The Q a pi network makes a designer choose

Everything about an L-section follows from one fact: its quality factor is $\sqrt{r - 1}$, where $r$ is the ratio of the resistances it joins, and nothing else can be chosen. The match with no knob measured the band that fixed Q leaves; the efficiency a fixed Q costs measured the loss, twice $Q_s$ over the parts’ Q; and the part that decides the match separated the two parts and found the loss to be $Q_s(1/Q_L + 1/Q_C)$, nearly all of it in the coil. Two components, two resistances, one answer.

Add a third component and the answer acquires a degree of freedom. A pi network — a shunt capacitor at each end and a series inductor between them — can match the same two resistances at any loaded Q above the L-section’s, and so can its dual, the tee: a series inductor at each end and a shunt capacitor between. Transmitters use them for exactly that freedom, because a higher Q rejects the harmonics an output stage makes. The question this essay measures is the price. Every unit of Q is paid for in two currencies at once — loss, and bandwidth — and the essay draws the exchange rates, compares the two three-part topologies, and finds the one direction in which the freedom does not extend.

A Q chosen, and paid for twice #

The match is 50 ohms to a kilohm at a megahertz, a ratio of twenty, with coils of Q 100 and capacitors of Q 1000. A pi network is designed by choosing its loaded Q at the high-resistance end: the shunt capacitor there has a reactance of the load resistance over Q, which transforms the kilohm down to a virtual resistance of $R_L/(Q^2 + 1)$ in the middle of the network, and the rest of the network is an L-section from that virtual resistance up to the source’s fifty ohms. The virtual resistance must sit below both ends, which is the condition $Q > \sqrt{r - 1}$.

At its least Q, 4.36, the pi network’s source-end capacitor vanishes and it is the L-section, losing 4.58 per cent with a band of 4.7 per cent of the frequency inside which the input reflection stays under 0.1. Raise the design Q to 8 and the same match loses 9.47 per cent for a band of 2.3; at 30 it loses 28.7 per cent for a band of 0.6. Both currencies move monotonically with the Q at every step drawn, and in opposite directions to anything a designer wants.

The loss roughly doubles from the least Q to Q 8, and it grows faster than in proportion as the Q rises, because the energy the network circulates is set by the sum of the Qs on its two sides — the chosen one at the load end, and the one the virtual resistance forces at the source end — and both grow. At Q 30, a quarter of everything the source delivers is heat. Nobody builds that match with Q-100 coils; the figure is the reason.

The band the Q leaves #

The second price is the band. A network that stores more energy per watt delivered is a more selective resonance, and its match holds over a narrower range of frequency.

With lossless parts, so that only the topology is drawn, all three networks match perfectly at a megahertz. The band inside which the reflection stays under 0.1 is 4.7 per cent at the least Q, 1.7 at 10 and 0.7 at 25 — close to inversely proportional to the design Q, which is the relation resonance, and the bandwidth it sets exactly measured for a single resonator. A transmitter tuned for one frequency does not care. A receiver’s front end or a broadband amplifier’s output does, and for them the L-section’s least Q is the most band any single network of this kind can give — which the last section of this essay returns to.

What the Q is bought for #

If a higher Q costs loss and band, something must justify it, and in a transmitter it is the harmonics. An output stage driven hard enough to be efficient produces a current rich in harmonics of its frequency, and the matching network is also the last filter before the antenna. Every pi or tee network in the family is a low-pass filter, so the question is how much each Q attenuates the second and third harmonics.

At the L-section’s least Q the second harmonic arrives 16.0 decibels below the fundamental and the third 24.4. At a Q of 10 those become 29.8 and 41.7; at 30, 39.1 and 51.2. The gain from the least Q to 10 is about fourteen decibels at the second harmonic, bought for an extra seven per cent of loss; from 10 to 30 it is about nine decibels more, bought for an extra seventeen per cent. The exchange rate worsens as the Q rises: early decibels of rejection are cheap and late ones are expensive.

That is why a transmitter’s output network is designed at a Q of about ten to twelve with the best coil available, and why its harmonic specification is usually met by a separate low-pass filter after it rather than by pushing its Q higher. The pi network’s Q is the right knob for the first fifteen decibels and the wrong one for the next fifteen.

