TUBE OPERATION  1
SMALL SIGNAL AMPLIFIERS.

Contents of this page :-
Basic tube operation, cathodes, anodes and gids, diodes, triodes and pentodes.
Parameters of Ra, µ and gm.
Fig1. Schematic for a basic triode amplifier based on 6SN7.
DC flow for quiescent conditions, dc equilibrium,
Mutual effect of anode voltage and grid voltage on Ia electron flow,
Effect of cathode bypassing biasing on Ra, and gain.
The tube modelled as a generator.
Fig 2. Schematic of basic 6SN7 generator model for illustrating ac operation of a tube.
NFB in the triode, AC signal flow,
Cathode capacitor bypass impedances, tube gain formula, gain without capacitor bypassing.
Fig 3. Electrostatic effects in a 6SN7 triode.
NFB in the triode, how it reduces THD.
Fig 4. Electrostatic effects in the 6AU6 pentode,
pentode and beam tetrode operation, pentode Ra, µ and gm.
6AU6 triode connected and its amount of internal NFB.
NFB in 6SN7, and why ß = 1 / µ , with more NFB and gain equations.
The Miller effect.
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The history and operation of vacuum tubes is best described in many of the old text books that one may find in university
reference or archive libraries providing you don't mind the dust and smell of old books.
Rather than repeat all that has been written in the past I wish to give a very brief
discussion on what is happening in the common tubes we use today in our audio amplifiers.

The simplest vacuum tube is the diode, which consists of an electron emitting cathode and an electron absorbing anode,
also called the "plate".
The cathode is the the heart of the vacuum tube set up in a glass envelope with a pure vacuum within.
Tubes cannot work without a vacuum.
The cathode is usually a metal tube structure at the centre of the tube. This is heated sufficiently cause its special oxide coated surface to begin emitting electrons. The heat gives the electrons in the molecules of the cathode sufficient
energy to spin off into the vacuum where they circulate, then lose energy, and fall back into the cathode.
The tube will have millions of circulating electrons where the density reaches an equilibrium. As the electrons increase in number, the negative charge of the cloud of circulating electrons build up until further emission is prevented because
the negative charge within the cloud of electrons around the cathode prevent more than a certain amount
of electrons filling the tube since the charge tends to repel electrons trying to take off from the cathode surface.
The cloud of electrons is called the space charge.

The anode is a metal cylinder or open ended box around the cathode within the tube. Without any voltage applied to the anode, and while connected to the cathode, the effect on the space charge is about nil.
But suppose we apply a positive voltage charge to the anode by connecting a battery with the -ve terminal taken to the cathode and positive terminal taken to the anode.

The electrons swirling around inside the tube are attracted by the positive anode and absorbed by it, and there is then a current flow of electrons entering the anode and flowing into the +ve battery terminal then through the battery and out the -ve terminal and around the the circuit  formed by battery and diode.

Should we connect the battery the other way around and make the anode negative with respect to the cathode
then no current flow would occur because the negative charge at the anode repels electrons.
This property of diodes is useful for where we want a flow of current in one direction only as in a power supply
or peak detector circuit in a radio etc where there is an alternating ac signal moving +ve then -ve on each crest
and trough of the waves with respect to the centre voltage usually taken as being 0V, or earth or ground potential
which is the reference point for all measurements of voltage.

The current flow from cathode is possible because the electrons swirling in the tube that are attracted by a +ve anode are immediately replaced my more electrons from the cathode and the current flow from cathode to anode varies
at a rate proportional to a constant x square root of the anode voltage squared.
The diode has resistance while the current flow occurs and can be calculated using Ohm's Law, R = E / I.
In this case the resistance we might measure is called the anode resistance, Ra.

If we measured the currents at various positive Ea we would be able to draw a graph with Ia on a vertical axis, and Ea on t a horizontal axis and

Ia = K x sq.root of Ea cubed. The value of K is determined by the tube dimensions, but a line for Ra can be plotted
by measurement without knowing what K is.
The curves obtained are the curves whose slope ant any point indicate the anode resistance, Ra, of the tube under test.
When you view the triode anode curves, you are looking at a set of Ra diode lines for various values of grid voltage.

