Kirchhoff's own frequency
Every figure in this collection before this one rests on an assumption so basic that it is rarely stated: that a circuit is a set of components joined by wires, and that the wires do nothing. A wire has the same potential everywhere along it, at every instant, and a component’s current is the same at both ends.
That is the lumped-element model, it is what makes a netlist a complete description, and it has a frequency.
Where the assumption comes from and what breaks it
Kirchhoff’s current law is a statement about simultaneity. The current arriving at a junction equals the current leaving it at the same moment, which requires the effect of a change at one end of a conductor to be felt at the other end instantly.
It is not. A change propagates along a conductor at a speed set by the surrounding material: the speed of light divided by the square root of the relative permittivity. On ordinary circuit-board material, with a permittivity around 4.4, that is about 1.43 × 10⁸ metres per second, a little under half the speed of light in vacuum.
So a signal takes about seven hundred picoseconds to cross ten centimetres of track. At a kilohertz that is a millionth of a cycle and nothing whatever depends on it. At a gigahertz it is most of a cycle, and the far end of the track is doing something completely different from the near end.
The crossover is not a matter of opinion; it is a length compared with a wavelength, and it is what the figure is about.
The two numbers
One degree of phase — 3.97 MHz for ten centimetres. The track is one degree long. At this frequency the two models agree to three parts in ten thousand, which is far inside anything measurable, and the lumped model is entirely trustworthy.
A tenth of a wavelength — 143 MHz for ten centimetres. This is the conventional criterion, and by this frequency the two models differ by thirty per cent. That is worth saying plainly: the usual rule of thumb marks the point at which the model is already thirty per cent wrong, not the point at which it starts to fail.
A tenth of a wavelength is thirty-six degrees of phase. Nobody would accept thirty-six degrees of error anywhere else in this subject, and the reason it is tolerated here is that the alternative is a much more demanding piece of analysis. Stating the boundary as one degree rather than as a tenth of a wavelength puts the number thirty-six times lower, and that is the honest figure for when the model starts costing something.
The two scale together across the slider:
| Length | 1° long at | A tenth of a wavelength at |
|---|---|---|
| 5 mm | 79.4 MHz | 2.86 GHz |
| 2 cm | 19.9 MHz | 715 MHz |
| 10 cm | 3.97 MHz | 143 MHz |
| 30 cm | 1.32 MHz | 47.6 MHz |
| 1 m | 397 kHz | 14.3 MHz |
| 3 m | 132 kHz | 4.76 MHz |
The last row is the one that reaches into ordinary work. Three metres of cable — a loudspeaker lead, an instrument cable, a run between two boxes — is one degree long at 132 kHz and a tenth of a wavelength at 4.76 MHz. Anything digital on that cable is outside the lumped model.
What the other model does
The curve that keeps going where the lumped one stops is the exact input impedance of a transmission line, and it does something no combination of lumped elements can.
A line’s input impedance repeats. At a quarter wavelength it inverts the load — a high impedance at the far end appears as a low impedance at the near end, and the reverse. At a half wavelength it reproduces the load exactly. And it repeats every half wavelength after that, for ever.
That behaviour is not an extreme version of the lumped answer; it is a different kind of function. A network of finitely many lumped elements has finitely many poles and settles to an asymptote. A line has infinitely many resonances, evenly spaced, and never settles to anything.
Which is why the essay says the two models describe different objects above the boundary rather than describing one object to different accuracies. Below the boundary the lumped model is an approximation. Above it, it is a category error.
The quarter-wave inversion, which is useful and dangerous
The impedance inversion at a quarter wavelength deserves its own paragraph because it is simultaneously a design technique and a way for a circuit to fail mysteriously.
As a technique it is the quarter-wave transformer: a length of line of the right impedance matches one impedance to another with nothing but a length of track. As a failure it is the reason an apparently harmless stub of unterminated track can present a dead short at some frequency — an open circuit at the far end, inverted by a quarter wavelength of line, is a short at the near end.
Both are the same figure. The one at the top of this essay, dragged to a length where the frequency of interest is a quarter wavelength, is either a matching network or a fault depending on whether it was intended.
The lumped model predicts neither. Asked about the same stub it reports a small capacitance, which is a perfectly reasonable answer at low frequency and completely wrong at the frequency where it matters.
The refusal
The lumped model in this collection declines rather than extrapolating. Asked to apply Kirchhoff’s laws at a frequency where a stated circuit size is more than ten degrees long, it raises an error naming both the size and the frequency and saying that the current entering one end is not the current leaving the other at the same instant, so no lumped network describes it.
That refusal is exercised inside the figure — it is triggered, and its message checked — rather than described. A refusal that is never run is a comment, and an assertion that has silently stopped rejecting has silently stopped testing.
What the boundary is not
Three things it would be easy to read into this essay are not in it, and separating them keeps the number honest.
It is not about the frequency of the signal, but about its bandwidth. A one-kilohertz square wave with a one-nanosecond edge contains significant energy up to several hundred megahertz, and it is that content which decides whether the lumped model applies. Reading the boundary against a nominal clock rate is the single commonest way to conclude wrongly that a circuit is electrically small.
