TUBE OPERATION  1
SMALL SIGNAL AMPLIFIERS, edited 2012.

Contents :-
Basic tube operation, cathodes, anodes and grids, diodes, triodes and
pentodes.
Fig 1, Vacuum diode internal structure,
Fig 2, Triode internal structure,
Parameters of Ra, µ and gm.
Fig3. 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 modeled as a generator.
Fig 4. 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 5. Electrostatic effects in a 6SN7 triode.
NFB in the triode, how it reduces THD.
Fig 6. Pentode internal structure.
Fig 7. 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 stored in the basement. But many
institutions have thrown out many of their old pre-1960 books which are no longer
read by anyone. 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 first vacuum tubes were called electron valves, because this referred to the
way these electronic devices controlled current flow rather like a valve controls
gas or fluid flow. The first simplest "Valve" was a diode with only cathode and anode
within a sealed glass container called a "tube." Valves evolved to have control grids
and then more grids to give more complex operation uses but each such valve had
at least a cathode and anode. Over time more than one valve was placed within a
tube so twin triodes or triode-pentodes and many others wee developed.
The word "tube" describes the glass container with a certain type number printed
on the glass. So a 6SN7 is seen on the glass envelope of a tube two identical triodes. 


The simplest vacuum tube is the diode, which consists of an electron emitting
cathode and an electron absorbing anode, also called the "plate", both mounted
within a glass container, usually a glass tube sealed at each end.
The flow of electrons between cathode and anode require these electrodes operate
in a "hard vacuum" with extremely low amount of gas molecules present. In small
signal tubes the cathode is usually a 2mm to 5mm dia nickel tube heated from inside
with an inserted folded length of heating wire which is oxide coated for insulation and
to give very low emission. The nickel cathode tube is
coated with strontium oxide plus
calcium oxide, barium oxide, or other oxides and heated
to between to between 800C
and 1,000C. The heat gives the electrons in the molecules of the cathode structure
sufficient energy to spin off into the vacuum where they circulate, then lose energy,
and fall back into the cathode. The diode 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 builds 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 because
the charge tends to repel negatively electrons trying to take off from the cathode
surface.
The cloud of electrons is called the "space charge."

Fig 1.
diode-internals.GIF

The anode is a metal cylinder or open ended box around the cathode within the tube.
With the anode connected to the cathode so that both anode and cathode have the
same voltage potential, the presence of the anode has no effect on the space charge. 

Look at (A) above.
If the anode is connected to a positive voltage from a battery and cathode to a
negative voltage from a battery the anode is then charged at a higher voltage
potential than cathode and some electrons in the space charge are attracted to
the anode which absorbs them and an electron current flow circulates around a
circuit formed by diode, battery, and connection wiring. The magnitude of the current
depends on the battery or dc voltage supply value, and the electron emission
capabilities of the cathode and distance between anode and cathode.

If we measured the currents at various positive voltages between anode and cathode,
Ea, we would be able to plot a graph with Ia on a vertical axis, and Ea on a horizontal
axis. If we joined the dots we would have a curve and at any point along the curve we
might draw a tangent and this straight line will pass through an Ea point and Ia point.
Using Ohms' Law, R = E / I will give us the ANODE RESISTANCE, Ra, for the diode
which changes with Ea.

In fact, Ia = K x sq.root of Ea cubed. K is a constant, and determined by the tube
dimensions, but a plotted curve to show Ra may be done without knowing what K is.
The curves obtained are the curves whose slope at any point indicate the anode
resistance, Ra, of the tube under test. The curves for any tube may be found in data for
the particular tube. When you view most tube "Ea vs Ia anode curves" for diode, triode,
tetrode, pentode, you are looking at a set of Ra diode lines for various values of control
grid Vdc bias voltage. But here is a picture of a typical rectifier diode Ra curve -
Fig 2.
5AR4-GZ37anode-curves-1-diode.GIF

Now look at (B), Fig 1 above.
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.