Let us consider the triode. The triode is simply a diode with a helical coil of fine wire placed around the cathode
and concentrically aligned with the anode cylinder or box. If this grid is connected to the cathode, the tube acts like a diode
and the grid has little effect on the current flow, since electrons will stream past the fine grid wires of the grid to get the anode
if it has a +ve voltage.
But when the grid is supplied with a -ve voltage, Eg,  with respect to the cathode voltage, Ek, the resulting voltage field effect of the fine wires  repels electrons and prevents them from flowing to the anode as they do with Ek = Eg. The electrons which
do flow past the grid on their way to the anode which attracts them avoid contact with the grid since it is negative and repels
electrons, so while the grid remains negative there is no grid current flow and the grid is an extremely high impedance
input terminal of the triode which controls the anode current between anode and cathode.
If Eg is sufficiently -ve, the electron flow ceases no matter how +ve the anode becomes.
So the grid has the ability to control electron flow in the triode.
If the grid is ever  driven positive with respect to the cathode, then  the grid absorbs electrons and a grid current then flows.
And because the grid is a helical wire or even a mesh of wires, most electrons flow past the positive grid to the
more positive anode. In small signal amps and most power amps the grids are never driven positive and all current control
is done with a changing negative voltage applied to the grid.

There are three parameters of a tube which are important for design purposes :-

Ra, anode resistance is the dynamic output resistance seen by anything connected to the anode.
So should we have a resistor between anode and a low impedance signal generator, then the output resistance of the tube which is the Ra or anode resistance can be calculated from
Ra, in ohms,  = signal voltage at the anode / signal current in series R from voltage generator with a fixed grid voltage.

Amplification factor, known as µ, is just a number without units and is the voltage gain of a tube when a load with infinite ac resistance is connected.
It is directly related to dimensions between cathode and grid and cathode and anode. It is the least changing parameter of the tube due to changes in Ia and Ea for the operating condition.
It can be easily measured by connecting a dc supply to the anode which has an extremely high signal impedance
and simply measuring input and output voltages at low levels.
µ = anode signal output voltage / grid input signal voltage where the load is a constant current source.

Transconductance, gm, is measured in mA/V and is the ability of the grid to cause anode current change.
It is easily measured with the anode taken to a fixed supply voltage but without any load connected except a small resistance of 10 ohms to monitor the current flow. Gm can change considerably for various chosen values of Ea and Ia.
Gm is easily measured  by applying a small signal to the grid, say 1Vrms, and measuring the current at the anode at the current sensing resistor.  Gm = Ia rms / Vg  rms, so a tube which produced 2mA rms of anode current change for 1Vrms
of grid voltage has gm = 2mA / 1V = 2mA/V.

The three tube parameters relate to each other in the simple formula for all tubes :-

Gm = µ / Ra , where Gm is in amps per volt, Ra is in ohms, and µ is the amplification factor.

If only two parameters are known, the third can easily be calculated. For example, a triode with µ = 20 and Ra = 13k
has gm = 20 / 13,000 = 0.001538 amps per volt, or 1.54mA/V.
Another example is a pentode with gm = 3mA/V, and Ra = 500k ohms. µ = gm x Ra = 0.003 x 500,000 =  1,500.

In triode amplifiers which are commonly set up in class A, there is a basic circuit formed with a power supply,
load resistance and triode tube.
Let us examine the working of the circuit in Fig1.
Fig1.


People with a keen eye for detail and whose minds contain common-sense will see that electrons are emitted from the
cathode and travel upwards past the grid to the anode. Yet they see the 3.4mA of anode current flow indicated as
a flow downwards by the arrow beside the dc RL, 47k.
Before anyone was aware of electrons, people thought current flowed from positive to negative, ( like good favours from the king to his people but we all know which way the taxes flow! )
So *convention* has it that current always flows from +ve  to -ve and I for one will not rock the boat to change the conventions. We don't really need to be mindful of the trillions of electrons flowing each second from cathode to anode.
We need to be able to have a mental picture in the conventional sense and in the electron activity sense.
After a short time delving into basic electronics we realise quite a few things  don't seem to comply with commons sense notions. ( For example a resistance with lots of ohms is an easier load for a tube to drive than a low number of ohms..)
In the schematic, the "indirectly heated" cathode is heated to about 900 degrees C to cause it to emit electrons.
This is done with a tiny heating element coated in a special refractory clays or oxides which are insulating material so the heating power from a power transformer winding has no connection of effect effect on the signal functioning of the circuit.
Some cathodes are "directly heated" and are like the heating element but made of special metal and coated with special oxides and doped with thorium for good emission and have the heating current passing through them and use appropriate methods to
minimize the effects of noise from heater current supplies. 

