Reactances of synchronous machines#Two reactions theory

File:Salientpole axii.png

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.{{sfn | Gieras | Shen | 2022 | p=211}} 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 $X\_d$(with the index d), one for the quadrature axis $X\_q$ (with the index q). This is often using direct-quadrature-zero transformation.

In machines with a cylindrical rotor the air gap is uniform, the reactances along the d and q axes are equal,{{sfn | Deshpande | 2011 | p=315}} and d/q indices are frequently dropped.

The flux linkages of the generator vary with its state. Usually applied for transients after a short circuit current. Three states are considered:{{sfn | Machowski | Bialek | Bumby | 1997 | pp=102-103}}

1. the steady-state is the normal operating condition with the armature magnetic flux going through the rotor;
2. the sub-transient state ($X\text{'}\text{'}\_d$) 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 ($X\text{'}\_d$) the flux is still out of the field winding of the rotor. The transient state decays to steady-state in few cycles.{{sfn | Ramar | Kuruseelan | 2013 | p=20}}

The sub-transient ($X\text{'}\text{'}\_d$) and transient ($X\text{'}\_d$) states are cheracterized by significantly smaller reactances.

The nature of magnetic flux makes it inevitable that part of the flux deviates from the intended "useful" path. In most designs, the productive flux links the rotor and stator; the flux that links just the stator (or the rotor) to itself is useless for energy conversion and thus is considered to be wasted leakage flux (stray flux). The corresponding inductance is called leakage inductance. Due to the presence of air gap, the role of the leakage flux is more important in a synchronous machine in comparison to a transformer. {{sfn|Lipo|2017|p=67}}

The synchronous reactances are exhibited by the armature in the steady-state operation of the machine.{{sfn|Das|2017|p=181}} 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.{{sfn|Park|Robertson|1928|p=515}}

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 $X\_d$, 90° for the $X\_q$). In this case, the reactance {{mvar|X}} will be related with the flux linkage $\backslash Psi$ and the phase current {{mvar|I}} as $X\; =\; \backslash omega\; \backslash frac\; \backslash Psi\; I$, where $\backslash omega$ is the circular frequency. {{sfn|Prentice|1937|p=7}} 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 $\backslash omega\; \backslash Psi$. 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 $V\_\{OPEN\}$ to short-circuit current $I\_\{SC\}$: $X\_d\; =\; \backslash frac\; \{V\_\{OPEN\}\}\; \{I\_\{SC\}\}$. These current and voltage values can be obtained from the open-circuit saturation curve and the synchronous impedance curve.{{sfn|Prentice|1937|p=8}}

The synchronous reactance is a sum of the leakage reactance $X\_l$ and the reactance of the armature itself ($X\_a$): $X\_d\; =\; X\_l\; +\; X\_a$.{{sfn | Machowski | Bialek | Bumby | 1997 | p=104}}

When analyzing unbalanced three-phase systems it is common to describe a system of symmetrical components. This models the machine by three components, each with a positive sequence reactance $X\_1$, a negative sequence reactance $X\_2$ and a

zero sequence reactance $X\_0$.

Das{{sfn|Das|2017|pp=180-182}} identifies the following reactances:

• leakage reactance $X\_l$. Potier reactance $X\_P$ is an estimate of the armature leakage reactance;
• synchronous reactance $X\_d$ (also $X\_S${{sfn | Klempner | Kerszenbaum | 2004 | p=144}});
• transient reactance $X\text{'}\_d$;
• subtransient reactance $X\text{'}\text{'}\_d$;
• quadrature axis reactances $X\_q$, $X\text{'}\_q$, $X$_q, counterparts to $X\_d$, $X\text{'}\_d$, $X$_d;
• negative sequence reactance $X\_2$;
• zero sequence reactance $X\_0$.

{{Reflist}}

• {{cite journal | last1=Park | first1=R. H. | last2=Robertson | first2=B. L. | title=The Reactances of Synchronous Machines | journal=Transactions of the American Institute of Electrical Engineers | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=47 | issue=2 | year=1928 | issn=0096-3860 | doi=10.1109/t-aiee.1928.5055010 | pages=514–535}}
• {{cite journal | last=Prentice | first=B. R. | title=Fundamental Concepts of Synchronous Machine Reactances | journal=Transactions of the American Institute of Electrical Engineers | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=56 | issue=12 | year=1937 | issn=0096-3860 | doi=10.1109/t-aiee.1937.5057505 | pages=1–21}}
• {{cite web |last1=Heydt |first1=G. |last2=Kalsi |first2=S. |last3=Kyriakides |first3=E. |title=A Short Course on Synchronous Machines and Synchronous Condensers |url=https://documents.pserc.wisc.edu/documents/general_information/presentations/presentations_by_pserc_university_members/heydt_synchronous_mach_sep03.pdf |publisher=Arizona State University, American Superconductor |date=2003}}
• {{cite journal | last1=El-Serafi | first1=A.M. | last2=Abdallah | first2=A.S. | title=Saturated synchronous reactances of synchronous machines | journal=IEEE Transactions on Energy Conversion | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=7 | issue=3 | year=1992 | issn=0885-8969 | doi=10.1109/60.148580 | pages=570–579}}
• {{cite book | last=Das | first=J. C. | title=Short-Circuits in AC and DC Systems: ANSI, IEEE, and IEC Standards | publisher=CRC Press | year=2017 | isbn=978-1-4987-4542-0 | url=https://books.google.com/books?id=PjwPEAAAQBAJ&pg=PA180 | access-date=2023-07-02}}
• {{cite book | last1=Gieras | first1=J.F. | last2=Shen | first2=J.X. | title=Modern Permanent Magnet Electric Machines: Theory and Control | publisher=CRC Press | year=2022 | isbn=978-1-000-77700-0 | url=https://books.google.com/books?id=KvGaEAAAQBAJ&pg=PA211 | access-date=2023-07-03}}
• {{cite book | last=Deshpande | first=M.V. | title=Electrical Machines | publisher=Prentice Hall India Pvt., Limited | year=2011 | isbn=978-81-203-4026-8 | url=https://books.google.com/books?id=JKsH_3AzYMMC&pg=PA315 | access-date=2023-07-03}}
• {{cite book |last1=Klempner |first1=Geoff |last2=Kerszenbaum |first2=Isidor |title=Operation and Maintenance of Large Turbo-Generators |date=2004 |publisher=John Wiley & Sons |isbn=978-0-471-61447-0 |url=https://books.google.com/books?id=DPdkLp_crLAC&pg=PA144 |language=en}}
• {{cite book | last1=Machowski | first1=J. | last2=Bialek | first2=J. | last3=Bumby | first3=J.R. | title=Power System Dynamics and Stability | publisher=Wiley | year=1997 | isbn=978-0-471-95643-3 | url=https://books.google.com/books?id=L7o4TitajRIC&pg=PA104 | access-date=2023-07-04}}
• {{cite book | last1=Ramar | first1=S. | last2=Kuruseelan | first2=S. | title=Power System Analysis | publisher=PHI Learning | year=2013 | isbn=978-81-203-4733-5 | url=https://books.google.com/books?id=ahntULiMFgwC&pg=PA19 | access-date=2023-07-04}}

{{Characteristics of synchronous machines}}

• {{cite book | last=Lipo | first=T.A. | title=Analysis of Synchronous Machines | publisher=CRC Press | year=2017 | isbn=978-1-351-83272-4 | url=https://books.google.com/books?id=AglEDwAAQBAJ&pg=PA67 | access-date=2024-09-16}}

Category:Electrical engineering

Category:Electrical generators