(C) shows the diode schematically. From this we see there is no connection or electronic
function by the heater filament, and only cathode and anode are available to be used in
a diode use application.

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 some electrons will stream past the fine grid
wires of the grid to get the anode if it has a positive voltage.
Fig 3.
triode-internals.GIF
But when the grid is supplied with a negative voltage, - Eg,  with respect to the cathode
voltage, Ek, the resulting negative voltage field effect of the fine wires repels some electrons
and prevents them from flowing to the anode. The negative grid voltage applied to the grid
reduces electron flow to the anode. The electron flow is in a state of equilibrium depending
on the summed effects of charge fields from anode and grid. While the grid remains negative
there is no current flow
into or out of the grid 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 negative, the electron flow may be be suppressed no matter how positive
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, it absorbs electrons and a grid current
then flows, and it is less than the anode current flow.

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.
If the grid is connected to the anode, the triode
becomes a diode with different Ea/Ia curves.
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. Ra is not just calculated by measuring Ea across a working tube and then
measuring current, and calculating Ra = Ea / Ia.
Ra is measured by measuring a small change of Ea and resulting small change in Ia, and Ra
is then calculated Ra = change in Ea / Change in Ia. For example, if you have a 6550
connected as a triode and Ea = 500Vdc, and Ia = 50mAdc, then one might assume
Ra = 500V / 0.05A = 10,000 ohms. That is quite wrong.
But should we change Ea from +500Vdc to +550Vdc, we could expect to find that Ia would
change from 50mA to 95mA, and the Ia change = +45mAdc.
So Ra = 50V / 0.045A = 1,111 ohms, which is the correct answer. Such an observation
of a triode is made without change to voltage between cathode and grid. In fact, the triode
is the equivalent to an imaginary 3 terminal device within a "mystery box" which has 3
terminals, anode, cathode and grid, and where there is an imaginary low impedance signal
voltage generator plus a series resistance equal to Ra to the output anode terminal.
Ra can be calculated while applying a small AC voltage to the anode and measuring current
change :-
Ra  =  Va rms  / Ia rms
Where Ra is anode resistance in ohms,
Va = signal voltage change between anode and cathode,
Ia =  signal current change in low value series R from voltage generator and for
a fixed value of grid voltage.

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.


Amplification factor, known as µ, is just a number without units and is the voltage
gain of a triode or other 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 where there are changes to Ia and Ea dc
for the Vdc or Ia dc operating condition. It can be easily measured by connecting a
"constant current supply" to the anode which has an extremely high impedance at any
signal frequency simply measuring input and output voltages at low levels.
µ = Va anode signal output voltage / g1 grid input signal voltage where the load is a
constant current source.

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 Fig 4 :-
Fig 4.
schematic of basic
        triode signal amp.
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 favors from the king to his people but they knew the real direction of
taxation flow! )
So convention has it that current always flows from Positive to Negative 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 realize
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 amplifier to power 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/or
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.

Let us consider the DC operation of the triode in Fig4.


The above Fig 3 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 Vdc in the 1.5k so that about +5Vdc exists at the cathode.
The cathode Vdc, Ek, 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 negative 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 dc 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 less 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 cathode bypass 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 if R1 = 1.5k didn't exist or was bypassed with a large value capacitor.
So the measured resistance "looking into" the anode circuit of the triode is 44.5k,
not 13k. At Fig 4, 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 of 20%.
The R1 is an application of series 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
modeled this way with a 6AU6 having a generator producing 4,500Vrms output
for 1Vrms input because µ = 4,500. 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 analyzing circuit behavior 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 5.
Schematic for 1/2
        6SN7 generator model.
Fig 5 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 5 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.

Consider the ac signal operation of the triode in Fig 5.
Consider the tube with no signal is happy to work with dc to give the state of
equilibrium of dc voltages shown on the Fig 4 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 X in ohms at a given frequency of sine wave
signal =
XC  =  1,000,000               
           6.28 x C x F
where XC 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 signal F = 1 kHz, regarded as the mid frequency for audio amplifiers.

