The match with no knob

The same part written two ways took a capacitor with a loss in it and wrote the loss once as a resistance in series and once as a resistance across, and found the two descriptions exact at one frequency and wrong at every other, by an amount set entirely by the quality factor. It treated the conversion as something to be careful of: a way of moving a datasheet’s number from one convention to another without noticing that it had been moved to one frequency as well.

The same conversion is also a circuit. A resistance with a capacitor across it is, at one frequency, a smaller resistance in series with a reactance. Choose the capacitor so that the smaller resistance is a source’s own, cancel the reactance with an inductor, and a load that was twenty times the source’s resistance looks exactly like the source. That is an L-section match, and everything the first essay said about the conversion’s band becomes a statement about the match’s — with one difference that decides the whole design. The first essay’s Q was the capacitor’s. This one’s is not anybody’s to choose.

The conversion, run on purpose #

Run in the other direction, the conversion takes a load resistance $R_L$ with a capacitor across it and returns an equivalent series pair. The series resistance is $R_L/(1 + Q^2)$, where $Q$ is the ratio of the load resistance to the capacitor’s reactance. Setting that equal to a source resistance $R_s$ fixes the quality factor at

$$Q = \sqrt{\frac{R_L}{R_s} - 1},$$

and the series reactance left over is $QR_s$, which a series inductor of the same reactance cancels. Two parts, both decided by two resistances and a frequency.

There are two ways to arrange them and they are not interchangeable in everything. The form drawn here, with the inductor in series and the capacitor across the load, passes direct current and attenuates the harmonics of whatever drives it; the other, with the parts exchanged, blocks direct current and passes the harmonics. Both have the same Q and the same band around the design frequency, because both are the same conversion, and the choice between them is made on what else the circuit needs from the section.

The network agrees with the conversion to the arithmetic’s floor. Solved as a load and a capacitor, it is 50.000 ohms in series with 217.9 ohms of capacitive reactance at a megahertz; solved with the inductor in front, it reflects 3.6 × 10⁻¹⁶ of what arrives — nothing, to double precision, which is what an exact match is.

An exact match is also a maximum. The load that takes the most found that a load draws the most power a source can give when its resistance equals the source’s, and here the kilohm is made to look like fifty ohms at a megahertz, so at that frequency it takes the whole of the power the source has available. Connected directly, the same kilohm across the same fifty-ohm source takes $4 \cdot 50 \cdot 1000/1050^2$, which is 0.18 of it; through the section it takes all of it. The reflection coefficient is the measure of how much of that it misses, and the staircase in time is where the same quantity decides what a line sends back to its source.

The Q nobody chose #

In the Q the components allow a quality factor is a ceiling set by the worst part, and in resonance, and the bandwidth it sets exactly it is a design variable that buys selectivity. In an L-section it is neither. Matching fifty ohms to a kilohm fixes it at 4.3589 before a single part has been picked, and there is no second L-section that does the same match with a different one: the low-pass form above and the high-pass form with the parts exchanged both have $\sqrt{19}$. The designer who specified two resistances has already specified the Q, and with it the band.

The band comes out close to a simple expression and it is worth seeing why. At the design frequency the source sees its own resistance; a little way off, the reactances no longer cancel, and the mismatch grows as the frequency offset times Q. The reflection is half the fractional mismatch for small mismatches, so a tolerance Γ on it is reached at a fractional offset of about Γ/Q on each side, and the band between the two edges is 2Γ/Q: 0.2/Q for a tolerance of a tenth. For this match that is 4.59 per cent and the network gives 4.73.

Half the power reflected is a larger tolerance and a different regime — |Γ| of 0.707 is not a small mismatch — but the same shape holds more loosely: 2/Q gives 45.88 per cent and the network 48.53. That is the loaded quality factor showing through. The source and the load both damp the section, so it behaves like a resonator whose Q is Q/2, and a resonator’s half-power band is one over its Q.

