Log-domain filtering and the Bernoulli cell

Abstract

In this paper, the dynamic behavior of a nonlinear circuit element termed a Bernoulli cell is described, which is composed of a suitably biased bipolar junction transistor (BJT) and an emitter connected grounded capacitor. This cell has application in the synthesis of log-domain filters, since it facilitates the development of a low-level design approach in which a frequency-domain transfer function is decomposed into timedomain current product equalities that can be implemented by direct use of the translinear principle (TLP). Furthermore, the dynamic of a log-domain structure can be analyzed and its frequency response can be easily derived when the embedded Bernoulli cells are identified. An analysis and a synthesis example are presented.

Modelling the dynamics of log‐domain circuits

“…These circuits can lose external linearity entirely. We show that, contrary to standard assumptions of log-domain circuit analysis [1][2][3][4][5], parasitic capacitances in the bipolar junction transistors (BJTs) have a significant effect on the dynamic behaviour of these circuits, no matter how small they are with respect to the floating capacitors. We prove here that when, in agreement with general assumptions of log-domain circuit analysis, transistors are modelled as static exponential nonlinearities, the equations of these circuits are unable to describe any dynamic behaviour.…”

“…Consider the mathematical model of multi-input integrator , i.e. Equation (5). Using Equation (11), the mathematical model of the differential-output multi-input integrator takes the form…”

“…This section is divided into two parts. The first part proves that the equations of the floatingcapacitor circuits, based upon common assumptions from log-domain circuit theory [1][2][3][4][5], are unable to describe any dynamic behaviour. The second part detects the simplest BJT dynamic model that permits the derivation of equations qualitatively capturing the dynamics for zero input.…”

“…Log-domain filters [1][2][3][4][5] are externally linear, internally nonlinear current-mode circuits [6], in which analogue processing is performed on the voltages at the output of a compression stage fed by the input currents. The processed voltages are then converted back to the linear domain through an expansion stage.…”

Design of Current-Mode Square-Root Domain Band-Pass Filter with Reduced Voltage

“…Frey [2,3] utilized the exponential transformation of the state space description of a linear transfer function, in terms of bipolar circuits, to implement log-domain filters and in turn, was embedded in as forming part of a broader branch of structures by Tsividis [4]. Additionally, Drakakis et al [5][6][7] proposed a systematic synthesis method, based on "Bernoulli cell", to implement the log-domain filters.…”