The mismatch that the cable hides

The lossless transmission line is the model this field is taught in, and it makes one statement about a mismatch that everything downstream depends on: the magnitude of the reflection coefficient is the same everywhere along the line. Only the phase turns. So a load’s standing-wave ratio can be measured from the other end of any length of cable, and the length does not matter.

That statement is exactly true of a line with no loss and exactly wrong of every real one, and the departure is large enough to invert the conclusion of an acceptance test.

The closed form, and the solve that confirms it #

A line’s propagation constant is $\gamma = \alpha + j\beta$: an attenuation per metre and a phase shift per metre. A wave travelling to the load and back travels $2\ell$, so it comes back attenuated by $e^{-2\alpha\ell}$. The reflection coefficient measured at the input is therefore

$$|\Gamma_{\text{in}}| = |\Gamma_L| \cdot 10^{-2\alpha_{\text{dB}}\ell/20}$$

and the phase, which the lossless model also gets, rotates at $2\beta\ell$.

The figure does not evaluate that expression to draw its curve. It solves the line — the full complex hyperbolic tangent of $\gamma\ell$, which knows nothing about the expression above — and then asserts the two agree. Over six lengths spanning two orders they agree to better than a part in a billion, which is the arithmetic’s own floor.

The consequence is stated most usefully in decibels, and in decibels it is trivial: the return loss measured at the input is the load’s own return loss plus twice the one-way loss. Ten decibels of cable buys twenty decibels of apparent match.

What that does to a measurement #

Take a load that is genuinely four to one — two hundred ohms on fifty, a reflection coefficient of 0.6, a return loss of 4.44 dB. It is a bad load by any standard.

Put twenty metres of ordinary small coaxial cable in front of it, at a gigahertz, where its attenuation is half a decibel a metre. The one-way loss is ten decibels. The instrument at the near end reads

• $|\Gamma_{\text{in}}| = 0.060$
• standing-wave ratio 1.128
• return loss 24.44 dB

That would pass essentially any specification written for a connector, an antenna or a filter. Nothing about the load has changed. The instrument is looking at it through ten decibels of attenuator, twice.

The length at which it happens is the number worth carrying: the figure bisects the length at which the reading falls to 1.5, and at a gigahertz it is 9.5 metres. Nine and a half metres of cable between an instrument and a load turns a four-to-one mismatch into something a specification would accept.

The boundary is a loss, so it is also a frequency #

The three panels have the same shape and different scales, and the scaling is the second measurement.

The attenuation of a coaxial cable in the regime that matters here is set by the skin effect: the conductor’s resistance per metre goes as $\sqrt f$, and so does the attenuation in decibels per metre. So the length that hides a mismatch, being a fixed number of decibels divided by the decibels per metre, goes as $1/\sqrt f$.

Measured: the length at which the reading falls to 1.5 is 30.18 m at 100 MHz and 9.54 m at 1 GHz — a ratio of 3.164 against $\sqrt{10} = 3.162$, which the figure asserts to two parts in a thousand.

That is a more useful statement than the length itself, because it says which measurements are safe. An audio-frequency or a low-radio-frequency measurement through a laboratory cable is not affected at all: at ten megahertz the same cable is 0.05 dB per metre, and thirty metres of it costs 1.5 dB one way, which moves a return loss by three decibels. A microwave measurement through the same physical cable is affected severely. The cable did not change; the frequency did.

What it costs, which is the other half #

An engineer told that a long cable improves a match might reasonably ask what the problem is. The problem is that the improvement is entirely in the measurement and the loss is entirely real.

At twenty metres and a gigahertz the input accepts 99.6% of the power available from a matched source — which is what a return loss of 24 dB means — and delivers ten decibels less of it to the load, because ten decibels went into the cable. So the arrangement is superbly matched and delivers a tenth of the power. A shorter cable is badly matched and delivers far more.

The two quantities point in opposite directions and only one of them is what anybody wanted. That is worth stating as a rule, because match is so often used as a proxy for efficiency: a reflection coefficient measured at the wrong end of a lossy line is not a statement about power delivered. It is a statement about power accepted, and the difference between accepted and delivered is the loss.

The same mechanism used on purpose #

The effect is not only a nuisance. It is one of the standard tools of the trade, and recognising it as the same arithmetic is worth doing because it fixes the sign.

An amplifier that is conditionally stable will oscillate if it sees the wrong impedance at its input or output. A cable of unknown length with an unknown thing on the end of it can present any impedance at all, so the standard defence is a small attenuator between the amplifier and the outside world. Three decibels of attenuator improves the worst-case reflection the amplifier can see by six, and six decibels of margin on a reflection is the difference between “any load” and “any load that is not absurd”.

