The same part written two ways

Open two capacitor datasheets and the same physical fact is written down two different ways. One gives an equivalent series resistance at a stated frequency; the other gives an insulation resistance and a dissipation factor, which is a resistance in parallel. Every handbook prints the pair of expressions that converts one into the other. Almost none of them prints the frequency the conversion is true at, and the conversion is true at exactly one.

The two models #

A hundred nanofarads with a hundred milliohms in series with it. That is a description of a real part: the plates hold the charge, and the leads, the plates’ own resistance and the dielectric’s losses all appear at low frequency as a resistance the current has to pass through.

The other description puts a resistance across the same capacitance. It is an equally reasonable picture of the same physics — dielectric loss is often more naturally a conductance than a series resistance — and there is no experiment at one frequency that distinguishes them, which is exactly why both are used.

At a hundred kilohertz the series pair has an impedance of $1.5917 - j15.915$ Ω. The parallel pair that matches it is not obtained here by substituting into an expression. It is read off that number: the admittance is one over the impedance, its real part is a conductance and its imaginary part is $\omega C$, and those two are the definition of what a parallel model is.

That gives 2.533 kΩ across 99.996 nF. The handbook expressions — $R_p = R_s(1 + Q^2)$ and $C_p = C_s Q^2/(1 + Q^2)$ — give the same two numbers to nine figures, and they appear in this essay only in that role: as a second route that has to agree with what the solve produced.

Exact means exact #

At the design frequency the two networks are swept independently and compared. The difference between them is $8.7\times10^{-19}$ of the impedance.

That is not a tolerance and it is not an agreement. It is the number that comes out when two floating-point calculations produce the same value and the subtraction leaves whatever was in the last bits — the same kind of quantity the previous field’s Thévenin comparison returns, and for the same reason. There is no residual physics in it to measure.

And nowhere else #

Move away from a hundred kilohertz and the two part company. How fast is the whole content of the figure, and it turns out to have a closed form.

Writing $\alpha$ for the frequency as a multiple of the design one, the fractional difference between the two impedances is

$$\left|\alpha - \frac{1}{\alpha}\right| \Big/ Q$$

which is derived by expanding both impedances for large Q and is checked here against the two solved networks rather than substituted for them. At Q = 159 it is right to four figures; at Q = 8, the lowest the slider goes, it is 3.2% high, which is what an asymptotic expression does when it is asked about a case that is not asymptotic.

Two things follow immediately, and both are visible in the figure.

The cost is symmetric in the logarithm. $|\alpha - 1/\alpha|$ is unchanged by replacing $\alpha$ with $1/\alpha$, so the substitution is exactly as wrong an octave below the design frequency as an octave above it. That is not obvious from either model — the series pair’s impedance falls with frequency and the parallel pair’s does something else entirely — and it is checked to three parts in a million.

And the band is proportional to Q. Setting the expression to a hundredth and solving the resulting quadratic gives the edges, which agree with the bisected ones to a per cent.

The bottom row is the one to keep. A part with a Q of eight — an electrolytic at a hundred kilohertz, a small ceramic on a long lead, almost any capacitor near its own self-resonance — is interchangeable between the two descriptions over four per cent of frequency. Quote its parallel-model numbers at 100 kHz and use them at 150 and the answer is eighteen per cent out.

The departure at a stated distance #

The band is one way to read the closed form. The other is to fix a frequency ratio and ask what the substitution costs there, which is the question a designer actually has: the numbers were taken at 100 kHz, the circuit runs at 500 kHz, and how wrong is the conversion.

Every entry in the first column is $1.5/Q$ and every entry in the second is $4.8/Q$, which is what $|\alpha - 1/\alpha|$ evaluates to at those two ratios. The last row is the only one where the closed form is visibly off — 18.3% measured against 18.9% predicted — and the direction is the useful one to know: the expression is pessimistic at low Q, so a designer using it is not being told the error is smaller than it is.

The bottom-right entry deserves saying out loud. A part whose loss was characterised at 100 kHz, used at 500 kHz, with the model swapped from series to parallel along the way, has an impedance half as large as the arithmetic says — and every step in getting there was a step somebody would describe as standard practice.

The dissipation factor, which is the same number again #

There is a third way the same fact is written down, and it collapses onto the first two.

