• WARNING: Tube/Valve amplifiers use potentially LETHAL HIGH VOLTAGES.
    Building, troubleshooting and testing of these amplifiers should only be
    performed by someone who is thoroughly familiar with
    the safety precautions around high voltages.

Cascode with dissimilar tubes?

Let's consider the textbook cascode circuit with the lower tube in common cathode and the upper tube in common grid connection. Either can be a triode or a pentode.

What about the requirements for the lower and upper tube? Which one should be higher gm? In which position is a pentode preffered over a triode?
 
Assuming the OP meant triodes, I would say that the higher gm device would normally go on the bottom for best gm performance or overall gain (although not necessarily best linearity). A lower gm device up top will allow some voltage movement at the plate of the lower device, helping to linearize the 3/2 power gm response below. (cascodes are quite non-linear, but OK for small signal input) The grid of the top device is normally operated at a constant V for best gain, but it also provides a convenient point for N Fdbk application, if it is stable. (the very good HF performance in cascode may oscillate, a small pF cap, grid to upper cathode could help stop that.) (caution, ceramic caps can act as VHF resonators too)
 
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When the RF Cascode stage was designed, the 1/Gm cathode impedance of the top tube provided a much lower impedance to the bottom tube's plate impedance, rp.

That significantly reduced the voltage gain of the bottom tube, and thereby significantly reduced the Miller Effect Capacitance on the bottom tubes grid. Great for RF!

I expect that the RF noise performance of an all triode cascode stage is quite a bit better, versus using a Pentode(s) in either / or both the bottom tube and top tube positions.

The high impedance of the plate of the cascode stage was very well suited to the high impedance of a high Q parallel resonator that was used in those RF circuits. And in those relatively narrow band applications *, it was easy to create high Q circuits that reduced that high impedance output to a lower impedance if necessary to drive the following stage.
* Narrow band RF applications, such as a Radar IF amplifier that was centered at 30MHz, but only had 1 MHz bandwidth.
Compare that to an audio application, 20Hz to 20kHz, with the center at 9990Hz, and 19990kHz bandwidth (wide band application).

* Narrow band, and Broad Band, as defined by: Center Frequency / Bandwidth.

So, as to the performance of an Audio cascode stage . . .
Your Mileage May Vary

Note: Stephe seems to have shown and used one of the best applications of an Audio Cascode Stage that I have ever seen.
Thanks Stephe!
 
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