A RATIONAL FORM OF NOTATION FOR THE DECOMPOSITION OF MULTIPHASE SYSTEMS INTO SYMMETRICAL COMPONENTS

M. P. TABINSKY

(Recommended by the scientific seminar of the Electromechanical Faculty).

In [1] an exhaustive justification is given for the decomposition of a multiphase system into symmetrical components. To determine the phase quantities of a real unbalanced system by means of symmetrical components, the equations are given in final form as follows:

\[ \left. \begin{aligned} E_a &= E_{a0}+E_{a1}+E_{a2}+\ldots\\ E_b &= E_{a0}+r^{\,n-1}E_{a1}+r^{\,n-2}E_{a2}+r^{\,n-3}E_{a3}+\ldots\\ E_c &= E_{a0}+r^{\,n-2}E_{a1}+r^{\,n-4}E_{a2}+r^{\,n-6}E_{a3}+\ldots\\ E_d &= E_{a0}+r^{\,n-3}E_{a1}+r^{\,n-6}E_{a2}+r^{\,n-9}E_{a3}+\ldots\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \end{aligned} \right\} \tag{1} \]

The equations for finding the symmetrical components of each sequence are as follows:

\[ \left. \begin{aligned} E_{a0} &= \frac{1}{n}(E_a+E_b+E_c+E_d+\ldots)\\ E_{a1} &= \frac{1}{n}(E_a+rE_b+r^2E_c+r^3E_d+\ldots)\\ E_{a2} &= \frac{1}{n}(E_a+r^2E_b+r^4E_c+r^6E_d+\ldots)\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \end{aligned} \right\} \tag{2} \]

Here \(n\) is the number of phases or the number of specified vectors,

\[ r=e^{i\Theta_1} \]

is the rotation factor,

\[ \Theta_1=\frac{p\cdot360^\circ}{n} \]

is the characteristic angle of the first sequence,

\[ p=1,\ 2,\ 3\ldots \]

When the number of phases is more than twenty, the form of writing equations (1) and (2) becomes inconvenient, and the occurrence of accidental errors is not excluded. Therefore it is expedient to give a general formula for finding any sequence. For this purpose, let us denote the vectors of an unbalanced multiphase system by a lowercase letter with a subscript corresponding—

corresponding ordinal number of the vector, counted from the first in the clockwise direction (Fig. 1).

For denoting the vectors of systems of symmetrical components we shall use capital letters with a double subscript. The first digit of the subscript “\(k\)” is the sequence (\(k=0,1,2\ldots n-1\)). The second digit of the subscript, written after a stroke, denotes the ordinal number of the vector of the given sequence (Fig. 2).

The characteristic angle of the \(k\)-th sequence, \(\Theta_k\), on which the order of succession of the maxima of the phase quantities with increasing ordinal number depends, is found from the formula

\[ \Theta_k=\Theta_1\cdot k=\frac{p\cdot 360^\circ}{n}\cdot k . \tag{3} \]

Fig. 1.

Let us recall that the order of succession of the maxima should not be confused with the direction of rotation of the vectors, for which the standard direction is taken to be counterclockwise.

Fig. 2.

After applying the indicated notation, equations (1) and (2) may be replaced by the formulas

\[ a_i=\sum_{\kappa=0}^{\kappa=n-1} A_k\cdot r^{\kappa(i-1)(n-1)}, \tag{4} \]

\[ A_{\kappa/1}=\frac{1}{n}\sum_{i=1}^{i=n}\bar a_i\cdot r^{\kappa(i-1)} . \tag{5} \]

Here \(k=0,1,2\ldots n-1\) is the ordinal number of the sequence,

\(i=1,2,3\ldots n\) is the ordinal number of the phase of the unbalanced system,

\(n\) is the number of phases of the unbalanced system.

As an example, let us find the first symmetrical component of the fifteenth sequence of a twenty-phase unbalanced system

\[ A_{15/1}=\frac{1}{20}\sum_{i=1}^{i=20}\bar a_i\cdot r^{15(i-1)} = \]

\[ = \frac{1}{20}\left[ \bar a_1-\bar a_3+\bar a_5-\bar a_7+\bar a_9-\bar a_{11}+\bar a_{13}-\bar a_{15}+\bar a_{17}-\bar a_{19} + j\left(-\bar a_2+\bar a_4-\bar a_6+\bar a_8-\bar a_{10}+\bar a_{12}-\bar a_{14}+\bar a_{16}-\bar a_{18}+\bar a_{20}\right) \right]. \]

Conclusions

The proposed form of notation for the decomposition of multiphase systems into symmetrical components is convenient for use and memorization.

Literature

1. K. F. Wagner and R. D. Evans. The Method of Symmetrical Components, ONTI, 1936.