The edge of a chain

Every model has an edge put four of the models this subject relies on — the ideal amplifier, the small-signal transistor, the ideal capacitor and Kirchhoff’s lumped circuit — on one axis, each with the frequency at which it stops being true. A boundary is a model and a tolerance then found that each edge moves as a power of the tolerance it is quoted at: Kirchhoff’s laws as its first power, the amplifier and the small-signal model as its square root, and a full-power bandwidth not at all. The exponent identified the mechanism.

Both described one element. Real systems are chains of them: a signal passes through several gain stages, crosses several boards, is amplified and amplified again. Every stage in a chain can sit inside its own one-per-cent edge while the chain as a whole is well outside its own, because the errors accumulate. How fast the chain’s edge moves in as stages are added turns out to be set by exactly the exponent the earlier essay measured, with one exception that does not follow the rule at all.

Three chains, each solved as a whole #

Each chain is solved as a single system, not by adding up per-stage errors on paper.

The amplifier chain is N identical stages, each a closed-loop gain of ten taken from an amplifier with a one-megahertz gain–bandwidth product, so each has a single pole at 100 kilohertz. The chain is built as one netlist — a controlled source and an RC lag per stage — and its response is bisected for the frequency at which it is one per cent below the ideal answer. Each stage is drawn at unit gain with the same pole, because the gain is a scale factor that cancels from the relative error, and sixteen stages of ten would put $10^{16}$ into one matrix.

The board chain is a signal path through N ten-centimetre lengths of track in a dielectric of permittivity 4.4, and its edge is the frequency at which the whole path is one degree long — the tolerance at which every model has an edge located Kirchhoff’s laws failing for one board.

The small-signal chain is N junction stages, each with an exponential transfer characteristic whose fundamental gain at a drive amplitude is the Bessel-function ratio $2I_1(x)/x$ that defines the single-stage edge. The amplitude each stage sees is the amplitude the stages before it delivered, gain and gain error included, and the chain’s input amplitude is bisected for the point at which the overall fundamental gain is one per cent high.

The amplifier chain moves as the square root #

One stage is one per cent low at 14.2 kilohertz, a seventh of its 100-kilohertz bandwidth, as the earlier essay found. Two stages reach one per cent at 10.0 kilohertz, four at 7.06, sixteen at 3.53. The fitted exponent over sixteen stages is −0.502, and the measured edges agree to six figures with the exact expression for a cascade of identical poles: the chain’s error is $(1 + x^2)^{N/2} - 1$ where a single stage’s is $\sqrt{1 + x^2} - 1$, with x the frequency over the pole.

The square root has a direct reading. Each stage’s error grows as the square of frequency at low frequency, so the errors of N stages add to N times one stage’s, and reaching the same total needs a frequency lower by $\sqrt N$. Four stages halve the usable band; sixteen quarter it. A designer who checks each stage of a four-stage amplifier against a one-per-cent flatness specification and finds each one passing at 10 kilohertz has built an amplifier that fails it there.

The board chain moves as the first power #

The lumped model’s error is a phase length, and phase lengths add: a path through four boards is four times as long electrically, and is a degree long at a quarter of one board’s frequency. The exponent is −1 exactly. This is the practical content of the rule that the lumped approximation belongs to a whole signal path rather than to a board: a backplane that connects four cards, each of which was laid out to be lumped at its own clock, is a transmission line at a quarter of that frequency, and nothing on any one card’s layout says so.

The chain’s exponent is the tolerance exponent, negated #

The two chains moved with different exponents, and the earlier essay measured two different tolerance exponents for the same two models. They are the same pair of numbers.

The amplifier’s edge moves as the tolerance to the 0.502 and the chain’s as N to the −0.502; Kirchhoff’s moves as the tolerance to the first power and the chain’s as N to the −1. Both points lie on $y = -x$. The reason is that a chain of N identical stages whose errors add is the same problem as one stage held to a tolerance N times tighter: the chain shares one error budget N ways. So the exponent that a boundary is a model and a tolerance used to identify a mechanism also predicts how fast a chain of that mechanism closes in, with no further measurement.

