|
| $P_{\mathrm{out}}$ | kW | 3 | 2,7 | 2,1 | 1,2 | 5,3 | 3,75 |
| $f_2$ | Hz | 200 | 200 | 200 | 200 | 200 | 200 |
| n | rpm | 3000 | 3000 | 3000 | 3000 | 3000 | 3000 |
| $\cos\varphi_{\mathrm n}$ | — | 0,8 | 0,7 | 0,85 | 0,7 | 0,75 | 0,8 |
| $\eta_{\mathrm{agr}}$ | — | 0,72 | 0,52 | 0,585 | 0,577 | 0,71 | 0,52 |
| $\cos\varphi_1$ | — | 0,89 | 0,4 | 0,89 | 0,47 | 0,92 | 0,5 |
| Weight | kg | 96 | 330 | 130 | 36 | 140 | 73 |
| kg/kVA | 32 | 150 | 62 | 30 | 26,4 | 19,5 | |
| Price | rubles | 145 | 520 | — | 48 | 195 | 135 |
| rubles/kVA | 48 | 193 | — | 40 | 37 | 36 | |
| $\Delta U$ | % | 10 | — | — | 30 | 13 | 30 |
| $K_y$ | — | 1000 | — | — | — | — | — |
| T | s | 0,05 | — | — | — | — | — |
| Design | — | enclosed | explosion-proof contactless | explosion-proof contactless | protected | protected | enclosed contact |
Table 4
| BEMU | ALA-3.5 | ATT-2-400 | ATT-2-400 | FC | SFC | ||
|---|---|---|---|---|---|---|---|
| \(P_{\text{out}}\) | kVA | 3 | 4.38 | 2.5 | 8 | 1 | 1.57 |
| \(f_2\) | Hz | 400 | 427 | 400 | 500 | 400 | 400 |
| \(n\) | rpm | 3000 | 2850 | 3000 | 3000 | static | static |
| \(\cos\varphi_n\) | — | 0.8 | 0.8 | 0.8 | 0.8 | 1 | 0.7 |
| \(m\) | — | 3 | 1 | 3 | 3 | 1 | 3 |
| \(\eta_{\text{agr}}\) | — | 0.71 | 0.55 | 0.515 | 0.75 | 0.5 | 0.7 |
| \(\cos\varphi_1\) | — | 0.88 | — | 0.85 | 1 | — | 0.7 |
| \(B\) and \(C\) | kg | 95 | 170 | 152 | 325 / 40.7 | 55 / (663 C) | 30 |
| kg/kVA | 31.7 | 39.7 | 60.7 | — | 55 | 28 | |
| Price | rub. | 135 | 255 | — | — | — | 2300 |
| rub./kVA | 45 | 58.2 | — | — | — | 1460 | |
| \(\Delta U\) | % | 10 | — | 2 | — | 25 | \(\pm 5\%\) |
| \(K_y\) | — | 1000 | — | — | — | — | — |
| \(T\) | sec. | 0.05 | — | — | — | — | — |
As is evident from the techno-economic comparison, in the frequency range up to 400 Hz BEMUs can successfully compete, especially when operating with a sharply variable, predominantly motor-type load, with known contactless converters, including those with ferromagnetic frequency multipliers (FC) and static semiconductor converters (SFC) [9, 10]. The latter are significantly inferior to the BEMU in cost and reliability.
Conclusions
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The developed calculation method—the search for optimal variants of contactless electromechanical amplifiers of increased frequency, 100–400 Hz—made it possible to design amplifiers with good weight and energy indices, as well as high dynamic and control properties.
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The developed amplifiers can successfully compete with known contactless generators and converters of increased, unregulated frequency, especially when operating with a motor load.
Literature
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