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66 The PCB Magazine • July 2016 The thermal gradient in the thickness e is negligible with regards to the other dimensions. This is verified if thickness is less than one tenth of the other dimensions, which is clearly the case for our test sample. In other words, when distance x 1 between the heating track and the first line, and the distance d = x 1 – x 2 are large enough with regards to the sample thickness, the heat transfer is considered as mono-dimen- sional (no heat gradient along Z). As an order of magnitude, the typical PCB sample to analyze is ~100 to 200 μm thick, while x 1 and d can easily be around 10 mm. • The test sample is large enough to be con- sidered as semi-infinite during the measure- ment time. The evolution of temperatures T 1 (t) and T 2 (t) at respective distances x 1 and x 2 from the copper track edge, is recorded by the IR sensor (could be also thermocouples or any other tempera- ture measurement system, with appropriate time constant and accuracy). The output of this pro- cess is the experimental curves (in time domain). With Laplace transform formalism, q 2 (p) and q 1 (p). are the Laplace transforms of T 1 (t) and T 2 (t). H(P) is the transfer function between q 2 (p) and q 1 (p). To determine H(P), we use the quadrupole formalism: With And: Where: Pe = 2(e + l) is the perimeter S = el is the heat flux flow section l is the thermal conductivity, a is the ther- mal diffusivity and E = is the thermal effu- sivity We can write: With It shows that q 2 is only a function of q 1 , p, a and , and thus: With To come back in the real space (time do- main) and find T 2 (t) function, we have to make the convolution product of T 1 (t) by the inverse Laplace transform of H(p): NB : L -1 [H(p)] is the inverse Laplace trans- form of H(p). From above, the model (shape of the curve) of T 2 (t) is known. With appropriate parameters, it can be adjusted to the experimental curve. An optimization algorithm is then used to de- termines the best a and h values minimizing the difference between the experimental and theo- retical curves. The next graphs represent this optimization. Note: Everything done under Matlab ® . Measurement System Setup The transient fin samples have to be pre- pared first. A square shape, approximately 100 x 100 mm 2 is sufficient, with the heating cop- A THERMAL CONDUCTIVITY MEASUREMENT METHOD, ADAPTED TO COMPOSITE MATERIALS