The mismatch no network can remove

What happens where the medium changes found that a wave reflects wherever the impedance changes, and that the reflected fraction depends on nothing but the ratio of the two impedances. Everything after it has been about making that reflection go away. The layer that makes a reflection vanish cancelled it at one wavelength with a quarter-wave layer of the intermediate impedance, and the taper that matches every note removed it at every wavelength shorter than the taper by never letting the impedance jump at all.

Both matched a resistance to a resistance — two media that absorb or carry away energy but store none. The taper ended by noting that its success depends on that, and that for a load which stores energy, matching over a band has a hard limit, paid for somewhere else. The limit was found by Hendrik Bode in 1945 and made general by Robert Fano in 1950, and it is one of the few results in engineering that says not how to build something but that it cannot be built.

A load that stores energy #

The simplest load with the property is a resistance $R$ with a capacitance $C$ across it. It is the input of a transistor, the electrical side of a piezoelectric transducer, a photodiode, the feed of a small antenna. At low frequency the capacitance draws almost no current and the load looks like $R$; at high frequency it shorts the resistance out. In between, every cycle charges the capacitance and discharges it again, and that energy, stored in the field between its plates, comes in from the source and goes back to it without ever reaching the resistance.

To deliver power to $R$, a network between the source and the load has to deal with that sloshing: supply the capacitance’s charging current from somewhere other than the source, at every frequency in the band at once. An inductor can supply it at one frequency, by resonating with the capacitance, and that is the whole idea of a matching network. The question is how well it can be done across a band.

Four attempts at one load #

The figure takes a load whose time constant is two in units of the band — the capacitance shorts out the resistance well before the top of the band — and asks a numerical optimiser to find the network of inductors and capacitors that keeps the worst reflection across the band as small as possible. With nothing between source and load, the reflection reaches 0.707 at the band edge. One inductor brings the worst case to 0.392, two elements to 0.320, three to 0.289.

Every improvement has the same shape. The reflection inside the band is flattened into a ripple at a roughly constant level, and outside the band it climbs steeply towards one. The networks are not reducing the load’s mismatch; they are choosing where it goes. Every network buys its flatter band with total reflection just beyond the band edge, and the more elements it has, the more sharply it can make that trade. The dashed line at 0.208 is the level no network of any size gets below across the whole band.

A fixed amount of match #

Bode and Fano’s theorem is a statement about all frequencies at once. Measure the quality of the match at each frequency by $\ln(1/|\Gamma|)$ — zero for total reflection, large for a good match, infinite for a perfect one. Then for a resistance shunted by a capacitance, and any network of lossless parts between it and a resistive source,

$$\int_0^\infty \ln\frac{1}{|\Gamma(\omega)|}\,d\omega \le \frac{\pi}{RC}.$$

The two areas in the figure are the same to four decimal places, and both are $\pi/RC$. The bare load has an infinitely good match at zero frequency, where the capacitance is invisible, and a poor one everywhere else; the network gives up most of that and spreads it evenly across the band, leaving almost nothing outside. Neither has more match than the other. A lossless network cannot create match; it can only move it about, or waste it.

The reason is the same analyticity that ties what a medium absorbs to how it delays. A passive, causal network’s reflection coefficient is an analytic function of complex frequency in the half-plane where signals grow, and the logarithm of an analytic function obeys an integral constraint: its average over all real frequencies is fixed by its behaviour at infinity. At infinite frequency the capacitance shorts the load, and the rate at which the reflection approaches one there is set by $RC$ and nothing else, because no network added on the source’s side can change what happens right at the load’s terminals. That rate is the integral.

Why the budget is π over RC #

The integral is less mysterious than its reputation, and following it through shows exactly what the capacitance is doing. A passive network cannot give out more energy than it receives, and it cannot respond before it is driven. Together those make its reflection coefficient, treated as a function of complex frequency, analytic and no larger than one in magnitude throughout the half of the complex plane in which signals grow. The logarithm of such a function is analytic too, wherever the function has no zeros, and the integral of an analytic function around a closed contour is zero.

