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Superposition Theorem for Linear AC Circuit Using Triangular Fuzzy System

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Superposition Theorem for Linear AC Circuit Using Triangular Fuzzy System International Journal of Grid and Distributed Computing Vol. 13, No. 1, (2020), pp. 1695–1699 Superposition Theorem For Linear AC Circuit Using Triangular Fuzzy System 1J. Hellan Priya 2G.Veeramalai 1&2 Assistant Professors, Department of Mathematics, M.Kumarasamy College of Engineering (Autonomous), Karur [email protected] , [email protected] Abstract In this paper, a new approaches to solve linear network using triangular fuzzy superposition theorem of AC circuit and also establish the superposition theorem based problems on linearly independent current source or voltage source of linear network. A new idea is apply to find the optimal current source or voltage source in the given linear circuit using linear fuzzy systems and also numerical illustration are given. Keywords: Triangular Fuzzy system, AC Circuit, Linear Network, etc. INTRODUCTION: All the electrical problems cannot be solved by linear system. Many capabilities possess the nonlinear ones, which output and input are coupled nonlinearly. Even if the problem is solvable with a linear time variant system, the nonlinear device does it usually with a simpler structure in some cases just with several units. The Superposition theorem is applicable to linear network consisting of independent sources, linear dependent sources, linear passive elements and linear transformers. When the linear circuit has large number of sources is current or voltage source use superposition theorem to obtain the current or voltage in any branch of the circuit. Since the AC circuits are linear the superposition theorem applies to AC circuits the same manner to applied DC circuits. The theorem becomes important if the circuit has sources operating at different frequencies. In this case since the impedances depend on frequencies, we must have a different frequency domain circuit for each frequency the total response and must be obtained by adding the individual responses in the phasor or frequency domain. First defined the fuzzy concept 1965, Zadeh [1], the concept of fuzzy set and there after it has been developed by several authors through the contribution of the different articles on this concept and applied on different branches of pure and applied mathematics. In [4],[5], [7] ,[8],[9],[10], [19] &[20], Some applications of triangular fuzzy numbers depends on their similarity. Chen and Lin proposed the distance of two triangular fuzzy numbers and calculated the similarity of two triangular fuzzy numbers based on their distance between them, our interests are in the study of superposition theorem concepts in triangular fuzzy linear circuit. PRELIMINARIES The preliminary segment, The Fuzzy numbers are the great consequence of fuzzy systems. Regularly used the fuzzy numbers in applications of the triangular fuzzy number. The basic concept and the result to this paper are referred in [4],[5], [7] ,[8],[9],[10], [19] &[20], 1695 ISSN: 2005-4262 IJGDC Copyright ⓒ200 SERSC International Journal of Grid and Distributed Computing Vol. 13, No. 1, (2020), pp. 1695–1699 3. Proposed Method Triangular fuzzy superposition theorem for AC circuit A Triangular fuzzy linear circuit having two or more independent triangular fuzzy sources one way to determine the value of specific variable of triangular fuzzy source (voltage or current) is to use nodal or mesh analysis. Another way is to determine the contribution of each triangular fuzzy independent sources to the Specific variable and then add them up. It is denoted as triangular fuzzy superposition principle. Triangular fuzzy Superposition theorem is applied when we are to determine the current in one particular branch of a triangular fuzzy linear network containing several voltage fuzzy sources and fuzzy current sources. We consider one independent fuzzy source at a time while all other independent fuzzy source are turned off. This implies that we replace every voltage fuzzy source by zero volt, and every fuzzy current source by zero amps. The dependent fuzzy source are left intact because they are controlled by circuit variables. Steps to be followed in superposition theorem of fuzzy system The Following steps in order to find the response in a particular fuzzy branch using superposition theorem of AC circuit of fuzzy systems. Here especially we have use triangular fuzzy systems. Step 1 – Determine the current in one particular branch of a triangular fuzzy linear network containing several voltage fuzzy sources and fuzzy current sources. Consider one independent fuzzy source at a time while all other independent fuzzy source are turned off. Step 2 – Replace every voltage fuzzy source by zero volt, and every fuzzy current source by zero amps. The dependent fuzzy source are left intact because they are controlled by circuit variables. Step 3 – Find the resultant current through the resistance by the fuzzy superposition theorem considering magnitude and the direction of each current. 