Remark II.2. The question of the unities in fractional elements like coils and capacitors is an open subject [72], [73]. There are interesting realizations of differintegrators and applications to circuits [44], [48], [74]-[77]. An interesting generalization of the Kramers-KrĂ¶nig relations was presented in [21]. Example II.1. A device with fractional characteristics is the electrochemical capacitor for which several fractional models were proposed for its impedance. This subject was studied in [78], [79] and the following model was obtained and validated experimentally Zs R0 () where R0 cc 12 ,, CC and C 3c =+ ++c c 1$$ $ c2 c2 cc 12 + 1 sC sC sC 11 1 , c3 represents a series resistance, the symbols stand for capacitances of the electrochemical capacitor model, and 1c and 2c are the fractional orders. It is interesting to remark that 1) the model is a series or a resistor and 3 ideal capacitors; 2) the order of one ideal capacitor is the sum of the other two, assuming a value slightly greater than 1. III. Study of Continuous-Time FARMA Linear Systems A. The Commensurate Case The CT-FARMA LS are the most important class of fractional systems, since they constitute the base for the study of fractional electric circuits. This type of system was introduced in (11) by means of its TF that we reproduce here together with its partial fraction decomposition M Gs ()== = / () kn . k As Bs () () a a / / k=0 N k=0 The parameters ,, ,,pk 12 N kp Az = R =0 k az N k . Bz = f are called pseudopoles and are the roots of the characteristic polynomial () k The roots of () = R =0 k bz M k k are the pseudo-zeroes and Rk stands for the residues of the partial fraction decomposition of ()/( ).Bz Az Remark III.1. Any system defined by (13), with rational orders can be converted into the form (11), that is to commensurate. We only may have to join additional null coefficients. The same happens with (14) that gives (12). Remark III.2. Let p be a complex number, and the equation sp 0-=a 01 ,11a that has infinite roots. However, there is only one solution for this equation in the first Riemann surface, if 0 = ;1aa by sp e/( )(/ ) j arg p ; -ar 1 arg p # ar, and it is given . For example, if () , () arg p r= then the root of the equation is in the second Riemann surface. Therefore, it does not lead to a pole or zero of a system. 42 IEEE CIRCUITS AND SYSTEMS MAGAZINE as bs k ak k k=1 sp R a - k ak Np (22) (21) The case a 1= corresponds to the classic model (10). As it is well-known [56], [57] the IR corresponding to this TF has the form Np gt () =!! (23) k / Rk t k ,, ,, ut k=1 ()! nk - 1 eu t pt (), where pk 12= f are the poles, nk denotes their multiplicity, and Rk !! is the causal/anti-causal Heaviside unit represents the corresponding residues in the partial fraction decomposition of (10). The function () step used to make a clear distinction between the causal (right) and anti-causal (left) solutions. This is important if the variable t does not stand for time, but, for example, it represents space. For 01 ,11a fraction /( 1 sp).n a - n a sp - - () ()!( ) 1 n = dn-1 sp a dn-1 - p we only have to pay attention to the case: () Gs = a - sp 1 . (24) 1 nk-1 we need to invert a generic simple Since (25) We will treat the causal case only. There are two different ways of inverting (25). 1) Using the Mittag-Leffler Function Using the geometric series, we can write successively () Fs = a ps sp 01 - nn valid for ;;11a1 // (26) 1 == 33 -- -spsp s aa a == nn nn , . Choosing the ROC ()Re sps2;;awill arrive at the causal inverse of F(s): () 3 ft = / n-1 n=1 p t na-1 C() na f t (). normally expressed in terms of the MLF as () ft tE pta = a-1 aa, () ()f t ft et = sp s - , (27) This function also called alpha-exponential [31] is (28) and gives the IR corresponding to partial fraction (26). If a 1 ,= then we have the classic result () step response () of (26) can be obtained from 3 rtf L rt @ = a 6 () f leading easily to () 11 nn1 1 $ = /ps --a -1 (29) ptf(). The , we rt tEaa+ () () t f == - a p a ,,1 Eptt111 $$ pta ff (30) a 6 () (), @ SECOND QUARTER 2022

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