This paper analyzes the small-signal impedance of three-phase grid-tied inverters with feedback control and phase-locked loop (PLL) in the synchronous reference (d-q) frame. The result unveils an interesting and important feature of three-phase grid-tied inverters - namely, that its q-q channel impedance behaves as a negative incremental resistor. Moreover, this paper shows that this behavior is a consequence of grid synchronization, where the bandwidth of the PLL determines the frequency range of the resistor behavior, and the power rating of the inverter determines the magnitude of the resistor. Advanced PLL, current, and power control strategies do not change this feature. An example shows that under weak grid conditions, a change of the PLL bandwidth could lead the inverter system to unstable conditions as a result of this behavior. Harmonic resonance and instability issues can be analyzed using the proposed impedance model. Simulation and experimental measurements verify the analysis

Synchronous reference frame (SRF) phase-locked loop (PLL) is a critical component for the control and grid synchronization of three-phase grid-connected power converters. The PLL behaviors, especially its low-frequency dynamics, influenced by different grid and load impedances as well as operation mode have not been investigated yet, which may not be captured by conventional linear PLL models. In this paper, we propose a state-feedback quasi-static SRF-PLL model, which can identify and quantify the inherent frequency self-synchronization mechanism in the converter control system. This self-synchronization effect is essentially due to the converter interactions with grid impedance and power flow directions. The low-frequency nonlinear behaviors of the PLL under different grid impedance conditions are then analyzed, which forms the framework of evaluating the impacts of the large penetration level of distributed generation units, weak grid, microgrid, and large reactive power consumption in terms of the frequency stability of PLL. Specifically, the PLL behavior of the converter system under islanded condition is investigated to explain the PLL instability issues and the related islanding-detection methods in early publications and industry reports

The auditory system operates over a vast range of sound pressure levels (100 -120 dB) with nearly constant discrimination ability across most of the range, well exceeding the dynamic range of most auditory neurons (20 -40 dB). Dean et al. (2005) have reported that the dynamic range of midbrain auditory neurons adapts to the distribution of sound levels in a continuous, dynamic stimulus by shifting toward the most frequently occurring level. Here, we show that dynamic range adaptation, distinct from classic firing rate adaptation, also occurs in primary auditory neurons in anesthetized cats for tone and noise stimuli. Specifically, the range of sound levels over which firing rates of auditory nerve (AN) fibers grows rapidly with level shifts nearly linearly with the most probable levels in a dynamic sound stimulus. This dynamic range adaptation was observed for fibers with all characteristic frequencies and spontaneous discharge rates. As in the midbrain, dynamic range adaptation improved the precision of level coding by the AN fiber population for the prevailing sound levels in the stimulus. However, dynamic range adaptation in the AN was weaker than in the midbrain and not sufficient (0.25 dB/dB, on average, for broadband noise) to prevent a significant degradation of the precision of level coding by the AN population above 60 dB SPL. These findings suggest that adaptive processing of sound levels first occurs in the auditory periphery and is enhanced along the auditory pathway.

Small-signal stability is of great concern for electrical power systems with a large number of regulated power converters. In the case of dc systems, stability can be predicted by examining the locus described by the ratio of the source and load impedances in the complex plane per the Nyquist stability criterion. For balanced three-phase ac systems the same impedance-based method applies, for which this paper uses impedances in the synchronous rotating reference (d-q) frame. Small-signal stability can be determined by applying the Generalized Nyquist stability Criterion (GNC). This approach relies on the actual measurement of these impedances, which up to now has severely hindered its applicability. Addressing this shortcoming, this paper investigates the small-signal stability of a three-phase ac system using measured d-q frame impedances. The results obtained show how the stability at the ac interface can be easily and readily predicted using the measured impedances and the GNC; thus illustrating the practicality of the approach, and validating the use of ac impedances as a valuable dynamic analysis tool for ac system integration, in perfect dualism with the dc case.

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