The reactances of synchronous machines comprise a set of characteristic constants used in the theory of
synchronous machines.[1] Technically, these constants are specified in units of the
electrical reactance (
ohms), although they are typically expressed in the
per-unit system and thus
dimensionless. Since for practically all (except for the tiniest) machines the resistance of the coils is negligibly small in comparison to the reactance, the latter can be used instead of (
complex)
electrical impedance, simplifying the calculations.[2]
Two reactions theory
The
air gap of the machines with a
salient pole rotor is quite different along the pole axis (so called direct axis) and in the orthogonal direction (so called quadrature axis).
Andre Blondel in 1899 proposed in his paper "Empirical Theory of Synchronous Generators" the two reactions theory that divided the
armaturemagnetomotive force (MMF) into two components: the direct axis component and the quadrature axis component. The direct axis component is aligned with the magnetic axis of the rotor, while the quadrature (or transverse) axis component is perpendicular to the direct axis.[3] The relative strengths of these two components depend on the design of the machine and the operating conditions. Since the equations naturally split into direct and quadrature components, many reactances come in pairs, one for the direct axis (with the index d), one for the quadrature axis (with the index q). In the machines with a
cylindrical rotor the air gap is uniform, the reactances along the d and q axes are equal,[4] and d/q indices are frequently dropped.
States of the generator
The
flux linkages of the generator vary with its state. Three states are considered:[5]
the steady-state is the normal operating condition with the armature
magnetic flux going through the rotor;
the sub-transient state is the one the generator enters immediately after the fault (short circuit). In this state the armature flux is pushed completely out of the rotor. The state is very brief, as the current in the
damper winding quickly decays allowing the armature flux to enter the rotor poles only. The generator goes into transient state;
in the transient state the flux is still out of the
field winding of the rotor. The transient state decays to steady-state in few
cycles.[6]
The sub-transient and transient states are cheracterized by significantly smaller reactances.
The synchronous reactances are exhibited by the
armature in the steady-state operation of the machine.[8] The three-phase system is viewed as a superposition of two: the direct one, where the maximum of the phase current is reached when the pole is oriented towards the winding and the quadrature one, that is 90° offset.[9]
The per-phase reactance can be determined in a mental experiment where the rotor poles are perfectly aligned with a specific angle of the phase field in the armature (0° for , 90° for the ). In this case, the reactance X will be related with the
flux linkage and the phase current I as , where is the
circular frequency. [10] The conditions for this mental experiment are hard to recreate in practice, but:
when the armature is short-circuited, the flowing current is practically all reactive (as the coil resistance is negligible), thus under the short-circuit condition the poles of the rotor are aligned with the armature
magnetomotive force;
when the armature is left open-circuit, the voltage on the terminals is also aligned with the same phase and is equal to . If saturation is neglected, the flux linkage is the same.
Therefore, the direct synchronous reactance can be determined as a ratio of the voltage in open condition to short-circuit current : . These current and voltage values can be obtained from the
open-circuit saturation curve and the
synchronous impedance curve.[11]
The synchronous reactance is a sum of the leakage reactance and the reactance of the armature itself (): .[12]
Park, R. H.; Robertson, B. L. (1928). "The Reactances of Synchronous Machines". Transactions of the American Institute of Electrical Engineers. 47 (2). Institute of Electrical and Electronics Engineers (IEEE): 514–535.
doi:
10.1109/t-aiee.1928.5055010.
ISSN0096-3860.
Prentice, B. R. (1937). "Fundamental Concepts of Synchronous Machine Reactances". Transactions of the American Institute of Electrical Engineers. 56 (12). Institute of Electrical and Electronics Engineers (IEEE): 1–21.
doi:
10.1109/t-aiee.1937.5057505.
ISSN0096-3860.
El-Serafi, A.M.; Abdallah, A.S. (1992). "Saturated synchronous reactances of synchronous machines". IEEE Transactions on Energy Conversion. 7 (3). Institute of Electrical and Electronics Engineers (IEEE): 570–579.
doi:
10.1109/60.148580.
ISSN0885-8969.