Unstable PLL-Controller as FM Modulator and Detection of Modulating Self-Oscillations

Abstract

Phase locked loops are well known nonlinear feedback control circuits extensively used in different electronic systems, particularly in communication and control. Generally they are used in stable mode of operation where loop error magnitude is small and system nonlinearity can be replaced by linear approximation. However, PLL designers over the years observed that due to variation of signal and loop parameters, dynamics of a PLL enters in a region which is not stable in conventional sense, even though predictable and controllable through parameter tuning. Hence attention has been given to explore so called unstable mode of PLL dynamics. PLLs operating in such mode produce periodic or chaotic self-oscillatory signals. In this chapter we focus on generation of self-oscillations in PLLs operating in their unstable mode and examine the possibilities of using these PLLs as modulators of periodic as well as chaotic oscillations. We consider a third order PLL with resonant type loop filter since in continuous time domain third order system is susceptible to chaotic self oscillations and gain as well as phase shift of a resonant filter could be tuned through a single parameter. We estimate the influence of loop design parameters in determining stable operating zone and hence find conditions of self-oscillation of third order PLL. With the variation of a design parameter, PLL is found to undergo a sequence of period doubling oscillations and ultimately chaotic oscillation of control signal results. In this condition, PLL is treated as an FM modulator of self-generated chaotic signal. We also report the effectiveness of PLL-based demodulators in detecting chaotically modulated signals. We consider second order and third order PLLs as chaos detecting loops for this purpose and obtain their relative responses. Detection of chaotically self-modulated signal is found to be difficult compared to that with periodically self modulated signal. We describe a circuit modification algorithm to PLL in order to enhance its response as a demodulator. Modified loop is shown to have increased stable zone of operation and faster transient response which makes it a better FM demodulator. Besides analytical and numerical simulation results, we have reported hardware experimental results on this problem.