The same construction appears as the isolation between two ports of a splitter, as the pad on a mixer’s local-oscillator port, and as the reason a directional coupler’s directivity is quoted as a number rather than assumed infinite. In every case it is $|\Gamma|$ multiplied by ten to the minus twice the loss over twenty, and the only difference from this essay’s cable is that somebody chose the loss.

What it costs is also always the same and always in the same place. Every decibel of attenuation ahead of a stage is a decibel on the noise figure of everything after it, exactly, by the cascade formula. So the trade is completely specified: buy two decibels of match with one decibel of noise, and the exchange rate is fixed by physics rather than by the component.

Getting the load back out again #

If the loss is known, the load’s own reflection is recoverable — the expression is invertible — and it is worth being explicit about how well, because the answer decides whether the measurement was worth making.

Multiply the measured $|\Gamma_{\text{in}}|$ by $10^{+2\alpha\ell/20}$ and the load’s magnitude comes back exactly. The difficulty is not the arithmetic; it is that the correction amplifies the instrument’s own errors by the same factor. Through twenty decibels of round-trip loss, a directivity error of −35 dB in the instrument becomes an effective −15 dB at the load’s plane, which is a reflection uncertainty of 0.18 on a load whose true reflection might be 0.06. The correction has taken a precise measurement of the wrong thing and produced an imprecise measurement of the right one.

That is the general shape of every de-embedding problem and the reason a return-loss specification is written at a stated reference plane. Two rules follow and both are worth stating flatly.

Measure at the plane the specification is written at, or move the specification. A number measured at the bottom of a feeder is a statement about the bottom of the feeder, and is a perfectly good specification if that is what it is called.

The correction is bounded by the instrument’s directivity, not by its dynamic range. An instrument that can measure 60 dB of return loss cannot recover a load through 20 dB of round-trip loss to better than its directivity plus 20 dB, and directivity is the number that gets left off a data sheet summary.

The Smith chart becomes a spiral #

There is a graphical consequence that is the clearest single picture of the difference.

On a lossless line, moving away from the load traces a circle of constant $|\Gamma|$ on the reflection plane, one full turn per half wavelength. That circle is the whole content of the lossless model: the mismatch is a radius and the length is an angle.

On a lossy line the radius shrinks as the angle turns, so the locus is a logarithmic spiral into the origin. Long enough, and every load looks matched, because the spiral has reached the middle.

That also explains a measurement artefact worth naming. Sweeping frequency on a mismatched lossy cable gives a reflection whose magnitude wobbles — the ripple of the standing wave — riding on a curve falling as the attenuation grows. Reading the average of that as the load’s mismatch is wrong by the attenuation, and reading the ripple is a much better estimate, because the ripple depends on the mismatch at the far end and comparatively little on the loss in between.

Where this changes what a measurement means #

Three places, and they are not exotic.

Acceptance testing through an installed feeder. An antenna at the top of a mast is tested from the bottom, through the feeder that goes up. The feeder’s loss is the whole reason the test is generous, and the correction is arithmetic — add twice the one-way loss back — but it requires knowing the loss, which is itself frequency dependent and degrades with age and water ingress. A feeder that has got worse makes its antenna look better.

Cascaded stages inside an instrument. An attenuator in front of a mixer improves the match seen by everything upstream by twice its attenuation, which is the standard reason for putting one there. That is a legitimate use of exactly this effect, and it costs exactly its attenuation in noise figure.

A time-domain reflectometer. The reflection from a fault at the far end of a long cable is attenuated twice on its way back, so a distant fault reads as a smaller discontinuity than an identical near one. Correcting for that is what the instrument’s “cable loss” setting is, and getting it wrong misjudges the fault rather than its position.

Where the model itself stops #

This essay has been about a model failing — the lossless line’s claim that a mismatch travels unchanged — and it is worth saying where the replacement stops, because it does.

The expression $|\Gamma_L|\,10^{-2\alpha\ell/20}$ assumes the attenuation is the only thing the line does to the wave’s amplitude, which requires the line to be uniform and its characteristic impedance to be real. Neither is exactly true. A real cable’s characteristic impedance has a small negative imaginary part wherever its loss is resistive rather than dielectric, so it is not fifty ohms but fifty ohms with an angle of a fraction of a degree, and a load that is exactly fifty resistive ohms is therefore not exactly matched to it. At the loss levels drawn here that correction is a few parts in a thousand of the reflection and is buried under everything else; at the very low frequencies where $\alpha$ is dominated by the conductor and $\beta$ is small it is not, which is the regime a telephone line lives in and the reason its characteristic impedance is quoted with a phase angle.

