Abstract
To study very fast transients in transformers, computing the winding inductance matrix at very high frequencies (MHz) is required. The air-core approximation formulas cannot be used for this purpose because at very high frequencies the core behaves as a magnetic insulating wall and so the distribution of the magnetic field is quite different from that of air-core inductors. This paper presents new analytical methods for computing the winding inductance matrix at very high frequencies considering the presence of the core. A simple method, based on numerical integration of the vector potential functions, is described for the calculation of inductance outside the core window. For the region inside the core window, two different analytical solutions are developed and inductance formulas are extracted. The final expressions are simple and fast convergent. Comparisons with finite-element method simulations prove the high accuracy of the technique.
I. INTRODUCTION
THE DETAILED model of winding consisting of inductive, capacitive, and loss components has been used widely for the analysis of fast and very fast transients in transformers [1]–[15]. For very fast transients, such as those caused by switching operations in gas-insulated substations (GIS), using one segment per turn in order to achieve the required detail of the winding model [16] is needed. In very fast front transients, the flux penetration into core is negligible. The core acts as a flux barrier at very high frequencies and the distribution of the magnetic field is quite different from that of air-core inductors. This is more significant in the core window. At high frequencies, the core window shapes the distribution of the magnetic flux, affecting the values of self and mutual inductances. Very large errors have been reported when the self and mutual inductances are computed with the commonly used air-core approximations [17]. The consequence of this is that the transient response of a winding is not properly computed when the core is represented as air.
VII. CONCLUSION
New analytical methods have been established for computing the winding inductance matrix at very high frequencies. For the region outside the core window, a simple method has been developed wherein only two well-behaved integrals need to be evaluated numerically in order to determine the inductances. For the region inside the core window, accurate formulas for the self and mutual inductances have been derived from the direct solution of Poisson’s equation in two different approaches. The final expressions are simple and fast convergent. The accuracy of the proposed formulas has been verified by comparisons with finite-element analyses.
