Study of Helix Functions and Multi-Source Rauch Filters

This paper presents a study of helix functions and multi-source Rauch filters using complex functions. Complex functions which imaginary unit j represents a 90-degree phase shift are used to express both analogue signals and transmission networks. Helix waves are proposed to analyze the behaviors of analogue signals such as sinusoidal signals, differential signals, and complex signals in both time and frequency domains. A general superposition formula is also introduced for deriving the transfer functions of multisource transmission networks such as fully differential amplifiers, fully differential low-pass filters, polyphase filters, complex filters, and quadrature signal generation circuits. In addition, the advantages of Rauch filters are an easiest selection of circuit components and a simplest design in fully differential forms and complex topologies. As a high-order feedback system operates in the under-damping region, ringing occurs and makes the network unstable. Therefore, the Nichols chart of self-loop function is used to do the stability test of a 4
^th
-order fully differential Rauch low-pass filter because it can be easily integrated in Network Analyzers. The simulation results of the phase margin at unity gain of the self-loop function in a proposed design of this filter are 74 degrees (over-damping region), 68 degrees (critical damping region), and 59 degrees (under-damping region). With the proposed superposition formula, the transfer function and the image rejection ratio of a 4
^th
-order Rauch complex filter are easily derived and investigated in both positive and negative frequency domains. The image rejection ratio of the Rauch complex filter is 32 dB. It's shown that complex functions play an important role in mathematical models of motion in multisource networks. The innovation of this work is the Nichols chart of self-loop function which is a useful tool for the stability test of feedback networks. The observation of the phase margin at unity gain of the self-loop function helps us determine the operating regions of high-order systems quickly. In addition, the use of the superposition formula is a fast network analysis method for high-order multi-source systems.