Regulating characteristic of synchronous electrical machine requires to specify the dependence of excitation current on the load current. The optimisation of technical and economic indices of synchronous electrical machines, operating alone or in parallel, is an important problem in the exploitation of synchronous machines. Due to these reasons, the analytical expressions of cylindrical rotor synchronous machine obtained previously for V- curves and Compounding curves are reconsidered to be described fully in a different way in two new coordinate systems: Butterfly Coordinate Systems (BCS-P & BCS-L). BCSs provide short definition of analytical equations while facilitating easy commenting and drawing of V- curves and Compounding curves which are only families of hyperbolas. Traditional and inverse properties of BCSs are also given.