A band rather than an edge

Every boundary this collection has drawn so far is one-sided. An ideal amplifier is ideal below a frequency. A small-signal model is small-signal below an amplitude. A lumped element is lumped below a frequency set by a size. In each case there is a good region and a bad one and a number between them, and the essay is about finding the number.

A switch is the first element here whose model fails at both ends of an axis, and it fails for two different reasons that belong to the same part. It is worth an essay for that shape alone, and then for what happens to the shape as the frequency rises.

Three numbers, and two of them are the same part #

An ideal switch is a short circuit and an open circuit. A real one is three numbers: an on-resistance, an off-resistance, and a capacitance across it that is there whichever state it is in. Half an ohm, a hundred megohms and five picofarads is an ordinary small analogue switch and is what is used throughout.

Both edges are measured on solved networks rather than computed from a divider expression. The closed state is a source, the on-resistance in series, and the load: the fraction of the drive that fails to arrive is what is measured. The open state is the same source with the off-resistance and the off-capacitance in parallel replacing the switch: the fraction that arrives anyway is what is measured. Both are nodal solves of three or four elements, and both are held against the closed forms they should equal.

The lower edge. With the switch closed, the load gets $R_L/(R_L + R_{on})$ of the drive, so it loses one per cent when $R_L = 99R_{on}$. Half an ohm gives 49.5 Ω, confirmed by solving at exactly that load and finding the loss to be 0.010 000 to a part in $10^{12}$.

The upper edge. With the switch open, the load gets $R_L/(R_L + Z_{off})$, so one per cent leaks through when $|Z_{off}| = 99R_L$. At direct current $Z_{off}$ is the hundred megohms and the edge is 1.01 MΩ.

Between them, 4.31 decades. That is a lot of room and it is why a switch is normally treated as ideal without comment: almost every signal a designer switches sits comfortably inside it.

What frequency does to the top of the band #

The off-capacitance is what makes this more than a table. Five picofarads across the open switch is in parallel with the hundred megohms, and it takes over at $1/2\pi R_{off}C_{off}$ — 318 Hz, which is low.

Above there $|Z_{off}| \approx 1/\omega C_{off}$ and falls a decade per decade, so the upper edge of the band falls with it. The band does not degrade gently at the top; it slides downwards at twenty decibels per decade while the bottom stays exactly where it is.

The last row is the point of the essay. The two edges meet, at

$$f = \frac{1}{2\pi C_{off}\,\frac{1-\epsilon}{\epsilon}\sqrt{\frac{1}{\epsilon^{2}}-1}\;R_{on}}$$

which for one per cent and this part is 6.43 MHz, with the off-resistance ignored — it moves the answer in the ninth figure. Above that frequency there is no load resistance whatever for which a switch is within one per cent of being a switch at both ends. A load small enough to make the off-isolation adequate is small enough that the on-resistance is losing more than one per cent, and the other way round.

The figure does not evaluate that expression to make the claim. It scans every load from a hundredth of an ohm to a hundred megohms at 1.3 times the closure frequency and reports the best error achievable — 1.17%, at the load that balances the two — and then scans again at 0.7 times and finds a load that manages 0.875%. At the closure itself it bisects for the balancing load and finds it wrong by one per cent in both states, to six figures. A closed form for a boundary is worth having; a boundary checked by asking every point on the other side is worth more.

What the square root in that expression is doing #

The product of $(1-\epsilon)/\epsilon$ and $\sqrt{1/\epsilon^2-1}$ is the interesting factor — very nearly $1/\epsilon^2$ — and it is a square for a reason worth stating.

Both edges are separated from the same quantity — the load resistance — by very nearly the same factor, once in each direction. The lower edge is $99R_{on}$. The upper is $|Z_{off}|/99.995$ rather than $|Z_{off}|/99$, and the difference is not rounding: above the corner the current through the off-capacitance is at right angles to the voltage across a resistive load, so the two add in quadrature, as $\sqrt{100^2 - 1}$, where a resistance would have added in phase. The edges meet when $|Z_{off}| = 99 \times 99.995\,R_{on}$. Tightening the specification from one per cent to a tenth of a per cent multiplies that by a hundred and divides the closure frequency by a hundred: the same part is not a switch above 63.7 kHz if a tenth of a per cent is wanted.

That is a much sharper penalty than the linear one people carry in their heads for “a bit more accuracy”, and it comes from the requirement being two-sided. A one-sided limit costs a factor of ten in frequency per decade of accuracy; this one costs a hundred.

The two failures do not look alike #

They are worth separating because the symptoms are completely different and a bench meets them in different circumstances.

Below the lower edge the switch is a resistor in series with the signal, and the fault is a gain error. It is stable, repeatable, frequency-independent below a megahertz and — the part everybody runs into — it varies with the signal voltage, because an analogue switch’s on-resistance depends on the gate-to-channel voltage. So the error is not just a gain error, it is a distortion, and it is worst in the middle of the supply range.

