The efficiency a fixed Q costs

The match with no knob found that an L-section has exactly one free parameter and it is not free. A series inductor and a shunt capacitor matching $R_{low}$ to $R_{high}$ have a quality factor of $\sqrt{R_{high}/R_{low} - 1}$, fixed by the two resistances with nothing left to choose, and everything else about the section follows: the band over which it holds its reflection under a tenth is 0.2/Q to within three per cent, whatever the parts are made of.

That essay’s last section named three things it had not measured, and the first of them was this one. Give each component a quality factor of its own and the match starts to dissipate, by an amount the section’s Q multiplies — so the same fixed Q that decides the band decides the loss, and the two are not independent choices.

The measurement turns out to put the threshold at a resistance ratio nobody would call demanding.

The circulating current, and why the loss is twice what it looks #

The mechanism is the same one that makes a resonator’s Q a property of its components, and the Q the components allow is where it was first measured: a reactive network circulates current that the load never sees, and the circulating current is what the losses are dissipated by.

In an L-section the shunt capacitor carries $Q_s$ times the load’s current and the series inductor carries the input current, which is also larger than the load’s by the same factor at the high-impedance end. Each of them has a resistance equal to its reactance over the components’ own quality factor $Q_u$. So each dissipates $Q_s/Q_u$ of what is delivered, and there are two of them:

$$\text{loss} \approx \frac{2Q_s}{Q_u}$$

The two is the part that is easy to lose. A derivation that accounts for the inductor and forgets that the capacitor carries the same circulating current comes out a factor of two optimistic, which on a design at one per cent is the difference between passing and not.

The figure measures the factor rather than deriving it. Over every ratio where the loss is under five per cent, the solved loss against $2Q_s/Q_u$ agrees to 0.03 per cent — so the expression is not a rule of thumb, it is the leading term of something the solve confirms to four figures wherever the leading term is supposed to hold.

One per cent at a ratio of a quarter #

Now put a number on it, and the number is the surprise.

Setting $2\sqrt{r-1}/Q_u = 0.01$ gives $r = 1 + (Q_u/200)^2$. With parts of quality factor a hundred — an ordinary air-cored inductor at a few megahertz, or a decent surface-mount one — that is a resistance ratio of 1.253, which is fifty ohms to sixty-three.

Fifty to sixty-three is not a match anybody thinks about. It is the sort of mismatch a connector introduces, and correcting it with an L-section built from ordinary parts costs a per cent of the power.

The threshold is bisected on the solved efficiency and agrees with $1 + (Q_u/200)^2$ to within a few per cent at every entry, so the closed form is being tested rather than evaluated.

Read the table as a statement about what a matching network can do. With components of quality factor a hundred, any match of more than about four to one loses several per cent; with quality factor a thousand — which means air-cored coils, silver-plated, at a frequency where they are well behaved — a match of twenty-six to one is still at one per cent. The achievable matching ratio is set by the components’ quality factor and by nothing else, and the section’s own Q is the exchange rate.

Splitting the match, and the ratio below which it hurts #

Several sections, and the band they buy established that splitting an L-section into several lowers each one’s quality factor and therefore widens the band: 4.73 per cent for one section, 14.98 for two, 30.34 for three and 30.83 for four, at a ratio of twenty.

The efficiency ought to follow, since each section’s Q is smaller and the loss goes as the Q. It does, and only above a ratio that has to be measured.

At a ratio of twenty, one section loses 8.03 per cent and two lose 7.14 — the split helps. At a ratio of three, one section loses 2.76 per cent and two lose 3.35 — the split hurts. The crossing, bisected on the two solved efficiencies, is at a ratio of 10.02.

The reason is arithmetic about counting. Splitting a ratio $r$ into $n$ steps gives each section a quality factor $\sqrt{r^{1/n}-1}$, and the total loss is $n$ times twice that over $Q_u$. For large $r$ the per-section Q falls faster than the count rises and splitting wins; for small $r$ the count wins, because two nearly-lossless sections still have four lossy components in them and one has two.

Read across each row and the minimum moves right as the ratio grows: one section up to about ten, two from ten to about a hundred, three or four beyond. The number of sections that is best for efficiency is not the number that is best for band, which keeps improving with every section, and the two answers differ by one or two sections over most of the useful range.

That is the answer to the question the essay before it left open, and it is a qualified yes: splitting buys efficiency as well as band, but only above a ratio of ten, and only by a few per cent of a few per cent.

