Power, and the part that does no work

The capacitor that is not where the load is

The rung below this one connects two capacitors to a load through nothing, and says so. Put three nanohenries of ordinary copper in — one of mounting loop per part and two of plane between the bank and the load — and the anti-resonance moves down to 5.63 megahertz and up to 1.29 ohms, a probe touching the ceramic reads a twelfth of what the load sees at 16.7 megahertz and three and a half times too much at 8.35, and the twentieth capacitor is worse than the second.

14 min read 8 figures The parasitic is the component The schematic is a label Measured, not tabulated

Assumes: The pair that is worse than either · Where the current comes back · The capacitor that is an inductor

The pair that is worse than either ends with a list of what is not in its model, and the first item on it is this:

No plane. Both capacitors here connect to the load through nothing. A real board has spreading inductance between the part and the load, of the same order as the part’s own, and it is in series with the branch.

This essay puts it in. The addition is three nanohenries — one of mounting loop for each capacitor, two of plane between the bank and the load — which is less inductance than either part already has inside it, and none of it appears on any schematic. It moves every number in the rung below, and it changes one conclusion from more is better with a caveat to more is worse past two.

Three nanohenries, and where they are

A decoupling capacitor is connected to the thing it decouples by copper, and the copper is a loop: down a via to the plane, across the plane, up a via to the load’s pin, and back. The loop has an area and therefore an inductance, and where the current comes back is the essay about how that inductance is decided by the return path rather than by the outward one.

Two of those loops matter here and they are in different places.

The mounting loop belongs to one part: the vias and the short run of plane between that capacitor’s own pads and the bank. It is in series with that part and with nothing else, so twenty capacitors have twenty of them in parallel and it divides by twenty. A nanohenry each is a conventional figure for a small ceramic on two vias.

The plane inductance is shared: it is the copper between wherever the bank is and wherever the load is, and every capacitor’s current flows through it. It does not divide by anything. Two nanohenries is a couple of centimetres of ordinary board.

That distinction — one in series with each part, one in series with all of them at once — is the whole of what follows.

Three nanohenries of copper move the peak to 5.63 MHz and raise it to 1.29 Ω. computed by solving, not by drawing. The same two capacitors, with and without the inductance of the way to them: one nanohenry of mounting loop per part and two nanohenries of plane between the bank and the load. The dashed curve is the bank as the rung below drew it, peaking at 1.187 Ω at 6.52 MHz; the solid one is what the load sees, peaking at 1.293 Ω at 5.63 MHz. The peak moves down because the branch that is inductive at that frequency got more inductive, and it rises for the same reason. Above about twenty megahertz the two part company entirely: the bank is still falling toward its parts' own resistances and the load is rising on two nanohenries that no capacitor is across. — Fig. 1 The same two capacitors with and without the copper. The dashed curve is the bank as the rung below drew it; the solid one is what the load sees.

What it does to the peak

The rung below found a parallel resonance between the bulk part’s inductance and the ceramic’s capacitance, peaking at 1.187 Ω at 6.52 MHz where either part alone presents about 0.20, with 5.87 amps circulating between them for every amp the load draws.

Adding the copper moves all three. The peak goes to 1.293 Ω at 5.63 MHz and the circulating current to 6.10 amps. Both directions are what the mechanism predicts: the resonance is an inductance against a capacitance, and the inductance is the one that grew.

The two contributions are not the same size and they do not do the same thing. The mounting loop alone takes the peak to 1.279 Ω at 5.64 MHz — nearly all of the movement, because it is in series with the bulk part’s own 5 nH and adds twenty per cent to it. The plane alone takes it to 1.205 Ω at 6.49 MHz — barely moving the frequency, because it is outside the resonant loop, and yet it takes the impedance ratio against the better part from 5.87 to 9.53, because it raises what the load sees without raising what either part presents at its own terminals.

That second number is the first sign of the real problem. The plane does not change the resonance. It changes who is measuring it.

One solve, two places to stand

The network is driven by a one-amp source at the load, so the potential at any node is an impedance per amp the load draws. Reading it at two nodes in the same solve — the load, and the ceramic capacitor’s own pads — costs nothing and answers a question no second measurement could, because the two readings are simultaneous rather than two conditions.