The two edges are not placed symmetrically either, and the reason is in the section’s behaviour far from its design frequency rather than near it. Far below, the capacitor is gone and the inductor is a wire, so the source sees the load itself and reflects $(R_L - R_s)/(R_L + R_s)$ of what arrives — 0.905 for fifty ohms and a kilohm, which is where the reflection tends at direct current. Far above, the capacitor shorts the load and the reflection climbs to one. So the half-power edge below the design frequency, 727 kHz, sits further out than the one above, 1.21 MHz, and a match of a small enough ratio has no lower half-power edge at all: $(R_L - R_s)/(R_L + R_s)$ equals 0.707 at a ratio of 5.83, so below that ratio the reflection at direct current is under 0.707, and it never reflects half the power at any frequency below the one it was designed for.

At two hundred to one both approximations close on the measurement — 1.43 per cent against 1.42, 14.25 against 14.18 — because the larger Q makes the band narrow enough for “a little way off” to stay a little way off across the whole of it. A match of fifty ohms to ten kilohms has a band of one and a half per cent, and that is the whole of what can be had from two parts.

The band against the ratio #

The measured band sits above 0.2/Q at every ratio and closes on it from above. At two to one the section’s Q is one, the band is 28.72 per cent and the product of band and Q is 0.287, forty per cent more than the small-mismatch argument allows; at a thousand to one the Q is 31.6, the band is 0.64 per cent and the product is 0.2011. The expression is a good account of any match whose ratio is large enough to need one, and a pessimistic one of a match whose ratio is small.

What is not on the axis at all is the impedance level. A match of five ohms to a hundred has the same Q and the same band as fifty to a thousand, and a match of five kilohms to a hundred kilohms has them too, because multiplying every resistance and reactance by one number changes no ratio in the network. The same filter a thousand times larger is that invariance for a filter, where it holds exactly for the design and breaks for the parts; here it says the band belongs to the ratio alone.

Splitting the match #

The only way to lower an L-section’s Q is to ask it for a smaller ratio, and the only way to do that without changing the match is to split it: step from the source’s resistance to an intermediate one, and from there to the load. Two sections through the geometric mean of the two resistances each match $\sqrt{20}$ to one at a Q of 1.863 instead of one section at 4.359.

Two sections triple the band, from 4.73 per cent to 14.98, and three double it again to 30.34. Then the count stops paying: four sections give 30.83 and six 40.11. The first splits lower every Q, and a lower Q is a slower departure from the match on both sides. After that the sections start to interact — each is exact only when it sees the resistance the next one presents, and away from the design frequency none of them does — so the reflection develops humps between which it dips back under the tolerance, and where those humps fall decides the band more than how many sections there are.

At four to one the interaction is starker. Three equal sections give a band of 72.78 per cent, and four give less, 62.71, and six less again, 54.80 — more parts, narrower band.

Read as a design rule for the twenty-to-one match, the figures say how many sections a band costs. A band of five per cent under a tenth of reflection is one section and no more; ten per cent needs two; thirty needs three. Forty per cent is six equal sections, and fifty cannot be had from equal sections at all, however many are used — past a few, the count stops buying band and the design has to place the intermediate resistances deliberately. That is a limit on a method rather than on the match, which is the distinction the lines field’s equal-ripple designs make concrete. Nothing is wrong with the fourth section. It is placed where equal ratios put it rather than where a designed response wants it, and the humps its extra resonance adds fall inside a band the three-section version had kept clear.

That is the same lesson several sections, and the band they buy drew from lines. A 4:1 quarter-wave transformer holds |Γ| under a tenth over 17.1 per cent of its frequency, and several sections whose reflections are made to cancel take that to 47.7, 67.9, 82.1 and 92.6 per cent — but only when the section impedances are designed as a response rather than stepped by equal ratios, and an equal-ripple design found by search buys a further 35 to 46 per cent on top. The lumped L-section at the same four to one, 13.43 per cent, is slightly narrower than a quarter wave, and it has the same property the line has: one section’s band is a fact about the ratio, and anything wider is design.