The operation of a diode or triode seems quite simple so far. But to understand the triode we must consider the
electrostatic effects within the tube acting between the electron space charge immediately surrounding the cathode
and anode voltage and grid voltage.
We must consider the dc flow of current first.

The above schematic is a very basic triode amplifier using a half section a 6SN7 twin triode.
Dc flows around the circuit from the +ve supply terminal through the anode dc RL, 47k, through the tube, and in the
R1, 1.5k "cathode resistor". The current flow of 3.4 mA generates a voltage in the 1.5k so that about +5V
exists at the cathode. The cathode dc voltage is what is also known as the quiescent cathode bias voltage which makes the grid negative in respect to the cathode. If the voltage at the cathode increased beyond 5V, there would be an increased -ve grid bias voltage which would oppose any increase in Ia and limit the rise of cathode voltage, Ek.
Therefore the tube is automatically biased for quiescent operation and dc current flow is fairly well regulated by the current feedback and the tube will settle in a state of equilibrium for its possible 20,000 hour life.

The current in the triode is regulated by the cathode resistor. To work out the value of the cathode bias resistor as a starting point which can then be adjusted or trimmed to attain the real world anode quiescent voltage we can say that

Rk     =      ( Ea / Ia ) - Ra
                          µ + 1
Where Ea = wanted anode to cathode dc voltage,
Ia is the idle current in amps,
Ra is the anode resistance for the value of Ia from the anode curves,
µ is the amplification factor from the data,
and 1 is a constant for all equations to work.

For the 6SN7 above,  Ea = 140V, Ia = 0.0034A, Ra =  13k for 3.4mA, µ = 20.
So Rk = [( 140 / 0.0034 ) - 13,000 ] / 21 = 1,341 ohms.
Its not the same as I have indicated above with 1.5k for Rk, but then why don't YOU try the above circuit
on a breadboard to see what you get?

If we had the supply voltage, and knew the dc RL value then we can say
Rk    =     ( B+ / Ia ) - ( Ra + RL ) 
                         µ + 1

Starting with B+ = +300V, and with 3/4mA, Ra = 13k, RL = 47k,
the Rk = 1,344 as we calculated before.
If the result is a negative resistance value because the top line of the equation is a negative value then you have
selected an impossible situation for the tube, so reduce Ia.

If we knew all the values on each side of the equation except for one, we can work it out by substitution.
So if Rk was known but not Ia, it would be easy to calculate.

The anode and grid voltages BOTH have an effect on the current flow to the anode.
The anode voltage plays a crucial part in the dc equilibrium. The current flow also generates a voltage across the DC RL, 47k, and with 3.4mA, the voltage at the anode = Ea = B+supply - voltage across 47k = 300V - ( 47,000 x 0.0034 )V = 140V.
The anode voltage has an attraction effect on electrons gathered between the grid and cathode, and the electron flow
in the triode is the direct result of the net electrostatic or voltage field effects of both anode and grid.
Where the cathode is taken directly to 0V and the grid has a separate fixed negative supply adjusted for the wanted
dc voltage conditions shown in the schematic, the Ra is the data value of the value we measured for the Ea and Ia operating conditions which I have listed on the schematic as µ = 20, and Ra = 13k for Ea = 140V and Ia = 3.4mA.
If for any reason  someone were to raise the Ea voltage 140V to 150V the rise in Ea would provoke an increase of
Ia change = 10V / 13k = 0.769 mA.