That makes the earlier essay’s re-ordering of the axis a statement about chains as well. It found that the board fails before the capacitor below a tolerance of 0.424 per cent because the two edges move with different exponents; by the same arithmetic, a long enough chain of boards fails before a chain of capacitors of the same length, because the first-power mechanism closes in faster. Every edge on the original axis moves left as a system grows, and the ones that move as the first power move fastest.

A chain with gain belongs to its last stage #

The small-signal chain is the exception, and it fails the rule for a reason that is worth seeing.

At unity gain per stage the small-signal chain behaves exactly like the amplifier chain: each stage sees the same amplitude, each contributes the same error growing as the square of the amplitude, and the edge moves as N to the −0.502. That is the rule.

With gain it collapses. A chain of stages with a gain of ten each delivers ten times the amplitude to every stage after the first, and the junction’s error grows as the square of the amplitude, so the last stage contributes a hundred times the error of the one before it. The last stage takes nearly all the error, and the chain’s input edge is the single stage’s edge divided by the gain of everything in front of the last stage: 7.30 millivolts for one stage, 726 microvolts for two, 7.26 microvolts for four. The rule does not fail because the mechanism changed; it fails because the stages are not identical in the only respect that matters — the amplitude they see.

This is the familiar design rule that the last stage of an amplifier determines its distortion, derived rather than remembered. It also says what the chain of amplifiers above would do if it were limited by amplitude instead of by frequency: nothing like $\sqrt{N}$, and everything like the gain. A designer budgeting linearity through a chain must budget the last stage, at the amplitude it sees, and the stages in front of it are almost free.

What a designer does with the exponents #

The practical rule is short. Before checking any stage against a system specification, divide the specification’s tolerance among the stages that contribute, and re-read each stage’s edge at its share. If the mechanism’s edge moves as the square root of its tolerance, a chain of N identical stages costs a factor of $\sqrt N$ in the edge; if as the first power, a factor of N. If the stages differ in the amplitude or the bandwidth they see, the budget does not divide evenly, and the stage that sees the most takes most of it.

For the amplifier chain that rule has a familiar form. A multi-stage amplifier meant to be flat to a given frequency needs each stage’s bandwidth to exceed the target by more than one stage alone would need, and the classic result for N identical stages — that the chain’s half-power bandwidth is the stage’s times $\sqrt{2^{1/N} - 1}$ — is the same arithmetic applied at the three-decibel tolerance rather than at one per cent. For the four stages drawn here that factor is $\sqrt{2^{1/4} - 1} = 0.435$, so the chain’s half-power bandwidth is 43.5 kilohertz against each stage’s 100 — a smaller loss than the one-per-cent edge’s halving, because at the half-power level the error is no longer small and the square-root law has begun to bend. The looser the tolerance, the less a chain costs in proportion, which is the same statement as the exponent made from the other side: a quantity specified loosely forgives a long chain, and one specified tightly does not. The instrument’s own rise time measured the time-domain twin of that result, where rise times of cascaded stages add in quadrature.

For signal integrity it reads as a warning about connectors and backplanes. The lumped edge belongs to the whole path, and a system assembled from individually well-behaved cards can be a transmission-line problem as a whole. The boundary that is a rise time measured what a single track does once it is past that edge; a chain of tracks reaches it sooner by exactly the number of tracks.

Where the exponent comes from, once more #

The exponents have a common origin worth stating plainly. Every model in this subject is the first term of an expansion, and its error is the first term it leaves out. For the amplifier and the small-signal junction the first neglected term is second order — the response is even in the departure, so the linear term vanishes — and the error grows as the square of frequency or of amplitude. For the lumped circuit the first neglected term is first order, a phase, and the error grows in proportion. The edge that is a region found two mechanisms of the first kind adding as magnitudes rather than in quadrature, because they are lags on the same waveform; the chain is the same addition of like with like.

The order of the first neglected term is therefore the one number that predicts how an edge moves with tolerance and how it moves with system size, and it is visible in a model’s form without any measurement: an even function’s error starts at second order, an odd one’s at first. A model whose error is of order p in its small quantity has an edge that moves as the tolerance to $1/p$ and, in a chain of identical stages, as N to $-1/p$.

