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== Two reactions theory ==
[[File:Salientpole axii.png|thumb|A diagram of a salient pole machine with direct (d) and quadrature (q) axes]]
The [[air gap (electrical machine)|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 (electrical)|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 <math>X_d</math>(with the index d), one for the quadrature axis <math>X_q</math> (with the index q).
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. == States of the generator ==
The [[flux linkage]]s 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}}
# the ''steady-state'' is the normal operating condition with the armature [[magnetic flux]] going through the rotor;
# the ''sub-transient state'' (<math>X''_d</math>) 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'' (<math>X'_d</math>) the flux is still out of the [[field winding]] of the rotor. The transient state decays to steady-state in few [[cycle (AC current)|cycles]].{{sfn | Ramar | Kuruseelan | 2013 | p=20}}
The sub-transient (<math>X''_d</math>) and transient (<math>X'_d</math>) states are
==
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 (magnetic)|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}}
Das{{sfn|Das|2017|pp=180-182}} identifies the following reactances:▼
* leakage reactance <math>X_l</math>. [[Potier reactance]] <math>X_P</math> is an estimate of the armature leakage reactance;▼
* synchronous reactance <math>X_d</math> (also <math>X_S</math>{{sfn | Klempner | Kerszenbaum | 2004 | p=144}}); ▼
* transient reactance <math>X'_d</math>;▼
* subtransient reactance <math>X''_d</math>;▼
* quadrature axis reactances <math>X_q</math>, <math>X'_q</math>, <math>X''_q</math>, counterparts to <math>X_d</math>, <math>X'_d</math>, <math>X''_d</math>;▼
* [[negative sequence]] reactance <math>X_2</math>;▼
* [[zero sequence]] reactance <math>X_0</math>.▼
== Synchronous reactances ==
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The synchronous reactance is a sum of the leakage reactance <math>X_l</math> and the reactance of the armature itself (<math>X_a</math>): <math>X_d = X_l + X_a</math>.{{sfn | Machowski | Bialek | Bumby | 1997 | p=104}}
== Sequence network reactances ==
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 <math>X_1</math>, a [[negative sequence]] reactance <math>X_2</math> and a
[[zero sequence]] reactance <math>X_0</math>.
== List of reactances ==
▲Das{{sfn|Das|2017|pp=180-182}} identifies the following reactances:
▲* leakage reactance <math>X_l</math>. [[Potier reactance]] <math>X_P</math> is an estimate of the armature leakage reactance;
▲* synchronous reactance <math>X_d</math> (also <math>X_S</math>{{sfn | Klempner | Kerszenbaum | 2004 | p=144}});
▲* transient reactance <math>X'_d</math>;
▲* subtransient reactance <math>X''_d</math>;
▲* quadrature axis reactances <math>X_q</math>, <math>X'_q</math>, <math>X''_q</math>, counterparts to <math>X_d</math>, <math>X'_d</math>, <math>X''_d</math>;
▲* [[negative sequence]] reactance <math>X_2</math>;
▲* [[zero sequence]] reactance <math>X_0</math>.
== References ==
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