A quarter wave, and the path the current takes back

Two facts about a quarter-wavelength of line are usually stated together and are worth separating, because one of them is an identity and the other is a technique with a bandwidth.

The identity is that a line a quarter of a wavelength long inverts whatever is on its far end about the square of its own characteristic impedance: $Z_\mathrm{in} = Z_0^2/Z_L$. A short circuit becomes an open circuit, an open becomes a short, and a load becomes something else entirely.

The technique is what falls out of that: to join two lines of different impedance without reflection, put a quarter-wavelength of line between them whose impedance is the geometric mean of the two.

Exact, and only there #

The reflection at the design frequency is 4.6×10⁻¹⁷. That is not a small number in the sense of being negligible; it is the residue of a complex division and it means the match is exact.

Twenty per cent below the design frequency it is 0.14. Twenty per cent above it, 0.13. Half the design frequency and it is 0.36 — most of the way back to the 0.60 the bare junction reflects with nothing between the two lines at all.

So the useful statement about a quarter-wave transformer is not the identity but a band, and the figure measures it rather than quoting the standard approximate expression for it. Scanning the solved reflection and bisecting for the two frequencies where it crosses a tenth gives 914 MHz and 1,086 MHz for a design frequency of 1 GHz: a fractional bandwidth of 17.1%.

That band depends strongly on how large a transformation is being attempted, which is the slider’s argument.

It is worth saying what “reflection” costs, because a coefficient between zero and one is easy to read as a percentage of something and it is a percentage of the wrong thing. Γ is a voltage ratio, so the power reflected is Γ². A bare fifty-to-two-hundred junction reflects 0.600 of the voltage, which is 36% of the power; a bare fifty-to-a-thousand junction reflects 0.905, which is 82%. The transformer takes both to nothing at the design frequency and to Γ = 0.1 — one per cent of the power — at the edges of the band this page measures.

A tenth is a common enough threshold that it deserves a name: it corresponds to a return loss of 20 dB, which is what a specification usually asks for and what the shaded regions in the figures mark.

The band narrows as the transformation grows #

Matching fifty ohms to seventy-five — a 1.5:1 step, and the mismatch between two common cable impedances — gives a bare reflection of 0.200 and a band of 65.6%. Matching fifty to a hundred is 0.333 and 36.7%. Fifty to two hundred is 0.600 and 17.1%. Fifty to a thousand is 0.905 and 6.0%.

The pattern is clear enough to be worth naming: the harder the transformation, the narrower the band over which one section achieves it. A single quarter-wave section is a good match over a wide band when there was not much wrong in the first place, and a narrow-band trick when there was.

What is done about that in practice is to use several sections in series, each making part of the transformation, with impedances chosen so that their individual reflections cancel over a band rather than at a point. That is a genuinely different design problem — it is filter design, with the reflection as the quantity being shaped — and it is named here and not built, because it needs machinery this site does not have.

The other exactness, and what breaks it #

The identity also says that a quarter-wave line with a short circuit at its far end presents an open circuit at its near end. That is exact in a lossless line, and it cannot be exact in a real one for a reason that needs no arithmetic at all: an open circuit carries no current, and a line that dissipates power must be carrying some.

Putting attenuation into the line model and evaluating the same expression with a complex propagation constant gives the size of it. A quarter-wave shorted stub in a lossless line has an input impedance limited only by how exactly the length is cut. With 0.2 dB per metre of attenuation the same stub presents about 38 kΩ, and with 1 dB per metre about 8 kΩ.

Those are large impedances and for most purposes they are open circuits. They are also finite, and the value is set by the loss rather than by anything about the geometry — which means the quantity that decides how good a resonator a length of line makes is the one property the ideal model leaves out entirely.

This is the same shape as the capacitor that is an inductor: a component behaves as its symbol says up to a point set by something the symbol does not show, and the number is computable from the thing that was omitted.

Where the current goes back #

The second half of this essay is about a question the first half did not have to ask, because a coaxial cable answers it by construction: where does the return current flow?

On a circuit board it is not obvious. A track runs over a plane, and the plane is a sheet of copper that the current could return through anywhere. What decides is a competition between two costs.

Resistance is minimised by spreading out. A wide path has more copper in parallel and less resistance, so at low frequency the return current fans out across the plane.

