On the Choice of the Air Gap Size in a Direct Current Machine
Abstract Generated abstract
This paper examines how the air gap under the main poles of a direct current machine should be selected when vibroacoustic performance is considered alongside electrical characteristics and mechanical reliability. Using known relations for tooth pulsations of the magnetic field and magnetic attraction forces, the authors relate vibration-producing forces to the ratio of slot opening width to air gap, then test this dependence on a P-40 type DC motor with air gaps of 1, 1.5, and 2 mm while maintaining the main flux by adjusting excitation current. The experiments show that increasing the air gap substantially reduces frame and foot vibrations, especially from 1 to 1.5 mm, although the reduction is not proportional over the full range. The study concludes that permissible vibration level should be included as a design criterion in choosing the air gap.
Full Text
ON THE CHOICE OF THE AIR GAP SIZE OF A DC MACHINE
E. E. Dutlov, K. A. Khorkov
(Presented by the scientific seminar of the departments of electrical machines and apparatus and general electrical engineering)
In designing DC machines, the choice of the size of the air gap under the main poles is determined by calculation of the electrical characteristics and by the requirements of the mechanical reliability of the machine. To reduce losses in the excitation winding and improve the commutation of the machine, it is expedient to reduce the air gap \(\delta\). At the same time, to increase the stability of machine operation and its mechanical reliability, an increase in \(\delta\) is required. The generally accepted method is to determine the size of the air gap from the condition that the field under the pole arc, under machine load, does not overturn, i.e., does not change its direction. In this case the air gaps obtained are considerably larger than those required to ensure the mechanical reliability of the machine.
At present, important indicators of the quality and reliability of electrical machines are the vibroacoustic characteristics of the machine. In the spectrum of vibration accelerations, vibrations at the tooth frequency have a considerable amplitude. The cause of these vibrations is phenomena associated with pulsations of the magnetic flux under the main poles during rotation of the toothed armature. It is known that the amplitude of the pulsations of the magnetic flux depends on the width of the openings of the armature slots \(b_0\) and on the size of the machine air gap \(\delta\). Consequently, in those cases when restrictions are imposed on the permissible vibration level, it is necessary to coordinate the choice of \(\delta\) with the permissible level of tooth pulsations of the magnetic flux.
The relation between the amplitude of the periodic component of the induction in the air gap of the machine \(B_0\) and the geometry of the region under consideration, i.e., with the quantities \(\delta\) and \(b_0\), is established by means of the coefficient \(\beta\), obtained from consideration of the field in the gap by the method of conformal mappings [1].
\[ \beta = \frac{B_0}{B_\delta} = \frac{1 - 2u + u^2}{2(1 + u^2)} . \tag{1} \]
\[ u = \frac{b_0}{2\delta} + \sqrt{1 + \left(\frac{b_0}{2\delta}\right)^2}, \tag{2} \]
where
\(B_\delta\) is the amplitude of the induction curve in the air gap of the machine.
If one uses the well-known equation for the magnetic attraction force, then the mean amplitudes of the radial and tangential components of the magnetic-attraction forces caused by the periodic component of the magnetic field under the pole, per unit length of the active part of the machine, in the absence of skew of the armature slots, according to [2], will be equal to
\[ P_0=\frac{B_\delta^2 \cdot Z \cdot \alpha \cdot 10^7}{16 \cdot \pi \cdot p}\cdot R_k,\ \text{N/m}, \tag{3} \]
\[ Q_0=\frac{B_\delta^2 \cdot Z \cdot \alpha \cdot 10^7}{16 \cdot \pi \cdot p}\cdot S_k,\ \text{N/m}, \tag{4} \]
where
\(P_0\) is the specific radial component of the force;
\(Q_0\) is the specific tangential component of the force;
\(Z\) is the number of armature slots;
\(\alpha\) is the pole-arc coefficient;
\(p\) is the number of pole pairs;
\(R_k,\ S_k\) are the machine-geometry coefficients.
For a uniform air gap under the poles,
\[ R_k=\frac{4\cdot \delta^2 \cdot \beta^2 \cdot k_\delta^2 \cdot b_z}{4\delta^2+(b_0-\varepsilon t_z)^2}, \tag{5} \]
\[ S_k=\frac{\beta^2 \cdot k_\delta^2 \cdot (b_0-\varepsilon t_z)^2 \cdot b_z}{4\delta^2+(b_0-\varepsilon t_z)^2}, \tag{6} \]
where
\(\delta\) is the value of the air gap under the center of the pole;
\(k_\delta\) is the air-gap coefficient;
\(\beta\) is the ratio of the amplitude of the alternating component of the magnetic flux to \(B_\delta\);
\(b_z\) is the width of the armature tooth;
\(b_0\) is the slot-opening width;
\(t_z\) is the tooth pitch;
\(\varepsilon t_z\) is the difference between the pole-shoe width and an integral number of armature tooth divisions.
