Symmetrical Components: The Commonwealth and International Library: Applied Electricity and Electronics Division
By L. J. Myatt and P. Hammond
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Symmetrical Components - L. J. Myatt
L.J.M.
CHAPTER 1
The h Operator
Publisher Summary
This chapter provides an overview of h operator. When performing calculations on single-phase circuits an operator was used, which, when multiplied by a vector, had the effect of rotating the vector through an angle of 90°, and for this purpose the j operator was used. In dealing with three-phase quantities, and in particular when applying the method of symmetrical components, there are definite advantages to be gained by having an operator that is capable of rotating a vector through an angle of 120°, because of the angular displacement of balanced three-phase vectors. The chapter further discusses symmetrical three-phase voltages, positive-phase sequence, negative phase sequence, zero-phase sequence, and Star-delta impedance transformations.
In a three-phase system, the three-phase voltages or currents may be said to be balanced if they are sinusoidal and, when represented by vectors, are of equal magnitude and displaced from each other by equal phase angles of 120°. A three-phase circuit is also balanced, or symmetrical, when each of the three phases contain equal impedances. Furthermore, if a balanced system of three-phase voltages is applied across the terminals of a balanced three-phase network, then the currents flowing in each of the three phases will also be balanced.
The solution of a problem involving a balanced three-phase system may be found by considering only one phase in which the voltage, or current, is taken as the reference vector and solving for that phase alone as in the case of a single-phase problem. The magnitude of the currents and voltages in the other two phases will then be the same as in the reference phase but there will be a corresponding phase displacement of ±120° as appropriate.
If now the applied voltage is unbalanced, or the impedances in each phase are no longer identical, the three-phase currents will also become unbalanced and the problem cannot be solved by considering a single phase as in the above method. It would be possible in a simple problem of unbalance to find a solution by applying Kirchhoff’s laws, but in the more involved type of problem, such as that involving a power network, a less laborious method is necessary.
Considerable thought was given to the problem of unbalance, at the beginning of the century, with particular reference to the unbalanced loading of three-phase machines, and L. G. Stokvis showed that, under such conditions, the armature m.m.f. of a three-phase generator may be treated as two separate components, equivalent to the effect of two balanced loads. It remained for C. L. Fortesque to develop the method, now known as the method of symmetrical components, after working on problems associated with the use of phase balancers in the single-phase electrification of railways. Following on from this, he studied the problem of unbalance in general when, in 1918, he published his very important paper The method of symmetrical coordinates applied to the solution of polyphase networks
. In this paper he showed how it was possible for a set of unbalanced polyphase currents to be resolved into a number of component systems of balanced currents equal to the number of phases. A further development of the theory provides considerable simplification in solving problems involving unbalanced faults on power