On the Chaotic Behavior of a Third-Order Log-Domain Filter

Abstract

Operation of log-domain filters revolves around the large-signal forward-active mode exponential current-voltage relationship of the bipolar junction transistor, which is used to map the input currents to the logarithmic domain, where the analog processing takes place, and to convert the filtered voltage waveforms back to the linear domain. Sometimes, the inherent internal nonlinearity of such circuits is the cause of unwanted externally-nonlinear effects. In this paper a differential third-order log-domain Chebyshev low-pass filter is designed by applying the method of operational simulation of doubly-terminated LC ladders. When the six grounded shunt capacitors of this filter are properly replaced with three half-sized floating capacitors, as is common practice in internally-linear fully-differential capacitively-loaded circuits, under particular conditions the resulting system exhibits limit-cycle oscillations, period-doubling bifurcations and chaotic behavior for zero-input. The presence of chaos is confirmed by extracting the spectrum of Lyapunov exponents of the system from the time series of the output voltage.