One solve, two places: the load sees 11.6× what a probe on the capacitor reads. computed by solving, not by drawing. The same network solved once and read at two nodes — the load, and the ceramic capacitor's own pads — with a one-amp source at the load so that each reading is an impedance. Below a megahertz they agree to -0.8 per cent, because a nanohenry is nothing there. At 16.7 MHz the load sees 0.307 Ω and the probe reads 0.0265 — a factor of 11.59, and the difference is two nanohenries of plane that the probe is on the wrong side of. A network analyser touching the capacitor is measuring the capacitor, which is not the quantity the specification is about. — Fig. 2 The impedance at the load and at the ceramic’s pads, from one solve. They agree below a megahertz and disagree by more than an order of magnitude twice, in opposite directions.

Below a megahertz they agree to eight parts in a thousand, which is the check: a nanohenry is nothing at those frequencies and the two nodes are the same node.

At 16.7 MHz the load sees 0.307 Ω and the probe reads 0.0265 — a factor of 11.6. The ceramic is at its own self-resonance there, where it is a 30 mΩ resistor and looks magnificent, and the two nanohenries of plane between it and the load are 0.21 Ω. A network analyser touching the capacitor measures the capacitor. The specification is about the load.

At 8.35 MHz the disagreement reverses: the load sees 0.0565 Ω and the probe reads 0.192, so the bench measurement is three and a half times pessimistic. The plane inductance and the bank form a divider that has a null of its own, and at the null the load is better off than anything measured at the parts would suggest.

So the error is not a bias to be calibrated out. It is a factor of eleven one way at one frequency and a factor of three the other way at half of it, and which one applies depends on where the load’s current spectrum happens to sit.

This is the same fault as two terminals measure the leads as well, one field over: a four-terminal measurement exists because a two-terminal one includes the copper between the instrument and the thing. What is different here is that the copper is not an error in the measurement — it is part of the circuit, and the measurement that excludes it is the wrong one.

The instruction that stops working

The advice after fit two kinds is fit more of the small ones, and the rung below measured what it buys and what it costs: twenty ceramics take the floor from 16 mΩ to 1.5 mΩ and move the anti-resonance down by a factor of 3.8.

Both of those are properties of the bank. The load’s question is different and is usually written as a target impedance: keep the rail below some number of milliohms, flat, from direct current up to whatever frequency the load’s current step contains. So the number to measure is the highest frequency at which the target is met.

Through two nanohenries of plane, the twentieth capacitor is worse than the second. computed by solving, not by drawing. The highest frequency at which the load stays under a 50 mΩ target, against how many ceramics are fitted. With the parts connected to the load through nothing, the ceiling rises from 19.8 MHz to 162 MHz — very nearly in proportion, because N inductances in parallel are one N-th. With one nanohenry of mounting loop each and two nanohenries of plane shared between them, it peaks at 8.05 MHz with two parts and falls to 4.96 MHz with twenty, heading for 3.98 MHz — which is the target impedance divided by 2π times the shared inductance and contains no capacitance at all. Parts are in parallel; the copper is in series with all of them at once. — Fig. 3 The highest frequency at which the load stays under 50 mΩ, against how many ceramics are fitted, with and without the copper. The two curves go in opposite directions.

With the parts connected to the load through nothing, the ceiling rises from 19.8 MHz with one ceramic to 162 MHz with twenty — very nearly in proportion, because N inductances in parallel are one N-th and the ceiling is set by inductance.

With one nanohenry of mounting loop each and two of shared plane, it peaks at 8.05 MHz with two parts and falls to 4.96 MHz with twenty. The twentieth capacitor is worse than the second, and what the curve 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}

a frequency with no capacitance in it at all. Above it, no arrangement of parts on the far side of that plane can hold the load below 50 mΩ, because the current has to get through the plane and the plane is an inductor.

The reason the curve is not monotonic is worth stating rather than smoothing over. With one ceramic the anti-resonance breaches the target and nothing above it comes back under; with two the deeper ceramic dip pokes back below and the ceiling jumps to 8 MHz; from there on, adding parts pushes the anti-resonance down in frequency faster than it lowers the high-frequency floor, and the ceiling walks back down toward the plane’s own number. There are three mechanisms in that curve and the maximum is where two of them change places — the same shape of result as the edge that is a region, where a boundary made of two mechanisms has a corner that neither one predicts.

The mounting loop is a loop, and its area is a decision

A nanohenry is not a property of a capacitor. It is a property of the path its current takes to the load and back, and that path is drawn by whoever placed the part.

