Modeling of the submicron CMOS differential ring oscillator for obtaining an equation for the output frequency

A symbolic expression that approximates the output frequency of the submicron differential ring oscillator, using the detailed transient behavior of the MOSFETs, is presented in this article. The circuit of the oscillator is simulated from 3-stage till 21-stage, with the range of output frequency from 0.3756 GHz till 2.6925 GHz. Later on, for verifying the similar functionality with different Beta ratios, a 7-stage differential ring oscillator is utilized. The average difference between the computed and the simulated values of the output frequency is found to be 1.98%, with TSMC 180 nm technology, when the value of Beta ratio was 2.3. The expression indicates that the output frequency is inversely proportional to the square of the device length. By including an empirical constant in the derived equation, the mathematical expression can be utilized for the hand calculations, for obtaining the output frequency of the differential ring oscillator.