What the loss is in decibels, and where it stops mattering #

Percentages are the wrong unit for the last part of this and decibels are the right one, because what a matching network costs is compared against what it was fitted to save.

A match exists to stop power being reflected. A ratio of $r$ matched by nothing reflects $|\Gamma|^2 = ((r-1)/(r+1))^2$ of the available power, so the mismatch loss without any network is $-10\log_{10}(1 - |\Gamma|^2)$:

The right-hand column is the price and the middle one is the benefit, and above a ratio of about one and a half the benefit is several times the price. At a ratio of three the network saves 1.249 dB and costs 0.122; at twenty it saves 7.4 and costs a third of one; at a hundred it saves fourteen and costs three quarters.

The two cross below the bottom of the table. Bisecting on the same pair of expressions puts the crossing at a ratio of 1.2124 with components of quality factor a hundred, and at 1.0411 with components of a thousand — so the honest statement is not “the match costs one per cent at a ratio of 1.253” but the match stops being worth fitting below a ratio of about 1.21. Below that, two lossy components have been added to correct a mismatch that was costing less than they cost, and the correct design decision is to leave them out.

That threshold is a good deal lower than the one-per-cent one, which is the reassuring half of the answer: an L-section is worth fitting well before it is a good L-section, because what it is competing against gets bad faster than it does. Both thresholds are set by the same $2\sqrt{r-1}/Q_u$ against functions of $r$ that differ, and the one a designer needs is this one, because it answers whether to fit a network at all rather than how bad the fitted one is.

The loss the reflection does not show #

There is a trap in this arrangement that that essay’s figure makes visible by not showing it.

A lossy L-section still presents a good match. The input impedance at the design frequency is still very close to the source’s resistance — the loss adds a small real part in the right place — so the reflection coefficient stays tiny and a network analyser measuring return loss sees nothing wrong. A match that is dissipating sixteen per cent of the power can have a reflection of a fraction of a per cent.

That is why insertion loss and return loss are two measurements. A network with a perfect return loss and a terrible insertion loss is an attenuator that happens to be matched, which is a perfectly good description of an L-section made of poor components, and is exactly what the loss that depends on what it causes is about from the transmission line’s side.

The practical consequence: tuning a matching network by watching the reflection converges on a network that reflects nothing and may be losing a great deal. What has to be measured is the power arriving at the load, and if that cannot be measured the components’ quality factor has to be known — which is the quantity the table above turns into a ratio.

The same arithmetic, where the match is a transformer #

An L-section is not the only lumped way to change a resistance, and the comparison is worth one paragraph because it explains why the other way is used at low frequencies and not at high ones.

A transformer changes a resistance by the square of its turns ratio with no resonance and therefore no circulating current beyond the magnetising current. Its loss is its copper and its core, neither of which is multiplied by anything resembling $Q_s$ — so a transformer’s efficiency is roughly independent of the ratio it transforms, where an L-section’s degrades as the square root of it. At a ratio of a hundred an L-section built from components of quality factor a hundred loses 16.6 per cent and a decent wideband transformer loses one or two.

What the transformer costs instead is bandwidth at both ends and a physical part that gets larger with the power, which is the band a turns ratio holds over’s subject. The two are complementary rather than competing: a transformer is the answer for a large ratio over a wide band at a frequency where one can be wound, and an L-section is the answer for a modest ratio at a frequency where a transformer cannot be.

The crossing between them, in ratio and in frequency, is the sort of question that could be answered on one axis and has not been. What can be said from here is where the L-section’s own curve goes: it is a square root in the ratio and a reciprocal in the components’ quality factor, and both of those are slower than the exponential improvements a designer would like.

Where the Q comes from and what it costs to raise #

$Q_u$ is almost always the inductor’s. A capacitor of any reasonable dielectric at radio frequencies has a quality factor of several hundred to several thousand; an inductor has to have wire in it, and wire has resistance that rises as the square root of frequency once skin effect sets in — which is the resistance that grows with frequency’s subject.

So the two components’ quality factors are wildly unequal in practice, and the “twice” in $2Q_s/Q_u$ is really $Q_s(1/Q_L + 1/Q_C)$ with the second term negligible. The figure uses equal quality factors because the claim being tested is the mechanism rather than the component mix, and the generalisation is obvious: reciprocals add, exactly as they do for a resonator, and the worst component decides.

