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Chapter 7. Deriving the 2nd-Order Butterworth LPF Transfer Function & Coefficients How do we mathematically derive the maximally flat response of a Butterworth filter, and how does it translate into discrete digital parameters? IntroductionThe Butterworth filter, introduced in 1930 by the British physicist Stephen Butterworth, is designed to have a frequency response that is as flat as mathemati..

Chapter 6. How to Fix Frequency Warping: The Concept of Pre-warping To achieve the exact digital cutoff frequency we desire, how should we intentionally manipulate the analog filter beforehand? Introduction...In our previous post, we discussed the Bilinear Transformation and how it maps the continuous s-domain to the discrete z-domain. We also discovered a critical side effect: Frequency Warping..

Chapter 5. The Necessity of Bilinear Transformation and Frequency Warping in Digital Control Why must we map the s-domain to the z-domain, and how does the Bilinear Transformation perfectly preserve system stability? Introduction: The Analog vs. Digital DilemmaThe filter design methodologies we learn in classical control engineering are heavily rooted in the continuous-time domain, represented b..

Stability Margin: Relative Stability, Phase/Gain Margins, and System Uncertainties Why is simply knowing whether a system is "stable" or "unstable" not enough in real-world control systems? Introduction: The Concept of Relative StabilityIn theoretical control engineering, checking absolute stability (whether poles are in the left-half of the s-plane) might seem sufficient. However, in practical ..

Nyquist Plot, Cauchy's Theorem, and Stability Criterion Why do we use the Nyquist Plot to analyze closed-loop stability using only open-loop information? Introduction...When designing control systems, evaluating the stability of a closed-loop system directly can be mathematically tedious. The Nyquist Plot offers an elegant graphical solution: determining closed-loop stability solely by analyzing..

Frequency Response and Bode Plot: Stability and System Characteristics Why do we analyze systems in the frequency domain, and how do Bode plots help us determine stability? Introduction...When analyzing the behavior of dynamic systems, the time domain gives us a direct view of how a system responds over time. A system's total response is always the sum of its transient response and steady-state ..