The dissipation factor is the ratio of the real part of the impedance to the imaginary part — equivalently the tangent of the angle by which the current in the part falls short of ninety degrees from the voltage. At the design frequency it is $1/Q$: 0.00628 for the hundred-milliohm part above, which a datasheet would print as 0.6%.

Both models give that same number at that frequency, because both models are that impedance there. Either side, they give different ones — the series model’s dissipation factor rises in proportion to frequency and the parallel model’s falls in inverse proportion — and the ratio between the two is $\alpha^2$. At five times the design frequency that is a factor of twenty-five between two descriptions of one part.

So a dissipation factor with no frequency attached is not an incomplete specification in the way a resistance with no frequency attached is incomplete. It is an ambiguous one: the number is a different property of the part depending on which two-element model the person who measured it had their instrument set to, and the two answers diverge as the square of how far the reader is from where they were standing.

What Q is doing here #

It is worth saying why one number controls this, because the reason is not that Q is a general figure of merit.

At the design frequency the reactance is Q times the series resistance, by definition. The impedance is therefore almost entirely reactive, and the two models agree about both parts of it. Move off in frequency and the reactive parts of the two models move differently — the series pair’s reactance goes as $1/\omega$, the parallel pair’s admittance goes as $\omega$ — but the discrepancy that introduces is a fraction of the resistance, which is $1/Q$ of the impedance. So the departure is a small thing divided by a large one, and the large one is Q.

That also explains the shape. At small $\alpha$ the term $1/\alpha$ dominates and the error grows in proportion to how far below the design frequency the model is used; at large $\alpha$ the term $\alpha$ dominates and it grows in proportion to how far above. The minimum between them is not a minimum of a smooth curve — it is a zero, because at $\alpha = 1$ the two are the same network.

Where this bites #

Reading a filter’s loss from the wrong model. A dissipation factor quoted at a kilohertz, used for a part running at a megahertz, is a parallel resistance a thousand times off its correct value at the working frequency. Whether that matters depends on what the resistance is doing — in a timing network it is a leakage and irrelevant; in a resonator it is the Q.

Simulating with the wrong one. A simulator asked for a capacitor with a parallel resistance builds exactly that, and it is the right network only at the frequency the numbers were converted for. The series model is the safer default at high frequency and the parallel one at low, and the crossover is the design frequency itself.

Measuring on an instrument that has already chosen. An impedance analyser reports Cs and Rs, or Cp and Rp, at a frequency the operator sets, and the two sets of numbers are different numbers, not two displays of one. An instrument set to the parallel model reporting 2.533 kΩ and one set to the series model reporting 0.1 Ω are agreeing perfectly.

And comparing two parts characterised at different frequencies. This is the one that produces arguments rather than errors. Two capacitors, one specified with a series resistance at 100 kHz and one with a dissipation factor at 1 kHz, cannot be ranked by putting both through the conversion — each conversion is exact only at its own frequency, and the two frequencies are two decades apart. What can be compared is the impedance of each at the frequency the circuit runs at, which requires knowing something about how the loss varies with frequency that neither datasheet says. The honest answer is usually that the two parts have not been characterised comparably at all.

The band, and how much of it there is #

One number is worth stating in the direction a designer would ask for it. Setting the closed form to a hundredth gives a quadratic in $\alpha$, and for any Q worth having the answer is that the substitution holds up to about $0.01Q$ times the design frequency and down to the reciprocal of that. So the band, expressed as a ratio, is about $(0.01Q)^2$: a factor of 2.6 at Q = 100, a hundred at Q = 1000, and 1.1 — which is nothing — at Q = 10.

Turned round: to use a conversion across a whole decade of frequency and stay within one per cent, the part needs a Q of about 285. Ceramic capacitors of reasonable dielectric manage that at a hundred kilohertz; electrolytics do not manage it anywhere.

What this essay does not do #

It does not model the frequency dependence of the loss itself. Both resistances are constants here. In a real dielectric the loss is a function of frequency, usually a mild one, and where it is not the correct statement is that neither two-element model fits at all and a third element is needed.

