Gas turbines are for most of them natural gas fired machines that are widely used in the energy industry due to their reliability, high availability, flexibility and their relatively lower GHG emissions compared to other fossil fuels. Researchers are continuously trying to make gas turbine more performing and less emitting. With the emergence of carbon capture and storage in order to reach carbon neutrality, the gas turbine exhaust gas recirculation (EGR) is grabbing even more the attention of researchers. Recirculating the exhaust of gas turbines increases the carbon dioxide concentration in the cycle and thus facilitates the carbon capture process. From a thermodynamic point of view, EGR in gas turbines leads to a change in the working fluid composition and also in the compressor inlet temperature. In addition to impacting the cycle, these changes impact also the compressor and turbine maps leading to a different matching point (pressure ratio). Many researchers tried to study the EGR impact on existing turbines with contradictory results on the performance level. The authors, in a previous paper tried to contribute to this debate by showing separatly the impact of the working fluid composition and the compressor inlet temperature. The compressor and turbine mismatch are highlighted. In this paper, a thermodynamic study is performed from a design perspective using a simple Brayton cycle. It shows that the composition change leads to a lower energy efficiency; however, a design of the compressor and turbine for a better matching point (pressure ratio) reduces the penalty.