The resistor that is right in size and wrong in angle

The resistor that is only a resistor asked when a resistor stops being its own value, and answered with a V. A small resistance is ended by its lead inductance and a large one by its shunt capacitance, so the frequency at which the part is ten per cent away from its value rises and then falls across the resistance axis, and its point sits at 91.02 ohms for the package drawn. Along the way it rejected an obvious guess: the characteristic impedance of the two parasitics, √(L/C), 141.4 ohms, is where the two effects balance and is not where the size is flattest.

That essay measured one thing about the impedance, its size. An impedance is a complex number, and three voltages that close on one is this collection’s reminder that a complex number is an arrow with a direction as well as a length. A resistor’s arrow is supposed to point along the real axis. Many of the jobs a resistor does care more about that direction than about the length: a current shunt in a power meter, the reference arm of a bridge, a divider in a phase-sensitive measurement, a feedback resistor whose phase is part of a loop’s margin.

Asked about the angle, the same three elements give a different answer in almost every respect.

Two orders of departure #

The algebra that found the size’s flattest resistance makes the difference plain. Writing k = L/(R²C) and x = ωRC, the impedance normalised to R is

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

so the tangent of its angle is x(k − 1 + kx²), which has a term in the first power of frequency. Its size, expanded for small x, is 1 + (k² − 2k − 1)x²/2, whose first departure is in the second power.

A quantity that departs as the first power always leaves before one that departs as the second, and the exchange rate between them is fixed wherever one parasitic dominates. The size is then off by about half the square of the angle’s tangent. So where the angle reaches one degree, the size is off by 152 parts per million; where the size is off by a tenth of a per cent, the angle is already 2.56 degrees; where the size is off by one per cent, the angle is about eight.

The figure is that arithmetic, solved. At ten kilohms the angle reaches a degree at 695 kilohertz, when the size is still 152 parts per million low. The size does not reach one per cent until 5.67 megahertz, eight times higher, and by then the angle is 8.11 degrees. A ten-kilohm resistor measured by an ohmmeter that reads only the size is within a per cent of its value to beyond five megahertz and is more than a degree away from being a resistance above seven hundred kilohertz.

An impedance analyser reports the same defect in a different vocabulary. Read as a series resistance and a reactance, a resistor a degree off carries a reactance of 1.75 per cent of its resistance — a quality factor of 0.0175, if the part were a capacitor or an inductor — and the same part written two ways is the reminder that the series and parallel readings of such a reactance agree at one frequency only. A size-only reading is the one reading such an instrument offers that cannot see the defect at all.

One time constant sets the degree #

The first-order term has a coefficient with a meaning. The tangent of the angle is ω(L/R − RC) to first order, a frequency times a time constant, and the time constant has the lead inductance working one way and the shunt capacitance the other. The angle reaches a given value at the frequency where that product reaches its tangent: a degree at tan 1°/(2π|τ|).

For ten kilohms, τ is −4.00 nanoseconds — the capacitance winning, which is why the angle is capacitive — and tan 1°/(2π × 4.00 ns) is 694.7 kilohertz, which is the figure’s 695. For a small resistance the inductance wins and τ is positive.

A ten-milliohm shunt has a time constant of 800 nanoseconds and is a degree away from being a resistance at 3.47 kilohertz, in the audio band, while its size is within a per cent of its value to 28.2 kilohertz. For a shunt whose job is to report a current’s size that difference hardly matters. For a shunt whose job is to report power it is the whole matter, because power is the current times the voltage times the cosine of the angle between them, and an error in that angle is an error in the cosine. At a power factor of one the cosine is flat and a degree costs almost nothing; at a power factor of a tenth, where the angle is already 84 degrees, one more degree changes the reading by about seventeen per cent.

That is the same currency the ammeter that is not in the circuit found a current transformer charging in, where half a degree of phase error at fifty hertz costs eighteen per cent of a power reading at a power factor of 0.05. The shunt was the instrument that essay set against the transformer, and the ammeter that is a resistor priced a shunt by its burden and its offset. Neither account gave the shunt a phase error, and at 3.47 kilohertz this one has a degree of it.

The time constant L/R − RC is also the quantity a precision resistor for such work is specified by, and its vanishing is a familiar condition in a different place. A divider with two ratios found that a divider with capacitance in it has one ratio at low frequency and another at high, and that they agree when the two arms’ time constants are equal. The resistor here is the same statement inside one part: its inductive time constant and its capacitive one cancel, and when they do the angle has no first-order term.