But in this case we have cathode bias, which is more complicated to explain.
Should we alter the Ea with +10V, the Ia change depends on the effective anode resistance, Ra', in the presence of the R1 cathode resistance.
For now, forget the action of capacitors in the circuit since these are only relevant to the ac signal operation which we will
later discuss in detail.
We have established the anode resistance, Ra, for the triode is 13k for 3.4mA, and if we increased the Ea by 10V
we could expect an increase in Ia = 10V / 13k = 0.77mA.
But we have a cathode resistance of 1.5k which increases the Ra to its effective Ra' value according to the following formula :-
Effective  Ra' with Rk =  Ra + [ ( µ  + 1 ) x Rk ]
This formula  like many I quote comes from the Radiotron Designers Handbook, 4th Ed, 1955, and derivation of the formula is all there in the text book.
So with the 6SN7, we get Ra' = 13k + [ ( 20 +1 ) x 1.5k ] = 44.5k, or about 3 times higher than were we not to have
R1 = 1.5k.
So the measured resistance "looking into" the anode circuit of the triode is 44.5k, not 13k.
At Fig1, there is 47k between the anode and B+ of 300V, and at dc, the Ra' is effectively 44.5k so a total Ra' + dc RL = 44.5k + 47k = 91.5k exists between the B+ supply and 0V.
A 10V rise in the B+ supply voltage will thus cause a change of Ia = 10V / 91.5 = 0.109mA.
Therefore the change in Ea would be 44.5k x 0.109 = 4.86V. ( from Ohm's Law, yet again..)
The advantage of the R1 cathode resistance gives us fairly good Ia regulation and freedom from
large variations should the B+ supply value rise or fall 20%. The R1 is an application of negative current feedback, because the voltage generated by the Ia in the 1k5 is in series with the grid input voltage.


The generator mental model of a tube.
This "looking into" description is vague to some, but suppose we didn't know what was inside the tube beyond the anode
terminal on the tube socket. We would be able to tell that there was a signal source with a certain value of source resistance,
otherwise known as generator resistance, or output resistance or impedance. For modelling purposes, beyond the anode pin is a series resistor equal to the Ra, or anode resistance, in series to an imaginary low impedance voltage generator with an
output = µ x Vg input voltage.
Every tube can be modelled as a very low impedance voltage generator whose output = µ x Vg and where there is a resistance between the generator output and what is the anode. For example, the model for a 6SN7 is a generator  producing
20Vrms output for 1Vrms grid input voltage. There is 13k between the 20Vrms output and the anode terminal.
If RL of 32k is connected, then 20Vrms flows across 13k + 32k so 14.2Vrms appears at the anode which also is the load voltage.
A pentode can also be modelled this way with a 6AU6 having a gene producing 4,500Vrms output for 1Vrms input
because µ = 4,500, then Ra = 1.5M, so with RL = 32k , 4.500Vrms  flows across 1,532k, so 94Vrms appears at the anode
which is also the load voltage.
The high voltage of the imaginary generator does not actually appear anywhere, but the imaginary model of the
tube as kind of generator works exactly the same as the tube, and can be a useful tool in analysing circuit behaviour in terms of resistances and impedances including negative feedback loops. We can have a better idea of circuit outcomes in the design process, and thus depend on our own brains rather than flying blind on a computer simulation program.
Here is a simple model schematic for a 6SN7 triode in a signal situation :-
Fig 2.


Fig 2 shows the ac working of the triode without having to worry about the dc conditions, although the
Ra and µ must be known for the Ea and Ia operating point, which is available from the Ra curves for the triode.
( Curves and loadlines are covered in Basic Tube Operation 2 ).


Any change in Ea regardless of how we cause it will cause a change in Ia because of the electrostatic effect
of the anode voltage on the electrons. If they feel a greater force of attraction due to higher Ea, more electrons flow,
and the grid must be made more negative to counter the effect of the increase in Ia to keep the voltage
at the anode from changing. So the cathode and anode BOTH have an effect on anode current flow.
The applied grid voltage needed to give the triode gain we see is that over comes the
effect of the anode voltage to oppose the Ia change. If Eg rises by 1V, Ia is increased and the load voltage increases so Ea falls by say 14V. This drop in Ea tends to cause less Ia to flow; the action of the anode voltage opposes what the grid voltage attempts to achieve. Put another way, if the effect of the anode voltage upon the Ia could somehow be screened off
then much less grid voltage would need to be applied to the tube to produce the same anode output voltage.

