Dynamic Errors of Butterworth Band-Pass Filters in Analog-digital Control and Monitoring Systems

The $\mathrm{n}^{\text{th}}$ order band-pass filters (BPF) included at the input of the analog-to-digital conversion (ADC) in the control and monitoring system (CMS) to extract a preset signal spectrum, consisting of the series-connected Butterworth $n^{\text{th}}$ order low-pass filters (LPF) with a cutoff frequency of the amplitude-frequency characteristics (AFCs) and Butterworth $n^{\text{-th}}$ order high-pass filters (HPF) with a cutoff frequency of the AFCs of the filter $\omega_{ss}$ are considered. Shows that the dynamic errors of the $n^{\text{-th}}$ order BPFs are determined by the dynamic error of the $n^{\text{-th}}$ order LPF with the cutoff frequency $\omega_{sH}$, as well as the frequency ratio $\omega_{\mathrm{ss}}/\omega_{\mathrm{sH}}$. During the simulation, the delay times of signals in the BPF are obtained which makes it possible to determine the BPF dynamic error when changing the filter order for $n=1,2,3$ and estimate the dynamic error due to the information delay in the BPF. It has been established that for the filters under consideration there is an exponential dependence (negative exponent) of the value of the maximum dynamic error on the ratio $\omega_{\mathrm{ss}}/\omega_{\mathrm{sH}}$, which makes it possible to obtain a simple approximating analytical equation for the value of the dynamic error of the random order BPF. An estimation of the error of the proposed approximation is given.

I. Introduction

In analog-to-digital control and monitoring systems (CMS), the signal time delay in the band-pass filter (BPF), included at the analog-digital conversion (ADC) input [1–23], is the cause of the dynamic error [4], which is added to the static error and actually reduces effective bit depth of the ADC [4].