The part that decides the match

The efficiency a fixed Q costs gave an L-section’s components a quality factor and measured what the match dissipates. The section circulates $Q_s$ times the load’s current through its own parts, $Q_s = \sqrt{r - 1}$ is fixed by the resistance ratio with nothing left to choose, and so the loss is twice $Q_s$ over the components’ Q — once in the inductor and once in the capacitor. With parts of Q 100, a match costs one per cent at a resistance ratio of 1.25 and nine per cent at a ratio of a hundred.

That essay gave both parts the same Q, which no pair of real parts has. A ceramic capacitor at a megahertz has a Q of several hundred to a few thousand; a small inductor on a ferrite or air core has a Q of thirty to two hundred. Its own continuation said what to expect: reciprocals add, so the loss should be $Q_s(1/Q_L + 1/Q_C)$ with whichever part is worse dominating, exactly as the Q the components allow found for a resonator. This essay measures that, and then asks the questions a designer with a budget asks next: where each watt of the loss goes, what improving one part buys, whether the arrangement of the parts matters, and how hot the coil gets.

Two quality factors, one sum #

The match is fifty ohms to a resistance $r$ times larger, at a megahertz: a series inductor from the source, a shunt capacitor at the load. Each part carries a series resistance of its own reactance over its own Q, and the efficiency is read from the solve as the power into the load over the power into the section. The coil’s Q is the slider; the capacitor’s is held at 1000.

The measured loss follows $Q_s(1/Q_L + 1/Q_C)$ to three per cent of itself wherever it is under two per cent. At a ratio of twenty, with the coil at 50 and the capacitor at 1000, the section loses 8.39 per cent — the leading-order rule says 9.15, because at that size the loss is no longer small and the section’s own dissipation reduces the current that causes it. Two parts of Q 95.2 each lose 8.40 per cent there, and within four tenths of a per cent of the unequal pair at every ratio drawn. That is the harmonic mean of 50 and 1000, and it is the content of the reciprocal law: the section hardly cares which part is lossy, only what the two reciprocals sum to.

Move the coil’s Q through 25, 50, 100, 200 and 400 and the loss at a ratio of twenty is 15.2, 8.39, 4.58, 2.55 and 1.50 per cent. The capacitor is the same part throughout, and from 25 to 400 the coil is responsible for the whole of the change. A sixteenfold improvement in one part cuts the loss tenfold, not sixteenfold, because the capacitor’s own contribution — the loaded Q over a thousand, 0.44 per cent — is a floor the coil cannot reach under.

Where each watt goes #

The reciprocal law says how much is lost. The solve can say where, by reading the current through each part’s own resistance.

The parts’ dissipation accounts for every watt the section loses, to a part in a million, and the two parts divide it in the inverse ratio of their quality factors. With a coil of 50 and a capacitor of 1000, 95 per cent of the loss is in the coil. With a coil of 100 and a capacitor of 1000 it is 91 per cent. With equal parts it is half each, at any Q. Turn the parts round — a coil of 1000 and a capacitor of 100, an unusual pairing but not an impossible one at microwave frequencies — and the coil takes 9 per cent, and the total is the same 4.58 per cent as the coil of 100 with the capacitor of 1000.

The reason is that both parts carry the same energy. In an L-section at its design frequency the series part’s reactance is $Q_s R_\text{low}$ and the shunt part’s is $R_\text{high}/Q_s$, and the current through the first and the voltage across the second are related so that each stores, on average, exactly the energy the other does. A part’s loss is its stored energy times the angular frequency over its Q, so equal stored energies make the dissipation proportional to $1/Q$. The second-order departure — half a per cent in the share — is the capacitor’s own resistance slightly changing the current the coil carries.

What improving one part buys #

Read as a purchasing decision, the reciprocal law has a sharp edge. Improving a part buys loss only until it is better than the other one.

With a capacitor of Q 1000, raising the coil’s Q from 50 to 100 halves the loss, from 8.39 to 4.58 per cent, and raising it from 100 to 1000 — a coil ten times better, and in practice a very different and much larger part — takes it to 0.86. With a capacitor of Q 100 instead, every curve flattens at 4.18 per cent, what the capacitor alone loses, and a coil better than about a hundred buys almost nothing.