A megohm is a degree out at 6.95 kilohertz. The resistor that is only a resistor put the same part’s ten-per-cent edge at 193 kilohertz and pointed out that a megohm feedback resistor in a transimpedance amplifier brings that corner with it whether or not it is drawn. Read as an angle, the corner a loop’s phase margin feels is twenty-eight times lower.

A divider made of two of them #

A divider’s ratio is two impedances divided, so its angle is neither resistor’s angle. To first order each resistor is its value times one plus jω times its own time constant, and the ratio’s angle comes out as ω times the top resistor’s share of the total times the difference between the two time constants, the bottom arm’s minus the top’s. Solved as a netlist of two three-element resistors in the same package, a ten-to-one divider of 9 kilohms over 1 kilohm has an effective time constant of 2.886 nanoseconds and a ratio one degree out at 963 kilohertz, where the first-order estimate says 962. The 9-kilohm resistor alone is a degree out at 772: the divider is better than its top arm, because the bottom arm’s time constant has the same sign and subtracts.

How much it subtracts depends on how alike the arms are. Two equal resistors have equal time constants, and their divider’s angle is zero to the arithmetic’s floor at a hundred megahertz, ten to the minus fifteen of a degree. A hundred-to-one divider of 99 kilohms over 1 kilohm keeps almost none of the cancellation — 71.6 kilohertz against 70.2 for its top arm alone — because a kilohm’s time constant is a hundredth of the top arm’s. And scaling a divider up by decades at a fixed ratio lowers the frequency by the same decades: 963 kilohertz at 9 kilohms over 1, 96.5 at 90 over 10, 9.65 at 900 over 100.

That is a divider with two ratios again, with the parasitics standing where the compensating capacitor stood, and it is the phase half of the divider, and the thing it does not know about: a ratio has no angle, and the two magnitudes chosen to make it bring one.

The guess that was right about the other question #

The size’s question was the resistance at which it is flattest, and the characteristic impedance was the wrong answer to that. It is the right answer to this.

At √(L/C) the time constant is exactly zero — L/R = RC — so the angle’s first-order term vanishes and what is left is the cubic, kx³. The angle stays under a degree to 730 megahertz, and for the first time on the axis the order of the two departures reverses: the size goes one per cent low at 283 megahertz, while the angle is still six hundredths of a degree.

The flattest angle is exactly where the two parasitics’ time constants are equal, which is why it is exactly the characteristic impedance with no other factor in it. The flattest size is at a value a factor of √(1 + √2) lower, because a second-order term collects contributions from both parasitics and their product in a different proportion.

At 91.018 ohms the size is flat to fourth order and stays within a per cent to 1.08 gigahertz. Its angle is not flat at all: the time constant is 51.5 picoseconds, and the angle reaches a degree at 53.9 megahertz, twenty times sooner. No resistance is flat in both. The two flat points are 1.554 apart, which is √(1 + √2), and between them every resistance trades one departure against the other.

At the six-millimetre body’s lumped ceiling, 138.8 megahertz, the two flat resistances read almost exactly opposite. The 91.02-ohm part is 2.959 parts per million from its value in size and 2.575 degrees from being a resistance; the 141.42-ohm part is 0.2428 per cent low in size and 0.00688 degrees in angle. Each is nearly perfect by one measure and visibly wrong by the other, at the highest frequency at which either measure means anything.

The widest bands, and the ceiling across them #

A flat point is a derivative. The widest band is a threshold, and for the size the two sit in different places: its widest ten-per-cent band sat near 93 ohms rather than at 91.02. The angle does the same.

The widest one-degree band is 1.13 gigahertz, at 150.69 ohms — above the flat point, because a threshold crosses the curve where the cubic term has grown rather than where the linear one vanishes. The widest one-per-cent band is 1.66 gigahertz at 95.806 ohms. Neither peak is at its flat point, and neither is at the other’s.

Both are above the flat line, and the flat line is not electrical. A six-millimetre body is one degree long, end to end at the speed of light, at 139 megahertz, and above that the three-element lumped model is not a description of anything. For the size that ceiling binds from 44 to 360 ohms. For the angle it binds only from 119 to 169 ohms: the angle’s V is so much deeper-sided than the size’s that only resistances close to √(L/C) reach the ceiling before their angle has gone. Everywhere else on the axis it is the angle, not the length, that ends the part.

