Optimal selection of the basic geometry of integrating asynchronous tachogenerators
Abstract Generated abstract
This paper addresses the selection of principal magnetic circuit dimensions for integrating asynchronous tachogenerators with hollow rotors, where high accuracy, compact size, material utilization, unification, and manufacturability impose competing design constraints. Adapting a method previously used for rotating transformers, it derives relationships between bore diameter, stack geometry, slot area, housing diameter, iron induction, current density, pole pairs, and winding parameters, then proposes determining the optimal geometry from the intersection of calculated functions for slot area. The article also gives practical ranges for permissible iron induction and excitation current density, and presents the method as a basis for designing tachogenerators with maximum accuracy and output steepness within specified dimensions and electromagnetic loading limits.
Full Text
OPTIMAL CHOICE OF THE BASIC GEOMETRY OF INTEGRATING ASYNCHRONOUS TACHOGENERATORS
V. F. KARAUSHEV
(Presented by the scientific-technical seminar of the Department of Electrical Equipment)
In designing any electrical machine, one of the most important factors ultimately determining all the basic characteristics of the future machine is the choice of the optimal values of the principal dimensions of the magnetic circuit—the bore diameter \(D\) and the stack length \(l\). In the calculation of asynchronous tachogenerators (AT) with a hollow rotor, this problem acquires special importance, since it is usually necessary to create a high-accuracy machine (relative amplitude error not more than \(0.01\text{–}0.1\%\)) in minimum overall dimensions, with the maximum possible utilization of active materials. In addition, it is important to satisfy the requirements of unification and manufacturability of the machine being designed. All these requirements are to a considerable extent contradictory; consequently, when determining the geometry of the magnetic circuit, it is necessary to seek an optimal variant, taking into account that the main requirement which the designed AT must satisfy is to ensure the specified accuracy.
In the known literature devoted to the calculation of asynchronous tachogenerators, questions concerning the choice of the optimal geometry of an AT are either not considered at all [1], or are considered in insufficient detail [2]. However, taking into account the considerations set forth above, it appears important to examine these questions in greater detail.
For the purpose of unification, it is recommended that the external dimensions of electrical micromachines be chosen in accordance with the \(R\text{–}10a\) series of preferred numbers (GOST 12126—66). Consequently, the problem of choosing the basic geometry of an AT consists in finding the optimal ratios between the diameter of the machine housing and the geometric dimensions of the magnetic-circuit punching, which ensure values of induction in the iron \(B_{\text{ж}}\) and current density \(j_0\) in the excitation winding that are permissible from the standpoint of accuracy and stability of the AT output characteristics.
As the basis for solving the problem posed, we shall adopt the method for selecting the optimal dimensions of the magnetic circuit, set forth in [3] for rotating transformers. We shall consider the essence of the method as applied to problems of designing integrating ATs with a machine having windings located on the internal stator. The corresponding sketch of the magnetic-circuit punching is given in Fig. 1.
Assuming that the induction in all sections of the magnetic circuit is approximately the same (which is practically the case for multipole low-saturation micromachines, such as the AT), for the geometrical dimensions of the magnetic core (Fig. 1) one may write
Fig. 1. Sketch of the magnetic-core punching
\[ 2h_j=\frac{D}{p\gamma}, \qquad b_z=\frac{\pi D}{\gamma Z}, \qquad d_1 \simeq \frac{\pi D\left(1-\frac{1}{\gamma}\right)}{Z+\pi}. \tag{1} \]
Here \(\gamma=\dfrac{B_{\text{ж}}}{B_\delta}\) is the ratio of the permissible value of induction in the iron to the maximum value in the air gap;
\(p\) is the number of pole pairs;
\(Z\) is the number of slots.
Further, from geometrical considerations one can approximately determine the slot area
\[ S_{\text{п}}\simeq \frac{\pi}{4}\left[D^2-\left(D_c+2h_j\right)^2\right] -\frac{b_z Z}{2}\left(D-D_c-2h_j\right) -\frac{\pi D-b_z Z}{2}\,d . \tag{2} \]
It follows from this that, other conditions being equal, the slot area is determined mainly by the value of the bore diameter \(D\). Therefore, in order to relate the slot area and the geometry of the punching to the outside diameter of the tachogenerator housing \(D_{\text{н}}\), we introduce the ratio
\[ y=\frac{D}{D_{\text{н}}} = \frac{1-\dfrac{2\delta}{D_{\text{н}}}}{1+\dfrac{1}{p\gamma}} \simeq \frac{1}{1+\dfrac{1}{p\gamma}}, \tag{3} \]
where \(D_{\mathrm{n}}=(D_{\mathrm{k}}-2\Delta_{\mathrm{k}})\) is the outside diameter of the magnetic-core stack; \(\Delta_{\mathrm{k}}\) is the wall thickness of the housing.