XC 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 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 Fig 4 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
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 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.
The amount of NFB applied and the voltage gain are usually expressed in dB by
engineers.
Gain in db = 20 x log ( Vout / Vin )
In this case, two lots of gain are considered, the "open loop gain", OLG, where there is
no NFB applied, and "closed loop gain", CLG where NFB has been applied.
In this example, dB of OLG = 20 x log 14 = +22.9dB. Gain could be a negative quantity
where output is less than input in an application.
In this example, dB of CLG = 20 x log 8.5 = +18.6dB. The difference in CLG and OLG
is the amount of dB gain reduction because of local current FB using the 1k5 unbypassed.
So NFB applied = 22.9dB - 18.6dB = -4.3dB.
An easier way to calculate dB of NFB = 20 x log ( CLG / OLG ) = 20 log x ( 8.5 / 14.0 )
= -4.3dB.

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

Fig 6.
Electrostatic
        fields in a 6SN7 triode.
The Fig 6 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. The space charge "feels" the effect of the anode voltage
because there is open space between grid wires; if the grid was a sheet of
metal the anode field would be interrupted completely. Some reduction of the
strength of field effect by anode is due to the grid's presence but the two
electrodes of grid and anode still have a summed effect on the space charge.

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 Va because more Ia in the RL means a lower
Va. 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
cathode space charge and thus 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 = K-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 such as pentodes and SS
devices 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 voltage caused NFB applied from
the anode to the electron stream, so the triode produces its highest amount of
distortion for a given current change. Consider the 6SN7 in the above example
and idle condition of Ea = 140V and Ia = 3.4mA. If the 6SN7 has an RL = 1k,
gain A = 0.6, and the maximum Va would be only about 2Vrms at clipping for a
3.3Vrms grid input signal and THD may be 10%. If however the same 6SN7 was
loaded with RL = a constant current source or some impedance in excess of
1Meg ohm, ie, 1 million ohms, then for 3.3Vrms input, Va would be 3.3 x 20 = 66Vrms
with virtually no current change because with a CCS load gain  = µ, and maximum
output voltage will be about 66 Vrms and the THD will be under 1%. Where the
load value is as high as possible, there is a maximum of applied internal NFB.

Applied voltage NFB reduces the gain which exists without NFB applied.
Now where
A = OLG = gain without NFB, A' = CLG = gain with NFB,
ß = fraction of output voltage fed back to input,

A'  = A / (1 + [A x ß] ) and 1 is a needed constant for the equation to work.

ß is determined by the ratio of small distance and large distance shown in Fig 6, above.
The distances act like resistances in a shunt FB circuit, input to grid = R1 and
grid to anode = R2. ß = R1 / ( R1 + R2 ).

