Some Issues in the Design of Electric Machine Amplifiers with a Smooth Armature
Abstract Generated abstract
This article addresses the design of cross-field electric-machine amplifiers with a smooth armature, for which conventional machine-constant methods do not give voltage-independent sizing curves. It argues that such amplifiers require a single-layer armature winding, limited air-gap induction, and high but thermally constrained current density, using experimental heating data to recommend practical electromagnetic loadings. The paper derives calculation formulas for armature conductor number, stack diameter, stator bore, nonmagnetic layer, and linear load, and provides recommended ranges for key design coefficients including length-to-pole-pitch ratio, pole coverage, and voltage coefficient.
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Some Issues in the Design of EMAs with a Smooth Armature
A. I. Skorospeshkin, A. I. Vilner
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
This article considers issues associated with the selection of electromagnetic loadings and principal dimensions in cross-field electric-machine amplifiers with a smooth armature.
Usually the principal dimensions of an electrical machine are selected on the basis of the machine constant [1]
\[ C_A=\frac{D^2 l_i n_{\mathrm{н}}}{P_{\mathrm{н}}\cdot K_E} =\frac{6{,}1\cdot 10^{11}}{B_\delta a\cdot A}. \tag{1} \]
The quantities \(B_\delta\) and \(A\) are taken with account of data from electrical machines that operate well in practice and are close to the machine being designed. In the absence of analogues, the principal dimensions are assigned approximately and then, by successive approximations, a machine constant \(C_A\) is obtained that ensures good calculated characteristics of the machine. In doing so, curves are constructed
\[ C_A=f\!\left(\frac{P_{\mathrm{н}}}{n_{\mathrm{н}}}\right) \quad \text{or} \quad C_A=f(P_{\mathrm{н}}) \]
for a specified speed of rotation. These curves determine the armature volume \(D^2 l_i\) for any ratios of \(D\) and \(l\) and do not depend on the rated voltage of the machine \(U_{\mathrm{н}}\).
Then, for the selected ratio \(\lambda=\dfrac{l_i}{\tau}\), the armature diameter is determined from the formula:
\[ D=\sqrt[3]{\frac{2p C_A P_{\mathrm{н}} K_E}{\pi \lambda n_{\mathrm{н}}}}. \tag{2} \]
When attempting to calculate the smooth armature of an electric-machine amplifier, we do not obtain a clear relationship \(C_A=f\!\left(\dfrac{P_{\mathrm{н}}}{n_{\mathrm{н}}}\right)\) independent of the voltage \(U_{\mathrm{н}}\); for each value of \(U_{\mathrm{н}}\) its own curve \(C_A\) is obtained.
Consequently, for an EMA with a smooth armature, a different approach is necessary for selecting the principal dimensions and electromagnetic loadings, as compared with machines of conventional design.
In designing electric-machine amplifiers, a number of additional requirements are imposed:
1) the amplifier must have a high quality factor;
2) the possibility of twofold and greater voltage forcing, to achieve which the magnetic system must be unsaturated;
3) the amplifier must provide the maximum possible co-
amplification coefficient, which limits the extent to which the control power can be increased.
Consequently, in an amplifier with a smooth armature the nonmagnetic layer should be as small as possible. In essence, this requirement leads to having to limit oneself to a single layer of conductors on the surface of the smooth armature. With a two-layer winding, the properties of the amplifier deteriorate considerably owing to the sharp increase in the air gap.
Thus, when designing an EMA with a smooth armature, it is necessary to choose a single-layer winding with the maximum permissible current density of the second stage \((\Delta_3)\) under heating conditions. Experience shows that the induction in the air gap \((B_\delta)\) should not exceed 3800 gauss.
The current density in the armature winding, owing to the direct cooling of the winding, is fairly high: in the tested amplifiers it was brought up to 21 A/mm².
