Engineering notes
Understand the behaviour.
Technical notes at engineering level: concise where possible, rigorous where necessary. Concepts, derivations and diagrams share one visual language.
Index
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The first notes establish the format. The library can expand without changing the structure of the site.
Physics
Propagation delay
A signal transition does not arrive everywhere at once. At sufficiently short rise times, conductor length becomes part of the electrical model.
Explanation
For an interconnect of length l and propagation velocity vp, the one-way propagation delay is:
When propagation delay is no longer negligible relative to the signal rise time, a lumped-circuit approximation becomes insufficient. The interconnect must then be treated as a distributed structure, with impedance and reflections included in the analysis.
The relevant comparison is therefore not simply signal frequency versus trace length; edge rate and propagation time determine when transmission-line behaviour becomes significant.
Diagram
Electronics
Phase-locked loops
A PLL is a feedback system that drives phase error toward a controlled condition while generating a related output frequency.
Explanation
A basic loop compares a reference phase with a divided output phase. The resulting error is filtered and used to control an oscillator. The divider ratio determines the nominal frequency relationship; loop dynamics determine how the system approaches and maintains that relationship.
Bandwidth and damping cannot be selected independently of noise, settling behaviour and stability. Practical design therefore requires the complete loop to be considered rather than treating the individual blocks in isolation.
Block diagram
Mathematics
Closed-loop response
The transfer function makes the consequences of feedback explicit and gives a framework for analysing stability and response.
Explanation
For a forward-path transfer function G(s) with unity negative feedback, the closed-loop transfer function is:
The denominator contains the characteristic equation of the closed-loop system. Its poles determine stability and transient behaviour; the loop gain determines how strongly disturbances and errors are corrected over frequency.
System view
Editorial principle
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