The resistor that is only a resistor

This collection has two essays about a component that stops being itself. A capacitor is a capacitance below a frequency its own lead inductance decides, and above it the part is an inductor. An inductor is an inductance below a frequency its own winding capacitance decides, and above it the part is a capacitor. Both are drawn, both have their edges computed from the part’s own numbers, and both belong to a thread called the parasitic is the component.

The third member of the family is missing, and it is the one everybody in the subject holds to be uninteresting. A resistor is a resistance. It is the element with no reactance in it, the one that appears in every netlist above without a story, and the last thing anybody checks when a measurement is wrong at ten megahertz.

It is also the only one of the three whose edge is not monotonic in its own value, and that is what earns it a figure.

Three elements, and two of them are not in the symbol #

The model is the smallest one that can be wrong in both directions. A resistance R; a capacitance $C_p$ across it, which is the end-to-end capacitance of the body and its two terminations; and an inductance $L_s$ in series, which is the leads and the current path through the element itself. Eight nanohenries and four-tenths of a picofarad here, which is a small axial part.

Both parasitics are read off a solved network rather than from a formula. The three elements are assembled into a netlist, an amp is driven into the terminal, and the voltage is read — the same impedanceBetween the whole site uses. The closed form for the same three elements,

$$Z(j\omega) = j\omega L_s + \frac{R}{1 + j\omega R C_p}$$

is computed beside it and the two agree to $3.3\times10^{-16}$ across eight decades of frequency, which is the site’s habit of measuring twice and is here doing no work beyond confirming that nothing is mis-stamped.

Which parasitic gets there first, and when #

The two parasitics do not compete. They act at opposite ends of the resistance scale, and which one decides the edge is settled by R alone.

A large resistance is shunted. At ten megohms, $1/\omega C_p$ falls to a tenth of R by nineteen kilohertz, and the part’s impedance drops below its own value there. The lead inductance is irrelevant: at nineteen kilohertz eight nanohenries is a milliohm.

A small resistance is added to. At ten ohms, $\omega L_s$ reaches a tenth of R at ninety-two megahertz and the impedance rises. The shunt capacitance is irrelevant: at ninety-two megahertz 0.4 pF is four kilohms across ten.

The hundred-ohm row is the one that does not fit the story, and it is the story’s point arriving early. At a hundred ohms neither parasitic dominates: the inductance is trying to lift the impedance and the capacitance is trying to drop it, and over a wide band they very nearly cancel. That row is not a resistor with unusually small parasitics — it has exactly the same eight nanohenries and four-tenths of a picofarad as the ten-megohm one — it is a resistor at which the two parasitics are of comparable importance and of opposite sign.

The right-hand half of that table is a clean 1/R: every decade of resistance costs a decade of bandwidth, measured as $-1.000$ decades per decade between a hundred kilohms and a megohm. The left-hand half is the mirror image, $+1.000$ decades per decade between three and thirty ohms. Two straight lines with opposite slopes, which is the shape that has a maximum between them.

Where the point of the V is, which is not where it looks #

The obvious guess for the best resistance is $\sqrt{L_s/C_p}$ — the characteristic impedance of the part’s own two parasitics, 141.4 Ω here. It is a good guess and it is wrong, and the reason is worth following because it is a two-line calculation that returns a number nobody would guess.

Write $k = L_s/(R^2C_p)$ and $x = \omega R C_p$. Then the impedance normalised to R comes out as

$$\frac{Z}{R} = \frac{1 + jx(k - 1 + kx^2)}{1 + x^2}$$

and expanding the magnitude for small $x$ gives

$$\left|\frac{Z}{R}\right| = 1 + \frac{k^2 - 2k - 1}{2}\,x^2 + O(x^4).$$

The second-order term vanishes at $k = 1 + \sqrt{2}$, so the impedance is flat to fourth order at

$$R = \sqrt{L_s/C_p}\ \big/ \sqrt{1 + \sqrt{2}}$$

which is 91.02 Ω for this package rather than 141.4. The figure does not take that on trust: it measures the curvature at a small $x$, bisects for the resistance at which it changes sign, and gets 91.017 9 Ω against a closed form of 91.017 97 Ω — six figures.

At $\sqrt{L_s/C_p}$ itself the departure is still second order and the coefficient is exactly $-1$: the impedance falls as $x^2$, and the part’s ten per cent band is 941 MHz. At 91 Ω the quadratic term is gone and the band is 2.00 GHz, more than twice as wide from a resistance one and a half times smaller.

Two things about that result deserve saying plainly. The condition contains no frequency, which is what makes it a property of the part. And it is the same maximally flat condition that names the Butterworth filter three fields over — a coefficient set to zero so that the first surviving term is as high an order as possible — arriving here in a component that is supposed to have no response at all.

