Abstract
Nowadays, the induction machine is by far the most commonly used electrical machine in electrical drives (for applications that require a highly dynamic behaviour or where energy efficiency or compactness is primordial the permanent magnet synchronous motor may be preferred, however). The main reasons for this are its straightforward and robust construction and its quite efficient energy conversion. Moreover, the last 30 or so years variable speed operation of induction machines using power electronic converters has ousted almost completely the DC commutator machine in variable speed applications. This chapter starts from the traditional transformer properties of an induction motor at standstill. Then, the operating principle of an induction machine is explained both intuitively and from a more mathematical point of view. In the subsequent sections we treat the energy conversion and torque, equivalent circuits and equations for an induction machine, the current locus and single-phase induction machines. Much attention is paid to per-unit values and scaling laws as these determine the behaviour of the machine to a great extent.
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Notes
- 1.
This is the emf in the reference winding 1U; the first index 1 indicates the winding where it is induced while the second index 1 indicates where the flux originates.
- 2.
Calculate the ratio of the (cyclic) magnetising field inductances. If the number of phases of stator and rotor is the same, would you indeed expect this result?
- 3.
One obtains the same transformation ratio for a rotation in the rotation direction of the field or opposite to it; it can be (e.g. graphically) shown, however, that dependent on the direction of the active current one or the other is slightly advantageous.
- 4.
A squirrel cage winding consists of (aluminium or copper) bars cast inside slots in the rotor core and that are short-circuited by short-circuit rings cast at the same time; it can be proved that a cage winding behaves as a normal multiphase symmetrical winding as to the fundamental.
- 5.
In fact, this rotor current will, in accordance with Lenz’ law, try to reduce the field and thus oppose the mmf by the stator current (which will have to increase again).
- 6.
Remark that we will use the notation X for a reactance referred to the primary frequency. \(X_{2\sigma }\) is thus the rotor leakage reactance referred to the primary frequency.
- 7.
Also the number of poles has a large effect on the magnetising inductance: with increasing number of pole pairs the magnetising inductance decreases as it varies approximately as \(1/\sqrt{N_{p}}\). As a result the leakage coefficient \(\sigma \) is also much higher for induction machines, the more so as the number of pole pairs is larger.
- 8.
If there was only a stator current and mmf, the field would be that resulting from the stator mmf; actually both stator and rotor mmfs contribute to the resulting field, except in no-load.
- 9.
In accordance with the M-convention we reversed the positive direction of \(I_{2}^{'}\).
- 10.
Remark however that a negative mechanical power does not necessarily implies generating! Conversely, however, generating always implies input of mechanical power.
- 11.
Prove that with non-zero stator resistance the pull-out slip slightly decreases and the pull-out torque decreases for motoring and increases for generating.
- 12.
In fact, the separate leakages of stator and rotor cannot be measured, see the remarks at the end of this section.
- 13.
Show that all these expressions are equivalent to those derived above for the original circuit.
- 14.
This is easily shown by introducing \(\theta \) and substituting \(s/s_{po}=\tan \theta \) in Eq. 4.106.
- 15.
Work out that the figure also gives the correct sign for generating and braking (all segments are oriented).
- 16.
As we know, this also is the case for a transformer.
- 17.
The notation n (for nominal) is used for rated values.
- 18.
Calculate the optimal power factor from Fig. 4.19 and compare to the power factor at rated slip.
- 19.
In the past additional rotor resistances were also used for speed regulation, but since the advent of power electronics this method for speed control all but disappeared.
- 20.
That the magnetising inductance decreases with the number of poles can be readily understood considering that a high number of pole pairs results in multiple crossings of the air-gap over the circumference.
- 21.
This minimum air-gap length also depends of the size, with an absolute minimum of usually 0.25Â mm.
- 22.
This may also be derived by considering the counter-rotating current layers and the counter-rotating fields corresponding to alternating current layers: from symmetry the torques resulting from these cancel out.
- 23.
This is similar to a transformer where the load current tries to decrease the main field; the primary current will of course increase trying to keep the total flux at the same level but because of the leakage inductance the main flux will decrease. For this case of an induction machine the equivalent rotor resistance RÂ /Â s for the positive rotating field is larger than the resistance \(R/s'\) for the negative rotating field and thus the remaining field for the former will be larger than the one for the latter.
- 24.
This single-phase machine is in fact equivalent with two mechanically connected machines that are electrically series connected, one for the left and one for the right rotation direction.
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Melkebeek, J.A. (2018). The Induction Machine. In: Electrical Machines and Drives. Power Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-72730-1_4
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DOI: https://doi.org/10.1007/978-3-319-72730-1_4
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