We know that the voltage gain of a grounded cathode amplifier with a bypassed cathode resistor depends, in a large part, on the load it sees at its anode. We usually express this as:
Gain = Rload * Mu / (Rload + Rp)
We also know that the amplification factor (Mu) of a tube is the product of its transconductance (Gm) and plate resistance (Rp). A little algebra gets us:
Gain = Rload * Rp * Gm / (Rload + Rp)
Gm * Rload || Rp
We know that if one parallel impedance is much, much larger than its partner, the final impedance value of the pair is roughly the value of the smaller of the two impedances in parallel. So, if Rp is really, really big, our gain is more or less the transconductance (Gm) multiplied by the load.
This is why pentodes, with their higher transconductance and large plate resistance, achieve so much more gain than triodes in many circuits. Pentodes, using the screen, shield the grid of the tube from the anode. This reduces the Miller Effect, meaning we don’t have to worry so much about trading our high frequencies for high gain. However, the screen in a pentode partitions the current through the tube and this makes the pentode generally noisier than a triode. The cascode circuit can achieve the same high gain as the pentode and enjoys the same reduction of Miller Effect. But unlike a pentode, the cascode does not suffer from partition noise, meaning the cascode can be a better choice where low noise is needed.
Think of the two tubes as a single device, an ubertube if you will. From the perspective of the output, the plate impedance of the ubertube is the Rp of the lower tube multiplied by “Mu + 1” plus the Rp of the upper tube. Yeah, it’s a much bigger number than you usually see with a single triode. Because the same current flows through both tubes, the transconductance of the ubertube (ie, the change in current through the tube due to change in grid voltage at the input) is the same as the transconductance (Gm) of the lower tube.
We can calculate the gain of the ubertube circuit in the same way we did for the single tridoe:
Gain = Gm * Rl || Rp
where Rp >> Rl, so close to
Gain = Gm * Rl
Because the plate resistance is so much larger than the load, we can also approximate the output impedance as roughly the load’s value. Finally, if the output is referenced to ground (as it usually will be), we know that pretty much any noise in the power supply will also appear across the output (think voltage divider with lower leg as that big ubertube Rp value). This is the 20% of the information that gives you 80% of what you need to know about the cascode.
If we dive a little deeper into the circuit, we find that the anode load seen by the lower tube is the Rl divided by “Mu + 1” of the upper tube. This means the lower tube is working into an almost vertical loadline, and so is just creating current fluctuation through the tube rather than a varying voltage output (note our gain equation depends on Gm, not Mu). The output voltage is created at the anode of the upper tube and the upper anode is shielded from the lower tube by the bias on the upper grid, which locks the upper cathode (and lower anode) at a fixed voltage.
This shielding effect from the perspective of the lower tube means the anode voltage of the lower tube is not changing. If the anode voltage of the lower tube is not changing relative to the grid voltage, we are not dealing with Miller Effect. This makes the cascode a very high bandwidth arrangement.
Finally, there is no rule that the upper and lower tubes need to be the same (although dual triodes are convenient). In fact, they don’t even need to both be tubes. Hybrid cascodes can save the complication of multiple heater supplies (note Vhk max ratings are important to not blowing up cascodes), and also allow the use of higher Gm devices in the lower position (more gain) or p-type devices in the upper position (better PSRR).
So do you need high gain and bandwidth and low noise in a single stage? Not too worried about output impedance or PSRR? The cascode may be a great option for you.