The effect of inherent finite delay which is not an integral multiple of the sampling period on the character of limit cycles in a second-order feedback control system with a relay type non-linearity is analysed. It is found that the addition of transportation log alters the character of the modes sustained by the system. All the modes that can be sustained for each fractional value of the delay which lies between zero and one and two and three sampling periods are determined by the discrete describing function method. In this connection the critical regions developed by the authors for periods ranging from 9Tto 20Thave been found to be very useful. The different sots of initial conditions which give rise to the various modes are also established via the state transition technique. It is observed that an increase in the transportation lag usually results in an increase in the amplitude as well as the period of oscillation. Square-wave dither is found to produce a large steady-state error for a step input. The larger the value of the delay, the larger the amplitude of dither signal required to quench the limit cycle. Furthermore, it is noticed that when the dither signal is unable to quench the limit cycle, it sometimes alters either the period or the amplitude of oscillation.