The rule is the one the Q the components allow drew for a resonator: spend on the worse part until it is no longer the worse part. In a match at a megahertz the worse part is nearly always the coil, so the coil is where the money goes, and a capacitor upgrade — from a general ceramic to a microwave-grade one — is worth something only after the coil has been brought up near it. The figure gives the exchange rate at a ratio of twenty, and the same shape holds at every ratio because the loss at every ratio is $Q_s$ times the same sum of reciprocals.

The arrangement does not matter #

An L-section can be built two ways. The low-pass form puts the coil in series and the capacitor in shunt; the high-pass form puts a capacitor in series and a coil in shunt. The two match the same resistances at the same frequency, and with a lossy coil it seems natural to expect the arrangement that carries the load’s full current through the coil to lose more.

They lose the same: 5.473 per cent against 5.472 at a ratio of twenty, and within a hundredth of a per cent of each other over the whole range. In both sections the series part carries the load current magnified by $Q_s$ and the shunt part the load voltage magnified by $Q_s$, and in both the two parts store the same energy, so moving a part from one arm to the other changes the current through it and the voltage across it in exactly compensating proportion. Which section to use is decided by everything else — whether direct current must pass, whether harmonics must be attenuated, which values are available — and never by the efficiency.

The coil runs warm #

A fraction lost is a design number. Watts are what a heatsink, a wire gauge and a ferrite’s Curie point are chosen against.

For every hundred watts the section delivers, the coil dissipates $100\,Q_s/Q_L$ watts and the capacitor $100\,Q_s/Q_C$. At a ratio of twenty, with a coil of Q 100 and a capacitor of Q 1000, that is 4.36 watts in the coil and 0.44 in the capacitor. A match delivering a kilowatt into an antenna puts 44 watts into a coil that might be a few centimetres long, and most of the engineering of a transmitter’s output network is the engineering of that coil — its wire, its form, its distance from anything metal.

The heat feeds back. A coil’s resistance rises with temperature at copper’s 0.4 per cent per kelvin and its skin effect changes with it, so its Q falls as it warms, so it dissipates more; a ferrite-cored coil adds a core loss with a temperature dependence of its own. Whether that loop settles is the fixed-point question the loss that depends on what it causes asks of a core, and here it is set by the same $Q_s$ that sets the share: a match at a higher ratio puts more of its power into the coil and needs a coil that stays further from its own runaway.

A match is a resonator, and its loss is a ratio of two Qs #

The reciprocal law is not peculiar to matching. It is the law every resonator obeys, and seeing the match as a resonator explains both its form and its size.

An L-section at its design frequency is a resonant circuit: the series coil and the shunt capacitor, looking from the low-resistance side, cancel each other’s reactance exactly, which is the condition that makes the input purely resistive. Loaded by the source and the load, that resonance has a quality factor — the ratio of the energy it stores to the energy it hands on per radian — and for an L-section it is fixed by the resistances, as $Q_s$ is. The parts’ own losses define a second, unloaded quality factor, the ratio of stored energy to energy dissipated inside the parts per radian, which is the harmonic sum of the coil’s and the capacitor’s.

The fraction lost is then the ratio of the two: energy dissipated over energy handed on, per radian, is the loaded Q over the unloaded Q. That is the general statement of which $Q_s(1/Q_L + 1/Q_C)$ is the L-section’s instance, and it is why the loss rises with the ratio: a larger ratio is a more selective resonance, which stores more energy per watt delivered, and stored energy is what the parts’ resistance acts on. Resonance, and the bandwidth it sets exactly measured the loaded Q as $f_0$ over the bandwidth; the Q the components allow measured the unloaded Q as the harmonic sum of the parts’. The match’s loss is one divided by the other.

That reading also says where the familiar rule of thumb comes from — that a filter or a match built from parts of Q 100 cannot usefully have a loaded Q above about ten, because its loss would then be a tenth of everything it passed. The rule is the reciprocal law with a loss budget of ten per cent written into it.

