Concept

Analyticity — where it appears

The property of a function of complex frequency that it has a derivative everywhere in a region, so its values on a line fix it everywhere else there. Causality makes a passive system's response analytic in one half-plane, and that is the source of the Kramers–Kronig relations and of bandwidth limits.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

$How close a network gets to a load that stores charge. The fraction of a wave's amplitude reflected from a resistance shunted by a capacitance, with RCωc = 2, against frequency in units of the band edge ωc, for the load alone and for networks of inductors and capacitors whose values, with an ideal transformer at the source, were optimised numerically to keep the reflection low across the band. The bare load holds it to 0.707 across the band; the 1-element network holds it to 0.392 across the band; the 2-element network holds it to 0.320 across the band; the 3-element network holds it to 0.289 across the band. The dashed line is Bode and Fano's floor, exp(−π/RCωc) = 0.208, which no network of any size can go beneath over the whole band; each added element brings the design closer to it, and each buys its flatter band with a reflection that climbs to total just beyond the band edge.$

The mismatch no network can remove

A quarter-wave layer or a taper can match a resistance to a resistance as well as anyone likes. Put a capacitance across the load and that stops being true for every network that could ever be built from lossless parts: Bode and Fano proved that the total amount of match available is fixed by the load's resistance and capacitance, so a network can only move it about — and a flat match across a band can never be better than e to the minus π over the load's time constant times the band.

waves · Impedance

The thickness fixes the area, and the design only moves it. How much a metal-backed resistive sheet 7.5 mm from the metal absorbs, measured as ln(1/|Γ|), against wavelength, for sheets of 150, 250, 377, 800 Ω per square. Rozanov's bound says the area under any such curve, over all wavelengths, can be at most 2π² times the thickness: 147.9 mm here. The areas computed are 147.9 mm for 150 Ω, 147.9 mm for 250 Ω, 147.9 mm for 377 Ω, 69.7 mm for 800 Ω. Every sheet up to the impedance of free space meets the bound exactly and only moves its area between wavelengths; the more resistive sheet wastes part of it.

The absorber whose bandwidth is its thickness

A metal plate reflects everything, and a surface that must swallow a radar wave instead has to be a match between free space and a short circuit. A resistive sheet a quarter-wave in front of the metal does it perfectly at one frequency. Widening the band takes more sheets and more depth, and a theorem says why that is not a failure of design: for any absorber backed by metal, the total absorption over all wavelengths is fixed by its thickness, and by nothing else except its permeability.

waves · Impedance

Named alongside it

The objects these essays reach for when they reach for this one.

Bandwidth Impedance Impedance-matching Capacitance Causality Permeability Quality factor Reflection Relaxation Standing wave

All concepts