Concept

Loss tangent — where it appears

The ratio of the real to the imaginary part of a capacitor's admittance, which is the reciprocal of its quality factor. It can be written as a resistance in series with the part or as one across it, and the two are equal at one frequency and nowhere else — which makes the same measured loss imply two very different noise densities.

Named by 4 essays across 4 fields — each of them below, with the objects they name alongside it.

The charge that comes back: a 0.2% dielectric, 10 s shorted, read at 900 s. computed by solving, not by drawing. The capacitor is charged to 10 V until every relaxation is complete, shorted for 10 seconds, then opened and watched. It climbs back to 20.00 millivolts — 0.2000 per cent of where it was — and the shape is the finding: it is a straight line on a logarithmic time axis, gaining 0.097 per cent of the charging voltage per decade. There is no time constant after which it is over, because there is no single time constant: one branch of the model comes to equilibrium per decade, for as many decades as the dielectric has. A decade before the reading it was at 0.1033 per cent.

The capacitor that remembers

Charge a capacitor, short it for ten seconds, open it, and it climbs back to a fifth of a per cent of where it was. Nothing leaked and nothing was gained: some of the dielectric had not finished discharging. The same defect measured as an admittance says the part is 0.593 per cent more capacitance at a tenth of a millihertz than at a kilohertz, and measured in a sample-and-hold it says a millisecond of hold costs a hundred parts per million — thirteen bits, on a part specified at nothing.

transients · Dielectric absorption

What is warm in a capacitor, by its two loss models. computed by solving, not by drawing. One 100 nF capacitor of loss tangent 0.02, written as a 3.183 Ω resistance in series with it and as a 7.958 kΩ resistance across it — the pair that converts exactly at 10.0 kHz and nowhere else. The noise at the terminals is 4kT times the real part of the impedance, so the two models give the same density at 10.0 kHz and are 33.0 dB apart at 100 Hz and 40.0 dB apart at a megahertz. The dots are the same quantity computed the other way — the resistor split out of the netlist, a source put in its place and the network re-solved — agreeing to 3.3e-16. The reactance itself contributes nothing at either end: a lossless capacitor has no real part and is not warm.

Only the real part is warm

Johnson's 4kTR is the special case of a statement about impedances: the noise across any passive two-terminal in equilibrium is 4kT·Re{Z}, so a reactance contributes nothing however large it is. That turns a modelling convenience into a noise figure. A 100 nF capacitor of loss tangent 0.02 written as 3.183 Ω in series and as 7.958 kΩ across it — the pair that converts exactly at 10 kHz — gives 0.226 and 10.10 nV/√Hz at 100 Hz, 33 dB apart, and 40 dB apart the other way at a megahertz.

noise · Johnson noise

At the same rise time a lossy edge draws 2.9–4.0 times more from a balanced via than a clean one, and the square law survives. computed by solving, not by drawing. The largest excursion of the reflection from a balanced via and from a 0.5 pF capacitive via against the 10–90 rise time of the edge that reaches them: behind 5 to 50 cm of lossy line, with a 20 ps edge sent in, and from clean raised-cosine edges of the same rise times on a lossless line. Behind the loss the balanced via's reflection falls as the −1.97 power of the rise time and the capacitive one's as the −0.82. Against clean edges of the same rise time the lossy edge draws 2.93 to 3.97 times as much from the balanced via and 1.07 to 1.54 times as much from the capacitive one.

The edge the loss delivers

A balanced via's reflection falls as the square of the edge's rise time, so the loss in front of it should hide it. A 20 ps edge sent down 20 cm of lossy line arrives with a 10–90 rise time of 194 ps, and the via's reflection falls to 9.1 mV. But a clean edge with that same 194 ps rise time draws only 2.29 mV from the same via. The lossy edge excites it 3.97 times as much, because loss makes an edge steep in the middle and slow at both ends, 3.07 times steeper at its centre than a raised cosine of the same 10–90 time. Across 10 to 50 cm the square law survives, an exponent of −1.97. It survives shifted: the 10–90 time overstates, by a factor of three or four, how slow the edge is to anything that reads its curvature.

lines · Reflections

The Q of a 10 mH coil with 1.2 Ω of winding and 8 pF across it, with cores whose own Q is 1000, 300, 100 and 50. computed by solving, not by drawing: the core's loss a resistance ωL·Qc across the inductance, constant loss tangent, the coil solved at each frequency. Without a core loss the Q peaks at 11340 at 325 kHz, the self-resonance over √3. With the cores: 825.6 at 136 kHz for Qc = 1000, 274.9 at 94.0 kHz for Qc = 300, 95.9 at 66.2 kHz for Qc = 100, 48.7 at 52.8 kHz for Qc = 50. The peak falls to just under the core's own Q for a poor core and further under it for a good one, and moves down from the self-resonance over √3.

The Q a core allows

A coil's Q, limited by its winding and its own capacitance, peaks at the self-resonance over √3. Put a core with a loss of its own across the inductance and the peak falls and moves: with a core whose own Q is 100, the 10 mH coil that peaked at 11,340 near 325 kHz peaks at 95.9 near 66 kHz. The peak sits where x³ = a/2 — x the frequency over the self-resonance, a the core's Q over the winding's — and reaches the core's Q less three times the two-thirds power of a/2 of it. At that peak the core carries 97 per cent of the loss, so a better winding hardly moves it. A core whose loss is a fixed resistance instead puts the peak at the geometric mean of two corners, as high as half the square root of the core's resistance over the winding's; a loss tangent that rises with frequency drags the peak down to the material's corner.

magnetics · Real inductor

Named alongside it

The objects these essays reach for when they reach for this one.

Equivalent series resistance Model range Core Design tradeoff Dielectric absorption Dispersion Eddy current Johnson noise Marching Parasitics Quality factor Real capacitor

All concepts