Inductance is minimised by staying close. The loop the current encloses is the area between the outgoing track and its return, and inductance is proportional to that area, so a return directly beneath the track has the least. At high frequency the reactance dominates the resistance and the current has effectively no choice.

A megahertz and a half, which is nothing #

The estimated crossover comes out at 105.5 kHz for that geometry, and the number is worth pausing on because it is so much lower than most readers expect. Solved across the plane rather than estimated from two paths, as the corner that is three decades wide does it, the change is a band rather than a corner: half the return has gathered beneath the track by 283 kHz and nine tenths by 1.42 MHz.

Above that band the return current runs directly beneath the outgoing track. That is still not a high-frequency phenomenon in any useful sense — a megahertz and a half is below the clock of almost anything digital — so for essentially everything a digital board does, the return path is determined by inductance and the current is directly under the track whether the designer intended it or not.

Two practical consequences follow, and both are about what happens when the current is prevented from taking that path.

A slot in the plane is a detour. Cut a slot across the plane under a track and the return current cannot go straight; it goes around, and the loop area becomes the area of the detour rather than the area of the track’s own height. A ten-centimetre detour around a slot turns a twenty-square-millimetre loop into something a hundred times larger.

A change of layer is a discontinuity. A track that passes through a via to another layer takes its return current with it, and unless there is a path for the return to change layers too — a nearby via between the two planes — it must find one somewhere else, which is another detour of the same kind.

Neither of those is visible on a schematic. Both are decided by the geometry of a copper sheet that the schematic represents by a symbol, and the site’s habit that a schematic is a label rather than a drawing has an unusually literal application here: the layout is the circuit.

What the loop radiates, and where the model stops #

The loop area matters because it is what radiates. A small current loop’s far field rises as the square of the frequency and in proportion to the enclosed area, which is why the two consequences above are stated in terms of area rather than in terms of length.

For the geometry drawn — twenty square millimetres, a milliamp, a hundred megahertz, three metres away — the field is −1.1 dBµV/m. Lift the track to 1.6 mm above the plane and the loop is 160 mm², eight times larger, and the field rises by the same factor to 16.9 dBµV/m. The Class B limit at three metres is 30 dBµV/m in that part of the spectrum, so the difference between a well-routed track and a badly routed one is a substantial fraction of the whole allowance, from geometry alone.

The expression used is the standard small-loop result, and it has a validity condition that is computed and drawn rather than assumed: the loop’s perimeter must be a small fraction of a wavelength. At a tenth of a wavelength, the two-hundred-micrometre case gives out at 1.68 GHz and the 1.6 mm case at 593 MHz. Above those the calculation is not slightly wrong, it is answering a different question, and the figure refuses to apply it there rather than extrapolating.

Why the two costs are the ones that compete #

There is a step in the return-path argument that deserves its own paragraph, because it is the part most often stated as a slogan.

The claim is not that current “prefers” one path. Current divides between every available path in inverse proportion to their impedances, always, and both paths carry some at every frequency. What changes with frequency is the ratio, and it changes because one path’s impedance is a resistance and the other’s is a reactance: one is flat and the other rises linearly.

Below the crossover the spread path’s 83.3 mΩ is smaller than the tight path’s ωL, so most of the current spreads. Above it the reactance exceeds the resistance and the split reverses. At the estimated crossover itself the two are equal by definition, and the two-path picture has the current dividing roughly evenly between paths that are physically far apart. Solved, it is not: at 106 kHz 14.0 per cent of the return is within one height of the track, against 48.7 per cent once it has gathered, and the rest is spread across a plane that carries some of it tens of millimetres out — which is still the least convenient case for anyone trying to predict what the board will radiate.

The number that decides it, R/2πL, has the form of a corner frequency because that is what it is: the return path is a one-pole divider between two impedances, and its crossover is the pole. Nothing about the phenomenon needs vocabulary the rest of this collection does not already have.

Two things follow that a slogan does not give. The crossover moves with the geometry rather than with the signal — a greater height lowers it, and a more resistive plane raises it in proportion — and the transition is gradual. Solved, it is more gradual than a decade either side: eight tenths of the gathering happens between 26.3 kHz and 1.42 MHz, so a board carrying signals anywhere from ten kilohertz to a megahertz has a return current that is genuinely undecided, and the proportion moves across the whole of that range.