In the Fig 1 schematic, strictly speaking, the Ea = the anode voltage - the cathode voltage = 140V -5V = 135V.
The actual Ea / Ia dc = 135V / 3.4mA = 39.7k ohms. Yet we see that when we change Ea slightly without a cathode resistance present, the change in apparent resistance at the anode = 13k, and so some mechanism is preventing the triode from
operating like a pure resistance. It is the negative feedback effect within the triode.
The NFB in every triode gives the triode its unique ability compared to all other devices to behave with a
lower anode resistance value than the load value without having an external loop of NFB connected.

Let us consider the ac signal operation.
Consider the tube with no signal is happy to work with dc to give the state of equilibrium of dc voltages shown on the schematic.
To cause a signal voltage change at the anode, we must apply a signal voltage change between the cathode and grid.
In this case we have the R1 bypassed with a large C value of 470uF.
Capacitor impedance or ac reactance in ohms at a given frequency of sine wave signal =
ZC  =  1,000,000               
           6.28 x C x F
where ZC is reactance or impedance at a frequency F in Hz,
1,000,000 is a constant for all equations,
6.28 = 2 x pye, or 2 x 22/7, a constant for all equations,
and C is in uF, and F is in Hz.

Let us consider the ac working at 1 kHz, regarded as the mid frequency for audio amplifiers.

ZC for 470uF at 1 kHz =    1,000,000              =  0.338 ohms    
                                       6.28 x 470 x 1,000
This is a tiny impedance compared to the R1 of 1.5k and the parallel input resistance "looking into the cathode".

Similarly, the capacitor C2 of 0.47 uF has an impedance at 1 kHz =338 ohms, and is quite negligible compared to the
resistance of the 100k ac RL to which the anode is coupled via C2.
The circuit which acts differently to ac signals as it does to dc signals.
This is no cause for alarm, and the schematic for the triode amp in Fig1 has been used for countless preamp stages.
The triode acts as if its cathode was connected directly to 0V, and with the anode connected to both
47k and 100k together which are thus effectively in parallel to make a an anode load = 32k.

All tubes give us the voltage amplification = µ x RL / ( Ra + RL )
All tubes obey this simple and universal gain formula. which applies only for where the cathode is grounded or
shunted to ground via a low impedance such as a high value bypass capacitor, usually an electrolytic type.
The RL in this case is the dc RL of 47k in parallel with the ac coupled load of 100k, so RL = 32k.
So we know Ra = 13k, RL = 32k, µ = 20, so gain = 20 x 32 / ( 13 + 32 ) = 14.2, so let us call that 14.
So we should get 14Vrms output from the 6SN7 with 1Vrms input set up as shown.

(( What happens if we disconnect and remove the C1 470uF "bypass" capacitor ?
The R1 1k5 would develop a signal voltage caused by the signal current. The input grid voltage would need to be increased
to still give 1Vrms between grid and cathode to cause a 14Vrms change at the anode.
The cathode signal voltage is is local current negative feedback with the same phase as the grid signal and is in series with the
grid signal.
The signal current in this case = 14Vrms / 32k = 0.435 mA, so we would see 1.5k x 0.435mA = 0.656Vrms at the cathode.
The input signal required to obtain  14Vrms at the anode is thus 1V + 0.656V = 1.65Vrms.
The overall gain has been reduced to 14 / 1.65 = 8.5, which means 4.3dB of NFB has been applied. ))

What evidence is there of NFB action within the triode at audio signal frequencies?
Fig 3.


The Fig 4 diagram shows the 6SN7 large enough to take a walk around inside.
The relative distances between cathode and grid, and cathode and anode are clearly shown; the latter is a larger distance.
The distances involved have a profound effect on the Ra and µ and gm of the triode, and the type of triode
one achieves in a factory depends on the relative distances.
But as you can see, the cloud of space charge electrons suspended in space around the cathode are subject to the effects of nearby voltage field effects from the grid, and the further away effect of the larger anode voltage.
The two fields have joint control of the electron flow.

Consider an undistorted pure sine wave signal applied to the grid from a suitable low impedance signal voltage source
shown coupled to the grid, and providing 1Vrms. As calculated above we will have 14Vrms output at the anode.