Take the contour along the whole axis of real frequencies and back round a vast semicircle. The contribution from the real axis is the Bode–Fano integral, doubled. The contribution from the semicircle depends only on how the reflection behaves at enormous frequencies, and there the capacitance has shorted the resistance completely, so the reflection coefficient approaches total reflection with a correction proportional to $1/(\omega RC)$. That single number, the coefficient of $1/\omega$, is all the semicircle sees, and it gives $\pi/RC$. No network on the source’s side of the load can change it, because at high enough frequency the capacitance is the first thing any network meets.

The inequality, rather than equality, comes from zeros. If the reflection coefficient has zeros in the growing half-plane, its logarithm has singularities there, each of which subtracts from the integral. A network that introduces such zeros throws away part of the budget — typically by adding delay without adding match — and the best networks have none. The two areas in the figure agreeing to four decimal places is the numerical sign that the optimiser found such a network rather than a wasteful one.

What a wider band costs #

If the whole budget is spent flat across a band from zero to $\omega$, with total reflection everywhere else, the best reflection is $|\Gamma| = e^{-\pi/RC\omega}$. That is the dashed line in every figure, and it turns a limit into a price list.

For a small capacitance or a narrow band the floor is negligible: at $RC\omega = 0.5$ an unlimited network could hold the reflection to 0.0019 and three elements manage 0.0146. For a large one it dominates: at $RC\omega = 8$ the floor itself is 0.675, and nothing that could ever be built will return less than two thirds of the amplitude somewhere in the band. In decibels the best return loss is $27.3/RC\omega$ — so doubling the band, or the capacitance, halves the achievable return loss in decibels, and a load that could be matched to 30 dB over one octave can be matched to only 15 dB over twice that range.

The difficulty is in the load, not in the ingenuity of the network, and that is the practical force of the theorem. A designer faced with a photodiode whose capacitance is too large for the bandwidth wanted is not short of a cleverer circuit; the answer is a smaller photodiode, or less bandwidth, or accepting the loss.

A floor for one load and none for another #

The floor is what an infinitely long network would reach, and the approach to it is slow: the first element removes nearly half the worst reflection, the third a tenth, the fifth almost nothing. Six elements still leave a gap of 0.062 above the floor, and Fano showed that closing it entirely requires infinitely many.

The contrast in the same figure is a plain resistance with no capacitance, a quarter of the source’s. On its own it reflects 0.600. Behind an ideal transformer, which is a lossless network of a particular kind, it reflects nothing at any frequency at all. A mismatch of resistance stores no energy, so nothing has to be sloshed back and forth, and there is no integral constraint; a practical transformer, a stepped line or a taper approaches the ideal one over a band as closely as its length allows. A step in resistance is a problem of length; a stored energy is a problem of principle.

The floor as a bandwidth for a given Q #

Engineers usually meet the theorem in a different form, for a load that has been tuned to resonance at the centre of a band. Its quality factor $Q$ measures how much energy it stores for each unit it dissipates per cycle, and for a band of fractional width $B$ the Bode–Fano limit becomes

$$B \le \frac{\pi}{Q\,\ln(1/|\Gamma|)}.$$

The numbers are sobering. A matching specification of a voltage standing-wave ratio of two — a reflected amplitude of one third, a tenth of the power — is a common requirement. For a load with $Q = 50$, typical of a compact antenna, the widest band that can be matched to that standard by any lossless network is $\pi/(50 \ln 3)$, under six per cent of the centre frequency. For $Q = 500$ it is under 0.6 per cent. A tighter match costs band logarithmically, a higher $Q$ costs it in proportion, and there is no third variable to trade.

Where the match goes when it is not in the band #

The figures draw what happens outside the band, and it deserves a sentence of its own. A network that holds the reflection low over a band has spent the load’s whole allowance there, so outside the band the reflection has to approach one. Power arriving at those frequencies is sent back towards the source.

For a transmitter that matters: a reflected wave returning into an amplifier at frequencies it was not designed to see can be absorbed as heat or, worse, drive it into oscillation, which is why wideband matching networks are often followed by isolators that absorb the returning power. And it means that matching and filtering are one design problem rather than two. The optimum networks Fano derived are Chebyshev filters with the load’s capacitance absorbed into their first element; a matching network for a reactive load is necessarily a filter, and choosing its band is choosing where it reflects.