4. Numerical Illustrations: The following numerical illustrations the proper use of fuzzy superposition of dependent fuzzy source. All the superposition equation are written by inspection using voltage division, current division, series parallel combinations, and ohm’s law. In case, it is simpler not to use superposition if the dependent sources remain active Consider the triangular fuzzy linear circuit to which we are going to determine the current I in the following circuit using superposition theorem for fuzzy system. Step 1 When the 50 900 V fuzzy source acting alone 1696 ISSN: 2005-4262 IJGDC Copyright ⓒ200 SERSC International Journal of Grid and Distributed Computing Vol. 13, No. 1, (2020), pp. 1695–1699 ((2, 3, 4) i(2,4,6)) ((4, 5, 6) i) Z (4, 5, 6) T (2, 3, 4) i(8, 9, 10) (4.30, 6.34, 10.71)(18.73, 23.21, 38.79) 50900 I T (4.30, 6.34, 10.71)(18.73, 23.21, 38.79) (4.67, 7.89, 11.63) (51.210 , 66.790 , 71.270 ) By Current Division rule 0 0 0 (4, 5, 6)i I (4.67, 7.89, 11.63) (51.21 , 66.79 , 71.27 ). (2, 3, 4) i(6, 9, 12) (1.49, 4.8, 11.05) (70.640 , 85.220 , 89.700 ) Step II When the 50 00 v source is acting alone (4, 5, 6)(2, 3, 4) i(2, 4, 6) Z (4, 5, 6)i T (6, 8,10) i (2, 4, 6) (0.47, 2.5, 11.40) i (4.24, 5.25, 11.40) (4.27, 5.81, 16.12) (45.00 , 64.540 , 83.670 ) 1697 ISSN: 2005-4262 IJGDC Copyright ⓒ200 SERSC International Journal of Grid and Distributed Computing Vol. 13, No. 1, (2020), pp. 1695–1699 50 00 I T (4.27, 5.81, 16.12) (45.00 , 64.540 , 83.670 ) (3.10, 8.61, 11.71) (83.670 , 64.540 , 45.00 ) By Current division rule 0 0 0 (4, 5, 6) I (3.10, 8.61, 11.71) (83.67 , 64.54 , 45.0 ) (6, 8, 10) i (2, 4, 6) (1.05, 4.82, 11.12)(114.630 , 91.110 , 63.430 ) Step III By Superposition theorem I I I (1.49, 4.8, 11.05) (70.640 , 85.220 , 89.700 ) (1.05, 4.82, 11.12)(70.640 , 85.220 , 89.700 ) (2.54, 8.36, 22.10) ( 70.640 , 85.220 , 89.700 ) 5. Conclusion: The superposition theorem of triangular fuzzy linear system used to find the optimal Current or voltage. The proposed method of triangular fuzzy systems is more accuracy for the normal calculus methods and also superposition theorem is very useful service only for linear circuits as well as the AC circuit which has more accuracy of supplies and also superposition theorem of fuzzy system is suitable for working on the principle of fuzzy linear system. REFERENCES 1. Zadeh, L.A., The concept of a Linguistic variable and its applications to approximate reasoning – parts I, II and III”, Inform. Sci. 8(1975) 199-249; 8 1975 301-357; 9(1976) 43-80. 2. E. Hansan and G. W. Walster, “Global optimization using Interval Analysis”, Marcel Dekker, New York, 2003. 3. Karl Nickel, On the Newton method in Interval Analysis. Technical report 1136, Mathematical Research Center, University of Wisconsion, Dec1971. 4. A. Nagoor Gani and S. N. Mohamed Assarudeen, “A new operation on Triangular fuzzy number for solving fuzzy linear programming problem”, Applied Mathematical Sciences, Vol.6, 2012, no.11, 525-532. 5. G.Veeramalai and P.Gajendran, ”A New Approaches to Solving Fuzzy Linear System with Interval Valued Triangular Fuzzy Number” Indian Scholar An International Multidisciplinary Research e-Journal , ISSN: 2350-109X,.Volume 2, Issue 3, (Mar2016), PP 31-38. 6. G.Veeramalai and R.J.Sundararaj, “Single Variable Unconstrained Optimization Techniques Using Interval Analysis” IOSR Journal of Mathematics (IOSR-JM), ISSN: 2278-5728. Volume 3,Issue 3, (Sep-Oct. 2012), PP 30-34. 7. G.Veeramalai, ”Unconstrained Optimization Techniques Using Fuzzy Non Linear Equations” Asian Academic Research Journal of Multi-Disciplinary,ISSN: 2319- 2801.Volume 1, Issue 9, (May2013), PP 58-67. 1698 ISSN: 2005-4262 IJGDC Copyright ⓒ200 SERSC International Journal of Grid and Distributed Computing Vol. 13, No. 1, (2020), pp. 1695–1699 8. J.J. Buckley, E. Eslami and T. Feuring (2002). Fuzzy Mathematics in Economics and Engineering. Physics - Verlag 9. D. Dubois, H. Prade (1980). Fuzzy sets and systems. Theory and application. Academic, New York. 10. R. Goetschel, W. Voxman (1986). Elementary Calculus, Fuzzy Sets and Systems 18:31-43 11. J.E. Dennis, R.B Schnabel (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Jersey. 12. J. Wright, Stephen, Nocedal, Jorge (2006). Numerical Optimization, 2e.pp(614) 13. Eldon Hansen, Global optimization using interval analysis- Marcel Dekker, 1992 14. K. Ganesan and P. Veeramani, On Arithmetic Operations of Interval Numbers, International Journal of Uncertainty, Fuzziness and Knowledge - Based Systems, 13 (6) (2005), 619 – 631.
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