Oscillations of Coupled Windings

P. Α. Α Β Ε Τ Τ I G. Ε. A D A M S F. J. M A G I Ν Ν I S S A S S O C I A T E M E M B E R A I E E A S S O C I A T E M E M B E R A I E E M E M B E R A I E E

USUALLY, S I N G L E - W I N D I N G T H E O R Y has been applied to multiwinding transformers for the deter­

mination of the natural frequencies and, in some cases, yields results which agree very well with observation.

Previously, 1 , 2 it has been shown that all mutual flux linkages of a coil or winding must be taken into account for natural-frequency calculations. The mutual-induct­ance function M{x,s) between two elements of the primary winding located at distances χ and s, respectively, from one end of the winding depends upon both χ and s when a secondary winding is present and short-circuited, since the

i\ - ( 2 )

Λ \ \\

1 \ \ 1 \\ !\\ ι \ \ V ^ ( 3 )

V,

Fig. 1 ( left) . M u t u a l I n ­ductance funct ions at center of an exper imenta l coil.

7—Air Core, no second-ary; 2—Iron core, no secondary; 3—Iron-core short-circuited secondary; 4—Air or iron-core copper screen

0 10 2 0 3 0 4 0 5 0 6 0

PER CENT WINDING FROM MIDPOINT

Fig. 2 ( b e l o w ) . F u n d a ­m e n t a l vo l tage dis tr ibu­tions of pr imary w i n d i n g wi th isolated ne ut r a l for large va lues of g / R ( g a p / a v e r a g e rad ius ) , short-c ircuited s e c o n d a r y

flux pattern is changed. Fig, 1 shows the mutual-in­ductance function at the center (x = 50 per cent) of coil.

By making use of the mutual inductance function as modified by the effect of the secondary, the integral-equation method 2 was extended to take into account the effect of the secondary upon the fundamental oscillation of the transformer. Fig. 2 shows the fundamental voltage distribution for a transformer with concentric windings having the primary line end grounded, primary neutral open, and secondary short-circuited and grounded.

Since the integral equation method is limited to the fundamental oscillation and to turn ratios of primary to secondary greater than about 5 to 1, the equivalent circuit method was used to determine the higher harmonics,. the effect of turn ratios close to unity, and the effect of relative winding direction of primary and secondary.

In conclusion, it has been found by application of the integral-equation and equivalent-circuit methods that : (1) when the turn ratio is considerably different from unity, short-circuiting the secondary affects the frequencies with primary neutral isolated, but not the frequencies with primary neutral grounded ; (2) in power transformers, the ratio of the first natural frequencies with grounded and isolated neutral usually varies between 5 and 12, if the secondary is absent or open-circuited, and between 1.1 and 1.6 depending upon the gap between primary and second­ary, if the secondary is short-circuited; (3) the values of the frequencies of two coupled windings with primary neutral isolated decrease gradually as the secondary ter­minal conditions change as follows: both terminals iso­lated, short-circuited and isolated, both grounded, one grounded and one isolated; (4) when the turn ratio ap­proaches unity, the effect of the secondary is more pro­nounced when the two windings are wound in opposite directions than when they are wound in the same direction ; and (5) the computation of impulse voltages in a power transformer may be done with a good approximation considering only the impulsed winding, provided the neutral of the impulsed winding is grounded and the turn ratio differs considerably from unity; if the neutral of the im­pulsed winding is isolated, or if the turn ratio is close to unity, both windings must be considered.

R E F E R E N C E S

1. Natural Frequencies of Coils and Windings Determined by Equivalent Circuit, P. A. Abetti, F. J. Maginniss. AIEE Transactions, vol. 72, pt. 3, 1953, pp. 495-504.

2. Fundamental Oscillations of Coils and Windings, P. A. Abetti, F. J . Maginniss. Ibid., vol. 73, pt. 3-A, 1954, pp. 1-10.

2 0 4 0 6 0 8 0 PER CENT WINDING FROM N E U T R A L

Digest of paper 55-29, "Oscillation* of Coupled Windings," recommended by the AIEE Committee on Transformers and approved by the AIEE Committee on Technical Operations for presentation at the AIEE Winter General Meeting, New York, Ν. Y., January 31-February 4, 1955. Published in the AIEE Power Apperstus and Systems, April 1955, pp. 12-21. P. A. Abetti is with General Electric Company, Pittsfield, Mass. ; G. B. Adams and F.J. Maginniss are with General Electric Company, Schenectady, Ν. Y.

1002 Abetti, Adams, Maginniss—Coupled Windings ELECTRICAL ENGINEERING