Circuit Response
every model has an edge, and the figure carries it
Where the models stop

The capacitance a third switch moves

A changeover's open channel leaks into the source of the channel that is closed, so its isolation into fifty ohms falls from 643 megahertz with a buffered source to 6.34 megahertz with a fifty-ohm one. Put a third switch to ground between two series switches and the leak lands on half an ohm of closed switch instead: 814, 643 and 2,240 megahertz from sources of nothing, fifty ohms and a kilohm, rising forty decibels a decade where a changeover's rises twenty. The price is the shunt switch's own capacitance, moved under the closed path, which makes the T one per cent wrong as a waveform at 6.59 megahertz beside the changeover's 6.61.
14 min read 8 figures The parasitic is the componentEvery model has an edgeTwo routes to a number

Assumes: A band rather than an edge · Kirchhoff's own frequency

Where an open switch leaks to found that a multiplexer’s open channels do not leak into the load. They leak into the node the closed channel holds, so their limit is set by the impedance of the source driving that channel, and a switch’s data sheet cannot state it. The smallest multiplexer, a changeover, isolates a fifty-ohm load to one per cent up to 643 megahertz when its sources are buffered, and up to 6.34 megahertz when they are fifty ohms — a factor of a hundred for a source impedance nobody chose.

That essay ended with the obvious response: if the isolation is decided by where the leaked current lands, make it land somewhere chosen. The usual way is a third switch. Between two switches in series, put one to ground, closed when the path is open and open when the path is closed. Built from three copies of the same half-ohm, hundred-megohm, five-picofarad part, that is a T, and this essay measures what it buys and what it moves.

From 0 Ω into 50 Ω: a T isolates to 814 MHz, a changeover to 643 MHz. computed by solving, not by drawing. The fraction of the drive that arrives with the path open, against frequency, from a 0 Ω source into 50 Ω, for the 0.5 Ω, 100 MΩ, 5 pF switch used three ways. A T reaches 1% at 814 MHz; a changeover reaches 1% at 643 MHz; one switch reaches 1% at 6.37 MHz.
Fig. 1 From a buffered source into 50 Ω, the fraction of the drive that arrives with the path open, for the same switch used three ways. The T reaches 1% at 814 MHz, the changeover at 643 MHz, a lone switch at 6.37 MHz.

From a buffered source the T is only a little better than a changeover — 814 megahertz against 643 — and both are a hundred times better than a lone switch. That is not where the T’s case is made. A changeover from a buffered source already has its leak landing on half an ohm, the closed channel’s on-resistance in front of a source of nothing, and a third switch can only offer another half ohm.

A node held at half an ohm

Open, the T’s first series switch lets a current through its off-capacitance, and that current meets a closed switch to ground. The node between the two series switches is therefore held by half an ohm, and the voltage on it is the on-resistance over the off-impedance of the drive: Rₒₙ·ωCₒff, whatever the source is, for as long as the source is small against the off-impedance.

A nodal solution of the T written independently of the figures shows that exactly. At ten megahertz the node sits at 1.571×10⁻⁴ of the drive from a buffered source and from fifty ohms, which is Rₒₙ·ωCₒff to the four figures printed, at 1.552×10⁻⁴ from five hundred ohms and at 1.498×10⁻⁴ from a kilohm, where the source has begun to be a visible fraction of the off-capacitance’s 3.2 kilohms.

The second open switch then has to carry that small voltage to the load, and it is another off-impedance in front of a fifty-ohm load: it divides again, by about the load over the off-impedance. So the T’s leak is the product of two small factors, each proportional to frequency, and it rises as the square. In the same independent solution, from a buffered source into fifty ohms, the T lets through 2.467×10⁻⁸ of the drive at one megahertz and 2.467×10⁻⁶ at ten, a hundredfold for a decade, while the changeover and the lone switch each rise tenfold.

Forty decibels a decade rather than twenty is the T’s real property, and it explains why the buffered comparison looks modest. The two curves are close near where they cross one per cent, and at every lower frequency the T is further ahead by a factor that grows with the square of the distance.