It follows from expressions (3)—(6) that the specific magnetic-attraction forces are a function of the coefficient \(\beta\), that is, ultimately, of the ratio \(\frac{b_0}{\delta}\). The influence of a change in the ratio \(\frac{b_0}{\delta}\) on the vibration level of the machine was investigated on an experimental setup built on the basis of a type P-40 dc motor. The armature of the machine under test has 31 slots with a skew of one tooth division and \(t_z=14\ \text{mm}\), \(b_z=10.5\ \text{mm}\), \(b_0=3.5\ \text{mm}\), \(l=60\ \text{mm}\). The number of machine poles is \(2p=4\). The pole geometry is determined by the parameters \(b_p=66\ \text{mm}\); \(\alpha=0.69\); \(\varepsilon=0.7\); \(\varepsilon t_z=10\ \text{mm}\). The air-gap value of the machine was taken equal to 1, 1.5, and 2 mm. In this case, for:
\[ \delta=1\ \text{mm},\quad \frac{b_0}{\delta}=3.5\ \text{and}\ \beta=0.25; \]
\[ \delta=1.5\ \text{mm},\quad \frac{b_0}{\delta}=2.3\ \text{and}\ \beta=0.18; \]
\[ \delta=2\ \text{mm},\quad \frac{b_0}{\delta}=1.75\ \text{and}\ \beta=0.12. \]
Table 1
| $\delta$, mm | $B_\delta$, T | $F_{\mathrm{st}}$, A | $F_\delta$, A | $\Sigma F$, A | $i_b$, A |
|---|---|---|---|---|---|
| 1 | 0.76 | 986 | 704 | 1690 | 0.85 |
| 1.5 | 0.76 | 986 | 1056 | 2042 | 1.02 |
| 2.0 | 0.76 | 986 | 1408 | 2394 | 1.21 |
Note: $F_{\mathrm{st}}$, $F_\delta$, $\Sigma F$ are the magnetomotive forces required to carry the magnetic flux through the magnetic circuit, the air gap, and the total mmf.
The amplitude of the magnetic pull forces when the air gap is increased to 1.5 and 2 mm decreases, according to (3–6), respectively by factors of 2 and 4.2. To keep the magnitude of the main magnetic flux unchanged—
Fig. 1. Vibrograms of the motor frame in the zone of the commutating poles (pt. 1), main poles (pt. 3), and motor feet (pt. 7) at different air-gap values: $\delta = 1;\ 1.5$ and $2$ mm.
... the gap, increasing the air gap in the investigations is accompanied by an increase in the excitation current. The calculated values of the excitation current are presented in Table 1.
For the above values of the air gap and the corresponding excitation currents, vibrations of the frame were recorded at two characteristic points: along the line of attachment of the main poles (point 3) and the commutating poles (point 1), and vibrations of the machine feet (point 7). The vibrograms were recorded while varying the rotational speed of the motor, operating under no-load conditions, from 1000 to 3000 rpm.
The instruments used for measuring vibration were an SM-211 vibrometer with a KD-13 pickup and a Marconi frequency analyzer. The results of the investigations are presented in the form of vibrograms in Fig. 1.
The largest vibration magnitudes occur at \(\delta = 1\) mm. At certain rotational speeds of the machine, owing to the resonance phenomenon, the vibration level of the frame reaches 87 dB, while the vibration level of the feet exceeds 90 dB. As the air-gap magnitude is increased, a steady decrease in the machine vibrations occurs. The reduction in vibration is especially significant when changing from a gap of 1 mm to \(\delta = 1.5\) mm. For both sections of the machine frame and the feet, the vibrations decreased on average by a factor of 3, and at individual speeds by a factor of 5 or more. When the machine gap was increased to 2 mm, the frame vibrations decreased by a factor of 4 in comparison with the variant \(\delta = 1\) mm. The foot vibrations (with the exception of vibrations measured at machine speeds in the range 2800–3000 rpm) also decreased by a factor of 4. Attention is drawn to the disproportionate decrease in machine vibrations with a reduction in the ratio \(\dfrac{b_0}{\delta}\); an increase in the air gap from 1 to 1.5 mm affects the machine vibration level much more strongly than an increase in the gap from 1.5 to 2 mm.
The investigations carried out show that, when selecting the magnitude of the air gap, it is necessary to coordinate it with the permissible vibration level of the machine.
References
- V. P. Shuyskii. Calculation of Electrical Machines. Moscow, “Energiya,” 1968.
- N. P. Ermolin. Magnetic noise of a direct-current machine. Izvestiya LETI, vol. 28, 1955.