Above the upper edge the switch is a capacitor in series with the signal, and the fault is crosstalk. It is not stable: it depends on frequency and on the edge rate of whatever is on the other side. A switch that leaks nothing measurable at a kilohertz passes a visible spike from a nanosecond edge, and the mechanism is the same five picofarads.

The two also fail in opposite directions with load resistance, which is the whole shape of the band. Everything a designer might do to fix one makes the other worse. Buffering the output with a high impedance improves the on-state error and ruins the off-state isolation. Terminating in fifty ohms does the reverse.

What is on the other side of each edge #

Neither failure is a place where the model gets gradually worse and then stops meaning anything. Both have something definite on the far side, and naming it is what keeps this from being a warning.

Below the lower edge the switch is not a switch, it is a divider, and the divider is the object to reason about. Its ratio is $R_L/(R_L + R_{on})$ and everything the networks field says about dividers applies — including that its output impedance is the two in parallel, which is what the next stage sees and which at a ten-ohm load is 0.48 Ω rather than 10.

Above the upper edge the switch is not a failed open circuit, it is a capacitor, and what gets through it is a differentiated version of whatever is on the other side. That is why the symptom is a spike on an edge rather than a level: the current through the off-capacitance is $C\,dV/dt$, so a five-volt edge with a one-nanosecond rise pushes 25 mA momentarily through five picofarads, and into a ten-kilohm load that is a spike of the full supply.

Both statements are quantitative rather than cautionary, and both are what the region diagram means when read outside its own boundary.

Where the band is not wide enough #

A multiplexer in front of a converter. Eight channels share one output, so seven switches are off and their capacitances sit together on the summing node. It is tempting to add them — forty picofarads rather than five — and divide the closure frequency by eight, to 804 kHz. That arithmetic puts the leak in the wrong place. The seven open switches do not leak into the load; they leak into a node the one closed channel is holding, through its on-resistance and its source. From buffered sources the eight-channel limit is 91.0 MHz, fourteen times above this switch’s own closure rather than eight times below it, and from fifty-ohm sources it is 901 kHz — close to the naive figure only because 50.5 Ω happens to be about a hundred on-resistances. What sets it is a source impedance that no switch’s data sheet can state, and Where an open switch leaks to measures it.

A sample-and-hold. The hold capacitor is the load, so the load impedance is deliberately made enormous — which puts the operating point hard against the upper edge of the band, and the leakage through the open switch is what sets the droop. The whole design is a negotiation with one end of this figure.

A relay replacing an analogue switch. A relay has a milliohm of on-resistance, a gigohm off and a picofarad across, so its band is 0.099 Ω to 10.1 MΩ at direct current and it closes at 16.1 GHz. That is eight decades of room and it is why a relay is still the right answer for a precision multiplexer, at the cost of everything else about a relay.

Why this shape is rare #

It is worth asking why almost every other boundary in this collection is one-sided, since two-sided ones ought to be common.

The reason is that most parasitics act in one direction. A capacitor’s lead inductance can only add impedance; an amplifier’s finite gain–bandwidth can only remove gain; a lumped model can only lose phase. There is no second parasitic pulling the other way, so there is no second edge.

A switch is different because the ideal it is measured against has two states, and the two states are opposite extremes of the same axis. The part is measured against zero in one and against infinity in the other, and one real number cannot be close to both. Anything with two ideal states has this shape: a diode is another — it is a short one way and an open the other, and the limits field’s diode-model essay is the same argument with a current axis instead of a resistance one — and so is a comparator.

The general form is worth stating because it says where to look for the next one. A model with two ideal limits has a band and not an edge, and the band’s width is set by the ratio of the two non-idealities. For the switch that ratio is $R_{off}/R_{on} = 2\times10^{8}$, and the band is its square root’s worth of decades either side of the geometric mean — 7.07 kΩ here, which is where a designer should put the load if there is any choice.

Where a designer actually sits #

The geometric mean of the two direct-current edges is 7.07 kΩ, and that is the load at which a switch has the most room on both sides — 2.15 decades of margin either way at direct current. It is not a number anybody quotes and it falls straight out of the shape: a two-sided band has a middle, and the middle is the geometric mean because both edges are ratios.

More usefully, it says what to do when the band is too narrow. Moving the load towards the middle buys margin on the side that is short and spends it on the side that has some, and the exchange rate is one for one in decades. A ten-kilohm load at a hundred kilohertz is already outside the upper edge; dropping it to a kilohm puts it back inside with half a decade to spare, at the cost of taking the on-state error from 0.005% to 0.05% — still forty times inside the lower edge.

That is the whole design procedure for the region, and it fits in a sentence because the region is two straight lines on logarithmic axes.

