The floor and the ceiling move apart

A decoupling bank is judged by two numbers, and they are quoted together as though improving one improved the other. The first is how low it goes — the least impedance anywhere on its curve, which is what a low-resistance part is sold on and what a bank of them is fitted to reduce. The second is how far up it holds — the highest frequency at which the rail is still under whatever the load tolerates, which is what a target impedance is a statement about.

Call them the floor and the ceiling. Every instruction in the subject is written as though they were the same instruction. Fit more capacitors, and the floor falls and the ceiling rises; fit a part with less series resistance, and the floor falls; fit a smaller package, and the ceiling rises.

The pair that is worse than either drew both of those for a bank connected to its load through nothing, and they behaved. Twenty ceramics beside a bulk part take the floor from 5.000 milliohms to 1.500 and the ceiling from 19.8 megahertz to 162, and neither number gives anything back. The capacitor that is not where the load is put three nanohenries of ordinary copper between the bank and the load — one of mounting loop for each part, two of plane shared by all of them — and found the ceiling reversing: it peaks at two parts and falls from there.

What nobody then asked is what the floor did. It is the other half of the same design decision, it is the number a designer is actually watching while adding parts, and it is computed by the same solve.

What is being solved, and what the two numbers are #

The netlist is the one the rung below built. Each capacitor is three elements — its capacitance, its series resistance, its series inductance — with a mounting inductance in front of it, and all of them hang from a bank node that reaches the load through a plane inductance. A one-amp source drives the load, so the potential at any node is an impedance in ohms per amp the load draws, and the same solve answers for every node at once. The third element is what makes any of this a question rather than a sum: the capacitor that is an inductor measured where a part stops being a capacitance, and a 100 nanofarad ceramic with a nanohenry of its own has done so above 15.9 megahertz.

Two of those nodes matter. The bank node is where a probe goes, because that is where the parts are and where two pads exist to touch. The load node is where the specification is, because a load experiences the potential at its own pins and nothing else. The rung below measured them at one part count and found them a factor of eleven apart at one frequency and a factor of three apart the other way at half of it.

The floor is the minimum of a reading over the sweep, found by locating a bracket on the grid and then golden-sectioning inside it, because a minimum read off a sweep is a minimum on a grid and the grid here is 115 points a decade. The ceiling is the highest frequency at which the reading is still under 50 milliohms, interpolated between the two grid points that straddle it rather than reported as the lower of them — that correction is two per cent, which is small enough to ignore and large enough to make two essays disagree about the same bank.

A target impedance is only a specification over a band, and the band has a bottom as well as a top. Below a few kilohertz the rail is held by a regulator’s feedback rather than by any capacitor, and that is a different mechanism with a different edge.

So the band this essay is about has a handover at each end, and neither is a component boundary. A source below a frequency is the lower one: a regulator is a voltage source while its loop still has gain, and above that it is whatever its output capacitance and its own copper make it. The upper one has no name because there is no part on the far side of it — above the ceiling the rail is held by nothing that appears on the bill of materials, and the question is only how high the ceiling was. Both of the numbers below are measurements of where a handover happens rather than of what a part does.

The floor, and who owns it #

The instruction “fit more of the small ones to lower the impedance” contains an unstated claim about which part the impedance belongs to, and the claim is false for the first six of them.

The bulk capacitor’s series resistance is 5 milliohms and each ceramic’s is 30. At its own self-resonance a capacitor is a resistor and nothing else, so the deepest point of the whole curve is whichever of those two numbers is smaller — and $30/n$ does not fall below 5 until the seventh ceramic. Measured, the floor sits at 4.9982 milliohms with six parts, at the bulk capacitor’s resonance, and jumps to 4.2850 milliohms with seven, at the ceramics’ — which is their 30 milliohms divided by seven. The floor does not improve gradually. It changes owner.

That is worth having on its own, because it means the first six ceramics buy nothing at all in the quantity they were fitted for, and it is true with no copper anywhere in the model. The mounting loop does not change it either: the depth of a series resonance is the resistance in it, so the mounting inductance moves where the bank’s floor is — 15.9 megahertz becomes 11.3 — and not how deep it is. Comparing the bank’s floor with the same bank’s floor with the copper deleted gives the same four figures at every count.

The load’s floor is a different quantity and it improves by 1.49 times where the bank’s improves by 3.33. It is also somewhere else: 2.83 megahertz against 11.3, because the plane’s inductance and the bank’s capacitance make a series resonance of their own, and the point at which the load sees least is a property of that pair rather than of any part in it.