But we also get a small harmonic distortion voltage, Vdn, at the anode and which will mainly be second harmonic,
and Idn, the distortion current in the triode and load.
This Vdn which appears at the anode has an effect on the electron stream so that +Vdn tends to cause a +Idn
current change in Ia which occurs in the 32k load to cause a -Vdn change in the Ea because more Ia in the RL means a lower Ea.
So the effect of the anode distortion voltage tends to oppose its own creation.
The method of delivery of the NFB in a triode is via the anode field effect upon the electron stream.
This effect is not linear, since at the beginning of this lecture we saw that changes to Ea caused changes
in Ia = a constant x ( the square root of Ea cubed ).
But despite the non-linearity of the internal mechanism of NFB delivery, the resulting signal voltage linearity of triodes is superior to any other known amplifier which all depend on external loops of NFB to ensure their use
results in a linear amplifier.

Some other wondrous facts about triodes need to be pointed out. If a triode is connected to a very low value
of load RL, say less than one tenth of Ra, the voltage gain will very low, and there will be virtually no NFB applied from the
anode to the electron stream, so the triode produces its highest amount of distortion.
Consider the 6SN7 in the above example and idle condition of Ea = 140V and Ia = 3.4mA.
If the 6SN7 has an RL = 1k, A = 1.42, and the maximum Vout would be only about 2Vrms with 10% THD.
If however the same 6SN7 was loaded with RL = a constant current source or some impedance in excess of 1M ohm, then there is virtually no current change, and we would see a gain about = µ, and maximum output voltage will be about 70Vrms
and the THD will be under 1%. Where the load value is as high as possible, there is a maximum of applied internal NFB.

At this point is seems appropriate to discuss pentodes and beam tetrodes to illustrate that in fact there is NFB in triodes while there is virtually none operating in multigrid tubes.

Long ago in the 1930s someone placed a second grid between the control grid and the anode using similar wire structure
for the helical winding. This was the screen grid, and usually connected to a fixed
positive voltage, Eg2, close to the Ea. Then they added a third fine wire grid between the anode and screen but with a
coarser pitch of turns and connected this to tha cathode. So there are 3 grids in a such a tube, called a pentode because of the
total of 5 electrodes.
The typical set up is shown in the schematic :-
Fig 5.


Electrons in the space charge around the cathode are only affected by the control grid voltage.
The screen grid which is at a fixed voltage prevents the anode voltage from having any major effect on Ia.
The screen grid wires are aligned to be in line with the control grid wires so that once the electrons have passed between the
control grid wires they are accelerated by the electrostatic field effect of the positive screen voltage but mostly do not strike the screen grid wires, but pass between them and continue on to be absorbed by the anode regardless of its voltage
which at all times will be positive. Some electrons striking the anode at high velocity cause other electrons
in atom orbits to be dislodged, or the electrons arriving bounce, and these will try to either return to the anode
or move towards the screen, which is also positive. This is called secondary emission, and occurs in all tubes
including triodes, but in pentodes the effect generates serious non linearity and dysfunction when the anode swings to a voltage
less than the screen voltage. Suddenly the screen attracts many electrons rather than the 10% to 30% of total cathode
current than it does during normal operation. The suppressor applies an electrostatic voltage field at cathode potential
between the screen and anode, and the secondary emission electrons which have much less velocity than the main electron
flow arriving at the anode will be turned back to the anode by electrostatic repulsion instead of moving to the screen.

The suppressor action was duplicated in the beam tetrode by using beam forming plates to concentrate the electron stream
into beams of electrons which themselves form such a concentrated stream of negative particles with a negative charge that
this repels any secondary emitted electrons and forces them to return to the anode only.
With suppression action, the pentode and beam tetrode offers stable operation.
There are very few small signal beam tetrodes, and most  small multigrids are pentodes.
The 6AU6 has a fairly high screen current % of total tube current but it matters not because of the low power operation and the
benefits gained by the designer when using a pentode. In much equipment made in the 1950s and 1960s, pentodes such as the 6AU6 and EF86 were favoured in audio circuits because they had more circuit gain than triodes and this allowed
more NFB to be used to make amplifier channel gains equal, and force distortion and noise and output resistance to be low
and all without such a large and expensive tube count.
Most beam tetrodes are power output tubes where they usually have screen currents less than 10% of anode currents
and less screen current of power pentodes.
The pentode or beam tube is best set up as a class A single tube with Eg2 at about 2/3 of the Ea value. Having said that
its is wise to adjust Eg2 to be the lowest value which still allows the maximum voltage swing at lowest THD
into the chosen load. 