From 50 Ω into 50 Ω: a T isolates to 643 MHz, a changeover to 6.34 MHz. computed by solving, not by drawing. The fraction of the drive that arrives with the path open, against frequency, from a 50 Ω source into 50 Ω, for the 0.5 Ω, 100 MΩ, 5 pF switch used three ways. A T reaches 1% at 643 MHz; a changeover reaches 1% at 6.34 MHz; one switch reaches 1% at 3.18 MHz.
Fig. 2 From 50 Ω into 50 Ω. The T reaches 1% at 643 MHz, the changeover at 6.34 MHz, a lone switch at 3.18 MHz. Fifty ohms of source have cost the changeover a factor of a hundred and the T a factor of 1.27.

Here is the case. The changeover’s leak lands on its closed channel’s on-resistance plus fifty ohms of source, and it has lost two decades of frequency for it. The T’s leak lands on its shunt switch, and the source is not in that path at all. Fifty ohms of source moves the T from 814 megahertz to 643, and the normalisation accounts for all of that and more: the fraction is of what would arrive with the path closed, which fifty ohms of source has already halved, and a leak rising as the square turns that halving into a factor of 1.41 in frequency. The T loses only 1.27, because at these frequencies the source has already begun to divide the drive before the shunt switch.

Better from a worse source

From 1 kΩ into 50 Ω: a T isolates to 2.24 GHz, a changeover to 318 kHz. computed by solving, not by drawing. The fraction of the drive that arrives with the path open, against frequency, from a 1 kΩ source into 50 Ω, for the 0.5 Ω, 100 MΩ, 5 pF switch used three ways. A T reaches 1% at 2.24 GHz; a changeover reaches 1% at 318 kHz; one switch reaches 1% at 303 kHz.
Fig. 3 From a 1 kΩ source into 50 Ω. The T reaches 1% at 2.24 GHz, the changeover at 318 kHz and a lone switch at 303 kHz.

From a kilohm the changeover has fallen to 318 kilohertz, almost exactly a lone switch’s 303, because its leak now lands on a source larger than anything else in its path. The T has gone the other way: it reaches one per cent at 2.24 gigahertz, nearly three times its buffered figure.

The reason is where the source sits. In a T the source is in series with the first open switch, in front of the half-ohm node. At the frequencies where the T finally reaches one per cent the off-capacitance’s reactance has fallen to tens of ohms — five picofarads is 49.5 ohms at 643 megahertz and 14 ohms at 2.24 gigahertz — and a kilohm of source in series with it divides the drive before the shunt switch is reached. The same division that ruins a changeover protects a T. The claim is checked at five hundred ohms too, where the T reaches one per cent at 1,479 megahertz against 814 buffered, and a changeover fed the same sources falls by a factor of about a hundred between a buffer and fifty ohms.

It is worth being plain about the gigahertz number, because it is the one most likely to be quoted and the one least likely to be true of a real circuit. It is a statement about a lumped network, and the last section of this essay says how small that network would have to be.

A band that shuts a hundred times later

The isolation figures fix the load at fifty ohms. The question A band rather than an edge asked of a lone switch — for which loads is the part within one per cent of ideal both open and closed — has an answer for the T too.

Three switches as a T: the band closes at 702 MHz rather than 6.43 MHz. computed by solving, not by drawing. The band of load resistance in which a T of three 0.5 Ω, 100 MΩ, 5 pF switches is within 1% of ideal in both states, from a 0 Ω source, against the band of one such switch. Closed, the T puts two on-resistances in series, so its lower edge is 99 Ω against 49.5 Ω. Open, it has no upper edge on the axis below 637 MHz, because the shunt switch holds the node between the two open ones and that node leaks the on-resistance over the off-impedance whatever the load (the fraction over 2π·Rₒₙ times the off-capacitance is 637 MHz). The T's band closes at 702 MHz, balanced at 99.6 Ω; the single switch's at 6.43 MHz.
Fig. 4 The band of load resistance in which the T is within 1% of ideal in both states, from a buffered source, against a lone switch’s. Closed, the T puts two on-resistances in series, so its lower edge is 99 Ω rather than 49.5 Ω. Open, it has no upper edge below 637 MHz, because the held node leaks Rₒₙ/|Zₒff| of the drive whatever the load. The T’s band closes at 702 MHz, balanced at 99.6 Ω; the lone switch’s at 6.43 MHz, 109 times lower.

The lower edge doubles, which is the first cost and the easiest to state: a closed T is two switches in series, so every error the on-resistance makes is twice as large. The upper edge has gone from the axis entirely until 637 megahertz — which is ε/(2πRₒₙCₒff), the frequency at which the half-ohm node itself reaches one per cent, and the same frequency that set a buffered changeover’s limit. And the band closes at 702 megahertz, a hundred and nine times later than a lone switch’s.

