For typical symmetrical systems with three-phase double-circuit transmission lines or two parallel three-phase double-circuit transmission lines, single real transformation matrix is applied. The objective is to determine Z (impedance) and Y (admittance) diagonal matrices in the mode domain. The proposed analyses are based on eigenvector and eigenvalue studies, using linear combinations of Clarke matrix elements. This methodology originates from techniques used for single three-phase transmission lines. Using the proposed extended techniques, Z and Y diagonal matrices are obtained for transposed cases of three-phase double-circuit line systems (6 order transformation matrix). For two parallel three-phase double-circuit lines (12 order transformation matrix), only two non-diagonal elements are not zero in mode matrices. For future development, the objective is to investigate the proposed single real transformation matrix application to non-transposed cases. © 2005 IEEE.2005 256260A.Semlyen and A.Dabuleanu, Fast and Accurate Switching Transient Calculations on Transmission Lines With Ground Return Using Recursive Convolutions, IEEE Transaction on Power Apparatus and Systems, Pas-94, pp.561-571, March 1975J.R.Marti, Accurate Modeling Of Frequency-Dependent Transmission Lines in Electromagnetic Transients Simulations, IEEE Transaction on Power Apparatus and Systems, Pas-101, pp.147-157, Jan. 1982Tavares, M.C., Pissolato, J., Portela, C.M., Mode Domain Multiphase Transmission Line Model-Use in Transient Studies (1999) IEEE Transactions on Power Delivery, 14, pp. 1533-1544Tavares, M.C., Pissolato, J., Portela, C.M., Quasi- Modes Three-Phase Transmission Line Model - Transformation Matrix Equations (2001) International Journal of Electrical Power Energy Systems, 23 (4), pp. 325-333Kurokawa, S., Tavares, M.C., Pissolato, J., Portela, C.M., Applying a New Methodology to Verify Transmission Line Model Performance - The Equivalent Impedance TestMorched, A., Gustavsen, B., Tartibi, M., A Universal Model for Accurate Calculation of Electromagnetic Transients on Overhead Lines an Underground Cables (1999) IEEE Transactions on Power Delivery, 14 (3), pp. 1032-1038Nguyen, H.V., Dommel, H.W., Marti, J.R., Direct Phase-Domain Modeling of Frequency-Dependent Overhead Transmission Lines (1997) IEEE Transactions on Power Delivery, 12 (3), pp. 1335-1342Noda, T., Nagaoka, N., Ametani, A., Phase Domain Modeling of Frequency-Dependent Transmission Lines by Means of an ARMA Model (1996) IEEE Transactions on Power Delivery, 11 (1), pp. 401-411Tavares, M.C., Pissolato, J., Portela, C.M., Six-Phase Transmission Line - Propagation Characteristics and New Three-Phase Representation (1993) IEEE Transactions on Power Delivery, 8, pp. 1470-1483A. Semlyen and M.H.Abdel-Rahman, State Equation Modelling of Untransposed Three Phase Lines, IEEE Transaction on Power Apparatus and Systems, Pas-103, No.11, pp.3402 - 3408, November 1984