Since the quantity \(\Delta_{\mathrm{k}}\) is selected structurally, the value of the diameter \(D_{\mathrm{n}}\) may be regarded as given. Then, using formulas (1) and (3), expression (2) can be written in the form:
\[ S_{\mathrm{n}}=\frac{\pi}{4}D_{\mathrm{n}}^{2} \left[ \left(\frac{p\gamma-1}{p\gamma+1}-y_{\mathrm{c}}\right) \left(1+y_{\mathrm{c}}-\frac{2p}{p\gamma+1}\right) -\frac{\pi p}{2(Z+\pi)} \left(\frac{\gamma-1}{p\gamma+1}\right)^{2} \right] =f_{1}(\gamma), \tag{4} \]
where
\[ y_{\mathrm{c}}=\frac{D_{\mathrm{c}}}{D_{\mathrm{n}}}\simeq(0.2 \div 0.3) \]
is the ratio of the diameters in accordance with Fig. 1.
On the other hand, the slot area is determined by the current density in the excitation winding of the AT. If it is assumed that the current drawn by the excitation winding from the network is the magnetizing current, which is quite permissible for integrating ATs, then the dependence of the slot area \(S_{\mathrm{n}}\) on the current density \(j_{0}\) can be established. For this we use the relations:
\[ \frac{1}{\mu_{0}}B_{\delta}\delta' = \frac{2\sqrt{2}}{\pi}\cdot \frac{w_{1}k_{w1}}{p}I_{0}; \qquad I_{0}=j_{0}q_{0}; \qquad \sigma_{s}S_{\mathrm{n}}k_{\mathrm{zp}}=2w_{1}q_{0}, \tag{5} \]
where \((w_{1}k_{w1})\) is the effective number of turns of the excitation winding;
\(\sigma_{s}\) is the ratio of the number of slots occupied by the excitation winding to the total number of slots;
\(k_{\mathrm{zp}}\) is the slot fill factor;
\(q_{0}\) is the cross section of the conductor copper.
It follows that
\[ \frac{S_{\mathrm{n}}}{D_{\mathrm{n}}^{2}} = \frac{a_{1}}{\gamma} = f_{2}(\gamma), \tag{6} \]
where
\[ a_{1}= \frac{\pi}{\sqrt{2}\mu}\cdot \frac{B_{\mathrm{zh}}\delta' p} {\sigma_{s}k_{\mathrm{zp}}k_{w1}j_{0}D_{\mathrm{n}}^{2}} \]
is the electromagnetic-load coefficient.
Figure 2 gives two groups of curves, calculated respectively from equations (4) and (6) for \(p=2, 3\), \(Z=12\), and \(y_{\mathrm{c}}=0.2 \div 0.30\).
To determine the optimal punching geometry by means of curves of this type, it is necessary to find the optimal value \(\gamma_{\mathrm{opt}}\) for the selected \(B_{\mathrm{zh}}\) and \(j_{0}\). For this, it is sufficient, with known coefficient \(a_{1}\) and number of pole pairs \(p\), to determine the abscissa \(\gamma\) of the point of intersection of the curves \(f_{1}(\gamma)\) and \(f_{2}(\gamma)\) and to use relations (1) and (3).
From the expression for the coefficient \(a_{1}\) it follows that, in order to find its value, and then the value \(\gamma_{\mathrm{opt}}\), it is necessary to choose the winding scheme and specify permissible values of \(B_{\mathrm{zh}}\) and \(j_{0}\), on the basis of the accuracy requirements imposed on the AT being designed.
Thus, the permissible value of the induction in the iron \(B_{\mathrm{zh}}\) is determined by the requirements for ensuring maximum magnetic permeability of the stack and linearity of the main magnetization curve. In small-size ATs, the value of \(B_{\mathrm{zh}}\) is often determined by the requirements of the technological feasibility of the punching. Depending on the material employed,
for ATs of different sizes one may indicate the following values of induction:
\[ \begin{aligned} &\text{Permalloy 79NM} &&—\ 0.3 \div 0.6\ \text{T};\\ &\text{Permalloy 50N} &&—\ 0.5 \div 0.9\ \text{T}. \end{aligned} \]
The permissible density is chosen approximately, proceeding from the permissible level of losses and the heating of the machine casing determined by them.
Fig. 2. Dependence
\[ \frac{S_p}{D_k^2}=f(\gamma) \]
Depending on the dimensions, the current density may be taken within the following limits:
\[ j_0=(3 \div 6)\cdot 10^2\ \text{A}/\text{cm}^2. \]
Thus, the article has set forth a method for choosing the optimal geometry of the magnetic circuit of an integrating tachogenerator with windings on the inner stator. In analogous fashion one can also solve the problem of choosing the geometry for ATs with a different arrangement of the windings.
The proposed method can serve as a basis for calculating an AT having maximum accuracy and steepness within specified dimensions under maximum electromagnetic loadings. With minor changes, the method can be applied to the calculation of damping ATs.
REFERENCES
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P. A. Zubtsov. Method for calculating an asynchronous tachogenerator. Energy Collection of the Chelyabinsk Polytechnic Institute, No. 2, 1959.
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G. S. Somikhina. “Choice of the basic dimensions of an asynchronous tachogenerator.” Proceedings of Higher Educational Institutions, Electromechanics, No. 8, 1963.
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Low-Power Electrical Machines. Edited by D. A. Zavalis̆hin. GEI, 1963.