1. Nenad Petrović, Velibor Pjevalica, Nebojša Pjevalica, Nikola Teslić

2. An ascending (a descending) SHLC, in the case of sinusoidal magnetic flux, can be represented by trigonometric polynomial (TP) (ref. [1], [2]) or by cosine polynomial (CP) (ref. [3], [4]). Both TP and CP are generated over the set of samples representing the Chebyshev nodes of the second kind (CHN_II) in the domain of electric angle (ref. [3], [4]). Left: Graph of TP or CP interpolation polynomial i 0 (  ) over 11 CHN_II nodes in the electrical angle domain  [- ,0].

3. By virtue of mapping i 0 (cos  ), for the sinusoidal magnetic flux case, the ascending part of ac excitation current i 0 (  ),  - , 0  is directly mapped into the appropriate SHLC i 0 (x), x= φ/Φ m  -1, 1  (ref. [3], [4]). Consequently, the CP of i 0 (  ) is directly transformed into the algebraic interpolation polynomial (AIP) of i 0 (x) with the ordinates remaining unchanged (ref. [3], [4]). Left: Graph of the cosine interpolation polynomial i 0 (  ) over 11 CHN_II nodes in the electrical angle domain  [- ,0]. Right: Graph of the algebraic interpolation polynomial i 0 (x) over 11 CHN_II nodes in normalized flux domain x  [-1,1].

4. In the domain  - , 0  a TP generated over n CHN_II nodes has the form After mapping the electric angle domain  - ,0  by function cos  into the normalized magnetic flux domain x= φ/Φm =  -1,1 ], the ascending part of ac excitation current i 0 (  ), as it previously has been shown, will be transformed into an appropriate SHLC i 0 (x). whereas the corresponding CP has the next form: At the same time, the above represented TP, will get the form of algebraic function (AF) whereas the corresponding CP will be transformed into the more convenient form of an algebraic interpolation polynomial (AIP): Note: Term T k (x) in the above expressions denotes a k degree Chebyshev polynomial of the first kind.

5. Since an AIP is more favorable for the SHLC modeling toward an AF, it is necessary to comparatively examine the error behavior of the SHLC approximation by using of these two functions. In other words, it is necessary to answer the question: how much an approximation of SHLC by using CP is accurate in relation to the approximation of SHLC by using TP? Due to the fact that the mapping i 0 (  ),  - , 0  into the SHLC i 0 (x)  i 0 (cos  ), is performed without changing the value of the ordinates (i 0 (  )  i 0 (cos  )), a CP and the corresponding AIP maintain an equivalent approximation errors. The same holds true for a TP and the corresponding AF. Therefore, the analysis of the error behavior will be implemented over the domain of electrical angle  [- , 0], instead of the normalized flux domain φ /Φ m  [-1,1], or the actual flux domain φ  [-Φ m, Φ m ].

6. In order to compute the coefficients of a CP by using DFT, it is necessary the set of 11 (n+1) samples to be extended onto the set of 20 (2n) samples, so that the next condition is met: i 0p (-  -  )= i 0p (-  +  ). In order to compute the coefficients of a TP by using DFT, it is necessary the set of 11 (n+1) samples to be extended onto the set of 20 (2n) samples, so that the next condition is met: i 0 (-  -  )=-i 0 (-  ). (This is an actual signal of ac excitation current for a single phase transformer.) The coefficients of the CP can now be computed on the semi closed interval  (-2 , 0  by applying DFT: The coefficients of the TP are computed on the semi closed interval  (-2 , 0  by applying DFT: -π/2 -π-π 0-3π/2-2π The extension for the condition i 0p (-  -  ) = i 0p (-  +  ) The extension for the condition i 0 (-  -  ) = -i 0 (-  ) 0-π/2 -π-π -3π/2 -2π