The return under 2.00 cm of track, 200 µm above the plane. computed by solving, not by drawing, as an estimate from the geometry: a low-frequency path assumed to be three track-widths wide, 16.7 mΩ, and the parallel-plate inductance µ₀h/w, 25.1 nH, which is the limit for a track much wider than its height. The estimated resistance and reactance are equal at 106 kHz. Solved across the plane instead of assumed, the return does not change path at one frequency: it gathers beneath the track across a band about three decades wide, and this corner falls inside that band. Above the band the loop is the track's length times its height, 4.0 mm², and a milliamp round it at 100 MHz radiates -15.1 dBµV/m at three metres. — Fig. 4 Where the return current actually flows, against where a schematic suggests it flows. The inductance of a decoupling connection is the area between the two, and nothing on the schematic sets it.

Two vias close together and a plane immediately beneath enclose almost no area and give a few hundred picohenries; the same part with its ground via a centimetre away encloses a long thin loop and gives two or three nanohenries. That is the same factor as between one ceramic and five, so where the part is placed is worth as much as how many parts there are — and on the curve above it is worth more, because placement moves the shared plane term as well and that one does not divide.

The millivolts in the wire is the same quantity in its other role: an impedance shared between two circuits, turning one’s current into the other’s voltage. Here both circuits are the same one, and the shared impedance is between a capacitor and the thing it was fitted for.

The cure still works, and it costs more

The rung below’s repair for the peak is a third capacitor whose series resistance is chosen rather than minimised, golden-sectioned on the peak itself. It survives the copper, and the numbers move in the direction the mechanism says they should.

The peak is one over the resistance you were told to minimise. computed by solving, not by drawing. The same pair with both series resistances scaled together, and the peak re-found each time. A parallel resonance is damped by loss and by nothing else, and the only loss in a decoupling network is the equivalent series resistance a data sheet quotes as a figure of demerit. The fitted exponent is -0.906: quartering the resistance takes the peak from 1.187 Ω to 4.699 Ω. A lower-resistance part is a better capacitor at its own resonance and a worse network at the pair's. — Fig. 5 The peak against the series resistance of both parts. The fitted exponent is close to minus one: the peak is one over the resistance that every data sheet asks to be minimised.

Bare, the peak against the parts’ own resistance fits an exponent of −0.95. With the copper it fits −0.92 — still very nearly one over, and slightly less so because the added inductance raises the quality factor a little faster than the resistance damps it.

The damping capacitor’s best resistance rises from 94.5 mΩ to 123.2 mΩ, and what it achieves falls: the residual peak goes from 137 mΩ to 198 mΩ, so the same cure is 1.44 times less effective through three nanohenries of copper. Both move, and they move for different reasons — the optimum resistance is a number about the resonance, and the achievable floor is a number about the whole network.

The cure is a capacitor bought for its resistance: 94 mΩ. computed by solving, not by drawing. A third part, 1 µF with a deliberately chosen series resistance, added to the same pair. There is an optimum and it is not zero: too little resistance and the part joins the resonance instead of damping it, too much and it is not connected at high frequency. Golden section on the peak itself puts it at 93.9 mΩ, which takes the peak from 1.187 Ω to 0.1373 Ω — a factor of 8.6, bought with loss. — Fig. 6 A capacitor added for its loss rather than its capacitance, bisected on the peak. The optimum is not zero, and it is not the smallest resistance available.

What is still being solved, and what is not

Nothing here is an approximation added on top of the rung below. The netlist is longer by five elements and the same solver answers it, so the branch currents come back as before and the energy check runs on every solve.

At the peak the two parts exchange 5.87 A for every amp the load draws. computed by solving, not by drawing. The branch currents at 6.52 MHz, from the same solve the impedance came from, per amp drawn by the load. The two capacitors carry 5.87 and 5.78 amps between them and almost none of it reaches the load: it is a current going round a loop made of one part's inductance and the other's capacitance. That number is the same as the 5.87 by which the pair is worse than either part alone, because both are the quality factor of the same resonance. — Fig. 7 The branch currents at the peak, per amp the load draws. The copper raises this too, because it is the same quality factor.

What has changed is which node the answer is read at, and that is a modelling decision rather than a refinement. The rung below is not wrong about the bank; it is right about a quantity that nobody experiences. A load experiences the potential at its own pins.

