Circuit Response
every model has an edge, and the figure carries it
Field

Lines, where a wire has a length

On the far side of the frequency at which Kirchhoff's laws give out. What a source drives into is decided by geometry before the load has any say; what comes back one delay later decides the rest. The wave picture is checked against a lumped ladder that has never heard of a wave — and the useful result is how badly the ladder does.
00.50011.5002468time, in propagation delaysvoltsthe divider: 1.0000 V0.833 V: the source against the linethe far endthe near endfirst arrival, 4.83 nssolved, then checked — a series against a dividernothing at the far end before 4.83 ns

The staircase in time

A source driving a metre of cable does not know what is on the far end of it for 4.83 nanoseconds. What it drives into during that time is decided by the cable's characteristic impedance and nothing else — and when the far end finally answers, the answer comes back as a staircase whose limit is the resistive divider the circuit was going to be all along.

 00.50011.50202468time, in propagation delaysvolts at the far endthe wave answer20 sectionsone delaysolved, then checked — a ladder against a latticeplateaus 0.08 V out at 20 sections

A ladder is not a line

A transmission line is usually introduced as the limit of a chain of inductors and capacitors as the number of sections goes to infinity. That is true, and it gives entirely the wrong impression of how close a finite chain gets. Forty sections still ring through every plateau by five per cent, and extrapolating the fitted convergence, reaching one per cent would need about nine hundred and sixty.

 00.2000.4000.6000.8000.50011.50frequency ÷ the frequency it was cut forreflection |Γ|the bare junction: 0.600|Γ| = 0.1, or 20.0 dB return lossexact here only17.1% of a bandsolved, then checked — the band scanned, not approximatedexact at one frequency, 17% of band

A quarter wave, and the path the current takes back

A line a quarter of a wavelength long, whose impedance is the geometric mean of the two it joins, matches them exactly — reflecting 5×10⁻¹⁷ of what arrives, which is the arithmetic's floor. At one frequency. Seventeen per cent either side of it the reflection is back to a tenth, and that band is the whole of what the technique is worth.

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