This work is devoted to a theoretical investigation of the current crowding problem in flat conductors bent at arbitrary angles. Using conformal mapping techniques, we succeed in obtaining an analytical expression for current density distributions in such conductors. It is shown that the current density increases in a small vicinity of the corner and approaches to infinity at its top. In order to eliminate the infinity, the vertex is replaced by an arc of a circle with a small radius. The method has been developed for an arbitrary angle; several specific examples are considered.
2. Shao, T. L., Y. H. Chen, S. H. Chiu, and C. Chen, "Electromigration failure mechanisms for SnAg3.5 solder bumps on Ti/Cr-Cu/Cu and Ni(P)/Au metallization pads," J. Appl. Phys., Vol. 96, 4518, 2004.
doi:10.1063/1.1788837
3. Lai, Y.-S. and C.-L. Kao, "Characteristics of current crowding in flip-chip solder bumps," Microelectronics Reliability, Vol. 46, 915-922, 2006.
doi:10.1016/j.microrel.2005.02.007
4. Lai, Y.-S. and C.-L. Kao, "Electrothermal coupling analysis of current crowding and Joule heating in flip-chip packages," Microelectronics Reliability, Vol. 46, 1357-1368, 2006.
doi:10.1016/j.microrel.2005.08.009
5. Zhang, K., L. Gong, and R. Unbehauen, "Field analysis of a circular thin-film/foil resistor," Archiv fur Elektrotechnik, Vol. 76, 423-426, 1993.
doi:10.1007/BF01576021
6. Gerasimenko, T. N., V. I. Ivanov, P. A. Polyakov, and V. Y. Popov, "Application of a conformal-mapping technique to a boundary-value problem of current distribution in plain conductors," Journal of Mathematical Sciences, Vol. 172, No. 6, 761-769, 2011.
doi:10.1007/s10958-011-0219-8
7. Smythe, W. R., Static and Dynamic Electricity, McGraw-Hill Book Company Inc., New York, Toronto, London, 1950.
8. Trefethen, L. N., "Analysis and design of polygonal resistors by conformal mapping," Zeitschrift fur angewandte Mathematik und Physik, Vol. 35, 692-704, 1984.
9. Kober, H., Dictionary of Conformal Representations, Dover, 1957.
10. Hagedorn, F. B. and P. M. Hall, "Right-angle bends in thin strip conductors," J. Appl. Phys., Vol. 34, No. 1, 128-133, 1963.
doi:10.1063/1.1729052
11. Lavrentiev, M. A. and B. V. Shabat, Methods of Theory of Functions of Complex Variable, Nauka, Moscow, 1987 (in Russian).
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