Raising an inductor’s quality factor costs size. A coil’s Q at a given inductance goes roughly as the square root of its volume, so a factor of two in quality factor is a factor of four in volume, and the table above says a factor of two in quality factor buys a factor of four in matching ratio. Those two fours are the exchange rate between a matching network’s physical size and what it can do, and they come from the same square root twice.

Where the leading term stops holding #

That $2Q_s/Q_u$ holds at large loss. It is a leading term and it is checked only where the loss is under five per cent. At a ratio of a thousand the solved loss is 38.7 per cent and the expression gives 63, because the “loss” is now a large fraction and the linearisation of a product of two efficiencies has stopped being valid.

That the components’ quality factor is frequency-independent. It is not, and the whole table therefore describes one frequency. A matching network designed at a band edge with the quality factor measured at band centre is optimistic, and by more than the band is wide if the inductor is near its own self-resonance.

That splitting is free in anything but loss. Each section is two more components, two more tolerances and two more parasitics, and several sections, and the band they buy already found that the fourth section buys almost no band. A split that improves efficiency by half a per cent and adds two parts is not obviously a good trade.

That an L-section is the only topology. A pi or a tee network has a quality factor that is not fixed by the two resistances — that is the whole reason they exist — so a designer who needs a different Q than $\sqrt{r-1}$ uses one, at the cost of a third component and a second thing to tune. The loss of those follows the same circulating-current argument with a different Q, and this figure does not measure them.

Efficiency off a solve, and the perfect parts that refuse the comparison #

The loss is read off a solve — power into the load over power into the network, on a netlist with each component’s resistance in it — and never from an expression.

The $2Q_s/Q_u$ rule is checked against that solve at every ratio where the loss is under 2.5 per cent, to five per cent, and comes out at 0.03.

The one-per-cent ratio is bisected on the solved efficiency and checked against $1 + (Q_u/200)^2$ to six per cent at every quality factor on the slider.

The crossing at which splitting starts to pay is bisected on the two solved efficiencies, and required to lie between a ratio of two and three hundred so that it is a measurement rather than an assumption about the direction.

And the whole comparison is refused with perfect parts: at a quality factor of ten million every number of sections delivers everything to five decimals, which is what says the ordering above belongs to the loss and not to the topology.

One Q, three consequences #

That essay’s finding was that an L-section has no free parameter: the ratio fixes $Q_s$ and $Q_s$ fixes the band. This essay adds the third consequence and it comes from the same place.

$Q_s$ fixes the band at $0.2/Q_s$. $Q_s$ fixes the loss at $2Q_s/Q_u$. So the band and the loss are locked to each other — a narrower match is a lossier one, in exact proportion, and no choice of components changes the trade because $Q_u$ scales both the loss and nothing else.

Written as a single statement: an L-section’s fractional bandwidth times its efficiency deficit is $0.4/Q_u$, a constant of the components alone. A designer cannot have a wide match and a low-loss one from the same parts; they can have either, and the product is bought with component quality.

That is the useful form because it says what to spend money on. Component quality factor is the only variable in it, the topology is not, and the resistance ratio merely chooses where on the fixed hyperbola the design sits. It is the same shape of answer as the gain–bandwidth product — the ideal amplifier, and where it stops being one — and it deserves to be as well known.

Still open: the unequal quality factors, the load that already has a reactance, and the network with a free Q #

The two quality factors separated. Reciprocals add, so the loss should be $Q_s(1/Q_L + 1/Q_C)$ with whichever is worse dominating — exactly as a resonator’s does. Solving with a good capacitor and a poor inductor would confirm the reciprocal addition in a matching network rather than in a resonator, and would say how much of the loss is recoverable by improving only the coil.

A load with a reactance of its own, which is the second thing the essay before it left open. A resistance with a capacitor already across it cannot be matched over an arbitrary band by any lossless network, and the bound depends on the product of its resistance and capacitance. Measured against an L-section that absorbs the load’s capacitance into its own, it would say how close two parts come to the bound — and now, with the loss measured, how much of the gap is the bound and how much is the components.

And a topology whose Q is a choice. A pi network’s quality factor is set by the intermediate impedance rather than by the end resistances, so it can be made higher than $\sqrt{r-1}$ but not lower. Whether there is any arrangement of lumped components that matches a given ratio at a Q below $\sqrt{r-1}$ is a question with a known answer in the abstract and no worked number here, and the answer decides whether the hyperbola above is a property of L-sections or of lumped matching.