Figure 1 presents the experimental dependence of the temperature rise of the armature winding on the current density. It is clear from the figure that at \(\Delta_3 > 20\) A/mm² the temperature rise increases very rapidly (operation occurs in the region of the critical current density).
Fig. 1
It is evident that, when designing an EMA with a smooth armature, the recommended current density in the armature winding may be in the range of 18–19 A/mm².
In this case the diameter of the bare copper can be obtained as
\[ d = \sqrt{\frac{2J_n}{a\pi\Delta_3}}, \tag{3} \]
where
\(I_n\) — current of the second stage, A,
\(\Delta_3\) — current density of the second stage, A/mm\(^2\),
\(a\) — number of pairs of parallel branches of the armature winding.
The fill factor of the armature stack with insulated wire is determined by the expression:
\[ K_z'=\frac{d_{\mathrm{iz}}N}{\pi\left(D+2h_{\mathrm{vn}}+d_{\mathrm{iz}}\right)}, \tag{4} \]
where
\[ \begin{aligned} D&\text{ — diameter of the armature stack, mm,}\\ h_{\mathrm{vn}}&\text{ — height of the insulation between the armature stack and the winding conductors, mm,}\\ d_{\mathrm{iz}}&\text{ — diameter of the standard insulated wire, mm,}\\ N&\text{ — number of conductors of a single-layer armature winding.} \end{aligned} \]
For simplification, one may take
\[ K_z'=\frac{d_{\mathrm{iz}}N}{\pi D}. \tag{5} \]
The fill factor \(K_z\) is determined from the graph (Fig. 2).
Fig. 2
The fill factor with bare copper is regarded as
\[ K_z=K_z'\frac{d}{d_{\mathrm{iz}}}=\frac{dN}{\pi D}. \tag{6} \]
The quantity \(K_z\) is determined from the graph (Fig. 3).
As the winding wire, wire of grade PETV-1 is selected, as the most acceptable in terms of its thermal and mechanical properties. This wire is used by us for EMAs with a smooth armature. In principle, any other wire may also be used, provided it ensures the required thermal and mechanical properties with a minimum insulation thickness.
Proceeding from this definition of the fill factor of the active layer of the armature and from known formulas, we determine the number of conductors of the armature winding:
\[ N=\frac{60aU_{\mathrm{n}}K_E10^8}{pn_{\mathrm{n}}\Phi q}, \tag{7} \]
where
\[ U_{\mathrm{n}}\text{ — voltage of the second stage, V,} \]
Fig. 3
\(\Phi_q\) — flux of the transverse circuit,
\(\alpha\) — pole overlap.
\[ \Phi_q = B_\delta \alpha \cdot \tau \cdot l_i, \tag{8} \]
where
\[ \tau=\frac{\pi D}{2p}\quad \text{and}\quad l=\lambda \tau \]
therefore,
\[ N=\frac{60aU_nK_E\cdot 10^8\cdot 4p^2}{pn_nB_\delta\alpha\cdot \lambda \pi^2D^2}. \tag{9} \]
Substituting
\[ D=\frac{Nd}{\pi K_z}, \tag{10} \]
we obtain
\[ N=\sqrt[3]{\frac{24\cdot 10^9p\cdot a\cdot K_z^2U_nK_E}{n_n\alpha\cdot \lambda\cdot B_\delta d^2}}, \tag{11} \]
where \(d\) — (cm), \(U_n\) — (V), \(B_\delta\) — (G), \(n_n\) — (rpm).
The expression for the diameter of the armature stack can be written in the form:
\[ D=\frac{l}{\pi}\sqrt[3]{\frac{24\cdot 10^9p\cdot a\cdot U_nK_Ed}{n_n\alpha\cdot \lambda\cdot B_\delta K_z}}. \tag{12} \]
Thus, this expression gives the smallest armature diameter \(D\), for the selected ratio of \(l\) and \(D\), on which \(N\) conductors can be placed, ensuring, at the specified induction \(B_\delta\), the rated voltage at the amplifier output. Such a diameter also ensures attainment of the maximum amplification factor.