From real sources the closure falls, and it falls for a reason worth noticing. From fifty ohms the T’s band closes at 89.8 megahertz, and from a kilohm at 4.53 megahertz — against a lone switch’s 6.35 megahertz and 318 kilohertz from the same sources. The isolation from a kilohm reached gigahertz, so it is not the open state that closes the band there. It is the closed one, and that is where the third switch’s price is paid.

The first power, once more

Three switches as a T: the band closes at 70.0 MHz rather than 63.7 kHz. computed by solving, not by drawing. The band of load resistance in which a T of three 0.5 Ω, 100 MΩ, 5 pF switches is within 1.00e-1% of ideal in both states, from a 0 Ω source, against the band of one such switch. Closed, the T puts two on-resistances in series, so its lower edge is 999 Ω against 499 Ω. Open, it has no upper edge on the axis below 63.7 MHz, because the shunt switch holds the node between the two open ones and that node leaks the on-resistance over the off-impedance whatever the load (the fraction over 2π·Rₒₙ times the off-capacitance is 63.7 MHz). The T's band closes at 70.0 MHz, balanced at 1000 Ω; the single switch's at 63.7 kHz.
Fig. 5 The T’s band at a tenth of a per cent, from a buffered source. The lower edge is 999 Ω against 499 Ω, the held node reaches the fraction at 63.7 MHz, and the band closes at 70.0 MHz — a tenth of its one-per-cent closure — while the lone switch’s closes at 63.7 kHz, a hundredth of its own.

The T’s closure moves as the first power of the tolerance and the lone switch’s as the second, exactly as the multiplexer’s did. A boundary is a model and a tolerance reads an exponent as the name of a mechanism, and the name here is the same one: the leak no longer depends on the load, so there are not two edges trading against each other to exhaust, only one error growing until it reaches the tolerance. At a tenth of a per cent the T’s band closes eleven hundred times later than a lone switch’s, where at one per cent it was a hundred and nine.

That exponent is the whole of the T’s value to anything asked for precision. At a tenth of a per cent a lone switch’s band has shrunk to tens of kilohertz while the T’s is still tens of megahertz, before its closed state is counted. And a lone switch asked for sixteen bits, about fifteen parts per million, has no band at all at any frequency: that is below the floor described at the end of this essay, where the off-resistance rather than the off-capacitance sets the limit.

What the shunt switch hangs on the closed path

With the path closed the shunt switch is open, and an open switch is not nothing. Its five picofarads, with the hundred megohms beside them, now connect the middle of the signal path to ground. In front of that node are the source and one on-resistance, and the two together with the capacitance are a low-pass.

Closed, the T is 1% wrong at 6.59 MHz as a waveform and at 89.7 MHz as an amplitude. computed by solving, not by drawing. The error of the closed path against frequency, from a 50 Ω source into 1 kΩ. The T's open shunt switch puts 5 pF on the signal path: its vector error reaches 1% at 6.59 MHz and its shortfall in amplitude at 89.7 MHz. The changeover, whose open channel hangs the same capacitance on its output, reaches 1% as a vector at 6.61 MHz. A single switch is 4.76e-2% wrong at every frequency drawn, which is its on-resistance.
Fig. 6 The error of the closed path against frequency, from a 50 Ω source into 1 kΩ. The T’s open shunt switch puts 5 pF on the signal path: its vector error reaches 1% at 6.59 MHz and its shortfall in amplitude at 89.7 MHz. The changeover, whose open channel hangs the same capacitance on its own output, reaches 1% as a vector at 6.61 MHz. A lone switch is 4.76×10⁻² per cent wrong at every frequency drawn, which is its on-resistance.

As a waveform the closed T is one per cent wrong at 6.59 megahertz, and the changeover at 6.61. The third switch has bought nothing here at all, and the reason is the finding Where an open switch leaks to ended on: the capacitance that leaks when a path is open is the capacitance that loads it when the path is closed, and in both the T and the changeover exactly one open switch’s capacitance hangs on a node driven through about the source impedance. The T moved the capacitance from the output to the middle of the path. It did not remove it, and a capacitance anywhere on the path costs the same.