A match worked through #

A hundred-watt transmitter at a megahertz is to feed a load of a kilohm through an L-section: a ratio of twenty, so a section Q of 4.36. The capacitor is a good ceramic with a Q of a thousand at this frequency. The coil is the variable.

With a small toroid of Q 50 the section loses 8.39 per cent: 8.4 watts of the hundred are heat, 95 per cent of it — eight watts — in the coil, and the antenna receives 91.6 watts, 0.38 decibels down. With a larger toroid of Q 100 the loss is 4.58 per cent, 4.2 watts in the coil, 0.20 decibels. With an air-cored coil of Q 200, 2.55 per cent and 0.11 decibels; with an exceptional coil of Q 400, 1.50 per cent. The capacitor’s own share is 0.4 watts throughout.

Two observations come out of the arithmetic that the percentages hide. The first is that the decibels look negligible while the watts do not: a third of a decibel is inaudible on a receiver and still eight watts inside a part the size of a coin. The second is that the coil’s temperature, not the efficiency, is often the binding constraint, and it scales with the absolute power. A match that is acceptably efficient at a hundred watts is the same match at a kilowatt, with eighty watts in the same toroid; the load that takes the most is a reminder that the match exists to deliver power, and the power delivered sets every other number.

Splitting, revisited #

The efficiency a fixed Q costs found that splitting a match into two sections, each matching the square root of the ratio, pays for itself only above a ratio of about ten: below it, the second section adds two lossy parts for less than it saves in Q. That was measured with equal parts. With unequal ones it might move.

It does not. With parts of 100 and 100 splitting pays above a ratio of 10.02; with 50 and 1000, above 10.07; with 300 and 3000, above 9.17. Every section’s loss is its own $Q_s$ times the same $1/Q_L + 1/Q_C$, so the comparison between one section and two is a comparison of $\sqrt{r - 1}$ against $2\sqrt{\sqrt r - 1}$, and the parts’ Qs cancel out of it. The small movement in the third pair is the second-order term, which is larger when the loss is small enough that a crossover sits in a flatter part of the curve. The crossover belongs to the topology. A designer can decide whether to split a match before choosing a single part.

How the measurements were made, and what they assume #

Every number is a netlist solved at the design frequency: a one-volt source behind fifty ohms, the section’s parts each with a series resistance of its reactance over its Q, and the load. The efficiency is the power into the load over the power into the section, both taken from the solved node voltages; each part’s dissipation is the square of the voltage across its own resistance over that resistance, and the sum of the parts’ dissipations is held against the total loss to a part in a million. The leading-order rule is never used to compute anything; it is the second route the solves are compared with.

The parts’ quality factors are taken at the design frequency and held there. A real coil’s Q varies with frequency — rising with it while the winding’s resistance is direct-current resistance, falling once skin and proximity effect and self-resonance set in — and a capacitor’s equivalent series resistance has its own frequency dependence, as the same part written two ways measured. At the design frequency, which is where the loss is drawn, the Q is the Q. The parts are linear, so a ferrite coil’s loss that rises with flux density, and so with power, is not in it; nor is the coil’s self-capacitance or any coupling between the two parts.

Still open: Q that moves with frequency, the network that chooses its Q, and power-dependent loss #

Quality factors that change across the band. A coil whose resistance is the skin-effect resistance has a Q rising as the square root of frequency; a capacitor’s Q often falls. A match is designed at one frequency and used across a band, and the loss at the band’s edges then depends on how each Q moves as well as on the mismatch. Whether a broadband match should be designed around the coil’s Q at the band’s bottom or its top is a sweep of this solve with frequency-dependent resistances.

A network whose Q is not fixed. An L-section has no choice of Q, and everything here follows from that. A pi or tee network adds a third part and can be designed at any Q above $\sqrt{r - 1}$; it buys narrower band and harmonic rejection, and it pays in loss. The Q a pi network makes a designer choose measures that trade.

A coil whose loss depends on its current. Ferrite cores lose more per cycle at higher flux, so a coil’s Q at a kilowatt is not its Q on a bridge. The share of the loss in the coil then rises with power, and the power at which a ferrite-cored match’s efficiency has fallen by a stated amount is a solve with the core’s loss law in the coil — which the match with no knob and its successors have all held at its small-signal value.