Let us consider some quantities entering formula (12). On the basis of calculations for a series of machines, it may be concluded that in amplifiers with a smooth armature the value \(K_E\) is somewhat greater than in series amplifiers and lies within the range \(1.2\text{–}1.5\), whereas in series—
for these it is equal to 1.05–1.15. This is explained by the high current density in the armature winding and, consequently, by the large voltage drop in the armature winding of an EMA with a smooth armature.
The value of \(\lambda\) in amplifiers with a smooth armature may be taken as 0.7–1.5, since the amplifiers have good commutation conditions. In series amplifiers \(\lambda = 0.4\text{–}0.7\).
The pole-coverage coefficient \(\alpha\) may be taken as 0.6–0.8 (in a series amplifier \(\alpha = 0.55\text{–}0.65\)) and is determined, respectively, by the openings of the large and medium slots. When choosing the opening of the large slot, one is guided only by the possibility of placing in it the control windings and the compensating winding, and not by the width of the commutation zone, as in ordinary amplifiers. Since, with a diametral winding on a smooth armature, the width of the commutation zone \(b_{\mathrm{kz}} \approx b'_{\mathrm{sh}}\) (\(b'_{\mathrm{sh}}\) is the brush width reduced to the armature diameter) and the opening of the large slot
\[ b_q = (1.5 \div 1.8)b_{\mathrm{kz}} \tag{13} \]
is provided automatically, the openings of the large and medium slots are taken to be the same.
The stator bore diameter is determined as
\[ D' = D + 2\delta, \tag{14} \]
where
\[ \delta = \delta' + \delta_{\mathrm{iz}} + d_{\mathrm{iz}}, \]
\(\delta'\) is the air gap between the poles and the armature winding,
\(\delta_{\mathrm{iz}}\) is the total height of the insulation of the armature winding.
The nonmagnetic layer is determined mainly by the diameter of the insulated conductor \(d_{\mathrm{iz}}\) and may be represented as
\[ \delta = K d_{\mathrm{iz}}. \tag{15} \]
In turn, \(d_{\mathrm{iz}} = I_{\mathrm{n}}\); consequently, the air-gap value is proportional to the armature current and is practically not selected in the calculation.
Next, an expression is obtained for the linear load by substituting \(N\) and \(D\) from (9) and (10) into (16):
\[ A_z = \frac{N I_{\mathrm{n}}}{2a\pi D}. \tag{16} \]
We obtain
\[ A_z = \frac{I_{\mathrm{n}} K_z}{2d}, \tag{17} \]
or, substituting (3), we find the expression for the linear load in the form
\[ A_z = 0.5 K_z \sqrt{1.5 a I_{\mathrm{n}}\Delta_z}. \tag{18} \]
Consequently, for a specified current density the linear load depends only on the current. For example, at \(\Delta_z = 20\ \mathrm{A/mm^2}\), the dependence \(A_z = f(I_{\mathrm{n}})\) is represented by the curve in Fig. 4. With a more detailed investigation of the heating of the armatures of amplifiers with a smooth armature, it will be possible, at the very initial stage of amplifier calculation, to determine \(A_{z\max}\), choosing the permissible linear load.
Overall, the results of the work carried out may be formulated as follows:
- Calculation formulas have been obtained for determining the principal dimensions of an EMA with a smooth armature.
- Recommendations have been given for selecting electromagnetic loadings.
- The choice of the values \(\lambda\), \(\alpha\), and \(K_E\) has been recommended.
Fig. 4
References
- I. M. Postnikov. Design of Electrical Machines. 1960.
- I. N. Rabinovich, I. G. Shubov. Design of Direct-Current Machines. 1967.
- F. A. Goryainov. Electric-Machine Amplifiers. 1962.