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Abstract

This paper presents a study of behaviors of high-order multi-source Rauch filters based on the characteristics of complex functions. The advantages of Rauch’s connections are an easier selection of circuit components and a simpler design in fully differential forms and complex topologies. A general superposition formula is also introduced for deriving the transfer functions of multi source transmission networks such as fully differential amplifiers, fully differential low-pass filters, polyphase filters, complex filters, and quadrature signal generation circuits because the shapes of any motions and the characteristics of any transmission networks are analyzed based on superposition principle. The proposed Nichols chart of the self loop function is used to do the ringing test for high-order Rauch low-pass filters because it can be easily calculated using MATLAB, simulated using SPICE, and measured in Network Analyzers. The innovation of Nichols chart of self-loop function is a useful tool for the ringing test for high-order networks. The simulation results of the phase margin at unity gain of the self-loop function in a proposed design of the 4th -order Rauch low-pass filter are 74 degrees (over-damping region), 68 degrees (critical damping region), and 59 degrees (under damping region). The pass-band gain and the image rejection ratio of the proposed design of a 4th -order Rauch complex filter are 12 dB and 32 dB, respectively. It’s shown that complex functions play an important role in mathematical models of motion in the multi-source networks.

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