Proof of Bode-Fano criterion - Page 1
Forum section: electronics / rf-microwave

author: promach
author_rank: Frequent Contributor
timestamp: April 28, 2019, 12:58:39 pm
content: I was readingthis thesisand it mentions Bode-Fano criterion almost in every chapter of this thesis.Could anyone advise about the?Note: Theoriginal proofis so tough to digest and comprehend.

author: vk6zgo
author_rank: Super Contributor
timestamp: April 28, 2019, 02:09:07 pm
content: It's way above my pay grade, but I can sort of make sense of it if I squint my eyes up, & hold my head "just so".It is basically saying that not all loads, even some resistive ones, can be matched using physically realisable coupling networks.I must say, I thought it was a fantasy when I first saw reference to "Bode-Fano" in the write up on using a piezo electric crystal as a physically small VLF antenna ( see my posting in reference to this).

author: promach
author_rank: Frequent Contributor
timestamp: April 28, 2019, 03:19:03 pm
content: Quote@vk6zgoWhat posting ?

author: rfeecs
author_rank: Frequent Contributor
timestamp: April 28, 2019, 05:52:14 pm
content: The expression you pictured is the special case of an RC in parallel.It was derived by Bode in his book "Network Analysis and Feedback Amplifier Design":https://archive.org/details/NetworkAnalysisFeedbackAmplifierDesign/page/n1His derivation starts in Chapter XVI, page 360 and that equation appears on page 367, equation (16-15).It looks pretty straightforward, but he does refer to earlier concepts from the book. You may have to read the previous 360 pages to follow it.It looks like he starts with the equation for mismatch loss in terms of the impedances and reflection coefficient in equation (16-10) and ends up with that equation two pages later.

author: vk6zgo
author_rank: Super Contributor
timestamp: April 29, 2019, 01:49:04 am
content: Quote from: promach on April 28, 2019, 03:19:03 pmThis one"Piezo-electric antenna? I think bollocks, how about you?"« on: 17-04-2019, 12:26:39 »this sub-forumIt was in reference to this item in "Nature Communications"https://www.nature.com/articles/s41467-019-09680-2It isn't really relevant, it is just the first time in my non-sheltered life that I heard of it.

author: ejeffrey
author_rank: Super Contributor
timestamp: April 29, 2019, 04:55:47 pm
content: This is how a friend of mine explained Bode-Fano criteria qualitatively:If you look at the smith chart of a simple mismatch with some reactance (say an RC network), the reflection coefficient (S11) always goes clockwise with increasing frequency. If you want to match that perfectly, you need a matching circuit where the phase goes counter-clockwise. Such a circuit would violate causality: outputs would come before inputs. What you can do is add resonators: resonances create loops on the smith chart, so they go counter-clockwise for a short frequency bandwidth. There is a tradeoff: the higher Q your resonance the more clockwise phase you can cancel, but the narrower bandwidth.A concrete example: consider a 50 ohm load (so perfectly matched by itself) that is attached to a 1 nanosecond segment of transmission line with Z=75 ohm. The transmission line is not matched to the load so there is a reactance you need to compensate. The matching network you want looks like -1 nanosecond of 75 ohm transmission line.The Bode-Fano criteria is basically writing down an integral that expresses the causality requirement of the matching network and relating that to the band limit.It comes with a whole bunch of assumptions: that the load and the matching network are passive, linear, and time invariant. So it is largely noted for the number of ways people find to get around it.

author: promach
author_rank: Frequent Contributor
timestamp: May 01, 2019, 01:18:23 pm
content: @rfeecsI am having problem deriving the Resistance Integral theorem which is needed by Bode-Fano criterion.

author: rfeecs
author_rank: Frequent Contributor
timestamp: May 01, 2019, 05:44:20 pm
content: Quote from: promach on May 01, 2019, 01:18:23 pmHe derives it in general starting in Chapter XIII. He uses Cauchy's Theorem plus various conditions on the nature of physical network impedance functions (real component is an even function of frequency, imaginary component is an odd function, etc.)He talks about Cauchy's theorem in Chapter VIII. From Wikipedia:https://en.wikipedia.org/wiki/Cauchy%27s_integral_theoremHe then confirms it directly for the case of a resistor in parallel with a capacitor:

author: promach
author_rank: Frequent Contributor
timestamp: May 02, 2019, 03:23:55 am
content: @rfeecsRight side of expression (13-6) is not zero, so this is really not cauchy integral theorem ?

author: rfeecs
author_rank: Frequent Contributor
timestamp: May 02, 2019, 05:35:59 am
content: Quote from: promach on May 02, 2019, 03:23:55 amThe derivation starts at section 13.4, equation (13-3) the right side equals zero.Even better, start at the beginning of section XIII.