What this essay does not claim #

That the switch model has a gate in it. It does not. The on-resistance here is a fixed half ohm, and a real analogue switch’s on-resistance varies by a factor of two or three over the signal range, which turns the lower edge into a distortion specification rather than a gain one. Modelling that needs a device rather than a resistance, and the semiconductors field is where it would go.

That the off-resistance matters much. It sets the upper edge only below 318 Hz, and almost no switching happens there. The number in the specification that a designer should read is the capacitance; the hundred megohms is a leakage specification in disguise and belongs with the sample-and-hold droop rather than with the isolation.

That the one per cent is fundamental. It is a criterion, and the essay is partly about how sharply the closure frequency depends on it — a hundredfold per decade. Anything quoted here is quoted with its criterion beside it for that reason.

That the two edges are independent. They are not: they are both properties of one part, and choosing a switch with a lower on-resistance almost always means a larger device and therefore more off-capacitance. The product $R_{on}C_{off}$ is roughly a constant for a given process, and that product is exactly what sets the closure frequency. So the band’s width in decades at direct current can be bought, and the frequency at which it shuts largely cannot.

The other boundaries this one is drawn against #

A two-sided boundary is unusual in this collection and it is worth naming the one-sided ones it is drawn against. Every model has an edge puts four of them on one frequency axis, and every one of those is a ceiling with nothing underneath it. The divider, and the thing it does not know about is the nearest relative — a divider is wrong when its load is too small and never when it is too large — and The capacitor that is an inductor is the other kind again, a component that stops being itself above a frequency and is exactly itself below one. What makes a boundary two-sided is that two mechanisms bound the same quantity from opposite directions, and a switch is the clearest case in the collection.

Why the region is a region #

The optimum inside this band is the same shape as two other results in the collection. The ammeter that is a resistor has an interior optimum for the same reason — two errors, one rising and one falling with the same parameter — and its answer contains neither the resistance nor the current. The edge that is a region is where the collection’s other boundaries are asked whether they are lines or bands, and most of them turn out to be narrow bands.

How the edges are measured #

Both edges are solved at the edge value itself — the closed switch losing 0.010 000 of the drive at 49.5 Ω, the open one passing 0.010 000 at 1.01 MΩ — rather than derived from a divider and quoted.

The upper edge is solved where the open switch is a capacitance, not only at direct current. At direct current an edge of $R_{off}/99$ is exact, and for as long as that was the only place the edge was checked, an upper edge of $|Z_{off}|/99$ passed at every frequency — and put the closure at 6.50 MHz. At 45.2 kHz the solved edge is 7.04 kΩ and passes exactly one per cent; the in-phase edge is 7.11 kΩ and passes 1.00998 per cent. The difference is small and it is the whole difference between an edge that was solved and one that was assumed.

The closure frequency is checked by scanning every load, not by evaluating the expression that produced it. At 1.3 times the closure the best achievable error over twenty-four decades of load is 1.17%, and at 0.7 times a load exists that manages 0.875%. Those two together are the claim, and at the closure itself the balancing load is bisected and is wrong by exactly the stated fraction.

The scan stops at a hundred megohms and the reason is recorded in the figure, because it is not tidiness. A one-ohm switch feeding a gigohm is a resistance spread of $2\times10^9$, and the site’s own current-law tolerance refuses the solve there — which is the singular-network essay’s measured number turning up in an unrelated figure and is the kind of coincidence worth writing down rather than working around.

And the closure frequency scales with the off-capacitance across the slider’s range: a decade of capacitance for a decade of frequency, from 64.3 MHz at half a picofarad to 684 kHz at forty-seven.

The other two-sided boundaries, and what makes a boundary two-sided #

A band bounded at both ends by the same part is unusual enough on this site to be worth comparing with the three other instances, because between them they say what produces one.

A resistor made of a clock is the closest relative and has three edges rather than two: a capacitor ratio under 0.0201, a signal below half the clock, and a clock below the frequency at which the charge stops arriving. Two of those want the clock high and one wants it low, so the design is squeezed in one variable from both sides — and the squeezing is done by the same switch this essay is about, through its on-resistance at one end and its charge injection at the other.

Two requirements pulling one capacitor has a window in a parasitic resistance, and its two edges turn out not to be the same kind of thing at all: the lower one is a stability edge bisected at 939 milliohms and the upper one is not a stability edge but a transient optimum, so the “window” a data sheet prints is two measurements with one name.

Stable, and unstable with less gain is the strangest, because both edges are the same mechanism and the safe region is in the middle of an axis where every instinct says one direction is safe: a loop stable between $1.81\times10^6$ and $3.36\times10^7$, whose phase crosses −180° at three frequencies rather than one, and which is broken by turning its gain down.

What the four have in common is that in each case two different consequences of one component are being required simultaneously — an on-resistance and an off-capacitance here, a settling time and an image there, a zero’s placement and a step’s droop, a phase and a gain. A boundary with one side is a model failing in one way. A boundary with two is a component being asked for two things.