Robust Colpitts and Hartley oscillator design

This paper explores the robust design of Colpitts and Hartley oscillators. Considering the uncertainty and unmodelled dynamics for Colpitts and Hartley oscillators, the period and amplitude of the limit cycle will change. The uncertainty for these nonlinear oscillators is due to the fact that the capacitance and resistance will be changed in practical applications, especially for integrated circuit (I.C.) process. The problem of Colpitts and Hartley oscillators can be transformed into a nonlinear Lur'e problem. Then, by the application of describing function method, the limit cycle for Lur'e problem is explored. With Barkhausen criterion, a Barkhausen characteristic polynomial is proposed in this paper. It can be found that for Barkhausen characteristic equation, as the closed-loop poles are clustered, the system is extremely sensitive to parameter's change. This means the oscillation frequency of Colpitts and Hartley oscillators will be easily changed. A robust configuration of Colpitts and Hartley oscillators is shown in this paper. Also, a formula to determine the limit cycle is locally stable and unstable is presented. Finally, the robustness of crystal oscillator is also investigated in this paper. Simulation and experimental examples will verify the above-mentioned results.