The rejection read here assumes that the network’s capacitors are still capacitors at the harmonics, which at three megahertz they comfortably are. It stops being true far higher up: the capacitor that is an inductor found a hundred-nanofarad part turning inductive above 14.5 MHz, and a pi network’s shunt capacitors passing their own self-resonance would stop shunting the harmonics they were placed to remove. That is a second reason the late decibels belong to a separate filter built for the band where they lie.

An output network, worked #

Put watts on the numbers. A hundred-watt stage at a megahertz drives a kilohm through a pi network, with coils of Q 100 and capacitors of Q 1000, and its output current carries second and third harmonics comparable with the fundamental — say each at the same level, as a hard-driven stage nearly gives.

At the L-section’s least Q, the network delivers 95.4 watts at the fundamental, sheds 4.6 as heat, and passes the second harmonic sixteen decibels down: about 2.4 watts of two-megahertz power reaching the antenna for every hundred of the fundamental. At a Q of ten it delivers 88.3 watts, sheds 11.7, and passes the second harmonic 29.8 decibels down — about a tenth of a watt — and the third 41.7 down, under seven milliwatts. At Q 30 it sheds 28.7 watts in its parts and passes the second harmonic 39.1 decibels down, about nine milliwatts.

Two things follow. The first is that the harmonic numbers improve far faster than the loss worsens over the first factor of two in Q — from 2.4 watts of second harmonic to a tenth of a watt for seven more watts of heat — and then the exchange turns against the designer: from Q 10 to Q 30 the second harmonic falls from a tenth of a watt to nine milliwatts while the heat rises by seventeen watts. Spurious-emission limits of forty to sixty decibels below the carrier are common, and no design Q in this family meets the strict end of that range at the second harmonic with these parts. The second is where the seventeen watts go: almost all of it into the one coil, which at Q 30 is dissipating more than a quarter of the transmitter’s output in a part designed to dissipate nothing.

So the practical design is the one transmitters actually use. The matching network runs at a modest Q, ten or twelve, and buys the first thirty decibels cheaply; a separate low-pass filter, terminated at both ends and designed for its stopband rather than for its match, buys the rest. The zeros that buy an order is the kind of filter that does that job — its transmission zeros put rejection exactly at the harmonics without raising the loaded Q at the fundamental at all.

Why the loss grows faster than the Q #

The pi network’s loss does not rise in proportion to its design Q: it roughly doubles from the least Q to 8, and nearly triples again from 8 to 30. The construction explains it. Choosing a Q at the load end forces a virtual resistance of $R_L/(Q^2 + 1)$ in the middle, and the source end then has to match fifty ohms down to that, at a Q of its own:

$$Q_1 = \sqrt{\frac{R_S (Q^2 + 1)}{R_L} - 1}.$$

At the least Q this second Q is zero. As the design Q rises it rises too — at Q 10 it is 2.0, at Q 30 it is 6.6 — and the energy the network stores is set by the sum of the two. The loss is that energy over the parts’ Q, so it grows as $Q + Q_1$, which rises faster than $Q$ alone, and the coil carries the current both halves of the network circulate. A tee’s construction is the mirror image, with a virtual resistance above both ends and a second Q on the load side, and it gives the same sum — which is the arithmetic behind the next section’s finding that the two topologies lose the same.

The reactance cancelled, and the resonance it buys met the same structure in a power feeder: cancelling a reactance at one frequency builds a resonance, and the resonance’s stored energy is what the network then has to carry. A matching network is that statement made deliberately, with the stored energy chosen in advance.

Two coils lose no more than one #

The pi network has one coil and two capacitors; the tee, its dual, has two coils and one capacitor. With coils ten times lossier than capacitors, the obvious expectation is that the tee loses nearly twice as much.

It does not. At a loaded Q of 10 the pi loses 11.68 per cent and the tee 11.77; at 30, 28.73 and 28.80. Across the whole range the two agree within eight tenths of a per cent of themselves. The loss follows the energy the network must circulate to present the chosen Q, and both topologies circulate nearly the same energy at the same loaded Q; the tee’s two coils share between them what the pi’s one coil carries alone. The arithmetic of the part that decides the match applies network by network: each coil’s dissipation is its stored energy over its Q, and the stored energy is what the Q fixes.

So counting coils predicts the wrong answer. A designer choosing between the two is choosing on other grounds — the pi has both its capacitors to ground, where a variable capacitor’s rotor is conveniently earthed, and one coil to wind; the tee has one capacitor and presents the source a coil rather than a capacitor, which matters to the output stage in a way the last section of this essay describes; one may give more convenient part values than the other at a given frequency — and the efficiency, at equal Q, is not one of them.

