Reactances of synchronous machines

The reactances of synchronous machines is a set of characteristic constants used in the theory of synchronous machines. Technically, these constants specified in units of the electrical reactance (ohms), although they are typically expressed in the per-unit system and thus dimensionless.

Two reactions theory

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. The relative strengths of these two components depend on the design of the machine and the operating conditions. Since the equation 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, and d/q indices are frequently dropped.

List of reactances

Das identifies the following reactances:

• leakage reactance $X_{l}$;
• subtransient reactance $X''_{d}$;
• transient reactance $X'_{d}$;
• synchronous reactance $X_{d}$;
• quadrature axis reactances $X_{q}$, $X'_{q}$, $X''_{q}$, counterparts to $X_{d}$, $X'_{d}$, $X''_{d}$;
• negative sequence reactance $X_{2}$;
• zero sequence reactance $X_{0}$;
• Potier reactance $X_{P}$.

Sources

• 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. ISSN 0096-3860.
• 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. 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. 7 (3). Institute of Electrical and Electronics Engineers (IEEE): 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.