Counted as an amplitude the same closed T looks thirteen and a half times better, one per cent short only at 89.7 megahertz. That is the gap The width no load can change measured for a switch charging a hold capacitor — a single pole departs first in phase, so its vector error grows as the frequency and its shortfall as the square — and it is the reason a T’s insertion loss in decibels can look excellent up to a frequency at which the waveform through it is already several per cent wrong. Where the mechanisms are one mechanism is the general statement: whether an error is a magnitude or a vector is a property of its direction, and a tolerance stated without saying which is not yet a tolerance.

Closed, the T is 1% wrong at 636 kHz as a waveform and at 8.83 MHz as an amplitude. computed by solving, not by drawing. The error of the closed path against frequency, from a 1 kΩ source into 1 kΩ. The T's open shunt switch puts 5 pF on the signal path: its vector error reaches 1% at 636 kHz and its shortfall in amplitude at 8.83 MHz. The changeover, whose open channel hangs the same capacitance on its output, reaches 1% as a vector at 636 kHz. A single switch is 2.50e-2% wrong at every frequency drawn, which is its on-resistance.
Fig. 7 The same comparison from a 1 kΩ source into 1 kΩ. The T’s vector error reaches 1% at 636 kHz and its amplitude shortfall at 8.83 MHz; the changeover’s vector error reaches 1% at 636 kHz as well.

From a kilohm the price is plainer still. The T’s closed path is one per cent wrong as a waveform at 636 kilohertz, the same as the changeover’s to three figures, while its open path isolated to 2.24 gigahertz from the same source. Those two numbers are 3,500 apart, and it is the smaller that decides where the T is useful: its band from a kilohm closes at 4.53 megahertz, with the load pushed as high as it goes to reduce the closed path’s error, and above that no load makes the closed T good enough.

There is a second, smaller price at low frequency, and a fifty-ohm load shows it. The closed T is two on-resistances in series with the load, so into fifty ohms from fifty it is 0.990 per cent short at direct current, where a lone switch is 0.498 — the T is already at its own lower edge before any capacitance is involved, and its vector error into fifty ohms reaches one per cent at 1.77 megahertz rather than the 6.59 it reached into a kilohm. The doubled on-resistance and the moved capacitance are two separate costs, and a fifty-ohm system pays both.

A claim about a package

2.20 cm of track, solved as a lumped circuit and as a line. The two agree to 0.030% at 18.0 MHz, where the track is one degree long, and to 30.1% at 650 MHz, where it is a tenth of a wavelength. Above that the lumped model is not approximately right; it is describing a different object.
Fig. 8 2.20 cm of track in a laminate of relative permittivity 4.4, solved as a lumped circuit and as a line. The two agree to 0.030% at 18.0 MHz, where the track is one degree long, and differ by 30.1% at 650 MHz, where it is a tenth of a wavelength.

Every number above 600 megahertz on this page belongs to a network whose connections have no length. Kirchhoff’s own frequency puts a frequency on that assumption for any circuit of a given size, and the figure puts it on 2.2 centimetres: at 650 megahertz a track that long is a tenth of a wavelength, and a lumped solution of it is wrong by thirty per cent. So the T’s 814 and 643 megahertz, its 702-megahertz closure and above all its 2.24 gigahertz are statements about three switches and their connections all inside a couple of centimetres, and a good deal less for the gigahertz figure.

That makes the T’s advantage a length as well as a frequency, which is the kind of boundary the edges that are lengths collects: nobody chooses it at the schematic, it appears in no netlist, and it is set by whoever places the three switches. Three switches spread across a board do not form the node this essay depends on, because the half ohm that holds it is at the far end of a piece of track with an inductance of its own. What the lumped solution does establish is the mechanism and its exponents; it does not establish the gigahertz.

What a third switch trades

Put together, the T is a clear exchange rather than an improvement.

It buys independence from the source. The open path’s leak lands on a switch rather than on whatever drives the channel, so the isolation that a changeover loses to fifty ohms of source the T keeps, and a kilohm of source improves it.

It buys a second order. The leak crosses two open switches and rises forty decibels a decade, and the band it leaves closes as the first power of the tolerance, so its advantage over a lone switch grows tenfold with every decade of precision asked for.

It costs a doubled on-resistance, which doubles the lower edge and puts a fifty-ohm load at the edge of what the closed T can serve.