Tighten the angle to a tenth of a degree and the widest band moves to 143.16 ohms, closer to √(L/C), as it should: the tighter the threshold, the smaller the frequency at which it is crossed, and the more exactly the first-order term decides. The range over which the length still binds shrinks to within about three ohms of the characteristic impedance, and at the size’s flat point the angle is out by a tenth of a degree at 5.40 megahertz.

Scanned separately on the same closed form and a finer grid of resistance, the widest band’s resistance walks towards the characteristic impedance as the threshold tightens: 162.31 ohms for three degrees, 150.82 for one, 145.52 for three tenths, 143.31 for a tenth and 142.22 for three hundredths, against 141.42, while the band itself narrows from 1.60 gigahertz to 348 megahertz. (The figure’s coarser grid puts the one-degree peak at 150.69.) A flat point is the limit of the widest band as the threshold goes to zero, and it is never the widest band at a threshold anybody specifies.

A longer body pulls the ceiling down to 33.3 megahertz and widens both ranges it binds, but the ordering survives: the angle’s range stays inside the size’s, from 71 to 280 ohms against 12 ohms to 1.7 kilohms. A resistor is limited by its length only near its own characteristic impedance, and in angle only very near it.

Where the angle is the specification #

A current shunt for power. The shunt’s time constant is almost all lead inductance — the current path through the element and the loop the sense connections enclose — and it sets a phase error proportional to frequency from the first hertz. Correcting it is possible exactly because it is a single time constant: a matching time constant in the sense path, the same structure as a compensated divider, cancels the first-order term. What remains is the cubic, which is the √(L/C) resistor’s behaviour arriving in a circuit.

A bridge. A bridge balanced in size at one frequency is unbalanced in angle at another if its arms have different time constants, and a phase-sensitive detector reads the angle. The bridge that is linear near one point measured the size of a bridge’s departure; an arm whose resistance is a decade away from √(L/C) carries an angle error that no adjustment of the size removes.

A termination. Fifty ohms sits between the two flat points, nearer the angle’s, and in this package it is a degree away from being a resistance at 19.84 megahertz while its size stays within a per cent to 164.6. A termination’s job is to present a real impedance to a line, and a reflection coefficient is a complex number: the degree is a reactive mismatch that the resistor at the wrong end and the termination essays beside it, which take the terminating resistor to be exactly its value, have no place for.

A resonant circuit’s loss. A related thread runs through the capacitor that is an inductor and the Q the components allow: parasitics that turn one kind of element into another. An angle error in a resistor is the first sign of the same thing — a resistance acquiring reactance — and at 8 degrees its reactance is a seventh of its resistance — the proportion of a capacitor with a quality factor of seven, turned upside down.

What the three elements leave out #

The package is the one the size was measured with: eight nanohenries and four-tenths of a picofarad, a small axial part. A surface-mount chip has less of both and moves every frequency here upward by a similar factor, and √(L/C) with them. What does not move is that the angle departs as the first power and the size as the second, that the exchange rate between them is fixed wherever one parasitic dominates, and that the flattest angle is exactly at the characteristic impedance.

The shunt capacitance is one element here. In a spiral-cut film resistor it is distributed along a helix, and the part is a short line whose angle need not follow the cubic at all near √(L/C). And a current shunt’s real inductance includes the loop its sense wires enclose, which is a property of the board rather than of the part — and which, as the time constant shows, is the quantity the shunt’s phase is actually specified by.

Where the real-resistor ladder goes next: the helix, the sense loop, and a resistor that is warm #

The resistor that is a line. A spiral film resistor’s capacitance and inductance are distributed along its cut, and near √(L/C) — the only place its angle is small enough for the distinction to matter — a distributed part and the three-element part should disagree. Solving the helix as a chain of distributed sections and finding the resistance at which its angle is flattest would say whether the characteristic impedance of a lumped model is still the answer for a distributed one.

The shunt whose inductance is a loop. A four-terminal shunt’s phase error is set by the mutual inductance between the current path and the sense loop as much as by the element. That makes the time constant a geometry, of the kind Kirchhoff’s own frequency and the essays on transmission lines measure, and solving it for a shunt and its sense connections would put a phase error on a layout decision.

A resistance that moves with its own current. A shunt carrying a large current warms, its resistance rises and its time constant falls with it, so its phase error is a function of the current being measured. That is a second nonlinearity in a part chosen for its linearity, and it would be measured against the thermal fixed points the magnetics field already solves.