Circuit Response
every model has an edge, and the figure carries it
Concept

Cascode — where it appears

A second transistor standing between the first and its load, its base held at a fixed voltage so that the lower device's collector barely moves. It removes the input capacitance's multiplication by the stage gain and raises the output resistance to rₒ times whichever of β and gₘrₒ is smaller, and it costs two volts of supply that appear in neither expression.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

A cascode multiplies rₒ by β, not by gₘrₒ — and the two are 21× apart. computed by solving, not by drawing. The output resistance of a cascode stage, measured by driving the output node with a current source and reading the voltage, against the current gain of the upper device. The plain stage's is 80 kΩ — rₒ and nothing else. The cascode's is 11.5 MΩ at β = 150, which is βrₒ to within a tenth and is 21 times below the gₘrₒ² every reference gives. The reason is in the netlist rather than in the algebra: the upper device's base draws current, so its rπ sits from the lower device's collector to signal ground and shunts the node the feedback works through. What the arrangement buys therefore scales with β and stops when β does, and the curve is the two expressions drawn against the measurement.

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.

semiconductors · Cascode A cascoded mirror is 90× the output resistance, and 43% of it goes back into the reference. computed by solving, not by drawing. The output resistance of a two-transistor mirror and of the same mirror with a cascode on each branch, measured by moving the output a little either side of its operating point and reading the current, against the current gain of every device. The plain mirror sits at rₒ = 89 kΩ and does not move. The cascoded one reaches 7.39 MΩ at β = 150 and rises with β until β stops being the smaller of the two quantities, where it saturates on gₘrₒ² = 268 MΩ. The third curve replaces the diode-connected upper device with a held voltage at the same potential and recovers 1.76 times the resistance, which is the upper device's base current being charged a second time — to the reference branch, where it moves the mirror's own bias.

The source that holds to the supply

Putting a second transistor on each branch of a current mirror is always described as buying output resistance and costing headroom, and both halves of that are measured here rather than repeated. The resistance goes from 82 kΩ to 7.39 MΩ, the floor rises by 0.71 volts — and the range over which the current is actually what it was set to goes from 1.70 volts to 9.09, because a plain mirror's current never stops climbing. Forty-three per cent of the resistance that should be there is missing, and it is in the reference branch.

semiconductors · Cascode Held one drop and 300 mV up, the cascode's floor falls from 0.832 V to 0.403 V. computed by solving, not by drawing. The output current of two cascode mirrors against their output voltage near the bottom of the supply, each divided by its own value at 5 V, on a model whose transistors saturate. The stacked mirror holds its upper device's base two drops up and stays within one per cent down to 0.832 V; the wide-swing mirror holds it one drop and 300 mV up, which leaves the lower device 0.302 V of collector-emitter voltage, and stays within one per cent down to 0.403 V. Its output resistance at 5 V is 13.2 MΩ against 7.39 MΩ, and its copy ratio 0.9758 against 0.9747.

The drop the lower device did not need

A cascode mirror's floor is a whole base-emitter drop plus the upper device's own minimum, because the stack that biases the upper base holds it two drops above ground. The lower device needs only a few hundred millivolts. Hold the upper base one drop and 300 mV up instead, and on a model whose transistors saturate the one-per-cent floor falls from 0.832 V to 0.403 V. The output resistance does not fall with it: it rises from 7.39 to 13.2 MΩ, because the upper device's base current no longer comes out of the reference. The cost has moved into the margin itself. Below about 150 mV the lower device saturates, and at 50 mV the resistance is 0.51 MΩ and the copy ratio 0.7665.

semiconductors · Cascode The cascode device's collector noise is cancelled below 7.22 MHz and its base noise never is. computed by solving, not by drawing. The output noise current of a cascode stage at 1 mA and β = 150, driven from 1 kΩ, into a short, source by source, with 20 pF at the node between the two devices. At low frequency the upper device's base shot noise contributes 0.667% of the lower device's collector noise power, 1/β, and its collector shot noise 4.88 × 10⁻⁵, about 1/β². The collector's contribution overtakes the base's at 7.22 MHz, where the admittance reaches gₘ/√β — 7.21 MHz from the lower half's solved admittance, against 25.1 MHz from the node capacitance alone, because the lower device's Cμ, seen from its collector, is multiplied by its own gain; it passes the lower device's own collector noise near 316 MHz.

