Cutoff Frequency
The cutoff frequency is defined as the frequency at which the the ratio of the output to input has a magnitude of 0.707, or -3dB for all the people who drive Geek Squad vans,
now we can take the formula for inductive reactance, which is XL = 2 pi F L and isolate the F, which is frequency,
F = XL / 2 pi L.
we can call F the Cutoff Frequency just for kicks,
lets say we have an 8 ohm speaker winding with 100 milli-henries inductance,
well inductance is directly proportional to the impedance of the transformer taps, so if we have 100 milli-henries speaker tap, and we have a plate winding of say 8 K ohms, we will have 8K/8 or 1000 times the inductance on the speaker winding, which will be 1000 times 0.100 henries = 100 Henries of primary inductance for the plate of the vacuum tube.
lets plug some of these numbers into the previous formula,
using the transformer tap impedance for XL,
F = XL / 2 pi L so
F = 8 Ohms / 6.28 times 0.1 Henry,
F = 8 / .628 = 12.7 Hertz.
and just for the heck of it, the primary F will be
F = 8,000 Ohms / 6.28 times 100 henries,
F = 8,000/628 = 12.7 Hertz.
this helps show that XL and L are proportional to each other.
but we can show something else with this same formula,
that is the relationship between Plate Resistance of a vacuum tube and the low cutoff frequency,
so F = XL / 6.28 times L, what happens if we keep everything constant and lower our XL which we are calling our primary tap impedance,
if we use a triode with 8 K ohm plate resistance, it gets put across the primary tap Z,, so 8 K ohm plate in parallel with an 8 K ohm primary tap will lower XL to 4 K ohms,
look what that does to the cutoff frequency,
F = 4,000 / 6.28 times 100 H = 6.37 Hertz.
so if we raise the plate resistance by using a pentode, the cutoff freq goes up, because the 8 K primary is in parallel with a much higher resistance, which means we use 8 K again as the XL in the formula.
Also, if we increase the secondary XL to 16 ohms and keep the inductance at 100 milli-henries, then the cutoff frequency will double as we cans see from the formula.
Now DC in a transformer lowers the L or inductance.
this means that since F = XL / 6.28 times L, if L goes down, then our cutoff frequency goes up.
so if we have DC in the transformer, we have to compensate.
Our max flux that the core will take will now have to include AC flux And DC flux,
they just add like this:
B max = B ac + B dc
so if we have too much B dc, we can saturate the core when the music is really crankin,
we need to have the core offer more resistance to being saturated, we can add resistance by adding a gap,
this increase in magnetic resistance will make it harder for the core to become saturated,
but how big do we make the gap?
well there are more formulas for this than there are dead volkswagons, so that means a good experiment can save a lot of brain cells,
usually a gap of 0.001 to 0.005 is the biggest you will see in audio transformers,
maybe 0.030 for a big power choke or pulse transformer, so not much space is needed to increase the magnetic resistance, a small gap really shorts out the field at that point,
here is some more stuff:
"A further clarification about Volt X seconds (Volts times seconds):
When a voltage is applied to an ideal inductor, the current and flux start at zero and increase linearly over time, until infinity.
We know that real inductors have limitations. One limitation is saturation at the core, which causes a drop in inductance and a much faster increase in current over time, until the maximum current is limited by the winding resistance.
Volt X seconds then becomes a measure of how close you are to saturating the inductor. Cored inductor and transformer vendors often specify how many Volt X seconds can be applied to an inductor until a pre-defined level of saturation occurs. This limits the maximum magnetic energy that can be stored in the inductor or transformer.
My power supply design coleagues at Linear Technology use VoltXseconds as a key figure of merit to select inductors for their designs.
High Pentode plate impedance
Most tube amplifiers have the high impedance of a pentode, or beam tetrode, driving the lower impedance of the primary of the output transformer. The plate impedance can be a factor of 10x higher (i.e. 40k) than the load impedance of the transformer primary (i.e. 4k).
We can think of the plate as a current source forcing current nearly instantaneously onto the transformer primary. This, in turn, means that the magnetic field created by this current is also formed nearly instantly, after the core material has had time to respond. The core material can change it's magnetization at rates faster than the audio range, so we will neglet any core magnetization delays.
In short: a current drive to the primary translates nearly instantaneously to a flux level in the core.
The concept of Volt X seconds does not apply in the case of current drive.
OK, so how do we apply the concept of Volt X seconds in a pentode plate circuit?
The answer is that the total impedance seen by the transformer includes the speaker load and, in this case, it is it's only significant load, if there is no negative feedback.
Now we can apply the concept of volt-seconds to the transformer primary impedance under output load.
The current that is forced by the plate first "sees" the reflected secondary impedance as a resistance (i.e. 4k), and a voltage appears nearly instantaneously at the primary, that is the current divided by the loaded transformer primary impedance. Then the internal current will increase, at first seemingly linearly, then exponentially, with an R/L time constant, where R is the primary impedance with the transformer under output load, and L is the primary unloaded inductance.
This is where saturation trouble will arise at low frequencies, if a half cycle takes long enough for the current to rise in the R/L time constant regime. A high DC standing current was the chief concern about using the 70V transformers for single ended use, because it brings the transformer much closer to saturation.
Negative feedback emulates a transformer load.
If the amplifier has significant negative feedback from the speaker output, or from the plate output, to the input, then the transformer voltage is regulated with negative feedback, and the concept of Volt X seconds can be applied again, even if there is no speaker load.
By significant negative feedback, it is meant that the output voltage follows the input voltage with the gain established by the feedback network. If the application of negative feedback to the amplifier lowered the net closed loop amplifier gain by 1/10, then the output follows the input within 10%, at the new closed loop gain.
This amount of feedback also lowers the open loop output impedance by a factor of 10. This means that a 40k plate impedance will look like a 4k impedance, which is comparable to the loaded transformer primary impedance."
"Part of what is understood as "triode sound', as you point out, is the high frequency roll-off from leakage inductance.
The reason to use the plate feedback is that the rest of the design can be conducted as if a triode were used.
The most important difference is the new closed loop pentode plate impedance would look like that of a triode. The simplest approximation of this impedance is that the closed loop plate impedance would be mu/gm, where mu is the inverse of the plate feedback attenuation, and gm is the transconductance of the pentode.
The reason why the plate efficiency is not lost with this approach, is that the Screen voltage remains at full pentode bias, as usual. In a triode connected pentode, the efficiency is lost because the screen voltage drops when the output swings downward.
A related approach is the Ultra-Linear circuit, where a tap from the output transformer drives the screen grid. This approach looses some of the pentode plate efficiency because the highest plate conductance is reduced by the simultaneous drop of screen voltage during the most negative plate swings.
I think I have seen direct feedback from the plate to the control grid of pentode in some radio design, but I can't remember where. Global feedback is certainly more popular, with the many ills that it corrects."
