Measurement of the Null Signal of Induction Tachogenerators with a Hollow Rotor Due to the Machine Manufacturing Technology
Abstract Generated abstract
This paper examines technological sources of the zero signal in hollow-rotor induction tachogenerators, with emphasis on magnetic asymmetry of the stator magnetic-core stacks. It derives approximate expressions relating zero signal to harmonics of magnetic reluctance nonuniformity and then evaluates the role of stator-stack asymmetry through experimental statistical analysis during successive assembly stages. The results indicate an approximately linear relation between zero signal and the measured magnetic asymmetry parameter, but with decreasing correlation as other machine errors are introduced. For the manufacturing process studied, stator-stack magnetic asymmetry accounts for only about 10 to 15 percent of the assembled machine’s zero signal, while air-gap nonuniformity and hollow-rotor electrical asymmetry are the dominant contributors.
Full Text
MEASUREMENT OF THE ZERO SIGNAL OF HOLLOW-ROTOR INDUCTION TACHOGENERATORS CAUSED BY MACHINE MANUFACTURING TECHNOLOGY
A. A. Kamenskii
The increased requirements for the accuracy of the output parameters of induction tachogenerators make it necessary to investigate and evaluate the technological factors that affect the accuracy characteristics, especially the zero signal.
Technological factors affect the errors in the shape, dimensions, and physical properties of individual units and parts. In-process inspection of machine windings makes it possible to completely eliminate errors associated with inaccuracies in the manufacture of the windings and with the presence of short-circuited turns. Therefore, the main influence on the zero signal is exerted by nonuniformity of the air gap, magnetic asymmetry of the magnetic-core stacks, and electrical asymmetry of the hollow rotor [1].
Theoretically and experimentally, of the entire range of errors, only the influence of air-gap nonuniformity on the zero signal has been studied sufficiently fully [2].
In this connection, it was of interest to determine the influence of the magnetic asymmetry of the stacks on the zero signal.
In the analytical consideration of the influence of certain factors on the zero signal, for the convenience of analysis the following assumptions may be adopted:
1 — capacitive couplings between the windings located on the stator are absent;
2 — the hollow rotor is symmetrical;
3 — the magnetic-core stacks are smooth;
4 — the field in the gap is plane-parallel.
Representing the magnetomotive force (MMF) by the fundamental harmonic and assuming that the magnetic reluctance affects only the spatial distribution of the magnetic flux, the magnetic induction along the stator bore can be written in the following form [3]:
\[ B(x)=\frac{F_1\cos Px-F_0}{R_0[1+\rho_k\cos(kx-x_{0k})]}, \tag{1} \]
where
\(F_1\) — amplitude of the fundamental harmonic of the magnetomotive force;
\(F_0\) — magnetic potential of the inner stack of the magnetic core;
\(P\) — number of pole pairs;
\(x\) — current linear coordinate along the gap;
\(R_0\) — mean magnetic reluctance of the circuit;
\(\rho_k\) — the relative amplitude of the \(k\)-th harmonic, accounting for the nonuniformity of the magnetic reluctance of the circuit along the stator bore, with
\[ \frac{\rho_k}{R_0} \ll 1; \]
\(x_{0k}\) — the spatial phase shift of the \(k\)-th harmonic relative to the axis of the excitation winding.
Analysis of expression (1) for the induction \(B(x)\) in the air gap, obtained by expanding (1) in a Maclaurin series and neglecting terms above the second order, shows that, owing to the nonuniformity of the magnetic reluctance, harmonics of the magnetomotive force appear, shifted relative to the fundamental harmonic of the excitation winding by 90 electrical degrees.
The magnetic potential \(F_0\) of the stationary core is found from the condition of continuity of the magnetic-field lines along the stator bore.