The noise the cascode device does add

A cascode's upper device is said to add no noise, because it is a common-base stage fed from a high impedance and its own noise current has nowhere to go but round itself. That is true of its collector noise, which reaches the output at 4.88 × 10⁻⁵ of the lower device's, about one part in β². It is false of its base noise. The partition of the emitter current into collector and base cannot cancel, so it arrives whole: one part in β, 0.667 per cent at a current gain of 150. The cancellation also needs the node between the devices to stay high-impedance. The lower device's own base-collector capacitance, seen from its collector, is multiplied by its gain, so the upper device's collector noise comes through above 7.2 MHz with 20 pF at that node, and at 100 MHz the cascode stage is up to 0.55 dB noisier than the plain one.

semiconductors · Cascode A 1 kΩ bias resistance at the upper base is attenuated to 7.97 × 10⁻⁶ at low frequency and passes the lower device's noise at 15.2 MHz. computed by solving, not by drawing. The output noise current of the cascode stage at 1 mA and β = 150, driven from 1 kΩ, source by source, with the upper device's base fed through 1 kΩ instead of held at signal ground. At low frequency the resistance's noise reaches the output as 7.97 × 10⁻⁶ of the lower collector's, about 2R/(gₘrₒ²): its voltage is followed to the node between the devices and becomes a current only through the lower device's rₒ. The upper base's own shot noise, flowing in the same resistance, is 0.683%. The resistance's noise passes the lower device's at 15.2 MHz, and at 100 MHz the stage's noise figure is 6.07 dB worse than with the base held.

The resistor the upper base is held through

A cascode's upper base is drawn at signal ground, and on a board it is held by a bias network with a resistance of its own. At low frequency that resistance's noise is harmless for the reason usually given — the upper device follows its base and turns the voltage into a current only through the lower device's output resistance — so 1 kΩ reaches the output at 8 × 10⁻⁶ of the lower device's noise. But the attenuation belongs to rₒ, and rₒ is shunted by the node's capacitance from 23.5 kHz, three hundred times below the frequency at which the upper collector's own noise stops cancelling. Unbypassed, 1 kΩ costs 6.07 dB at 100 MHz, more than 100 Ω or 100 kΩ does, and 100 pF across it makes any resistance free.

semiconductors · Cascode A mirror, cascoded or not, delivers two transistors' shot noise at low frequency — and cascoding it does not add a third. computed by solving, not by drawing. The output noise current of a 1 mA NPN mirror into a short, in units of one device's collector shot noise 2qI, against frequency, for a plain two-transistor mirror, the stacked cascode mirror and the wide-swing cascode with its upper base held by a separate diode at 100 µA. At 1 kHz they deliver 1.986, 1.960 and 1.965: the reference transistor's shot noise, copied, and the output transistor's own. At 100 MHz the mirror's own pole and the capacitance at the output transistor's collector have begun to take noise away, and the three read 1.681, 1.196 and 1.512.

The base current the mirror counts twice

A current mirror delivers two transistors' shot noise — the reference device's, copied, and the output device's own — and cascoding it does not add a third: 1.960 and 1.965 of one device's noise for the stacked and wide-swing cascodes against 1.986 for the plain mirror. What the cascode adds is its upper output transistor's base noise, and there the two arrangements differ by a factor of four. In the wide-swing mirror the base current goes to a separate bias and counts once, one part in β. In the stacked mirror it goes into the reference branch, which mirrors it back into the node it left, and it counts 3.82 times at β = 150 — the same base current that took 43 per cent of that mirror's output resistance. The reference resistor's noise, meanwhile, adds almost nothing even at 1.8 V, because it takes its place in the diode's node rather than adding to it.

semiconductors · Cascode A cascode is 15.9 times wider than the plain stage from 100 kΩ and 1.30 times from 10 Ω, with a 5 kΩ load. computed by solving, not by drawing, each bandwidth bisected on the solved magnitude. The common-emitter stage of 1 mA, Cπ 20 pF and Cμ 2 pF into 5 kΩ, alone and with a second device stacked on its collector, across four decades of source resistance. The plain stage falls from 12.1 MHz to 110 kHz as its Miller-multiplied input node slows. The cascode runs from 15.8 MHz, where its output node sets the bandwidth — Cμ2 against 5 kΩ, 15.9 MHz — to 1.75 MHz, where its input node does: rπ in parallel with the source, against Cπ and twice Cμ, 1.78 MHz. Neither single-node figure is ever below the stage's own bandwidth. The faint line is the cascode with the upper device's own Cπ left out of the middle node, and it is never below the solid one and at most 0.74% above it.

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.

semiconductors · Device bandwidth

Named alongside it

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

Output impedanceInput capacitanceCurrent mirrorCurrent noiseShot noiseCompliance rangeDesign tradeoffEarly effectHeadroomModel rangeNoise figureBandwidth

All concepts