Symmetrical component transformation

As is well known, all alternating current (AC) power systems are basically three-phase circuits. This fact makes voltage, current, and impedance a 3-D matrix form. A symmetrical component transformation (i.e., Fortescue and Clarke transformations) is well known to deal with three-phase voltages and currents. However, the transformation cannot diagonalize an n X n impedance/admittance matrix. In general, modal theory is necessary to deal with an untransposed transmission line. In this chapter, modal theory is explained. By adopting modal... 

For tensors of higher rank we must ensure that the bases are properly normalized and remain so under the unitary transformations that correspond to proper or improper rotations. For a symmetric T(2) the six independent components transform like binary products. There is only one way of writing xx xx, but since xx x2 = x2 xx the factors xx and x2 may be combined in two equivalent ways. For the bases to remain normalized under unitary transformations the square of the normalization factor N for each tensor component is the number of combinations of the suffices in that particular product. F or binary products of two unlike factors this number is two (namely ij and ji) and so N2 = 2 and x, x appears as /2x,- Xj. The properly normalized orthogonal basis transforming like... 

The data in the figure illustrate that for the Ih direct sum the orders of the first three nonvanishing multipole moments of the Ceo molecule by finding the first three F(/) to have a component transforming as the totally symmetric representation that are found at levels 1, 6 and 10. 

In octahedral symmetry the fictitious S operator follows the T irrep. Its third symmetrized power transforms as the components of the /-harmonics and also subduces a T irrep, as indicated in Table 7.1. Rewrite the p- and /-parts of the IT I operators for the /g quartet state as a spin Hamiltonian of the fictitious spin. 

In strong magnetic fields o is axially symmetric. When transformed into its principal reference system (PAS) by using rotation matrices, the tensor is described by three principal components [i = 1,2,3) ... 

The prime on the mean and anisotropy parameters, a and y, respectively, denote values obtained from the polarizability derivative - defined in the sense of Equation [5], but in components of the tensor daij 8Q t rather than a j itself. In the case of gases and liquids, Pi is lower than for vibrations that are totally symmetric (vibrations transforming under the totally symmetric representation of the molecular point group), but exactly for for other vibrations that lower the molecular symmetry, since a is then zero. 

The magnitudes of the six (or nine) elementary force systems (the moment-tensor components) transform according to standard tensor laws under rotations of the coordinate system, so there exist many different combinations of elementary forces that are equivalent. In particular, for a symmetric (six-element) moment tensor one can always choose a coordinate system in which the force system consists of three orthogonal linear dipoles, so that the moment tensor is diagonal. In other words, a general point source can be described by three values (the principal moments) that describe its physics and three values that specify its orientation. 

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

Separating the even and odd components of the function F, by means of the projection operators F- and F produces functions that transform according to irreducible representations Ag and A of the group Ci, which consists of symmetry elements E and i. An analogous technique could be used to con-stmct functions symmetric and antisymmetric with respect to a mirror plane or a dyad. 

Piperazine-2,5-diones can be symmetric or asymmetric. Symmetric DKPs are readily obtained by heating amino acid esters,1179-181 whereas asymmetric DKPs are obtained directly from the related dipeptides under basic or, more properly, acid catalysis, or by cyclocondensation of dipeptide esters.1182-185 As an alternative procedure hexafluoroacetone can be used to protect/activate the amino acid for the synthesis of symmetric DKPs or of the second amino acid residue for synthesis of the dipeptide ester and subsequent direct cyclocondensation to DKPs.1186 The use of active esters for the cyclocondensation is less appropriate since it may lead to epimerization when a chiral amino acid is involved as the carboxy component in the cyclization reaction. Resin-bound DKPs as scaffolds for further on-resin transformations are readily prepared using the backbone amide linker (BAL) approach, where the amino acid ester is attached to the BAL resin by its a-amino group and then acylated with a Fmoc-protected amino acid by the HATU procedure, N -deprotection leads to on-resin DKP formation1172 (see Section 6.8.3.2.2.3). 

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

This equation can be solved by Laplace transform techniques and Mt expressed as modified spherical Bessel functions [28]. However, because the boundary conditions on M are radically symmetric, only the / = 0 (i.e. S-wave) component is of interest. 

The selection rules for Raman vibrational transitions are also readily derived from group theory. Here, the transition probability depends on integrals involving the components of the molecular polarizability matrix a. Since a is symmetric, it has only six independent components axx aw axi axr ayi aMf These six quantities can be shown14 to transform the same way the six functions... 

Electron spin may interact directly 22,67,123 with the oscillating electromagnetic field of a photon. In this case the transition operator,, /(t) of Section IV, does not commute with the symmetric group S f, so that direct transitions between pure singlet and pure triplet states are allowed. Selection rules for this type of transition may be derived, as has been done by Chiu,22 by noting that f(t) has components which transform as different representations of the double group SF 0 (see Sect. VIII). 

In a linear theory, the kinetic coefficients Ly are independent of the forces. They are, however, functions of the thermodynamic variables. In view of the Onsager relations, not only is the L matrix of the transport coefficients symmetric, but the transformed matrix is symmetric as well if the new fluxes are linearly related to the original ones. This also means that the new Ly (i st j) contain diagonal components of the original set. 

The anisotropic form of Fick s law would seem to complicate the diffusion equation greatly. However, in many cases, a simple method for treating anisotropic diffusion allows the diffusion equation to keep its simple form corresponding to isotropic diffusion. Because Dtj is symmetric, it is always possible to find a linear coordinate transformation that will make the Dij diagonal with real components (the eigenvalues of D). Let elements of such a transformed system be identified by a hat. Then... 

---