The ceiling, and who is reading it #

The rung below compared the ceiling of a bank with copper against the ceiling of the same bank with the copper deleted, which is a comparison between a board and a board that does not exist. The comparison a bench makes is different and worse: the copper is there in both readings, and the only thing that changes is where the probe is.

At twenty parts the two numbers are 16.3 times apart, and both of them are correct measurements of the network that is on the board. A probe on the capacitors watches the ceiling climb from 13.2 to 82.4 megahertz as parts are added, very nearly in proportion, which is what $N$ inductances in parallel do and is exactly the improvement the parts were bought for. The load’s ceiling falls over exactly the same additions. Nothing about the bench measurement is wrong: it is repeatable, it is insensitive to the instrument, and a second probe beside the first agrees with it.

The frequency the load’s ceiling is heading for is

$$f = \frac{Z_{\text{target}}}{2\pi L_{\text{plane}}} = \frac{50\ \mathrm{m}\Omega}{2\pi \times 2\ \mathrm{nH}} = 3.98\ \mathrm{MHz}$$

which contains no capacitance at all. It is the same shape of quantity as the shared impedance in the millivolts in the wire: an impedance common to two things, turning one’s current into the other’s voltage, and belonging to neither of them.

Twenty capacitors, and the two ends at once #

Put the two readings on one sweep at the count where they are furthest apart and the mechanism is visible in a single picture.

Every ceramic added lowers the bank’s floor and moves it up in frequency, and moving it up in frequency is what makes it unreachable: the plane’s reactance rises linearly while the parts’ floor falls, so the gap between what the parts achieve and what the load receives widens at both ends at once. Ten parts give a factor of 47.2 and twenty give 94.3 — exactly double, because the bank’s floor halved and the load’s reading did not move, being the copper rather than the parts.

This is the failure the probe is part of the circuit inverts. There, an instrument disturbs the node it reads and the trace is a picture of a circuit that exists only while the probe is attached. Here the instrument reads its node perfectly and the node is the wrong one, which is the harder fault to notice because every check available says the measurement is good — the same complaint two terminals measure the leads makes about a two-wire resistance, arriving in a place where the extra copper is not an error but part of the circuit.

One in series with each, one in series with all #

The mechanism is a single distinction and it decides everything above. A mounting loop belongs to one part, so $N$ of them are in parallel and divide by $N$. A plane between the bank and the load is shared, so it divides by nothing. Where the current comes back is where those inductances come from: a loop enclosing an area, drawn by whoever placed the part and absent from every schematic.

The useful question is therefore not how much inductance there is but where it is, and that can be asked directly by holding the total and moving it.

The same three nanohenries are worth 4.8 times depending only on their arrangement, and the two part counts change places. Below 0.901 nanohenries on the shared path, twenty capacitors beat two; above it, twenty capacitors are worse than two, by a factor of 1.92 at the right-hand edge of the sweep. The crossing is not a soft one — it is a bisected point at which two banks differing by eighteen parts hold the identical frequency.

The reason is arithmetic rather than subtlety. What the load sees at high frequency is the shared inductance plus the per-part inductance divided by the count, and only the second term responds to buying parts. Once the first term dominates, the parts are competing for a path that does not widen, and each one added lowers the anti-resonance into a band that was previously clear without buying anything above it. Two mechanisms exchanging places at a point is the shape the edge that is a region collects, and here the point is a design rule rather than a boundary of validity.

Ten parts of one value, or one part of ten times it #

The distinction has an immediate consequence that the floor cannot see and the ceiling can. A microfarad of ceramic can be fitted as ten 100 nanofarad pieces or as one 1 microfarad piece in the same package — same series resistance, same series inductance, same mounting loop, same plane.

The floor cannot tell the two arrangements apart and the ceiling is 1.83 times different. A designer choosing on the first number is choosing on a quantity that is blind to the decision.

Why the two differ is worth stating exactly, because it is a theorem rather than an effect. $N$ identical branches in parallel are one branch whose capacitance is multiplied by $N$ and whose resistance, self-inductance and mounting inductance are each divided by $N$ — not nearly, but to the arithmetic. Solving the twenty-branch netlist and the single collapsed branch at six frequencies apiece agrees to a worst relative difference of $1.6\times10^{-14}$, which is rounding. The identity is asserted rather than assumed on every figure that uses it, and the tolerance is bracketed against the one mistake it exists to catch: dividing the capacitance and the two series parasitics while leaving the mounting loop whole, which is percent-level and would be caught immediately. A tolerance that is not bracketed is a test that either fires on nothing or accepts what it was written to reject, which is the discipline one step computed twice applies to a transient and applies here for the same reason: the collapsed netlist is a shortcut, and a shortcut checked against a round number is a shortcut nobody checked.