The second assumption is uniformity. A cable with a periodic imperfection — a regular variation in diameter from the manufacturing process, or a repeated bend — has a reflection that adds coherently at the frequency whose half wavelength matches the period, and the result is a narrow spike of reflection that no amount of loss between it and the load can hide, because it is not at the load. That is a structural return loss, it is specified separately, and it is the one fault in this field that gets worse with length rather than better.

What the loss changes about the rest of the field #

Nine essays in this field quote a delay and a reflection coefficient, and a lossy line makes both of them frequency-dependent. The delay that is not one number is where that is measured directly, with two exponents separating the two mechanisms. The staircase in time is the lossless picture every reflection argument in the field is drawn on, and A ladder is not a line is the other approximation to the same object, converging far more slowly. Several sections, and the band they buy is a design whose whole merit is a reflection coefficient, measured through cable that would hide it. And The resistor that is only a resistor is the component version of the same argument: an instrument sees the far end through everything in between.

What is checked #

Two assertions, and the second is the one that makes the first mean something.

That the reflection at the near end is the load’s own, attenuated by twice the one-way loss, over six lengths from half a metre to fifty. The line is solved with its loss in — a complex hyperbolic tangent that shares no arithmetic with the expression — and the two agree to better than a part in a billion. A closed form checked against the route that produced it would prove nothing; this one is checked against a route that never saw it.

And that the length that hides a mismatch falls as one over the root of frequency, measured as 3.164 times shorter per decade against $\sqrt{10}$. That is the claim that turns a fact about a length into a fact about a frequency, and it is what makes the boundary a boundary of this collection’s usual kind rather than an observation about one cable.

The square root, and the model it belongs to #

The one over root frequency is not an empirical fit; it is the skin effect, and it carries a condition that this essay’s loss model does not state. The delay that is not one number is where that condition is measured, and it puts a second boundary underneath this one: below the frequency at which a trace’s reactance overtakes its series resistance — 910 kilohertz for ordinary copper — the line is a diffusion rather than a wave, with a velocity proportional to $\sqrt f$ rather than constant. Above it the arrival is still exactly linear in the length and the rise time grows as the square of it.

So the loss whose decibels per metre hide a mismatch here has three regimes and this essay is drawn in the middle one. Below 910 kilohertz the line is not carrying a wave at all and the standing-wave argument has nothing to describe. Between there and wherever the dielectric’s own loss overtakes the conductor’s, the loss goes as $\sqrt f$ and the hiding length falls as $1/\sqrt f$, which is the measurement made here. Above that the dielectric loss goes as $f$ and the hiding length falls faster than the square root — so the numbers in the table are, at the gigahertz end, an underestimate of how short a cable has to be before it stops flattering its load.

That last point is the one worth carrying to an acceptance test. A quarter wave, and the path the current takes back is the other essay in this field where a good measured number conceals a design that is only correct at one frequency — a section reflecting $5\times10^{-17}$ of what arrives, at one frequency, with seventeen per cent either side of it back to a tenth. The two failures pair: one hides a bad load behind a lossy cable, the other hides a narrow match behind a single-frequency measurement, and both are discovered by sweeping.

An instrument that flatters, which is the rarer kind #

Most of the instrument errors this collection measures make a reading worse than the truth — a probe loads a node, a lead adds its own resistance, a shared return adds somebody else’s current. This one does the opposite, and that is what makes it dangerous rather than merely inconvenient.

A four-to-one load at the end of twenty metres of ordinary coaxial cable measures 1.13 at the near end, a return loss of twenty-four decibels, and passes an acceptance test the load could never pass. Nothing has gone wrong with the instrument, the cable or the measurement: the reflection really is that small where it was read, because the reflected wave made the journey twice and lost twice the one-way loss doing it. The reading is correct about the node it was taken at and wrong about the question it was asked.

That is the same distinction the probe is part of the circuit draws — a reading is two solves, the circuit and the circuit-with-the-instrument, and the quantity drawn is the difference — with the sign reversed. There the instrument’s presence degrades the thing being measured and the reading tracks the degraded circuit honestly. Here the instrument’s distance improves the number and the reading tracks a quantity nobody wanted.

The practical consequence is a rule about where to stand rather than about what to buy. A return-loss measurement is a measurement of a plane, and the plane is wherever the instrument’s reference is — so a specification that names a return loss without naming the plane it is measured at has left out the term this essay computes, which at ten decibels of intervening loss is a factor of a hundred in reflected power.