## Us [ Ua Ub UCT

Rs 0

is

d

's

0 Rr

ir

— dtt

It is assumed that the flux linkages and r^r in Eq. (3.73) are linearly related to the six currents ia, ib, ic, if, i1d, and i1g by a 6 x 6 inductance matrix

^a

Laa

Lab

Lac

Laf

Laid

Lalq

ia

^b

Lba

Lbb

Lbc

Lbf

Lbid

Lblq

ib

^c

Lca

Lcb

Lcc

Lcf

Lcld

Lclq

ic

^f

Lfa

Lfb

Lfc

Lff

Lf1d

Lf lq

if

^ld

L1da

Lldb

Lldc

L1df

Lldld

Lldlq

ild

. ^1? .

L1qa

L1qb

L1qc

Liqf

Llqld

Llqlq

 ^s Lss Msr is ^r Mrs Lrr ir

where self-inductances are denoted by two like subscripts, and mutual inductances are denoted by two unlike subscripts. In matrix form

where Lss represents the stator self-inductance matrix, and Lrr represents the rotor self-inductance matrix. The stator-rotor and rotor-stator 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, self-inductance 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 salient-pole generator

The self-inductance of any stator winding varies periodically from a maximum (when the direct-axis coincides with the phase axis) to a minimum (when the quadrature-axis is in line with the phase axis). The self-inductance 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 direct-axis 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 pole-pairs p as follows

The air gap of a salient-pole 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 sinusoidally-distributed windings, the variations of the stator mutual inductances, i.e. the off-diagonal 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 self-inductances and mutual inductance between any two circuits both in direct-axis (or in quadrature-axis). All the rotor self-inductances, 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 direct-axis (or both in quadrature-axis) is constant. The mutual inductance between any rotor direct-axis circuit and quadrature-axis circuit does not exist, thus

Lfld

= L1 df =

Mr

Lflq

= Liqf =

0

Lldlq

= Llq id =

This are the off-diagonal 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 space-fundamental component of the produced flux links the sinusoidally distributed stator, all stator-rotor 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 rotor-stator mutual inductance matrix is equal to the transpose of the stator-rotor mutual inductance matrix. Thus, Msr = MTs in Eq. (3.75). The dynamic behavior of a synchronous, salient-pole generator is thus described by Eq. (3.73) and the time-varying 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 coupled-circuit differential equations. The solution of these equations is complicated due to the fact that the inductances between the stator-phase 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 direct-axis, and the other with the quadrature-axis. Since the axes of the rotor windings are already along the direct and quadrature-axes, 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

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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