So the whole of the difference between ten parts and one is that the ten divide their mounting loops and the one does not. It is not the capacitance, which is identical; it is not the dielectric, which is the same; it is not the resistance, which the floor is set by and which the load never reaches. That is also why the identity is worth having as machinery: it makes a twenty-part bank cost one solve instead of twenty-one, which is what allows a sweep over part count to be drawn at all.

A third number, and it goes the other way #

The peak between the parts is the quantity the rung below this one is named for, and adding ceramics improves it: it falls from 1.293 ohms at 5.62 megahertz with one to 0.377 ohms at 1.58 megahertz with twenty. That looks like the one place where more parts are unambiguously better.

The circulating current does not follow the peak down. It rises by half over the first four parts and is still above its starting value at twenty. That current is the quality factor of a resonance nobody designed — resonance and its bandwidth measured the same quantity on a circuit built to be one — and here it is a current that does no work, heats both parts at a frequency where their series resistance is all of their loss, and flows in a loop of board copper whose area was decided by a layout rather than by a specification. Each figure’s own check is that the bulk part’s branch current and the ceramics’ agree to within six per cent, which is the statement that almost none of it reaches the load.

Three numbers, then, and adding parts moves them in three directions: the bank’s floor down, the load’s ceiling down past two parts, and the circulating current up before it comes back. No single figure of merit can be improved without reading the other two.

What it does not say #

It does not say that fewer capacitors are better. Below 0.901 nanohenries of shared inductance the ordinary instruction is right and more parts are worth almost exactly what they claim to be. What it says is that the instruction is conditional on a quantity nobody measures and nobody puts on a schematic, and that the condition is not obscure: two centimetres of plane is about two nanohenries, which is twice the crossing.

The 50 milliohm target is not incidental either. The ceiling is where a curve crosses a level, so a tighter target crosses it earlier and a looser one later. The plane’s own frequency scales with the target directly, so the 3.98 megahertz above becomes 7.96 at a 100 milliohm target and 1.99 at 25. What does not change with the target is the shape — the reversal, the crossing, and the disagreement between the two nodes are all ratios.

Four things are outside the model and each would make the result worse rather than better. The plane is a lumped two nanohenries standing in for a two-dimensional spreading inductance, and above the frequency Kirchhoff’s own frequency identifies the plane pair is a radial transmission line with resonances of its own. Mounting loops a few millimetres apart share flux, so $N$ parts do not give exactly $L/N$ and the division is worth less than the arithmetic above. The capacitance is the nameplate value, and the capacitance that is not one number measures a class II ceramic at a fifth of it under working bias. And every curve here rises without limit at the top of the sweep because no on-die capacitance stops it.

The number worth carrying #

The rule is a comparison rather than a quantity. The inductance every part shares sets a ceiling no number of parts can pass — 50 milliohms over 2π times two nanohenries is 3.98 megahertz — and parts are worth buying only while the inductance each one has to itself, divided by how many there are, is still a real fraction of the path. Once it is not, an added capacitor buys nothing above the ceiling and walks the anti-resonance further down below it, which is a loss rather than a wash.

Measured, with three nanohenries of copper in the bank and its target at 50 milliohms: twenty capacitors beat two while less than 0.901 nanohenries of the three is shared, and lose to them above that — 4.09 megahertz against 7.84 when 2.75 of the three is shared. At the crossing itself both counts reach 8.88 megahertz, and eighteen capacitors are worth nothing. Nine tenths of a nanohenry is about nine millimetres of that plane, which is the distance the whole rule turns on and is shorter than most parts sit from their load.

The habit that goes with it is the one every model has an edge asks for, applied to a rule rather than to an equation. “Fit more decoupling” is a model, it has a range, and the range is bounded by a quantity that appears on no bill of materials. The way to find out which side of it a board is on is to compute the impedance of what is fitted, at the node the question is about — which is a smaller ask than the layout rules usually offered instead, and it is the only measurement that distinguishes a bank that is working from a bank that is merely measuring well.