The analytical expression for the zero signal arising as a result of nonuniform magnetic reluctance, under the generally accepted assumptions [3], taking the 1st and 3rd harmonics into account, has the following form:
\[ E_0=\frac{2\pi}{\sqrt{2}} W_c K_{\mathrm{ob}} f\Phi_0 \frac{\rho_k^2}{6} Sm\,2x_{0k}, \qquad \text{for } P=K \tag{2} \]
\[ E_0=\frac{2\pi}{\sqrt{2}} W_c K_{\mathrm{ob}} f\Phi_0 \left( -\frac{4}{3}\rho_k Sm\,x_{0k} +\frac{4}{15}\rho_k^2 Sm\,2x_{0k} \right), \qquad \text{for } P=\frac{k}{2} \tag{3} \]
\[ E_0=\frac{2\pi}{\sqrt{2}} W_c K_{\mathrm{ob}} f\Phi_0 \frac{8}{63}\rho_k^2 Sm\,2x_{0k}, \qquad \text{for } P=\frac{k}{3} \tag{4} \]
\[ E_0=\frac{2\pi}{\sqrt{2}} W_c K_{\mathrm{ob}} f\Phi_0 \left( -\frac{8}{15}\rho_0 Sm\,x_{0k} +\frac{8}{63}\rho_k^2 Sm\,2x_{0k} \right), \qquad \text{for } P=\frac{k}{4} \tag{5} \]
\[ E_0=0 \quad \text{for } P=2k,\; 3k \text{ and a number of other ratios.} \]
The expressions for the zero signal (2–5) are valid when the nonuniformity of the magnetic reluctance is caused by an air-gap error. If the nonuniformity of the magnetic reluctance is due to magnetic asymmetry of the magnetic-core stacks, then, in a first approximation, the zero signal \(E'_0\) will be proportional to the coefficient \(K_\mu-1\), which indicates the ratio of the reluctances of the magnetic-core stacks and of the air gap,
\[ K_\mu - 1 = Z_\mu G_0, \]
where \(Z_\mu\) is the magnetic reluctance of the magnetic-core stacks;
\(G_0\) is the magnetic permeance of the air gap.
The zero signal \(E_0\) is equal to
\[ E'_0=(K_\mu-1)E_0. \]
As formulas (2–5) show, the zero signal depends on the ratio of the number of pole pairs \(P\) to the order of the asymmetry harmonic \(K\). The most characteristic air-gap error may be considered ovality (\(K=2\)), which should have the strongest effect on two-pole machines (\(P=1\)). Magnetic asymmetry of the stacks, as experiment has shown, is usually expressed by a harmonic of the first and sometimes the second order and, consequently, also affects the zero signal.
In a real machine all errors—namely, air-gap error, magnetic asymmetry of the magnetic circuit, and electrical asymmetry of the hollow rotor—act simultaneously. Therefore, the study of the influence of the magnetic asymmetry of the stacks on the zero signal was carried out by an experimental-statistical method (4): the degree and form of the influence of the parameter under study on the output were determined.
The magnetic-circuit stacks were manufactured by bonding under pressure. The stacks were fired on the press up to the stop, i.e., with height limitation. The pressure was about \(50\ \mathrm{kg/cm^2}\).
The stacks were then subjected to the required mechanical treatment. Before assembly, the stator stacks were divided into three groups depending on the magnitude of the magnetic asymmetry of the stack (the asymmetry of the back of the stator stack).
\[ \text{Group I} \;—\; \Delta\mu = 0 \div 0.145; \]
\[ \text{Group II} \;—\; \Delta\mu = 0.145 \div 0.29; \]
\[ \text{Group III} \;—\; \Delta\mu = 0.29 \div 0.435, \]
where
\[ \Delta\mu = \frac{\mu_{\max} - \mu_{\min}}{\mu_{\max} + \mu_{\min}} . \]
The influence of the magnetic asymmetry of the stator stack on the zero signal was determined during assembly after the following operations:
installation of the stack in the housing and adjustment with its “technological” core;
installation in the motor-generator housing of the standard stationary core;
installation of the hollow rotor.
Statistical analysis of the experimental data showed that the form of the influence of the magnetic anisotropy of the stator stack on the zero signal is practically weakly nonlinear and is described quite well by a straight-line dependence. The correlation coefficient between \(E_0\) and \(\Delta\mu\), which determines the degree of influence of magnetic asymmetry (without taking into account the phase shift of the harmonic of magnetic asymmetry relative to the axis of the excitation winding) on the zero signal, changes significantly as the acting errors increase.
The zero signal and the correlation coefficient in machines with a technological core and without a hollow rotor:
\[ E_0 = 32 + 128.6\,\Delta\mu\;(\mathrm{mV}), \]
\[ r = 0.5. \]
The zero signal and the correlation coefficient in machines assembled without a hollow rotor:
\[ E_0 = 31.3 + 110.6\,\Delta\mu\;(\mathrm{mV}), \]
\[ r = 0.38 \]
The zero signal and the correlation coefficient in assembled machines:
\[ E_0 = 53.21 + 91.9\,\Delta\mu\;(\mathrm{mV}), \]
\[ r = 0.24 \]
Analysis of the magnitude of the correlation coefficient shows that in the assembled machine, 10–15% of the zero-signal magnitude in this technological process is due to the magnetic asymmetry of the back of the stator stack. The main influence on the zero signal is exerted by the nonuniformity
...the nonuniformity of the air gap and the electrical asymmetry of the hollow rotor. The electromagnetic asymmetry of the stationary core has an insignificant effect on the zero signal of the machine.
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