It does not touch temperature. A capacitor’s series resistance moves substantially with temperature — an electrolytic’s by an order of magnitude between 20 °C and −20 °C — which moves Q, which moves every band above. Everything here is at one temperature, which is the general omission the edges that move with the room collects: the numbers are right and the condition attached to them was left off, and it is the same condition every time. Here it bites harder than usual, because the band this essay computes is a function of Q alone — so a part whose Q moves by an order of magnitude has a validity band that moves by three orders, and the conversion that was exact over an octave and a half at one temperature is exact over a few per cent at another.

It does not model the capacitance moving either, and for the part most likely to be described this way that is the larger omission. The capacitance that is not one number measures a class II ceramic at its rated voltage as 2.000 µF read as a slope, 5.814 µF as a charge average and 2.105 µF as a bridge reads it, on a part printed as ten — three answers to three questions, all correct. A quality factor is a ratio of a reactance to a resistance, so a capacitance uncertain by a factor of three is a quality factor uncertain by the same factor, and the band this essay computes from it moves accordingly. The coefficient that is about one reading closes the last escape: the third printed number is a coefficient of the one capacitance a bridge reports at zero bias, and two parts a bridge cannot tell apart differ by 1.80 at the voltage they are used at.

And it says nothing about which model is “correct”. They are the same part. What is being measured is the cost of writing it down one way and using it the other, which is a property of the arithmetic rather than of the capacitor.

Where the conversion is used #

A series form and a parallel form equal at one frequency is a conversion three later essays depend on. The Q the components allow is where the number this page converts becomes a resonator’s ceiling. The capacitor that is an inductor is where the series resistance is drawn as part of an impedance rather than as a quality factor. Two requirements pulling one capacitor is where the same resistance is asked for two incompatible values at once, and Resonance, and the bandwidth it sets exactly is what the quality factor is spent on.

The gate #

The equivalence is checked at the design frequency and asserted below $10^{-12}$, which is loose enough not to depend on rounding order and tight enough that no real disagreement could pass.

The handbook expressions are asserted against the solve, to nine figures, in that direction: the solve is the source and the expressions are the check.

The closed form is asserted with its regime stated. Half a per cent at Q ≥ 30 and five per cent below, because it is an asymptotic expression and a single tolerance across the whole slider would be either false at the bottom or vacuous at the top.

And the symmetry is checked separately, at half and twice the design frequency, to three parts in a million — because it is the claim a reader is least likely to expect and the one a plot on a logarithmic axis makes easiest to believe without checking.

Where the wrong conversion actually costs something #

A conversion exact at one frequency is harmless as long as the frequency it was made at and the frequency it is used at are the same, and three measurements in this collection are places where they are not.

Two requirements pulling one capacitor is the sharpest, because the quantity being converted is the whole design. A regulator’s loop is stabilised by the zero its output capacitor’s series resistance puts at $1/2\pi R_{esr}C$ — a series representation by construction — and the stability edge is bisected at 939 milliohms with the best transient sitting ten per cent inside the forbidden region. A part whose loss was characterised in the parallel convention, at the frequency a bridge happened to use, converts to a series resistance that is right at that frequency and wrong at the zero’s, by a factor this essay’s band tells how to compute.

The q the components allow is the one where the error compounds. The reciprocals of the component quality factors add, so a resonator’s figure is assembled from two numbers that were each measured somewhere and are each being used somewhere else — and the result sits below the smaller of the two, so an error in the worse component is an error in the answer at full weight.

And what actually fills a null turns the same quantity directly into a specification: a notch’s depth is its arm’s loss at twenty decibels per decade, exactly, so a quality factor taken from a parallel-model measurement at the wrong frequency is a null depth wrong by twenty times the logarithm of whatever this essay’s band says the conversion cost.

The common shape is worth naming. In all three the quantity is measured once, converted once, and used at a third frequency — and the conversion’s own condition travels with neither the measurement nor the use. A loss figure quoted with its frequency and its convention is two extra words and removes all of it.

That places this essay’s result in a family the collection keeps meeting. The constant that is a window is the same failure on a diode’s ideality factor — a number defined as a derivative, so it has a value at every current and no value anywhere, with eight one-decade fits to one curve returning factors from 1.23 to 1.98. A series resistance and a parallel one are two derivatives of one object taken under two conditions, and the condition here is a frequency rather than a current. In both cases the specification is exactly true about the measurement that produced it, exactly wrong about any other, and carries nothing that says which.