As I said, most electrons moving towards the anode miss the screen wires but some are absorbed and thus unlike the
control grid, screen current flows at all times unless the tube is cut off by a very negative control grid voltage.
There is always between about 5% and 35% of tube current flow devoted to the screen, but the screen signal current
is somewhat more non-linear than the anode current. The screen voltage supply to the above typical 6AU6
would usually simply have a low value electrolytic of about 10uF connected from screen to cathode and with about 150k
from the screen to B+ supply of say 300V.

When you have a screen structure that prevents the anode voltage from having any effect on the electron flow,
the control grid has a free hand to cause current flow irrespective of the load value connected.
A typical signal pentode such as the 6AU6 connected in the above schematic but with a fixed Eg2 supply of say 100V
and suitable R1 = 440ohms, Ek = 1.5V, Ia = 3.4 mA, Eg2 = 100V, Ra = 1.5M ohms at least, and gm = approximately 3mA/V. Since for all tubes µ = gm x Ra, then the µ for the typical 6AU6 pentode = 1,500,000 x 0.003 = 4,500.
This implies that if we could arrange the anode of the pentode to have a constant current source for its
dc supply, ie, a load with an infinite ohm value, then the voltage gain would be 4,500.
And were we to make changes to Ea or the B+, of  +/-10V, virtually no anode current change occurs.
We would also find that that gain would change almost proportionately to the RL value; the pentode
would appear to be a very high output impedance device compared to load value, ie, a current generator rather than a voltage
generator like a triode because the Ra of a triode is much lower than the load which is normally used.

The pentode will also produce much more THD than the triode of similar position in a given circuit because there is no
local electrostatic NFB from anode to grid. The action of distortion voltages produced at the anode have no effect on the electron stream.

Voltage gain of the pentode is much higher than triode. For the 6AU6, we can apply the gain formula for a 32k RL.
Gain = 4,500 x 32k / ( 1,500k + 32k ) = 94.
The more approximate formula for pentode gain is simply gm x RL, so if gm =  3mA/V, gain into 32k
= 0.003 x 32,000 = 96, just slightly more than the formula says which bothers to include Ra.
Where RL is less than 1/10 of Ra the pentode gain is approximately gm x RL.

Consider the same 6AU6 connected as a triode with its screen and suppressor connected to its anode.
The triode thus created behaves as if there was a solid metal anode placed where the screen is located and of the same
size as the screen dimensions. Current flow is affected by the change in screen voltage which is the same as the anode voltage.
the triode connected pentode has Ra = 12k approx, gm = 3mA/V, and µ = 36.
So with RL = 32k, gain = 36 x 32k / ( 12k + 32k ) = 26.
This is about a 1/4 of the gain of the pentode, but we would find the lower Ra and lower THD to be favourable.
The pentode would usually have to be enclosed by an external NFB resistor network to reduce its THD and Ra to that of a triode to be more useful in a given simple circuit for audio so we may as well stay with a triode and its simplicity.
This is 11.1dB of NFB acting in the 6AU6 when triode connected.

An equivalent model of the 6AU6 in triode can be depicted with a resistive shunt NFB loop used with the 6AU6 in pentode.
For this we would have 52k ohms between signal input and grid, and 1.9M between anode and grid ( after the output
anode de-coupling DC blocking cap ). Allow the Ea/Ia load for the example above and with RL = 32k.
So where we have -94V at the anode, there is +1V at the grid, there is 0.05mA of flow in the 1.9M.
There will be the same current in the 52k, so we have +2.6V across the 52k, so Vin = + ( 2.6V + 1V ) = +3.6V.
oerall gain is thus 94 / 3.6 = 26, the same as the triode produces.
ß, the fraction of output volotage fed back = Rin / ( Rfb + Rin ) = 52 / 1,952 = 0.0266.

Output resistance of the pentode with NFB applied = Ra / ( 1 + [ µ x ß ] ) = 1,500,000 / ( 1 + [ 4,500 x 0.0266 ] )
= 12.4k, which is very close to what the quoted data figure is for Ra in triode.

One may ask how much NFB is in a triode like a 6SN7? We could assume that if we had a similar tube with the same gm
of 1.5mA/V for the above Ea/Ia conditions, and with a screen that the Ra might be 2M. Therefore µ = 3000.