It is not about the length of the wire, but about the loop. Current flows out along one conductor and back along another, and the physical size that matters is the area enclosed by the pair. A long signal track over a ground plane immediately beneath it is a small loop; two short wires far apart are a large one. The figure’s parameter is called a length because that is the natural way to state it for a line, and the quantity underneath is a geometry.
It is not a threshold below which nothing happens. The two curves in the figure separate continuously from the very lowest frequency drawn. The one-degree mark is a place where the separation reaches a stated size, chosen because it is small enough to ignore; there is no frequency at which the distributed behaviour switches on.
A note on the permittivity
One number in this essay carries more weight than it looks: the relative permittivity of about 4.4, which sets the propagation velocity and therefore every frequency quoted.
It is a property of the insulating material, it varies between board types by a factor of about two across the ordinary range, and it varies with frequency as well — the same material that measures 4.6 at a kilohertz measures nearer 4.2 at a gigahertz. So the boundaries in the table are correct to about ten per cent and no better, which is entirely adequate for their purpose and worth stating rather than implying a precision the input does not have.
Two special cases are worth knowing because they sit at the ends of the range. A signal in free space or on an open wire travels at essentially the speed of light, so the boundaries are about twice as high as the table’s. A signal in a solid-dielectric coaxial cable travels at about two-thirds of it, close to the board figure. And a signal on a track at the surface of a board travels faster than one buried inside it, because part of its field is in the air — which is a real effect, is why the two kinds of track have different propagation delays on the same board, and is exactly the sort of detail that makes the ten per cent honest rather than pessimistic.
Reflection, which is the same fact in the time domain
Everything above is stated in frequency, and the time-domain version is the one most people meet first, so it is worth connecting them.
Send a step down a line whose far end is not matched to its impedance. The step travels at the propagation velocity, arrives at the far end, and finds a load that cannot absorb it — so part of the energy turns round and comes back. The returning wave arrives at the source after twice the transit time and adds to whatever is there, and if the source is not matched either, part of that reflects again. What comes out is a staircase: a series of steps separated by the round-trip time, converging on the answer the lumped model would have given immediately.
That is the same phenomenon as the resonances in the figure at the top of this essay, seen in the other domain, and the connection is exact: the frequencies at which the input impedance goes to infinity are those at which a round trip is a whole number of half-cycles, so the reflections add in phase.
The practical criterion follows without any new argument. If the round-trip time is short compared with the rise time of the signal, the staircase happens entirely within the edge and is invisible; if it is comparable, the edge has steps in it. Twice seven hundred picoseconds for ten centimetres of track means a signal with a one-nanosecond edge is right at the boundary — which is why the numbers in this essay have become ordinary rather than exotic.
Two ways to stay inside the model
Since the boundary is set by size and frequency together, there are exactly two ways to stay inside it, and both are used.
Make it smaller. The reason a decoupling capacitor is placed as close as possible to the pin it serves is not superstition; it is that the loop it forms with that pin has an inductance proportional to its area, and the resonance that inductance produces sits at a frequency that falls as the loop grows. Halving every dimension moves the boundary up by a factor of two.
Make it slower. A series resistor that deliberately slows an edge is the standard remedy for ringing on a short line, and it works by moving the signal’s own bandwidth below the boundary rather than by moving the boundary. It is the cheapest fix available and it is available only when the speed was not needed.
What is not available is a third option, and the reason is worth stating. There is no arrangement of lumped components that makes a circuit electrically small; size is a property of the layout, not of the netlist. That is the sharpest form of this collection’s claim about schematics — the drawing contains none of the variable that decides the answer, so no amount of care in the drawing can address it.
What this means for the rest of the collection
Everything else on this site is a lumped-element analysis, so it is worth being explicit about the scope this boundary defines.
The audio and instrumentation circuits in the earlier fields operate below a megahertz, on circuits a few centimetres across, and are three or four orders of magnitude inside the boundary. Nothing about them is affected.
The filter comparisons run to about thirty kilohertz and are equally safe. The capacitor’s self-resonance at 14.5 MHz is not — a hundred nanofarads with a centimetre of lead is about at the one-degree boundary for that lead at its own resonant frequency, which is a coincidence worth noticing: the two boundaries in this collection that are closest together are the two that both depend on the geometry of the connection rather than on the component.
Beyond that, the whole apparatus of nodal analysis stops applying and has to be replaced by something that treats a conductor as a medium rather than as an element. That is a different subject with different machinery, and the honest thing to do is to name the boundary and stop, rather than to keep drawing lumped figures at frequencies where they mean nothing.
The size that has already shrunk
One last observation, because it explains why this boundary keeps mattering more rather than less.
Nothing about the physics has changed since Kirchhoff, and nothing about circuit boards has changed much either — the propagation velocity on ordinary material is what it always was. What has changed is the frequencies. A circuit whose fastest edge takes a nanosecond contains significant energy up to several hundred megahertz whatever its nominal clock rate, and at 300 MHz the one-degree boundary on ordinary board material is about 1.3 millimetres.
That is smaller than most component packages. Which is why high-speed digital design is, in practice, transmission-line engineering wearing a schematic — and why the schematic, which shows none of the geometry, is at its least informative precisely where the geometry has become the whole answer.