. ^1? . 

L1qa 
L1qb 
L1qc 
Liqf 
Llqld 
Llqlq 

^s 

Lss 
Msr 

is 
^r 

Mrs 
Lrr 

ir 
where selfinductances are denoted by two like subscripts, and mutual inductances are denoted by two unlike subscripts. In matrix form
where Lss represents the stator selfinductance matrix, and Lrr represents the rotor selfinductance matrix. The statorrotor and rotorstator mutual inductances are represented by Msr and Msr respectively.
The synchronous generator is thus represented as a group of magnetically coupled circuits. The circuits are shown schematically in Fig. 3.25. Each of the windings has thus its own resistance, selfinductance and mutual inductances with respect to every other winding. Notice that most inductances in Eq. (3.75) depend on the angular position of the rotor [222].
Inductances of salientpole generator
The selfinductance of any stator winding varies periodically from a maximum (when the directaxis coincides with the phase axis) to a minimum (when the quadratureaxis is in line with the phase axis). The selfinductance Laa for example, will reach a maximum for 0e = 0°, a minimum for 0e = 90° and maximum again for 0e = 180° and so on. That is, Laa has a period of 0e = 180 electrical degrees and can be exactly represented by a series of cosines of even harmonics of angle [246]. Because of the rotor symmetry, the diagonal elements of the submatrix Lss are represented as

 Figure 3.25: Schematic representation of mutually coupled circuits.
where both Ls and Lm are constants (Ls > Lm), and 0e is the angle between the directaxis and the magnetic axis of phase a in electrical degrees as shown in Fig. 3.24. Since the angle included in one pole pair p is 360 electrical degrees, the angle de in electrical units is related to the mechanical angle 0m through the number of polepairs p as follows
The air gap of a salientpole synchronous generator varies along the inner circumference of the stator. Consequently the mutual inductances between any two stator phases are also periodic functions of the electrical angle 6e, and hence vary with time. It can be concluded from symmetry considerations that the mutual inductance between phase a and b should have a negative maximum when the pole axis is lined up 30° behind phase a, or 30° ahead of phase b, and a negative minimum when it is midway between the two phases. Thus, for a machine with sinusoidallydistributed windings, the variations of the stator mutual inductances, i.e. the offdiagonal elements of submatrix Lss can be represented as follows
Lab 
= Lba 
Lbc 
= Lcb 
Lca 
= Lac 
where \MS  > Lm [246]. Notice that the signs of the mutual inductance terms depend upon assumed current directions and circuit orientation.
The elements of the submatrix Lrr consist of rotor selfinductances and mutual inductance between any two circuits both in directaxis (or in quadratureaxis). All the rotor selfinductances, i.e. the diagonal elements of submatrix Lrr, are constant since the effects of stator slots and saturation are neglected. They are represented with single subscript notation
Lff 
= Lf 
ldld 
= Lld 
'lqlq 
The mutual inductance between any two circuits both in directaxis (or both in quadratureaxis) is constant. The mutual inductance between any rotor directaxis circuit and quadratureaxis circuit does not exist, thus
Lfld 
= L1 df = 
Mr 
Lflq 
= Liqf = 
0 
Lldlq 
= Llq id = 
This are the offdiagonal elements of submatrix Lrr.
Finally, consider the mutual inductances between stator and rotor circuits. Obviously, these are periodic functions of the electrical angle ee. Because only the spacefundamental component of the produced flux links the sinusoidally distributed stator, all statorrotor mutual inductances vary sinusoidally, reaching a maximum when the two windings in question align. Thus, their variations can be written as follows
Laf 
= Lfa ' 
= Mf cos ee 

Lbf 
= Lfb = 
= Mf cos (ee  
i n 
Lcf 
= Lfc = 
= Mf cos (ee + 
i n 
Laid 
= Llda 
= Mld cos ee 

Lbld 
= Lldb 
= Mld cos(ee 
 i n 
Lc1d 
= Lldc 
= Mld cos (ee 
+ 1 n 
Lalq 
= Llqa 
= Mlq sin ee 

Lblq 
= Llqb 
= Mlq sin (ee ■ 
 In) 
Lclq 
= Llqc 
= Mlq sin (ee 
It follows from above equations that the rotorstator mutual inductance matrix is equal to the transpose of the statorrotor mutual inductance matrix. Thus, Msr = MTs in Eq. (3.75).
The dynamic behavior of a synchronous, salientpole generator is thus described by Eq. (3.73) and the timevarying coefficients are given by Eqs. (3.76)(3.80). Analysis of the dynamic behavior can thus be accomplished by the solution of a set of simultaneous coupledcircuit differential equations. The solution of these equations is complicated due to the fact that the inductances between the statorphase windings and the rotor circuits are a function of the rotor angle 6e (and hence change with time). This complication can be avoided by using Park's transformation [211, 212].
Park's transformation
The key idea of Park's transformation is to express the stator flux linkages in the rotating d, q reference system instead of the normal stator fixed reference system. The stator windings are replaced with two fictitious windings which are fixed with respect to the rotor. One winding is chosen to coincide with the directaxis, and the other with the quadratureaxis.
Since the axes of the rotor windings are already along the direct and quadratureaxes, the transformation needs only to be applied to the stator quantities. In order to ensure that the rotor quantities remain unaffected, Park's transformation matrix T0dq is expanded with a 3 x 3 identity matrix I3 as follows
with
V2 cos(p6n sin(p 9n
Todq 0
0 I3

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