Abstract
Not only induction machines, but synchronous machines as well may exhibit dynamic problems when fed by a variable frequency supply. In this chapter we analyse and represent the dynamic behaviour of synchronous machines in an analogous way as we have done for the induction machine. The results are quite similar. However, for synchronous machines such scaling laws do not exist to the same extent as for induction machines.
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- 1.
You may also check the Wikipedia pages on the Clarke transformation (which is also called the \(\alpha \beta \)-transformation).
- 2.
In the literature, this is sometimes called doubly-coupled leakage.
- 3.
In fact, as for an induction machine or transformer, these turns ratios cannot be measured.
- 4.
In the URS, the load angle is positive if the stator voltage leads the emf.
- 5.
From the equality of the voltages, it follows that the number of effective windings should be the same as in the original machine. In contrast, from the equivalence of the mmf (cf. the inductances of the d- and q-windings compared to the original ones), the new number of effective stator windings should be larger by a factor of 3 / 2.
- 6.
However, by appropriately choosing the turns ratios, we may eliminate this one as well - prove this.
- 7.
The difference is usually small, however.
- 8.
These use AC excitation in the q-axis; however, how can stable operation be achieved with only q-axis excitation?.
- 9.
There are now 20 original parameters (13 electrical, 3 mechanical and 4 supply and excitation parameters) for the drive while the system order is 7.
- 10.
As there is only one rotor winding in the q-axis in our model, it is sometimes referred to as a transient time constant, although the term ‘subtransient time constant’ is more common.
- 11.
In fact, in a synchronous machine, the rotor excitation current is DC, but the AC rotor current in an induction machine can equally be regarded as an excitation current with slip frequency, inducing an emf in the stator with stator frequency and providing a torque with the flux.
- 12.
Please prove that the gain we used for the DC machine also corresponds to a dynamic gain.
- 13.
Over-excitation tends to decrease the stability via its effect on the open-loop zeros; note that even at rated resulting flux, a synchronous machine can be over-excited.
- 14.
I.e. more or less symmetrical in the two axes.
- 15.
Calculate both dynamic and static synchronous torques for \(\delta _{0}=0\); it will become clear that the static synchronous torque is obviously zero for \(\gamma _{dq}=1\).
- 16.
For the same value of \(e_{v}\) and \(|\delta |\), \(E_{p0}>E_{q0}\) for \(\delta >0\) and \(E_{p0}<E_{q0}\) for \(\delta <0\) from which the distinct shift of the complex zeros can be derived. You may verify this shift by calculating the low-frequency root loci for zero, positive and negative load angles, for example in Matlab.
- 17.
Verify this using a Matlab model.
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Melkebeek, J.A. (2018). Modelling and Dynamic Behaviour of Synchronous Machines. In: Electrical Machines and Drives. Power Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-72730-1_28
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DOI: https://doi.org/10.1007/978-3-319-72730-1_28
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72729-5
Online ISBN: 978-3-319-72730-1
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