The peak of the band is not the flat point either #

There is one more turn, and it is the kind this site keeps meeting.

The flat resistance is defined by a derivative: it is where the infinitesimal departure vanishes. The band is defined by a threshold: it is the first frequency at which the departure reaches ten per cent. Those are different questions, and they have different answers. Scanning the resistance axis for the widest ten per cent band puts the peak at about 93 Ω rather than at 91.02, and moving the threshold moves the peak again.

This is exactly the distinction the transients field met, where a settling time’s optimum is decided by which excursion is the last one outside the band rather than by any derivative, and where the answer therefore jumps as the band is changed. A criterion with a number in it is not a limit of the criterion without one.

It also produced a defect worth recording, because the first version of this figure had it. The departure frequency was found by bisection, which assumes one crossing. Near the flat resistance there are three — the impedance dips back inside the ten per cent band before leaving it for good — so the bisection reported whichever branch its bracket straddled and put a spurious 3.2 GHz spike on the curve at 112 Ω. It is found by scanning upward now, which is what “the frequency at which it stops being a resistance” actually means.

The two ends of the resistance axis fail for different reasons, and drawing them together is the only way to see that the boundary has a minimum in the middle rather than running monotonically one way.

The ceiling, which is not electrical #

The V has a lid on it, and the lid belongs to this site rather than to the part.

A six-millimetre body is one degree long — in the sense the limits field means, a phase shift of one degree across the component at the speed of light in air — at 139 MHz. Above that, “the impedance of this resistor” is not a well-formed quantity: the current entering one terminal is not the current leaving the other at the same instant, and the three-element lumped model that produced every number above has stopped being a model of anything.

So the useful band of a resistor is the smaller of two numbers. Below about fifteen ohms and above about 1.4 kΩ, the parasitics bind and the lumped model is still good where they do. Between those two resistances, the part’s length binds first, and the whole point of the V — the 2.00 GHz at 91 Ω — is unreachable.

That is a boundary of a shape this site has drawn before and never quite in this direction: two independent limits, of different kinds, crossing, so that which one is quoted depends on where on an axis it is read. The slider on the second figure is the length of the body, and shortening it from twenty-five millimetres to one lifts the ceiling from 33 MHz to 833 MHz and narrows the band of resistances the ceiling binds over from 3.6 Ω–5.8 kΩ to 66 Ω–160 Ω.

Where this is met #

A megohm feedback resistor. A transimpedance amplifier with a megohm in the loop has a feedback network whose own corner is 193 kHz here — before the amplifier, before the photodiode, before anything anybody designed. It is the reason that topology needs a deliberate feedback capacitance: there is one whether it is drawn or not, and the alternative to choosing it is inheriting it.

A current-sense shunt. A ten-milliohm shunt is on the far left of the axis, and its edge is about a megahertz for eight nanohenries — a resistance whose whole purpose is measuring a fast current, departing from its own value at a frequency in the band that current occupies. The sense-lead geometry is not a detail there; it is the specification.

A terminating resistor. Fifty ohms into a transmission line sits close to the flat point of its own package by accident, which is one of several reasons terminations behave better than the rest of the parts on a board. It is the only value in ordinary use anywhere near the bottom of the V.

A gain-setting pair in an amplifier. Two resistors of the same ratio can be chosen in kilohms or in megohms, and the frequency field’s first essay is about what that choice does to the loading. This is the other half of it: the megohm pair brings a corner at a couple of hundred kilohertz with it, and the kilohm pair does not. A ratio has no edge; the two numbers that make it do.

What this essay does not claim #

That the parasitic values are universal. Eight nanohenries and 0.4 pF are one part in one package. A surface-mount chip has perhaps a fifth of the inductance and a third of the capacitance, which moves $\sqrt{L_s/C_p}$ to a hundred and ten ohms and lifts every frequency on the axis by about a factor of three. What does not move is the shape of the curve, the closed form for the flat resistance, or the existence of the ceiling.

That the shunt capacitance is a single lumped element. It is not, particularly for a spiral-cut film resistor, where the body has a distributed capacitance along a helix and behaves as a small transmission line of its own. The three-element model is the first one that is wrong in both directions, which is the criterion this site uses for a model; it is not the last word.

That a resistor has no self-resonance. It has one, at $1/2\pi\sqrt{L_sC_p}$, which is 2.81 GHz here and above the lumped ceiling for every body size the slider offers. That is why the figure draws a departure rather than a resonance: the resonance is real and unreachable, and drawing it would be drawing a feature of the model rather than of the part.