In this case  we have Ra' = Ra / [1 + ( µ x ß )] = 2,000k / 1 + ( 3000 x ß ), since we know
everything except ß. ( We can safely neglect the figure 1, which becomes relevant in triode amps with low µ.)
Therefore Ra' = 2,000k / 3000 x ß
ß  is the fraction of output voltage fed back in shunt with the input and is determined by the the ratio of distances of the grid to cathode and screen/anode to cathode. For 6SN7 we get ß = 0.05, ie, the ratio of electrostatic field effects from anode and control is about 1:20, or 1 / µ of the triode.
Applying ß = 0.05, we get Ra = 2,000k / ( 3000 x 0.05 ) = 13.3k, very close to what we actually measure.
If we apply any value of Ra for a hypothetical pentode, it could be 10M, and if gm is still 1.5mA/V then µ = 15,000,
so for ß = 0.05 Ra' = 10,000k / ( 15,000 x 0.05 ) = 13.3k.

It cannot be known what the Ra and µ would be for a hypothetical 6SN7 pentode would be. What we do know is that the real 6SN7 is just like a pentode but with an electrostatic shunt NFB in place where ß = 1 / µ.

If we consider that the the triode is merely some high Ra device with a shunt NFB loop fitted internally,
then voltage gain with NFB, A'  = gm x RL / ( 1 + [ gm x RL x ß ] ), because for a pentode gain without NFB = gm x RL.
So for a 6SN7 in the above case we know gm = 1.5mA/V and ß = 1 / µ,
so A' = gm x RL  / ( 1 + [ gm x RL / µ ] ).
For RL = 32k, A' = 0.0015 x 32,000 / ( 1 + [ 0.0015 x 32,000 / 20 ] ) = 14.1.

The simple formula for gain A = µ x RL / ( RL + Ra )
We have a 6SN7 where Ra = 13k, µ = 20 and RL = 32k, so gain
 = 20 x 32k / ( 32k + 13k ) = 640 / 45 =  14.22, very close to what the other formula says.

Both formulas very nearly agree with the load line analysis, and all three nearly agree with what we might measure,
since theory is one thing, but factories produce tubes which are not exactly all the same.

I rarely ever use pentodes in small signal amps or in power amps as input or driver tubes. When I do though they are usually triode connected.
I have more to say about expoiting the possible very high gain of a pentode in later some discussion of possible circuitry
which I know I could employ but do not because of the complexity involved. I have left the mention of using constant current source loading of triodes to a later stage of discussions on basic topologies for triodes.
See my page on 'various circuit topologies'.
In any line level preamp stage or power amp input stage I would rarely ever need more gain than say 15, or about what a 6SN7, 6CG7, or 12AU7 might give.
Such triodes are very linear at the listening levels which are well below their maximum output ability.
Where lower than line level signals are to be handled as in the case of a phono amp, I may use
12AX7, 12AY7, 12AT7, 6EJ7 in triode, and with 6CG7 or 12AU7 as cathode follower output buffers.

There is slight disadvantage with triodes due to what is called the Miller effect. There is always some capacitance between
the grid input and cathode, Cgk, and between grid and anode, Cga. The data for triodes usually gives the capacitance
of Cgk and Cga in a low amount of pF, measured when the tube is without any signal present, and usually both values are so low as to be negligible were it not for the effect of the gain of the triode.
In a normal common cathode amplifier as in Fig1, the Cgk is the data figure and usually less than 4pF.
but where there is gain, the Cga measured at the grid with signal present appears to be Cga x tube gain.
This is because if +1V is applied to the grid, and -14V appears at the anode almost instantly, it is as if you had to apply
1V input to 14 times the Cga, so 3pF Cga becomes 42pF with gain = 14. So if the source impedance driving the
grid of the triode was 100k, then you have an RC low pass filter with a -3dB point, or pole, at 15.9kHz,
which is lower than we would wish for.
A 12AX7 with gain set for 90 and Cga = 1.7pF will have Miller capacitance = 153pF, and to get a pole at 65kHz, the source resistance must be less than 16 kohms.

Readers should now move to Loadline Analysis for small signal tubes in Tube Operation 2.

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