Finite heat-pulse theory for transient thermal product sensing

Transient thermal product (TP) sensing methods are traditionally formulated using infinite step-input solutions of the one-dimensional heat equation. While mathematically convenient, this assumption is physically inconsistent with pulsed excitation techniques, in which heat input is applied for a finite duration. Classical step-input models describe only the forced heating phase and cannot represent the passive cooling behaviour that follows pulse termination. In this work, a finite heat-pulse formulation is developed for transient one-dimensional conduction into a semi-infinite medium. The governing equations are expressed in the Laplace domain, yielding a closed-form analytical solution that describes both the heating and cooling phases within a single expression. The classical infinite step-input solutions emerge naturally as limiting cases as the pulse duration tends to infinity. It is shown that applying the step-input model to finite-pulse cooling data introduces systematic errors in the extracted thermal product exceeding 200% within two pulse durations of termination, demonstrating the necessity of the finite-pulse formulation for cooling-phase analysis. The framework is extended to layered probe architectures incorporating finite-thickness high-conductivity substrates such as diamond, and augmented with a lumped thermal energy-balance model to account for sensor thermal mass and parasitic heat losses. Together, these contributions establish a rigorous and practically motivated analytical basis for modelling, probe design, and thermal product extraction in modern pulsed sensing systems.