And it costs the capacitance it moved. The shunt switch’s off-capacitance loads the closed path exactly as a changeover’s open channel does, so as a carrier of waveforms the T is no better than a changeover, and the frequency at which it stops carrying them is set by the source impedance after all.

What none of this contains is the switches’ own charge. The offset that knows the signal measures the charge a switch dumps as it turns off, and a T turns three switches at every change of state, one of them in the opposite sense to the other two; the order in which they move, and whether the path is ever briefly shorted to ground through the shunt switch, are questions about timing that three passive numbers cannot answer. And the source is a resistor throughout, where a real driver is a source only below a frequency — which matters less to a T than to any other switch circuit here, because the T is the arrangement built to stop caring about the source.

Still open: where no load is good enough

The floor below any load. Every switch model so far has taken the off-resistance to be negligible beside the off-capacitance, and at direct current it is not. There a lone switch’s two edges are ninety-nine on-resistances and a ninety-ninth of the off-resistance, and they meet when the tolerance falls to the square root of the ratio of the two resistances: 70.71 parts per million for this part, at a best load of 7.071 kilohms, which is 13.79 bits of resolution. Below that no load at any frequency makes a lone switch ideal enough. Whether a T, whose open node is held by a switch, removes that floor or merely moves it is the next distinct question about switches, and it is a question about resolution with no frequency in it.

A shunt switch that is not the same part. Everything here built the T from three identical switches. A larger shunt switch has a smaller on-resistance, which holds the open node harder, and a larger off-capacitance, which loads the closed path harder. The two effects pull one size in opposite directions, so there is a shunt switch that makes the T’s band as wide as it can be, and it depends on the source and on the tolerance in ways this essay’s results already suggest.

The π, and the T with a line in it. Shunt, series, shunt is the other three-switch arrangement, and it places its capacitance differently again. And a T spread across a board is three switches joined by lines, which the lines field already knows how to solve; putting the two together would draw the length at which the half-ohm node stops being a node.

Part 4 on ideal switch

One argument about Ideal switch, and one of 8 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

Where 8 channels leak to: 91.0 MHz from buffered sources, 901 kHz from 50 Ω. computed by solving, not by drawing. The frequency at which the open channels of a multiplexer built from the 0.5 Ω, 100 MΩ, 5 pF switch leak 1% of the signal onto the shared output, against the impedance of the source driving the selected channel, into 1 MΩ. 2 channels: 637 MHz buffered, 6.30 MHz from 50 Ω; 8 channels: 91.0 MHz buffered, 901 kHz from 50 Ω; 16 channels: 42.4 MHz buffered, 420 kHz from 50 Ω. Each falls as the reciprocal of the source impedance plus the on-resistance, and the single switch's own band closes at 6.43 MHz. Where an open switch leaks to Part 3 — A lone switch has a band whose width no load can change, because its open state leaks into the load. In a multiplexer the seven open channels leak into a node the selected channel holds, so the load leaves the answer and the source takes its place: one per cent of leak at 91.0 megahertz from buffered sources and 901 kilohertz from fifty ohms, moving as the first power of the tolerance rather than the second. Adding the channels' capacitance into one forty-picofarad switch puts it at 804 kilohertz, near the fifty-ohm figure by coincidence and a hundred and thirteen times low for a buffered one. One switch is never better than 70.71 ppm; a T of three reaches 10 ppb only into 200 MΩ. computed by solving, not by drawing, at direct current. The worse of a switch's two errors — closed, the fraction the load fails to receive; open, the fraction it receives anyway — against the load, for one 0.5 Ω, 100 MΩ switch and for a T of three, from a buffered source. The lone switch is best at 7.07 kΩ, the geometric mean of its two resistances, where both errors are 70.71 ppm, 13.79 bits: no load does better. The T has no best load. Its worse error falls with the load towards Rₒₙ/(Rₒₙ + Rₒff) = 5 ppb, the square of the lone switch's resistance ratio rather than its root; it is within twice that from 200 MΩ, it passes the lone switch's floor only above 14.1 kΩ, and into 7.07 kΩ it is 141.4 ppm, worse than one switch. Solved on the network up to 1000 MΩ and continued, dashed, from the closed form it matches. The floor below any load Part 5 — A switch of half an ohm closed and a hundred megohms open is within one per cent of ideal for loads between two edges, and the edges close on each other as the tolerance tightens. At direct current they meet at 70.71 parts per million, into 7.07 kilohms: no load makes that switch better, which is 13.79 bits and a boundary with no frequency in it. A T of three such switches has no best load at all. Its error falls with the load towards five parts per billion — the square of the lone switch's resistance ratio rather than its root — and reaches ten only into two hundred megohms. Into the 7.07 kilohms that suited one switch, the T is worse than one switch.

