Reactances of synchronous machines

A diagram of a salient pole machine with direct (d) and quadrature (q) axes

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 armature magnetomotive 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.

The flux linkages of the generator vary with its state. Three states are considered:[5]

1. the steady-state is the normal operating condition with the armature magnetic flux going through the rotor;
2. 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;
3. 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.

Das[7] identifies the following reactances:

• leakage reactance ${\displaystyle X_{l}}$;
• synchronous reactance ${\displaystyle X_{d}}$ (also ${\displaystyle X_{S}}$[2]);
• transient reactance ${\displaystyle X'_{d}}$;
• subtransient reactance ${\displaystyle X''_{d}}$;
• quadrature axis reactances ${\displaystyle X_{q}}$, ${\displaystyle X'_{q}}$, ${\displaystyle X''_{q}}$, counterparts to ${\displaystyle X_{d}}$, ${\displaystyle X'_{d}}$, ${\displaystyle X''_{d}}$;
• negative sequence reactance ${\displaystyle X_{2}}$;
• zero sequence reactance ${\displaystyle X_{0}}$;
• Potier reactance ${\displaystyle X_{P}}$.

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 ${\displaystyle X_{d}}$, 90° for the ${\displaystyle X_{q}}$). In this case, the reactance X will be related with the flux linkage ${\displaystyle \Psi }$ and the phase current I as ${\displaystyle X=\omega {\frac {\Psi }{I}}}$, where ${\displaystyle \omega }$ 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 ${\displaystyle \omega \Psi }$. 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 ${\displaystyle V_{OPEN}}$ to short-circuit current ${\displaystyle I_{SC}}$: ${\displaystyle X_{d}={\frac {V_{OPEN}}{I_{SC}}}}$. 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 ${\displaystyle X_{l}}$ and the reactance of the armature itself (${\displaystyle X_{a}}$): ${\displaystyle X_{d}=X_{l}+X_{a}}$.[12]

• Park, R. H.; Robertson, B. L. (1928). "The Reactances of Synchronous Machines". Transactions of the American Institute of Electrical Engineers. Institute of Electrical and Electronics Engineers (IEEE). 47 (2): 514–535. doi:10.1109/t-aiee.1928.5055010. ISSN 0096-3860.
• Prentice, B. R. (1937). "Fundamental Concepts of Synchronous Machine Reactances". Transactions of the American Institute of Electrical Engineers. Institute of Electrical and Electronics Engineers (IEEE). 56 (12): 1–21. doi:10.1109/t-aiee.1937.5057505. ISSN 0096-3860.
• Heydt, G.; Kalsi, S.; Kyriakides, E. (2003). "A Short Course on Synchronous Machines and Synchronous Condensers" (PDF). Arizona State University, American Superconductor.
• El-Serafi, A.M.; Abdallah, A.S. (1992). "Saturated synchronous reactances of synchronous machines". IEEE Transactions on Energy Conversion. Institute of Electrical and Electronics Engineers (IEEE). 7 (3): 570–579. doi:10.1109/60.148580. ISSN 0885-8969.
• Das, J. C. (2017). Short-Circuits in AC and DC Systems: ANSI, IEEE, and IEC Standards. CRC Press. ISBN 978-1-4987-4542-0. Retrieved 2023-07-02.
• Gieras, J.F.; Shen, J.X. (2022). Modern Permanent Magnet Electric Machines: Theory and Control. CRC Press. ISBN 978-1-000-77700-0. Retrieved 2023-07-03.
• Deshpande, M.V. (2011). Electrical Machines. Prentice Hall India Pvt., Limited. ISBN 978-81-203-4026-8. Retrieved 2023-07-03.
• Machowski, J.; Bialek, J.; Bumby, J.R. (1997). Power System Dynamics and Stability. Wiley. ISBN 978-0-471-95643-3. Retrieved 2023-07-04.
• Ramar, S.; Kuruseelan, S. (2013). Power System Analysis. PHI Learning. ISBN 978-81-203-4733-5. Retrieved 2023-07-04.