For a typical non-symmetrical system with two parallel three phase transmission lines, modal transformation is applied using some examples of single real transformation matrices. These examples are applied searching an adequate single real transformation matrix to two parallel three phase transmission line systems. The analyses are started with the eigenvector and eigenvalue studies, using Clarke's transformation or linear combinations of Clarke's elements. The Z C and parameters are analyzed for the case that presents the smallest errors between the exact eigenvalues and the single real transformation matrix application results. The single real transformation determined for this case is based on Clarke's matrix and its main characteristic is the use of a unique homopolar reference. So, the homopolar mode becomes a connector mode between the two three-phase circuits of the analyzed system. ©2005 IEEE.1 111118Dommel, H.W., (1984) Electromagnetic Transients Program - Rule Book, , Oregon(1992) Transients Analysis Program Reference Manual, , Vancouver, CanadaMorched, A., Gustavsen, B., Tartibi, M., A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables (1999) IEEE Trans. on Power Delivery, 14 (3), pp. 1032-1038. , JulyFaria, J.A.B., Mendez, J.B., On the modal analysis of asymmetrical three-phase transmission lines using standard transformation matrices 1997 IEEE Winter Meeting, , 97WMIEEE-PESWedepohl, L.M., Application of matrix methods to the solution of travelling-wave phenomena in polyphase systems (1963) Proceedings IEE, 110, pp. 2200-2212. , DecemberSemlyen, A., Deri, A., Time domain modelling of frequency dependent three-phase transmission line impedance (1985) IEEE Trans. on Power Apparatus and Systems, 104 (6), pp. 1549-1555. , JuneClarke, E., (1950) Circuit Analysis of AC Power Systems, 1. , Wiley, New YorkWedepohl, L.M., Nguyen, H.V., Irwin, G.D., Frequency -dependent transformation matrices for untransposed transmission lines using Newton-Raphson method (1996) IEEE Trans. on Power Systems, 11 (3), pp. 1538-1546. , AugustNguyen, T.T., Chan, H.Y., Evaluation of modal transformation matrices for overhead transmission lines and underground cables by optimization method (2002) IEEE Trans. on Power Delivery, 17 (1). , JanuaryGustavsen, B., Frequency-dependent transmission line modeling utilizing transposed conditions (2002) IEEE Transactions on Power Delivery, 17 (3), pp. 834-839. , JulyPrado, A.J., Pissolato Filho, J., Tavares, M.C., Portela, C.M., Modal transformation applied to transients in double three phase transmission lines (2001) 2001 IEEE Power Engineering Society Summer Meeting, , Vancouver, BC, Canada, 15-19 July