tailieunhanh - Sabatier Agrawal Machado Advances in Fractional Calculus Episode 8

Tham khảo tài liệu 'sabatier agrawal machado advances in fractional calculus episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | A DIRECT APPROXIMATION OF COLE - COLE SYSTEMS 267 N Y T E YcCk T . K 0 27 A formula for the determination of the coefficients for the approximation of a single Cole-Cole system is given by rcc k R sin a - 1 n eaTo . feaTk-aAT 2 cos na eaTo arctan ----- v Ị e2aT 1 cos2 na naỵje2aTo 1 cos2 na I eaTk aAT 2 cos na eaTo arctan ---- v cos2 na naye2aTo 1 cos2 na 28 If the distribution density 27 is used to evaluate 25 the result for the coefficients of the approximation is given by rk rccK k k 1 2 . M. K 0 The parameter rCCK k denotes the kth parameter of the approximation model of the Kth Cole-Cole system which contributes to the impedance 2 . According to 25 an integral has to be solved in order to determine the coefficients rk of the approximation. The exact evaluation of this integral is not necessary because AT is very small in order to justify the simplification 23 . An alternative method to determine rk is to use the Simpson-integration rule. Hence the parameters rk can alternatively be determined by AT _ _ rk Y Tk AT 2 4ỹ Tk Y Tk AT 2 . 29 6 Finally the most simple approach to determine the rk-parameters is the rectangular integration. This simplification leads to a larger error compared to the Simpson-rule if it is used to evaluate a finite integral. A rectangular integration for the computation of the model parameters rk gives rk AT Y Tk . These presented three approximation methods will be compared in the subsequent section. 268 Haschka and Krebs T Tk Fig. 4. Distribution density and coefficients of the direct approximation. 4 Simulation of a SOFC-Impedance in the Time Domain The fractional-order impedance 16 which is a typical one for SOFC is simulated in the time domain using the proposed direct approximation methods presented in this contribution. The Y T -distribution density of this impedance is depicted in Fig. 1. In order to cover at least q 99 of the area between distribution density and T-axis for each Cole-Cole system the lower bound TLB and the

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