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# On methods of calculation of microscopic integral inductance with high symmetry

Keywords:

V.G. Sapogin – Ph. D. (Phys.-Math.), Professor, Department физики, Academy of Engineering and technology of SFU (Taganrog) E-mail: sapogin@mail.ru N.N. Prokopenko – Dr. Sc. (Eng.), Professor, Head of Department, Don State Technical University (Rostov-on-Don) E-mail: prokopenko@sssu.ru

The diminution of geometric size of IC elements generates the problem of calculation of microscopic inductance. During last twenty years the methods of calculation of planar spiral inductances, loaded at silicon film (CMOS technology), are most actively developed. In models the integral inductance is substituted by equivalent circuit of replacement with lumped parameters. The models can’t explain physical reasons of negative inductance’s appearance and answer the question: how nulls of inductance can be moved by changing of spluttering technique? In the paper the review of valid physic-mathematical methods for calculation of parameters of integral inductance with high symmetry, which are deprived of such shortcomings, are proposed. The first method issues from physical definition of inductance as proportionality coefficient between the current and magnetic flux. The static inductance, calculated by such method, gives the true order of microscopic inductance’s values till cutoff frequencies by order of 1 GHz. Through the first method of calculation the next problems have been solved: 1) flux inductance of the ring with finite radial thickness has been calculated; 2) inductive properties of various spiral shave been researched; 3) running inductance of cylindrical conductors has been determined; 4) the model of coaxial cable’s integral inductance, which can be not only positive, but under the certain conditions, it can be large and negative, has been built. The second method permits to calculate the inductance, inserted by the substance, where Foucault currents are arisen. The external magnetic field and the field of substance’s response create the complete magnetic field in conductive cylinder. The complete field’s energy, averaged on the period of its changing, is partially taken at the heating of this substance and is partially reserved as magnetic field’s energy, associated with inductive properties of this substance. The redistribution of energy to a large extent depends of the frequency of external field. Nulls of inductance arise in the moment, when the whole energy of complete magnetic field consumes at the compensation of losses. Through the second method of calculation the problem for external magnetic field, which direction coincides with the axis of solid conductive cylinder, has been solved. The quadric frequency dependence of inductance, where its nulls divide the whole frequency band into three intervals: LF (low frequency), MF (medium-frequency) and EHF (extremely high frequency) one. At LF-interval the inductance is always negative. The scale and the first null of film’s inductance of RPGS inductor, agreed with observation, have been estimated. The solution of fundamental problem of cutoff frequency in strip coils is discussed.
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