Concept

Dominant pole — where it appears

The pole nearest the imaginary axis, whose time constant is the slowest and which therefore decides a response's bandwidth on its own. It decides the bandwidth alone only while the next pole is far above it, and the interval between them is what a phase margin is spent on.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

A common-emitter stage with 2.0 pF from collector to base. computed by solving, not by drawing. The stage's midband gain is 144.7 and its −3 dB point is at 504 kHz. The Miller approximation lumps 311 pF at the input and predicts 643 kHz — 21.6% high. The network's second pole is at 336 MHz and its right-half-plane zero at 3.08 GHz, both of which the approximation has no room for.

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.

semiconductors · Device bandwidth

A sum that is exact, and the bandwidth estimate that is not. computed by solving, not by drawing, at 28 spreads of the three capacitor values in a resistor chain. The sum of the open-circuit time constants — each capacitor's own value times the resistance seen at its terminals with the other two removed — is 600.00 µs here, and it equals the ratio of the first two coefficients of the denominator to 2.0e-9 and the sum of the negated reciprocal poles to 2.0e-9. That much is a theorem. What is an estimate is the bandwidth: one over 2πΣτ gives 265.3 Hz against a measured 309.2 Hz, low by 14.2%. It is low at every spread on the axis — the estimate is never optimistic — and comes within ten per cent only once one of the three time constants is 7.48 times the others.

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.

transients · Open circuit time constants

An inverting unity gain driving 2.2 nF, and the pole that is inside the loop. computed by solving, not by drawing. Two ten-kilohm resistors around a 10 MHz amplifier make a gain of 1.00, and a loop that closes against 2.01 — one plus the ratio, not the ratio. Hanging 2.2 nF on the output leaves the closed-loop gain at a kilohertz unchanged — 0.99998002 against 0.99997988 — and takes the phase margin from 90.0° to 30.1°. The mechanism is at the other end of the amplifier from the summing-junction case and the arithmetic is the same: the load works against the amplifier's own fifty ohms of output resistance, which puts a second pole in the forward path — inside the loop, where the feedback has to live with it — while the gain the loop closes against does not move at all. Forty-five degrees is reached at 905 pF, bisected on the netlist. The capacitance is not part of the signal path and does not appear in any expression for the gain.

The load that gets inside the loop

Hanging a capacitor on an amplifier's output changes nothing a reader can find in any expression for its gain, and takes the phase margin of a unity-gain inverter from ninety degrees to thirty. The mechanism is fifty ohms of output resistance that no data sheet page puts next to the stability page: the load works against it, the pole that results is in the forward path, and forty-five degrees arrives at 905 picofarads — which is a metre of coaxial cable.

feedback · Capacitive load

The common-emitter stage from a 100 Ω source: the solved response against the Miller estimate. computed by solving, not by drawing. The stage of 1 mA, 5 kΩ load, Cπ 20 pF and Cμ 2 pF has a gain from base to collector of 182.0. From 100 Ω the network's bandwidth is 3.41 MHz; the Miller estimate — Cμ moved to the input, multiplied by one plus the gain — says 4.23 MHz, 1.24 times it. The estimate has one pole, at the input; the network has two, and from a low source the second is the one that sets the bandwidth.

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.

semiconductors · Device bandwidth

A gain of ten round an amplifier with a second pole: its half-power bandwidth and its noise bandwidth, each over what the gain–bandwidth product predicts. computed by solving, not by drawing: the stage as a netlist round a two-pole amplifier of 1 MHz, the second pole placed at r times GBW/G. The half-power bandwidth agrees with the second-order closed form √r·g(√r/2) to two parts in a thousand at every placement: 1.000 at r = 0.5, 1.272 at r = 1, 1.414 at r = 2, 1.231 at r = 5, 1.110 at r = 10, 1.010 at r = 100 times GBW/G. It is 1.414 times the product's prediction at r = 2 and within ten per cent only beyond r = 10. The noise bandwidth is (π/2)·GBW/G to half a per cent at every r: the second pole moves the one and leaves the other where it was.

The bandwidth the product still gets right

Gain times bandwidth is a constant for an amplifier with one pole, and real amplifiers have a second. Put it at r times the closed-loop bandwidth the product predicts and the half-power bandwidth becomes √r·g(√r/2) times GBW/G — 41.4 per cent wide at r = 2, within ten per cent only beyond r = 10 — so a follower round a 1 MHz part with its second pole at 2.5 MHz runs 40 per cent wider than the rule says. The noise bandwidth does not move at all. A second-order loop's noise bandwidth is set by one coefficient the second pole cannot touch, and it stays (π/2)·GBW/G at every r and every gain, to half a per cent on the netlist. A third pole is what breaks it.

feedback · Gain-bandwidth

A stage with its second pole at 3 times the bandwidth the product predicts settles within 0.1% in 5.06τ, against 6.91τ for the one-pole stage the product describes. computed by solving, not by drawing: the step of a stage whose amplifier has a second pole at r = 3 times GBW/G (solid), against the one-pole stage of the same product (dashed). Time is in units of τ = G/(2π·GBW). Settled within 0.1% at 5.055τ, against ln(1000) = 6.908τ, with the netlist of a gain-of-ten stage agreeing at 5.057τ. The stage with the second pole is usually faster, whatever it does on the way.

The gain that stops making a stage faster

The gain–bandwidth product says a stage of gain G settles in time proportional to G, and a stage with a second pole does something else. Below r = 4 the closed loop's poles are a complex pair whose real part is half the sum of the open loop's, π times the second pole's frequency, with neither the gain nor the product in it: a 1 MHz amplifier with a 0.5 MHz second pole takes 4.10 µs to settle within 0.1 per cent at unity gain, 4.04 at two and 4.22 at five, where the product says 1.10 to 5.50. Above the knee at G = 4·GBW/f₂ the product returns. In between lies the fastest stage, r = 3.32 for a 0.1 per cent band, which settles in 3.44 time constants against the one-pole stage's 6.91: half the time, and not at the r that gives the widest band.

feedback · Gain-bandwidth

Named alongside it

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

Model range Gain–bandwidth product Bandwidth Capacitive load Closed-loop gain Damping ratio Half-power bandwidth Miller effect Open circuit time constants Phase margin Poles Decoupling

All concepts