Analytical Solutions for Transient Thermal Spreading Resistance of a 3-D Flux Channel

In microelectronic devices, the moment a high-power current is transmitted into the system, heat is simultaneously generated, and the thermal field keeps developing until it reaches a steady-state field after a period of time. In this work, transient analytical solutions for the temperature field and thermal resistance of a rectangular 3-D flux channel are obtained. The flux channel is assumed to have a small heat source on the top surface, whereas convective effects are considered on the bottom surface and lateral edges. The time-dependent solutions are presented explicitly as infinite Fourier series forms. In addition, the solutions are also presented in dimensionless forms as generalized solutions. Moreover, an existing, well-established simple model that represents the profile of the transient thermal spreading resistance for a semi-infinite flux tube is used to verify the presented forms of the analytical solutions, and the results compare very well when considering a flux channel of large thickness. Further, the solutions are used to study the behavior of temperature and thermal resistance over time for some dimensional and nondimensional problems.