The trade as a curve #

Each design Q gives one network with one loss and one rejection. Drawn as points in the plane of those two numbers, each topology traces a curve, and the curve is the useful object.

The L-section is a single point: 16.0 decibels of rejection for 4.58 per cent of loss, with no way to move. The pi and the tee each trace a curve leaving that point, and the two curves nearly coincide: for thirty decibels of second-harmonic rejection the pi loses 11.96 per cent and the tee 12.05. A designer reading the curve does not choose a Q at all. They choose a point — the rejection a specification demands, or the loss a coil can stand — and the Q is whatever that point needs.

The curve also moves with the parts. With coils of Q 200 every point’s loss would roughly halve, and the rejection would be unchanged, since rejection belongs to the topology and loss to the parts. A better coil therefore slides the curve down rather than along, which is the same statement as the previous essay’s — spend on the worse part — read in the plane of two specifications instead of one.

The freedom runs one way #

The pi and tee networks free the Q upwards. The efficiency a fixed Q costs left open whether any lumped network can match a given ratio at a Q below $\sqrt{r - 1}$, and the answer decides whether the L-section’s band is a property of the L-section or of lumped matching.

At every ratio, a pi network at one and a half times its least Q is more selective than the L-section; no choice of its design Q goes the other way, because the virtual resistance in its middle must sit below both ends, and the Q on each side of it is then at least the L-section’s. The only lumped route to a wider band is more sections, each matching a smaller ratio with a smaller Q: at a ratio of a hundred, the narrowness falls from 46.5 for one section to 5.2 for two. Several sections and the band they buy measured the same principle for quarter-wave transformers on a line.

So the hyperbola that $\sqrt{r - 1}$ draws is a property of single-resonance matching, not of lumped matching: any network with one resonance is at least as selective as the L-section, and anything less selective has more than one resonance. That is what Bode and Fano’s bound makes rigorous for a load with reactance; for two resistances it is this figure.

How the networks were solved, and what they leave out #

Each network is a netlist driven at the design frequency from a one-volt source behind fifty ohms: the pi as two shunt capacitors and a series coil, the tee as two series coils and a shunt capacitor, each part with a series resistance of its reactance over its Q. Design values come from the virtual-resistance construction, and every network is checked, with lossless parts, to match to better than one part in ten thousand at the design frequency. Loss is power into the load over power into the network; rejection is the power reaching the load at a harmonic against at the fundamental, from the same source; bands are bisected on the input reflection of the lossless network, so that they describe the topology rather than its parts.

The parts’ Qs are held at their design-frequency values at the harmonics too, which flatters the coil: a real coil’s resistance rises with frequency through skin and proximity effect, and its self-capacitance eventually resonates, which at the third or fifth harmonic can reduce the rejection a lumped model promises. The source is linear; a real output stage presents a source impedance that depends on its drive, and the harmonics it makes depend on the load the network gives it at each harmonic — which is how class-E and class-F amplifiers use the network’s harmonic impedances deliberately rather than merely attenuating through them.

Still open: the harmonic impedance, the coil at the harmonic, and the double-pi #

The impedance the output stage sees at each harmonic. The rejection here is a transfer from source to load. What an output stage’s efficiency depends on is the impedance the network presents back to it at each harmonic: a pi network shows the stage a near short at the second harmonic through its input capacitor, a tee shows it a high impedance through its input coil, and that difference — invisible in every figure above — changes the waveform the stage produces and so the harmonics it makes. The two topologies, equal on loss and rejection, differ there.

The coil’s own resonance. A coil with self-capacitance becomes a capacitor above its self-resonance, and a pi network’s series coil near its self-resonance at the third harmonic passes that harmonic rather than blocking it. Where the rejection curve turns over, for a coil whose self-resonance is a stated multiple of the design frequency, is a solve with the coil’s capacitance in it.

Two pi sections in cascade. A double-pi spends the same total Q in two resonances, and so should buy more rejection per unit of loss than one pi at the same overall Q — more decibels at the harmonics from a curve that leaves the same L-section point more steeply. How much more, and whether the gain survives coils of Q 100, is the comparison between one high-Q resonance and two moderate ones.