expressions of the coefficients as in eq 8 d 2 a r ja i d b r jb i c r jc i r f d r jd i 0 10 a r ja i a 1 u2212 a 2 d 1 b r jb i a 1 u2212 a 2 v sa u2212 v sb u2212 d 1 b 1 u2212 b 2 d 1 c r jc i b 1 u2212 b 2 v sa u2212 v sb u2212 d 1 d r jd i v sa u2212 v sb u2212 d 1 u2212 d 2 d 1 a 1 i sa a 2 i sb a 3 i sc and d 2 b 1 i sa b 2 i sb b 3 i sc 11 from the imaginary part of eq 10 the following second order polynomial equation can be obtained d 2 u2061 a r u2146 r u2146 i u2062 a i d u2061 b r u2146 r u2146 i u2062 b i c r u2146 r u2146 i u2062 c i 0 12 finally the fault distance d can be obtained by solving eq 12 note that this fault location equation based on the direct circuit analysis can be applied to any system which is balanced or unbalanced type three phase or three single phase systems fig2 shows a flow chart of.