A common-emitter stage with 2.0 pF from collector to base
Drawn above at its default parameters, which is almost never how an essay calls it. A placement states the numbers that essay is arguing about, so the figure a reader meets is about that argument rather than about the generator — 98% of the placements on this site pass one, and the phase that raised that number from 12% found eight captions describing a figure the page was not showing.
At those defaults the edge it states is Miller is 22% optimistic — the right-hand slot
of the caption strip, which on this site is never used for anything else, and which is read
back out of the drawing above rather than out of the code that wrote it.
It belongs to Devices, and where they stop being linear, which is to say a change to it is a
change to lib/figures/semi.js.
It takes a slider on collector-base capacitance (picofarads) with 7 settings,
and every one of them has passed the same assertions as the frame above — a figure whose
circuit stops doing what its caption says at any setting stops the build.
Called by 6 essays
which is the blast radius of changing it
The frequency a device sets for itself
A common-emitter stage's bandwidth is decided by two picofarads between its collector and its base. The Miller approximation says how — lump it at the input, multiplied by one plus the gain — and predicts 643 kHz where the solved network gives 504 kHz. Twenty-two per cent optimistic, and it has no room at all for the second pole or for the zero in the right half-plane that the network also has.
A sum that is exact, and the estimate that is not
Add each capacitor's value times the resistance seen at its own terminals with the others removed, and the total is the ratio of the first two coefficients of the denominator polynomial — a theorem, holding to a part in a billion at every spread tested. Divide one by two pi times it and you have a bandwidth estimate that is 14 per cent low with three equal capacitors and never once optimistic. Two settings of the slider have the same three time constants and bandwidths two per cent apart, which is why the sum can never be more than an estimate.
The device that never sees the swing
A second transistor standing between the first and the load does two things that every reference gives one expression each for, and one of the two expressions has no ceiling in it. The output resistance is not gₘrₒ² — that is 248 megohms here, and the measurement is 11.5 — it is βrₒ, because the upper device's base draws current and shunts the very node the feedback works through. The bandwidth really is fourteen times better, and what it costs is two volts of a five-volt supply.
The resistance that is below zero
An emitter follower's output resistance is 5.5 Ω at direct current and −21.9 Ω at 257 MHz, and the sign is not the transistor's: with an ideal source at the base there is no negative band at all, and a hundred nanohenries of wire between the source and the base produces one from 110 to 301 MHz. A capacitance resonating inside that band is a resonator with loss of the wrong sign, so 4.7 to 100 pF on the emitter oscillates while 1 pF and 470 pF do not — a band of load capacitance with quiet ground on both sides of it.
The node the bandwidth belongs to
The Miller estimate puts a common-emitter stage's bandwidth at its input node, and whether that is right depends on the source. From 10 kΩ it is within 1% at every load from a gain of 19 to 1,190; from 100 Ω it is 24% optimistic and from 10 Ω 3.41 times, because the input node has become fast and the output node sets the bandwidth. The stage's two poles split from 92 to 2,993 times apart as the source rises. A load capacitance the estimate cannot see takes the bandwidth from 12.1 MHz to 1.49 MHz from 10 Ω and moves it 6% from 100 kΩ. And the multiplier is the gain from base to collector, 182 — not the 144.7 from the source.
The nodes a cascode hands the bandwidth to
A cascode is said to remove the Miller effect, and into 5 kΩ it does: from a 100 kΩ source it is 15.9 times wider than the plain stage. From 10 Ω it is 1.30 times wider, because there the bandwidth was never the input's — it belongs to the output node, Cμ2 against the load, 10.0 ns from every source. From a high source it belongs to the input, which is now rπ against Cπ and twice Cμ rather than against a multiplied Cμ, 74.7 ns of Cπ beside 15.4 of Cμ. The node the cascode adds holds 0.545 ns and moves the bandwidth less than one per cent. Its three poles cross rather than split, the sum of time constants is worst — 24% low — at the source where two nodes are equally slow, and the removal is not unconditional: the middle node's gain is 1.01 at a 1 kΩ load, 12.3 at 1 MΩ, and above 3.11 MΩ the cascode is narrower than the stage it replaced.