Where the limit decides designs #

The load with a capacitance across it is not an abstraction chosen for convenience. A transistor’s input, a laser diode’s junction and a photodiode are all such loads, and the bandwidth of optical receivers and wideband amplifiers is routinely set by the Bode–Fano limit on their input. A piezoelectric ultrasound transducer is a resistance, standing for the power radiated into tissue, shunted by the capacitance of its electrodes, and the fractional bandwidth an imaging probe can achieve is decided by that capacitance.

Antennas are the most prominent case. An antenna much smaller than the wavelength stores far more energy in its near field than it radiates each cycle, so its radiation resistance is shunted, in effect, by a large reactance — the stored energy that a small antenna cannot avoid. A matching network can tune it to resonance at any one frequency, but the Bode–Fano integral then caps the bandwidth over which it can stay matched, and a resonance’s bandwidth is fixed by its quality factor. That is why a phone’s antenna is tuned between bands rather than matched across all of them at once.

The pattern in all of these is that the stored energy is fixed by the physics of the device before any circuit is attached: a transistor’s input capacitance by its geometry, a transducer’s by its electrodes and the permittivity of its ceramic, an antenna’s by its size in wavelengths. The Bode–Fano integral turns each of those into a bandwidth, and the design choice that moves the limit is always a change to the device rather than to the network in front of it.

The same analytic argument limits things that are not circuits. An absorber coating that must stop radar reflections over a band obeys an integral of the same form, and it says that a thin absorber cannot be broadband: the product of its thickness and the width of the band it absorbs well is bounded, whatever material it is made of.

Where the theorem stops #

The network was lossless. A resistor in the network can flatten the reflection over any band, by absorbing what would have been reflected. The match then looks good and the power still does not reach the load. The theorem is a statement about delivering power, and it is honest only for networks that do not burn it.

The network was passive. Circuits with gain can synthesise negative capacitances and inductances — non-Foster elements — that cancel a load’s reactance over a wide band, and so evade the integral. They work in principle and are hard in practice, because a circuit that presents negative reactance is close to one that oscillates.

The network did not change in time. The analyticity behind the integral belongs to linear, time-invariant systems. A network whose elements are switched during a pulse is neither, and a medium switched in time conserves different things from one at rest; switched matching networks have been proposed that deliver short pulses to reactive loads beyond the static bound.

The load was one particular shape. Each kind of reactive load has its own integral. A resistance in series with an inductance gives an integral of the same form weighted by one over frequency squared; loads with several reactive elements give several constraints at once, and the best achievable band is set by the most restrictive.

What the figures leave out #

The networks in the figures were found by a numerical optimiser, not by Fano’s closed-form design procedure, and an optimiser offers no guarantee of the global optimum. The best four-, five- and six-element networks it found improve on each other by less than a Chebyshev design of the same size would, and the true optimum for six elements is probably somewhat closer to the floor. The floor itself is not in doubt, and every design the optimiser found respects it, but the gap drawn at six elements is an upper bound on the gap, not a measurement of it.

The elements are ideal: inductors with no resistance, capacitors with no leakage, a transformer with no loss or bandwidth of its own. Real components have all three, and at microwave frequencies the lumped elements themselves are approximations to short lengths of line.

Still open: how far a switched network can beat the bound #

The Bode–Fano limit assumes a network that does not change during the signal. Networks whose elements are switched or modulated in time are not bound by it, and calculations since 2018 have shown that a reactive load can in principle absorb a short pulse with less reflection than any static network allows, by storing the incoming energy while it arrives and releasing it into the load afterwards.

What that costs is less clear. The switching itself requires energy and must be timed to the signal, and whether a time-varying matching network can outperform the static bound for continuous, unpredictable signals — rather than for a pulse whose arrival is known — with realistic switches and realistic modulation power, has not been established. Experiments have demonstrated the principle at radio frequencies for particular pulses.

The habit worth carrying away is to ask whether a limit belongs to the design or to the thing being designed for. The Bode–Fano integral is a property of the load, fixed before any network is chosen, and every network does no more than decide where the load’s fixed allowance of match is spent.