In the present work an improved analysis of the relay auto-tuning method is proposed to calculate the maximum value of the controller gain by considering the higher-order harmonics. The reported Fourier series analysis for a first-order plus time delay (FOPTD) unstable system (Vivek and Chidambaram, 2005) is modified to an unstable second-order plus time delay system (SOPTD). Improved results in the ultimate gain values of the controller are obtained. The effectiveness of the method is demonstrated by simulation on three SOPTD transfer function models. The different models considered are: (i) a system with two unstable poles and a negative zero; (ii) a system with one stable and one unstable pole; and (iii) a system with two unstable poles and a positive zero. A simple method is also proposed to calculate the minimum value of the controller gain. The proposed method gives enhanced closed-loop performances in all the cases. The method is also applied by simulation on the nonlinear model equations of an unstable bioreactor.