The two halves of the page #

The quarter-wave transformer and the return path look like separate subjects and they are the same one, which is why they share an essay.

Both are about the fact that above the frequency where a circuit is electrically large, the geometry is the circuit. A quarter-wavelength of line is a component whose value is a length. A return path’s inductance is an area. Neither has a symbol on a schematic, neither appears in a netlist, and both are decided by decisions that look like drawing rather than like design.

The frequency at which that transition happens is the one Kirchhoff’s own frequency computes, and this field is what is on the far side of it. What this page adds is that the transition is not a single frequency for all purposes: the return path crosses over at a hundred kilohertz, the lumped model gives out at four megahertz, and the small-loop radiation model gives out at one and a half gigahertz. Three boundaries, four decades apart, belonging to the same ten centimetres of copper.

One number this page does not compute #

The radiated field here comes from a closed-form expression rather than from a solve, and that makes it the only quantity in this field that is quoted rather than measured. It is worth being explicit about, because everything around it is solved and the difference is easy to miss.

The expression is the standard small-loop result: a loop of area A carrying a current I produces a field at distance r proportional to f²AI/r. Its derivation assumes the current is uniform around the loop, which is what “electrically small” means, and the figure carries the frequency at which that assumption fails. What the figure does not do is derive the coefficient, and it could not without machinery — a field solver — that is a different subject from the one this site is about.

The honest description is therefore that the loop area is computed from the geometry, the validity boundary is computed from the geometry, and the constant between them is taken. That is one borrowed number in a collection that computes everything else, and stating which one it is seems better than letting it blend in.

Seventeen per cent, and what buying more of it costs #

The band this section holds — seventeen per cent for a reflection under a tenth on a four-to-one transformation — is the whole of what one quarter-wave section is worth, and it is a narrow enough answer that the next question is immediate.

Several sections, and the band they buy is that question answered, and the shape of the answer is filter design rather than transmission-line theory: sections whose reflections cancel over a band take the 17.1 per cent to 47.7, 67.9, 82.1 and 92.6, and an equal-ripple design found by minimax search buys a further 35 to 46 per cent at the same worst reflection, with its ripples level to four decimals. The reflection has become the shaped quantity, which is exactly what the two resistors a ladder was designed between says a ladder filter’s synthesis is doing — the same arithmetic, the binomial and equal-ripple choices there being the Butterworth and Chebyshev choices here.

And the number that was wrong is the correction to that sequence, which is worth reading beside it because the failure was in the measurement rather than in the design. A five-section transformer’s band was printed as 97.3 per cent where the answer is 134: the objective was the worst of a grid rather than the worst of a band, the band was then measured by bisecting a function that crosses its threshold five times, and the assertion guarding all of it passed — because 97.3 is still more than the four-section figure it was required to exceed.

That is the useful pairing to end on. One section is exact at a point and narrow; several sections are approximate everywhere and wide; and the arithmetic that measures how wide is easier to get wrong than the arithmetic that designs them.

What “exact” is worth on a real section #

The $5\times10^{-17}$ this section reflects is the arithmetic’s floor, and it is worth saying what stands between that number and a built transformer, because three of the four obstacles are measured elsewhere in this field and none of them is the design.

The line has loss. The mismatch that the cable hides is the measurement of what that does to a reflection, and its direction is the surprising one: loss improves a measured mismatch, exactly, at twenty decibels of return loss for every ten of one-way loss — so a lossy quarter-wave section reflects less than a perfect one and is worse at its job, since what it is not reflecting it is absorbing.

The line has dispersion. The delay that is not one number finds a real trace’s velocity proportional to $\sqrt f$ below 910 kilohertz for ordinary copper, and a quarter wave is a statement about a velocity — so a section cut to be a quarter wave at one frequency is a different electrical length at another by more than the frequency ratio alone accounts for.

And the impedance is realised rather than specified. The two resistors a ladder was designed between is the same sensitivity in the filter field, where a source outside 0.886 to 1.137 of the design resistance costs half a decibel — and a transmission line’s characteristic impedance is set by a geometry with its own tolerance, so the geometric mean this section is cut to is met to whatever etching and laminate thickness allow.

Which does not weaken the result; it locates it. The exactness measured here is the design’s, and it establishes that nothing in the concept limits the match. Everything that does limit it is a property of the medium, and every one of those has its own essay.