That this is the same claim as the capacitor essay. It is the same construction and a different result. The capacitor’s edge is monotonic in its value and this one is not, and the whole content here is the interior maximum and the closed form for where it is.

The other two components with a band #

A resistor that stops being one above a frequency completes the set. The capacitor that is an inductor and The inductor that is a capacitor are the other two, each ending at its own self-resonance. The Q the components allow is where all three ceilings become one number, and Kirchhoff’s own frequency is the boundary above which none of them is a component at all.

The gate #

Two routes to the impedance, the solved three-element netlist and the closed form for the same three elements, required to agree to $10^{-11}$ across eight decades. They come out at $3.3\times10^{-16}$.

The flat resistance is bisected on the measured curvature and asserted against $\sqrt{L_s/C_p}/\sqrt{1+\sqrt{2}}$ to a part in $10^5$. The closed form is in the prose so a reader can check it; the number in the figure came from measuring.

The two asymptotic slopes are fitted, not assumed: $+1.000$ decades per decade between three and thirty ohms, $-1.000$ between a hundred kilohms and a megohm. Non-monotonicity is the whole claim of the figure, and asserting the two opposite slopes is what makes it a measurement rather than a description of a picture.

The ceiling is asserted to cut the curve twice, at every body size the slider offers, because “the useful band is set by the length between two resistances” is only true if the crossing exists — and at one millimetre the two crossings are 66 Ω and 160 Ω, which is a narrow interval and still an interval.

And the widest band is asserted to be above the lumped edge. That is the finding, and if a package were ever built for which it were false the assertion should fail rather than the caption quietly change meaning.

The third member of a family, and what the other two do #

The two components this essay is a completion of are measured in the same field and each has a shape worth setting beside this one.

The capacitor that is an inductor follows $1/2\pi fC$ for four decades and then turns round and climbs: above 14.5 MHz a hundred nanofarads is an inductor, and a decade past that its impedance is ninety-nine times what its capacitance predicts — all of it caused by about a nanohenry of lead and via that nobody chose and nobody drew. The inductor that is a capacitor is the same measurement with the components exchanged, and it comes out with the same structure and a different number: the ten-per-cent departure from $\omega L$ sits at $f_0/3.317$, and it sits there at every winding capacitance and every inductance tried, so the shape of the departure belongs to the resonance rather than to either part.

Read against those two, what is unusual here is not that a resistor has an edge but that its edge is not monotone in its own value. A capacitor’s self-resonance falls as the capacitance rises and an inductor’s falls as the inductance rises: in both cases more of the intended quantity is less bandwidth, monotonically, and a designer trading one against the other knows which way to move. A resistor has parasitics of both kinds in parallel and in series, so a large value is limited by its own shunt capacitance and a small one by its series inductance, and the best case is an interior optimum — 91.02 ohms for the package drawn here — with no design freedom in it at all, since the resistance is the quantity the circuit needed.

That is a boundary of a kind the collection has met once before in another field. The floor a circuit has finds a source resistance at which an amplifier’s two noise generators sum to a minimum — 6.67 kΩ, the ratio of the two, and not the resistance that transfers maximum power. Both are interior optima in a resistance, both are set by the ratio of two parasitic quantities rather than by either alone, and in both cases the value the circuit actually needs is decided somewhere else entirely.

The value is chosen for something else #

That last point is what makes this essay’s boundary awkward in a way the capacitor’s and the inductor’s are not. A designer who needs a hundred nanofarads and finds its self-resonance too low can fit two of fifty, or a different package, or a different dielectric — the capacitance is the requirement and the geometry is negotiable. A designer who needs ten megohms needs ten megohms, and this essay says that a ten-megohm resistor stops being one at 19 kilohertz whatever is done about the package.

Which is why the resistances the rest of this collection chooses for other reasons are worth checking against the curve. The floor a circuit has puts an optimum source at 6.67 kΩ, comfortably inside the flat region. Where the trouble is at the input puts a megohm in the feedback path of a photodiode amplifier, which is three decades above it and near enough to the top of the sweep that the shunt capacitance is part of the design — that essay’s feedback capacitance is “often most of it” the resistor’s own, which is this essay’s parasitic named from the other side. And the current that does not reach the input works with gigohms and teraohms, where the parasitic is not a shunt capacitance across the part but the board it is standing on: a teraohm across a board from a fifteen-volt rail is fourteen millivolts of error through a gigohm source, and a humid morning takes that resistance down two decades.

So the family this essay completes has three members and the third one behaves differently from the other two: its edge is not monotone in its own value, the value is not usually negotiable, and above a few megohms the dominant parasitic stops being a property of the component at all.