Where this goes next

Where 8 channels leak to: 91.0 MHz from buffered sources, 901 kHz from 50 Ω. computed by solving, not by drawing. The frequency at which the open channels of a multiplexer built from the 0.5 Ω, 100 MΩ, 5 pF switch leak 1% of the signal onto the shared output, against the impedance of the source driving the selected channel, into 1 MΩ. 2 channels: 637 MHz buffered, 6.30 MHz from 50 Ω; 8 channels: 91.0 MHz buffered, 901 kHz from 50 Ω; 16 channels: 42.4 MHz buffered, 420 kHz from 50 Ω. Each falls as the reciprocal of the source impedance plus the on-resistance, and the single switch's own band closes at 6.43 MHz. Where an open switch leaks to A lone switch has a band whose width no load can change, because its open state leaks into the load. In a multiplexer the seven open channels leak into a node the selected channel holds, so the load leaves the answer and the source takes its place: one per cent of leak at 91.0 megahertz from buffered sources and 901 kilohertz from fifty ohms, moving as the first power of the tolerance rather than the second. Adding the channels' capacitance into one forty-picofarad switch puts it at 804 kilohertz, near the fifty-ohm figure by coincidence and a hundred and thirteen times low for a buffered one. A hold capacitor's band closes at 6.43 MHz, where a resistor's does. computed by solving, not by drawing. The switch of 0.5 Ω closed, 100 MΩ open and 5 pF across it, driving a capacitor. The lower edge is the smallest capacitance onto which the open switch feeds through no more than 1%: 495 pF far above 318 Hz, rising as the reciprocal of frequency below it because the leakage charges the capacitor. The upper edge is the largest capacitance the closed switch tracks to 1%, counted as a vector. The two meet at 6.43 MHz; a resistor on the same switch closes at 6.43 MHz, and counted as a magnitude the capacitor's band closes at 91.6 MHz. At the closure the closed switch's phase is 0.57 degrees. The width no load can change A switch's band was drawn against a load resistor and closed at 6.43 megahertz. Put a hold capacitor where the resistor was and the band changes axis and shape — a diagonal below 318 hertz, a floor of 495 picofarads above it — and stays exactly as wide: 1.8083 decades at 100 kilohertz for both loads, closing at 6.43 megahertz for both. Count the capacitor's error as a magnitude, as the resistor's always was, and the band appears to stay open to 91.6 megahertz. That extra room is 8.11 degrees of lag the count cannot see. Where a switch is a switch: a band, and the 6.43 MHz at which it closes. computed by solving, not by drawing. A switch of 0.5 Ω closed, 100 MΩ open and 5 pF across it is within 1.0% of being ideal only for loads between 49.5 Ω and 1.01 MΩ — 4.31 decades, and both edges are the same part. The upper edge is a frequency as well as a resistance, because the off-capacitance shunts the open switch: it falls a decade per decade above 318 Hz and meets the lower edge at 6.43 MHz, where the band closes and no load at all will do. Checked by scanning every load at 1.3 times that frequency and finding the best possible error to be 1.17%. A band rather than an edge Every other boundary in this collection is one-sided: a model is true below a frequency, or below an amplitude. A switch is a switch only for loads between 49.5 ohms and 1.01 megohms — bounded at both ends by the same part — and the upper end is a frequency as well as a resistance, so the band narrows as the frequency rises and shuts completely at 6.43 megahertz, above which no load resistance at all will do. 10.0 cm of track, solved as a lumped circuit and as a line. The two agree to 0.030% at 3.97 MHz, where the track is one degree long, and to 30.1% at 143 MHz, where it is a tenth of a wavelength. Above that the lumped model is not approximately right; it is describing a different object. Kirchhoff's own frequency The current law says the current entering a node equals the current leaving it at the same instant, which assumes the signal crosses the circuit in no time. It crosses at about two-thirds the speed of light, so the law has a frequency of its own — set by nothing but the physical size of the board.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Design tradeoffElectrical lengthLoadingLumped-elementModel rangeOff isolationOn-resistanceParasiticsQuadratureSource impedance