A 100 nF capacitor, and what it is above 7.12 MHz. The dashed line is 1/(ωC), which is what the symbol means. The solid line is the same part with 30 mΩ of series resistance and 5.0 nH of series inductance, solved. They part company at 2.26 MHz and by a decade above resonance the part's impedance is 99.0× what its capacitance predicts. — Fig. 8 One capacitor: capacitive down, resistive at the bottom, inductive up. The mounting loop simply adds to the third of those, which is why it moves the resonance and not the floor.

The parameter that was accepted and did nothing

The function that builds this netlist has taken an lPlane argument since the day it was written, and until this essay it did nothing at all: it pushed an inductor from the bank’s node to a node called far that no other element touched — an open circuit — while the source went on driving the bank’s node regardless.

No figure was ever wrong, because no caller passed it. That is the only reason.

It is worth recording beside the resistance that grows with frequency and the other places this collection has met the same shape, because the shape is the point: an argument that is accepted and silently ignored is invisible to every check the site has. Every gate here asks whether a number that was drawn is right. None asks whether a parameter that was passed did anything, and the failure mode is a figure that looks like it is answering the question it was asked. The same absence produced two figures with no tick labels at all, and a whole family of generators that a sweep did not know existed.

The repair was to make the two nodes real — the bank at one, the load at the other, the source at the load — which is the model this essay needed anyway.

What a measurement of this would take

Measuring the load’s impedance rather than the bank’s means putting the ports where the load is, which on a real board means a pair of probe points at the die’s pins and not at the capacitors. That is usually impossible, which is why the measurement that gets made is the one at the parts.

The time-domain route is easier and is the one that finds this in practice. A load current step whose edge contains 16 MHz produces a rail movement at the load of eleven times what the impedance at the capacitors predicts, and the ring is at the anti-resonance rather than at anything the regulator’s loop can produce. A source below a frequency is where the regulator’s own output impedance runs out; everything above that belongs to the bank and to the copper, and only one of those two is on the bill of materials.

What is not in this model

No distributed plane. Two nanohenries is a lumped stand-in for a spreading inductance that is really a two-dimensional problem, and above a few hundred megahertz the plane pair is a radial transmission line with resonances of its own. Kirchhoff’s own frequency is where the lumped description stops, and for a ten-centimetre board that is around 40 MHz for a degree of phase — so the curves here are honest to about there and are a sketch above it.

No mutual inductance between the mounting loops. Two capacitors mounted a few millimetres apart share flux, and the coupling is positive, so N parts in parallel do not give exactly L/N. The measured departure from 1/N is a well-known few tens of per cent at close spacing, and it makes the ceiling story slightly worse rather than better.

And no on-die capacitance. Above the frequency at which the package inductance isolates the die, the only capacitor across the rail is the one inside the chip, and every curve here rises without limit because nothing in the model stops it. That is a statement about the model rather than about a supply, and it is the same caveat the rung below carries.

The habit this belongs to

The rung below drew a network property that is not a property of either part. This one draws a property that is not even a property of the network as anyone would draw it, because the schematic has no symbol for the two centimetres of copper that turn fit twenty capacitors into fit two.

The measurement that finds it is not a better instrument. It is one solve read at the node the question is about, instead of at the node the probe fits on.

Reading the wrong node, elsewhere

That last sentence is the instruments field’s whole subject arriving in the power field, and the pairing is worth drawing because the two essays diagnose the same failure from opposite ends.

The probe is part of the circuit is about an instrument changing the node it reads: a hundred and fifteen picofarads across a two-kilohm source is one per cent wrong at 6.8 kHz, and the trace on the screen is a picture of a circuit that only exists while the probe is attached. This essay is about an instrument reading a node faithfully and the node being the wrong one — a twelfth of what the load sees at 16.7 megahertz and three and a half times too much at 8.35, with nothing wrong with the probe at all.

The second failure is the harder one to notice, because every check available says the measurement is good. It is repeatable, it is insensitive to the instrument, and it agrees with a second instrument placed beside the first. What it does not agree with is the same measurement made two centimetres away, and nobody makes that comparison unless they already suspect the answer.

Which is the argument for the netlist over the bench in this particular case. A source below a frequency makes the same point about a regulator’s output impedance — 0.43 milliohms, below the milliohm at which a two-wire measurement measures its own leads — and reaches the same conclusion: some quantities are easier to compute correctly than to measure correctly, and a solver’s advantage is that it can be asked about a node rather than about a place a probe fits.