jackinnj,
There is now a reasonable amount of recent data available indicating that, for audio frequencies, the classic formulas for estimating the equivalent noise resistances of valves (tubes) should be taken with a pinch of salt. This reservation is as valid for the pentode formula as the triode formula: the first term of the formula you snipped is a variant of the classic triode estimation formula of rn eq=2.5/gm, while the second term corresponds to the estimated partition noise added by the pentode’s screen grid.
The classic noise analysis focuses on shot noise in valves operating in the zone limited by space-charge, in which the grid controls the output at the anode – in other words, the circumstances in which we generally use’em! The theory underpinning the classic formulas is in Chapter 5 of Aldert van der Ziel’s
Noise, Prentice-Hall, New York, 1954. The formulas are consistent with the set of historical experimental data widely cited for noise in triodes at 30Mhz, set out by T. E. Talpey in Table 4.1 on page 175 of his excellent paper ‘Noise in Grid-Control Tubes’, in the collection
Noise in Electron Devices edited by Louis Smullin and Hermann Haus and published by John Wiley and Sons, New York, in 1959. The data in Talpey’s table shows that the classic triode noise formula rn eq=2.5/gm holds fairly well at 30Mhz, which is at the top of the HF/bottom of the UHF radio band. The data is cited elsewhere (see for instance page 487 of Gewartowski, J.W., and Watson, H.A.,
Principles of Electron Tubes, Van Nostrand, New Jersey, 1965 which can be found on
Frank's Electron tube Pages) and the findings and formulas were widely known (see augmented formulae and data on pages 937 and 938 of the 4th edition of the
Radiotron Designer’s Handbook).
However, authors of the era do not specify experimental measurements of valve output noise at lower frequencies and modern experimenters have measured triode noise at audio frequencies that is higher than the classic models and formulas anticipate. To illustrate, take the results in Table 2 at page 58 of Frank Blöhbaum’s September 2010 article (‘A New Low-Noise Circuit Approach for Pentodes’ in Volume 0 of Jan Didden’s terrific
Linear Audio) and calculate rn eq=2.5/gm from the valve data sheets. The formula estimates equivalent noise resistances that are lower than the measured values.
Higher-than-expected noise measurements were considered in Burkhard Vogel’s subsequent article in the September 2012 edition of
Linear Audio, in which he developed formula to include 1/f noise. Both Vogel’s and Blöhbaum’s experimental data shows large contributions of 1/f noise, over and above the shot noise that was the basis of the classic triode noise formulas. 1/f noise is not well-understood and, if the late Per Bak’s studies are any indication, a good explanation would put you in contention for a Nobel Prize.
Frank Blöhbaum’s experimental results are particularly interesting in respect of pentodes. For instance, converting his Table 1 of measured equivalent input noise densities to RMS
output voltages shows that pentodes are between 3.4dB and 14.0dB noisier than the same valve strapped as a triode. This is consistent with the general dictum on pentode ‘noisiness’ cited by Morgan Jones (see page 94 of
Valve Amplifiers, 4th edition, Newnes, 2012), though it is important to reference this result to
output noise. Designing to optimise signal-to-noise ratio also requires reference to
input noise densities, and Frank’s Table 1 in his article shows that the pentode’s extra 6dB to 14dB of gain levels the playing field: once gain is accounted for, the equivalent input noise voltage densities of pentodes are comparable to those of the same valve strapped as a triode and, in some cases, are lower.
This points to the usefulness of pentodes as devices with low input capacitance capable of achieving a lot of gain in a single stage, while maintaining a signal-to-noise ratio comparable to that which the triode might achieve – assuming of course that the triode could approach the pentode’s gain! This is, of course, the original rationale for designing and manufacturing small-signal pentodes.
Frank’s results also allow the contribution of partition noise to equivalent input noise voltage densities to be estimated. For the small-signal pentodes he tested, partition noise was between 0.5dB and 3.1dB. Applying his BestPentode connection eliminates this and achieves more gain, to the extent that the triodes became markedly inferior performers against the parameters of gain and input noise voltage density. The caveats on this result are (of course) that we want a lot of gain in a single stage, that we have a clean power supply with a low output impedance across the audio band, and that we are comfortable with the pentode’s distortion spectrum, which can include more higher-order harmonics than the equivalent triode.
On that basis, useful applications could be the input stages of microphone pre-amplifiers, or of phono pre-amplifiers, or and all-in-one voltage and driver stage for a power amplifier (I’m building examples of the last two). By contrast, I’m also building a low-gain line stage for which triodes are better suited – configuring small-signal pentodes for low gain looks like a recipe for a poor signal-to-noise ratio!
Pentode noise