7. i CPodd (  ) i CPeven (  ) -π-π-π/2 i 0p (  ) 0-3π/2-2π Lemma I: For the coefficients of CP and TP holds c 2k-1  a 2k-1. Proof: Let the i 0p (  ) is decomposed by means of convolutional subtracting and convolutional adding that provides i TPeven (  ) i TPodd (  )i0()i0() -π-π-π/20 -3π/2 -2π Let the i 0 (  ) is decomposed by means of convolutional subtracting and convolutional adding that provides On the basis of the definition, it follows i CPodd (-  +  )  i CPodd (-  -  ) and i CPodd (-  /2+  )  -i CPodd (-  /2-  ) resulting with vanishing of all coefficients except the cosine with the odd indexes. Also, on the basis of the above definitions it follows that i CPodd (  )  i TPodd (  ) resulting in c 2k-1  a 2k-1.

8. Functions of error distribution due to approximation i 0 (  ),  - , 0  by using CP (i n0p (  )) and TP (i n0 (  )) are defined by the expressions for CP: The mutual relationship between these two functions of error distribution will be given by For the estimation of behavior e 0p (  ) with respect to e 0 (  ) it is necessary to analyze the behavior of the analytical term i n0 (  ) - i np0 (  ). Based on the previous lemma, this term has the form and for TP: Since i CPeven (  )  i TPeven (  ),  - , 0 , the analytical term e nCP-TP (  ) approximates the zero function i CPeven (  ) - i TPeven (  ) = 0 for  - , 0 . This means that the analytical term e nCP-TP (  ) gets zero values for all of the ChII nodes  j : e nCP-TP (  j )  0,   j  - , 0 . Since the location of the largest particular extrema of the analytical term e nCP-TP (  ) on the closed interval  - , 0  is of the especial interest.

9. 1.1 Common property of the analytical term e nCP-TP (  ) over the domain  - , 0  Being on the basis of e nCP-TP (  ) definition, follows: 1.2 Property e nCP-TP ( -  / 2 ) = 0 for an odd number of ChII nodes over the domain  - , 0  For an odd number of the samples on the domain  - , 0  n ChII  2k+1, will be n  2k, that imply  k  -  /2, k  n/2. This means that the sample i 0 (-  /2) belongs to the set of ChII nodes, and based on e nCP-TP (  ) definition follows

10. 1.3 Analysis of the distribution of extrema of the e nCP-TP (  ) function for an odd number of ChII nodes over the domain  - , 0  In order to simplify notation, terms of the e nCP-TP (  ) expression will be particularly marked as By virtue of Chauchy’s theorem, between every two consecutive nodes  j such that i n0p_even (  j )  I n0_even (  j ), there must be at least one point  jd such that i n0p_even (  jd )  i n0_even (  jd ). (Further analysis must rely on the assumption, supported by the experimental testing, that between any two consecutive nodes  j there exists only one such point  jd.) Due to the fact that e nCP-TP (  j )  0,  j  - , 0  it follows that i n0p_even (  j )  i n0_even (  j ). (i n0p_even (  ) is a cosine interpolation polynomial of the function i n0_even (  ),  - , 0  ) Note: The factor of increase the ordinates in relation to their actual value is 100 for i n0_even (  ) and i n0p_even (  ), 50 for i n0_even (  ) and i n0p_even (  ), and 2000 for e nCP-TP (  ) and e nCP-TP (  ).

11. With regard to can be concluded that for the n ChII  2k+1 and 1.3 Analysis of the distribution of extrema of the e nCP-TP (  ) function for an odd number of ChII nodes over the domain  - , 0  nodes  j there exists the same number of the nodes  jd such that i n0p_even (  jd )  i n0_even (  jd ), j = 0,...,2k = n. This means that the function i n0p_even (  jd ) is an interpolation approximation of the function i n0_even (  jd ) of the degree n also, but on the narrower closed interval  0d,  2kd  - , 0 . This fact imply the highest oscillation of the function e nCP-TP (  ) on the closed intervals  0  - ,  1  and  2k-1,  2k  0 . Consecutively, extrema of the function e nCP-TP (  ) on these two intervals will be achieved at the points  0d and  2kd. The conclusion of the preceding analysis is formulated by the following expression

