PYTHON · NUMPY

Bandwidth is a budget: Bode–Fano

Meet the conservation law that says a perfect match over a band is impossible.

01Challenge

Try to beat the Bode–Fano bound with real matching networks, fail, and then measure how much of the budget a two-element design already spends.

02Model

You cannot match perfectly over a band. Bode and Fano proved that for a reactive load the integral of ln(1/|Γ|) over all frequency is bounded by a number the load hands you — and no network, however clever, can exceed it. That turns matching into accounting: the budget is fixed, and if you spend it deeply in one band you have nothing left for another. Push the reflection down here and it rises there. Control theory has the same law under a different name — Bode's sensitivity integral, the waterbed effect — and the same shape shows up whenever people widen the randomisation band in sim-to-real and watch peak performance sag. Whether that last one obeys a provable integral of its own is, as far as we know, open.

03Guided practice

1. The budget. For a parallel RC load it is π/RC, and it is all you get. 2. The designs. Grid-search two-element networks for the best worst-case match over bands of growing width. 3. The tax. The wider the band, the worse the best possible match — and none of them beats the floor. 4. The receipt. Integrate ln(1/|Γ|) for one real design and see how much of its allowance is already gone.

04Feedback

PASS WHEN PASS when no design beats the Bode–Fano floor and the wider bands force worse matches.

If your run fails, check:

A cell errored — run it again.
No design should beat the bound; check the integral.

05Retrieve & space

What does Bode–Fano bound? (the integral of ln(1/|Γ|) over all frequency).
What happens to the best possible match as you widen the band? (it gets worse — the budget is fixed).
What is the control-theory sibling of this law? (Bode's sensitivity integral — the waterbed effect).

06Mastery & project

You can state why broadband matching is a budget and predict the direction of the trade.

The habit: when a fix seems free, look for the band where you paid for it.

GO FURTHER · THE LIVE PLAYGROUND

Smith Chart Navigator — the matching game ↗ — Load a .s2p measurement and the game runs broadband: push the curve down in one place and the band makes you pay in another.

← Channels, capacity, and the matched filter The physical price of a bit: Landauer →

PYTHON · NUMPY

Bandwidth is a budget: Bode–Fano

Try to cheat a conservation law, then read the receipt.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you