Chains this subject already contains #

Several earlier measurements were chain measurements without being called that, and the rule reads them back. The band that closes with the order found one Sallen–Key section inside a tenth of a decibel of its design over three decades of impedance level and the band shrinking as sections were cascaded: each section’s departure from its design is a small error that grows toward the edges of the impedance range, and a cascade adds them, so the usable range closes in as the order rises. The rule predicts how fast from the order of each section’s error, without solving the cascade.

A ladder network is not a cascade found the opposite behaviour in a doubly terminated LC ladder network, whose passband moved fifty-four times less than a buffered cascade’s for the same component tolerance. That is a chain whose stages are not independent: a ladder network’s elements interact, and at the ripple peaks the sensitivity to every element is zero to first order, so its errors start at second order and do not accumulate as a cascade’s do. It is the exception that confirms the rule’s premise — the rule applies when stages are isolated from each other and their errors simply add, and a ladder network is engineered so that they do not.

The small-signal exception has its own precedent too. The order that stops helping derived the amplitude at which a degenerated stage’s distortion reaches a target, each expression one order longer than the last. A chain of such stages inherits the last-stage rule unchanged: whatever the single-stage edge is, the chain’s input edge is that edge divided by the gain in front of the last stage, because distortion is a function of the amplitude a stage actually sees.

The general form is therefore a pair of rules for any cascade. Where stages are isolated and see the same conditions, the edge moves as $N^{-1/p}$ with p the order of each stage’s error. Where they see different conditions — a growing amplitude, a growing bandwidth demand — the stage that sees the worst condition takes the budget, and the edge is that stage’s. A system is usually a mixture of both, and the useful habit is to ask, for each model in it, which of the two rules applies before believing any single stage’s edge.

What the model leaves out #

The stages are identical and their errors add with one sign. Real stages differ, and some errors can cancel: a chain of amplifier stages with slightly different bandwidths has an error that is still the sum of their squared frequency terms, so the rule holds with an average stage in place of the identical one; but a chain that alternates a peaking stage with a drooping one can be flatter than either, which is what equalisation does deliberately. Errors that partially cancel do not share a budget; they spend it on each other.

The board chain treats phase as the only error. Real connectors add their own discontinuities, each a small reflection, and reflections from several discontinuities interfere: they add in proportion at some frequencies and cancel at others, which is a different and more irregular way for a chain to fail than the steady accumulation of phase drawn here.

And the small-signal chain uses the junction’s exponential alone. A real stage has emitter degeneration and feedback that change both its gain and the order of its first error term; two currents with one name and the degeneration measurements elsewhere in this field change the single-stage edge, and the chain’s last-stage rule carries over with the new edge in place of the bare junction’s.

The number to keep #

A chain of N identical stages shares one error budget, so its edge moves as N to minus the inverse of the error’s order: as $N^{-1/2}$ for the amplifier — four ×10 stages from a 1 MHz amplifier are 1% low at 7.06 kHz, not 14.2 — and as $N^{-1}$ for Kirchhoff’s lumped circuit — four boards are a degree long at 993 kHz, a quarter of one board’s edge. The chain’s exponent is the tolerance exponent negated, measured on the line $y = -x$. With gain between stages the rule gives way to a simpler one: the last stage takes the error, and four junction stages of ten are 1% off at 7.26 µV, a thousandth of one stage’s 7.30 mV.

Still open: stages that are not identical, errors that cancel, and the chain in time #

Unequal stages. A chain whose stages have different bandwidths or see different amplitudes shares its budget unevenly. Finding the allocation that maximises the chain’s edge for a fixed total of stage bandwidth — how to spend gain–bandwidth across stages — is an optimisation this measurement sets up and does not perform.

Errors that cancel. A chain that alternates stages whose errors have opposite signs can be flatter than any of its stages. How far deliberate cancellation can move a chain’s edge outward, and how sensitive that is to tolerance, is the equaliser’s version of this question.

The chain in time. A cascade’s step response settles to a tolerance more slowly than any stage’s, and settling time has its own exponent in the tolerance — logarithmic for a single pole. Whether a chain’s settling edge follows the same negated-exponent rule is measurable on the same netlists.