A transformer is the one lumped match whose band is not tied to its ratio. The band a turns ratio holds over finds a transformer flat between an edge its magnetising inductance sets and an edge its leakage sets, and neither edge is the square root of the ratio less one. An L-section spends no energy storage it does not need and pays for that economy with a band the ratio dictates; a transformer stores energy in a core and pays with two edges of its own.

What the Q means for the parts #

A fixed Q is not only a fixed band. Each reactance in a section stores, at the design frequency, Q times the energy the match delivers per radian, so the current in the series inductor and the voltage across the shunt capacitor are both larger than the delivered power alone would suggest, and both grow with the ratio. The match from fifty ohms to ten kilohms is not merely narrow; its parts carry fourteen times the reactive power the load dissipates.

That matters in two ways this page does not measure. A real inductor has a loss, and a loss that is small beside a part’s own reactance is not small beside a match whose Q multiplies it, so a high-ratio L-section trades efficiency against the parts’ own quality in the way the Q the components allow describes for a resonator. And the voltage on the shunt capacitor is the load’s voltage, which for a match into a large resistance is large: the part matching into ten kilohms has to be rated for the square root of the power times ten kilohms, whatever its reactance.

Put numbers on the fifty-ohm-to-kilohm section delivering a watt at a megahertz. The load’s voltage, and so the shunt capacitor’s, is $\sqrt{1\,\text{W} \times 1\,\text{k}\Omega} = 31.6$ volts; the source’s current, and so the series inductor’s, is $\sqrt{1\,\text{W} \div 50\,\Omega} = 141$ milliamperes; and the inductor’s reactive power at 217.9 ohms is 4.36 volt-amperes, which is Q times the watt delivered, as the section’s Q says it must be. A match of two hundred to one at the same watt puts 100 volts on its capacitor and 14.1 volt-amperes into its inductor for the same single watt of output.

How the numbers were obtained #

The parts come from the conversion: $Q = \sqrt{R_L/R_s - 1}$, a series reactance of $QR_s$ and a shunt reactance of $R_L/Q$. The load and its capacitor are then solved as a network at the design frequency and required to be the source’s resistance in series with the inductor’s reactance, and the whole section is solved at every frequency for the impedance it presents to the source, from which the reflection follows. Each band is found by scanning outward from the design frequency on a grid of a ten-thousandth of a decade and bisecting the first crossing of the tolerance on each side. The scan is not a nicety: several sections make the reflection dip back under the tolerance between humps, and a bisection started from the design frequency lands in the far dip and reports a band that is not contiguous.

What it does not say #

It uses ideal parts. Every inductor and capacitor here is lossless and exactly its value, so the reflection at the design frequency is the arithmetic’s floor and the band is the network’s alone; a real section’s match is limited by its parts’ losses and tolerances before it is limited by its Q.

It matches two resistances. A load with its own reactance — an antenna, a transducer, the input of a transistor — has a limit on how wide any lossless match to it can be, whatever the number of sections, and that limit is not the one measured here.

And its multi-section matches are not designed. Equal ratios are the obvious way to split a match and not the best, and the numbers for three, four and six sections describe that obvious way. A designed multi-section match would do better and would not show a fourth section narrowing the band.

Still open: the loss a Q multiplies, sections designed as a response, and a load that stores energy #

The efficiency a match costs. Give each inductor a quality factor of its own and the match starts to dissipate, by an amount the section’s Q multiplies. Solved against the ratio, it would say at what ratio an L-section made of ordinary parts loses more than a per cent of what it delivers, and whether splitting the match — which lowers every section’s Q — buys efficiency as well as band.

Sections designed as a response. The sections above step by equal ratios. Choosing the intermediate resistances so that the reflections of the sections cancel across a band, as the lines field does for quarter-wave sections, would turn the non-monotone counts into a designed equal-ripple band, and would put a number on how much of the gap between 30 per cent and the line’s 80 is design rather than lumped parts.

A load with a reactance. A resistance with a capacitor already across it cannot be matched over an arbitrarily wide band by any lossless network, and the bound depends on the product of the load’s 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 whether splitting the match helps a reactive load at all.