Natural frequencies of 3-phase transformer windings.

Abstract

The equations of the windings of a 3?phase transformer together with the terminal conditions completely specify the electrical behaviour of the windings. However, they are awkward and difficult to solve. These difficulties are due to the existence of capacitive and inductive couplings between the phase windings.

A better physical picture of the phenomenon and simplifications in the solution is obtained if these couplings are eliminated by a transformation of the instantaneous voltages and currents in the phase windings into substitute variables.

The method of transformation of the phase variables, like the conventional method, retains all the capacitive and inductive couplings between the component windings and does not eliminate them.

It is possible to eliminate all the couplings between the component windings by using a modified transformation. The pertinent equations carry a more clear notation for the variables and numerics.

A comparison of this method with the method proposed earlier (in simplified Sturm–Liouville form) reveals: (i) A hope for non?reciprocal capacitive and inductive couplings between the component windings. These couplings introduce many difficulties in the visualisation and subsequent solution of the phenomena. (ii) These couplings are eliminated only if symmetry of the core is assumed. In view of the fact that this condition is incorrect for core?type transformers of usual construction, this method may lead to incorrect results.

An equivalent network representation for any given set of terminal conditions of the windings of a 3?phase transformer can be obtained by the procedure outlined in this investigation. A number of examples have been worked out and general equivalent networks pertinent to star? and delta?connected windings also.

There is no cancellation (as exists in the case of autotransformer) between the component natural frequencies of the windings of a 3?phase transformer with grounded neutral and the short?circuit natural frequencies of those windings isolated; neutral wires on all three line terminals are connected and earthed. The number of natural frequencies that can be excited in the windings of a 3?phase transformer for any given set of terminal conditions of the windings (corresponding to the pertinent open? or short?circuit constraints) depends also on the point or points of application of the excitation. Some of the natural frequencies of the windings are rendered non?excitable from some of its terminals.

The natural frequency pattern of the windings of a 3?phase transformer has a fine structure. This fine structure is suppressed in the computed results if the capacitive and inductive couplings between the phase windings are ignored.

The theoretical predictions about the natural frequencies and transient responses have been borne out by the test results obtained with a 3?phase core?type transformer having single windings on each limb. Hence the validity of the method of analysis presented herein may be considered to be established.

The errors of the computed frequencies of the windings of the transformer for the various test conditions are mostly within 15%, which is of the order of the error in the computed frequencies that can be expected when the windings are represented by five?conductor equivalent circuits. There are mismatches between the magnitudes of the transient voltages in the excited windings for the various test conditions investigated. However, when only one phase winding is excited, the secondary voltages in the excited phase are approximately the same.

The voltages in the unexcited phase windings are proved; they are increased in both the line?to?line and the unexcited phase windings having located terminals, where large voltages of opposite polarity and shape similar to the crest values appear in these windings.

Comparison of the observed and the computed transient responses indicates they are in good agreement. Hence the method of analysis presented herein provides a satisfactory procedure for the estimation of these transient voltages.