12. Based on the above considerations, and the fact that e nCP-TP (-3  /2) = 0, it follows that the function e nCP-TP (  ) has at least n+2 zeroes over the domain  (-2 , 0 . At the other hand, any trigonometric polynomial in  (-2 , 0  of degree n which is not identicaly zero has the Haar property, i.e. it cannot has more than 2n zeros (ref.[6]). Therefore, the possibility of additional n-2 (n=2k) zeroes on  - , 0  exists. Being that for the first derivative of the function e nCP-TP (  ) holds it means that, if the additional zeroes of the function e nCP-TP (  ) exist on  - , 0 , their number must be even and these zeros must be symmetric toward  =-  /2. Since the greater number of the additional zeroes imply the better approximation, the assumption of existence of only n+1 zeroes on  - , 0 , imply the worst case for the error behavior.

13. 1. By increasing the number of nodes, the absolute values of extrema of the function e nCP-TP (  ) are reduced, but its behavior remains in accordance with the preceding analysis. This situation is shown by virtue of graphs of the function e nCP-TP (  ) for n  10 and n  20. Note: The factor of increase the ordinates in relation to their actual value is 5000.

14. TABLE 1 n ChII index sample i 0 (index) [A] coefficients for TP polynomial coefficients for CP polynomial 0 -0.123795498 c0c0 0.0276971868 1 -0.098412061a1a1 0.0896541475 c1c1 2 -0.044543009b1b1 -0.0220809109 c2c2 -0.0095414213 3 -0.003735111a3a3 0.0300749038 c3c3 4 0.013030176b3b3 -0.0011651403 c4c4 -0.0047526785 5 0.019154967a5a5 0.0036494057 c5c5 6 0.026337257b5b5 0.0027702397 c6c6 -0.0000148624 7 0.045335193a7a7 0.0004456793 c7c7 8 0.074312542b7b7 0.0009040244 c8c8 0.0004815688 9 0.107005978a9a9 0.0000286386 c9c9 -0.0000286386 10 0.123795498b9b9 -0.0001054122 c 10 -0.0000424001 2. Polynomials TP and CP approximate i 0 (  ) at the point  -  /2 in an identical manner for an odd number of the samples on the closed interval  - , 0 . However, at the points   -  and   0, where i 0 (  ) attains their extrema, and hence, where is i 0 (-  )  0  i 0 (0), an application of CP has a significant advantage over an application of TP. Namely, the first derivative of CP, at the points  -  and  0 attains zero values also, that is not the case with TP. This situation is shown by virtue of graphs of the function of error distribution: e 0 (  )  i 0 (  )-i n0 (  ), e 0p (  )  i 0 (  )-i n0p (  ), as well as e nCP-TP (  )  i n0p (  )-i n0 (  ) for n  10. (The factor of increase the ordinates in relation to their actual value is 5000. The coefficients for the CP i n0p (  ) and TP i n0 (  ) are given in Table 1.)

15. The first result of this work is to answer the question: How much an approximation of SHLC by using CP is accurate in relation to the approximation of SHLC by using TP? For the case when the set of samples on the closed interval  - , 0  has property of ChII nodes, and the number of samples is odd, accuracy depends on the number of samples. With a smaller number of samples, the CP achieves better accuracy due to the fact made under the second point of discussion. When the number of samples increases, the accuracies of these two approximation mutually converge. The second result of this work is formulated by the Lemma I, which determines the general property of samples i 0 (  ) that represents the Chebyshev nodes, regardless of their number and category to which they belong.

16. Furthermore, the TP method gives direct and natural relationsheep between the number of harmonics of an ac excitation current i 0 (  ) with the degree of its approximation CP i n0p (  ) and the degree of the algebraic polynomial I n0p (x) approximating corresponding SHLC with an eqivalent level of accuracy. From the point of view of metrology, the TP method (ref. [2]) and the CP method (ref. [4]) achieve an equivalent level of computing efficiency and almost an equivalent level of accuracy. But concerning SHLC modeling, the CP method (ref. [4]) is obviously more suitable over the TP method. This work was partially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, under grant number: TR32014.