Modal theory

This section discusses propagation constants and characteristic impedances and admittance matrices in the modal domain after reviewing the modal theory. 

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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... 

In this section, we have discussed the method that directiy applies eigenvalue theory. However, it is not efficient in terms of numerical computations as it requires the product of off-diagonal matrices. The method will be more complete with modal theory. 

Applying modal theory, the solutions are derived as explained in Section 1.3. With modal theory, since the coefficient matrix in Equation 1.141 is a diagonal matrix, the equation is also written as... 

Figure 2.8d indicates that aerial mode voltages exist even in the case of voltage application to all phases. This cannot be explained by conventional symmetrical component theory. Modal theory predicts aerial mode components at 28% on the upper phase, 7.6% on the middle phase, and 13% on the lower phase, which agrees well with the field test results of 20%, 4.3%, and 11%, respectively. 

In the case of a three-phase line, Equation 2.11 becomes a matrix, and we need to apply modal theory, described in Section 1.4. 

From the earlier results, it is clear that the product Z Y or the actual propagation constant matrix is diagonal and purely imaginary at the infinite frequency or in the perfect conductor case. This results in the fact that the attenuation is zero and the propagation velocity is a light velocity in free space on any phases. Also, it is noteworthy that the modal theory is not necessary as far as only the propagation constant is concerned, because it is already diagonal. 

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