The required amount of copper stabilizer to protect a YBCO conductor coil from damage caused by a hot spot due to a quench is studied in this paper. YBCO coated conductors which are made by deposition of thin YBCO film on high resistance metal substrates are highly resistive when they are quenched. Therefore, stabilization and quench protection are more important for YBCO conductors than Bi/Ag sheathed tapes which have low resistance silver matrix. In the work we numerically calculated maximum temperature rise of a YBCO conductor quenched by a local disturbance during the sequence of quench detection and energy dump. In the analysis, necessary amount of copper to keep the maximum temperature of the conductor below a threshold value is calculated for cases of conduction cooled coils of dry windings. Based on the analysis, optimum conductor design is discussed to protect the coil from a quench reliably.