The chaotic dynamics observed for the second-order phase-locked loop (PLL) operating as a frequency-modulated (FM) demodulator are studied theoretically as well as experimentally. The existence of horseshoe chaos in this circuit is proved using Melnikov's method. It is demonstrated that chaos can indeed occur in a practical FM demodulator circuit made of an integrated circuit module and using realistic parameter values. Explicit expressions that specify the possible region of chaos are derived. >

Chaos from phase-locked loops

The authors present a rigorous analysis and a detailed experimental study of a chaotic attractor observed from a widely-used practical circuit: namely, a second-order phase-locked loop as an FM demodulation. The existence of chaos in this system proven rigorously by using Melnikov's method, and explicit expressions are derived which specify the possible region of chaos, or, more accurately, the region of existence of the homoclinic orbits. Chaos has been demonstrated in actual experiments in which an IC is used as an FM demodulator. It has also been confirmed by computer simulation. Thus, it has been verified that horseshoe chaos occurs for a wide range of parameter values in practical phase-locked loop FM demodulator systems. >

http://ieeexplore.ieee.org/iel1/31/113/00001845.pdf

This paper presents a method which evaluates the harmonic distortion of the phase-locked FM demodulator in the absence of noise. An approach of the harmonic-balance method with the digital Fourier transform is described, and then it is applied to the phase-locked loop in order to calculate the steady-state phase error and harmonic distortion.

Harmonic Distortion of a PLL FM Demodulator for Periodic Signals

A widespread test of the phase-locked loop (PLL) FM demodulator under Gaussian modulation is simulated using the Monte Carlo method. Unified noise-free and noise performance analyses of the PLL FM demodulator are presented. Substantial reduction of the modulation limit by the input bandpass filter is reported in the region of the input bandwidth of practical interest. Bessel bandpass filters of order greater than two are shown to compare favorably with Butterworth filters in front of the PLL FM demodulator relative to the intermodulation (IM) distortion. A lower bound on the loop noise bandwidth is found by minimizing the output click rate for given IM distortion specifications. FM threshold of 4 and 7 dB for the root-mean-square (RMS) frequency deviation-to-message bandwidth ratio 0.1 and 1.0, respectively, is reported on the worst-case IM distortion of 45 dB.

PLL FM demodulator performance under Gaussian modulation

This contribution reports on devices that are of critical importance to the development of an accurate and phase-coherent optical synthesizer. In particular, we report on techniques to generate optical signals at accurate offset frequencies from a given optical signal, as well as techniques that can be used to link the optical and microwave frequency domains. A detailed presentation of potential noise sources of the various techniques enables a calculation of the likely frequency stability of the synthesizer. When those calculations are combined with experimentally measured results, we demonstrate that the long-hoped-for goal of optical synthesis at the fractional frequency level of 10−15 (≈ 1Hz) level is readily achievable with present-day technology.

Accurate Optical-Frequency Synthesis

Methods are presented for determining the modulation limits of phase lock loop (PLL) FM demodulators These limits establish the maximum deviation that the PLL can support (ie, remain "inlock") based on the loop design parameters The modulation limits are derived from experimental data on first and second order loops using sinusoidal and Gaussian noise modulation The analysis of each PLL is broken into two regions, low frequency and high frequency modulation It is shown that the deviation limit remains constant with modulation frequency for a first order loop in the low frequency region In the high frequency region the deviation limit increases with increasing modulation frequency for both first and second order loops The deviation limit in a second order loop increases with decreasing modulation frequency in the low frequency region and is a function of the loop damping

F. Gardner

Papers

Angle Modulation Limits of a Noise-Free Phase Lock Loop