Where R1 = 1/20 of R2, ß = 1 / 21 = 0.0476

We do not know what A, or µ would be if all the anode electrostatic field was 
prevented from having any effect on the space charge and electron flow.
But we do know µ = gm x Ra, and Ra of triodes is low because of the internal NFB,
and if there was no internal NFB we might expect Ra to be a very high number of ohms.
If the anode were completely screened off from the space charge then no current
change could occur for any anode voltage change so Ra might be calculated as
Ra = Va change / Ia change = Va / 0.0 = infinity, somewhat impossible in the real world.
So Ra must have a finite value for the simple formulas to work.
In fact, the simple formula µ = Gm x Ra probably is a simplified form of a more complex
equation and thus works easily and well enough for audio frequencies without having to
include capacitance effects. But from what simple formulas we have, and for 6SN7
and RL = CCS and with ß = 0.0476, A' = 20 which is the triode µ, or maximum triode
gain without any current change.
Now A' = A / ( 1 + [A x ß] ), and we can solve the equation to find A if we know all
other unknowns,
then 20 = A / / ( 1 + A x 0.0476 ).
then A = 20 x (1 + [ A x 0.0476 ] ),
A = 20 + 0.952A,
A - 0.952A = 20,
0.0476A = 20,
Therefore A = 20/0.0476 = 420.
In the 6SN7 we could suggest the gain without NFB would be 420, and is reduced to 20
with the NFB so the gain reduction factor = 20/420 = 0.0476 or approximately 1/µ,
where µ is the triode µ.
This reduction is -27dB, and a lot.
I am no expert but for a 12AX7 with µ = 100, the CCS load gives a gain reduction of 1/100,
and any THD would be very low. Triodes are never perfect though, and have voltage
range limits and the examination of the anode Ea/Ia curves do show triodes like 12AX7
and 6SN7 and many directly heated triodes to be extremely linear with a CCS load which
appears as a horizontal load line if it was plotted across the Ra curves.
The wonderful fact about any triode is that it can produce a very predictable and linear
voltage gain character and this linearity is highest when there is no Ia current change, ie,
the current flow is kept constant regardless of the change in Ea. The notion of the triode
grid having transconductance where a change in Eg causes change in Ia seems to not
apply when the triode has a constant current load. In effect, a CCS, or Constant Current
Source load is like having an infinitely high ohm resistance between anode and a B+ supply.
I = V / R, and if R is infinity, then I = 0.0 amps. Such considerations as these always confound
the novice, and always confuse those wanting simplistic explanations for something that
cannot be explained as simply as they wish. Nature made the Universe complex, so get
used to complexity, and nearly all cause and effect relationships are non linear.
Wherever a linear relationship is found, it is usually due to a negative FB effect.

Now very good equipment is needed to accurately draw a set of curves for a given triode.
Very good linearity is 1% THD for 100Vrms of output. I have measured such low THD in
a triode loaded by something approaching a CCS, but often the Ea / Ia curves printed
in old data and text books were drawn using equipment which had a lot of THD over 1%
so you cannot use the curves to accurately predict THD which may in fact be less or more
than the curves indicate. With the 6SN7 in the example in fig 3, with 32k load and
Vg = 3.3Vrms, Va = 46Vrms and THD would be approximately 4%. The effect of loading
the triode to cause substantial current change reduces applied internal NFB and increases
THD. Therefore to achieve low distortion in small signal amps the load should be kept high
or to a figure of over 10 x Ra. This is achievable using CCS loading for dc supply to anodes
or a choke with an added series R. Or with an extra high B+ with high value DC carrying
RL, or by means of having 2 triodes arranged as a µ-follower as in some of my preamp
schematics.

Pentodes and beam tetrodes.
At this point is seems appropriate to illustrate that although there is NFB in triodes, there is
virtually no NFB operating within pentodes and beam tetrodes.

Long ago in the 1920s someone placed a second grid between the control grid g1 and the
anode using similar wire structure for the g1 helical winding. This was the screen grid, g2,
and it was usually connected to a fixed positive voltage, Eg2, slightly less than Ea.
This caused the anode voltage change to have very little field effect on the space charge and
cathode to anode electron current flow so Ra became very high while gm stayed the same.
Now as µ = gm x Ra, µ could be thousands rather than the maximum of about 100 as it is in
a 12AX7.
But the 1922 tetrode has some really strange secondary emission effects where Va swing
moved well below Eg2 so the pure 4 electrode tetrode wasn't much used and is never used
now. But late in 1920s Phillips engineers added a third grid, g3 between the anode and
screen and this had a smaller wire pitch and was kept at a low voltage equal to the cathode :-
Fig 7.
pentode-internals.GIF
The extra grid g3 was called the suppressor grid, because it placed a slight
negative voltage field between the anode and screen and prevented electrons
bouncing off the anode and being absorbed by the screen when the anode
voltage swung to a lower voltage than the screen. So there are 3 grids in and
it is called a pentode because of the total of 5 electrodes. Fig 7 above shows
the pentode structure :-
Fig 7.
Electrostatic
        fields in pentodes.
Electrons in the space charge around the cathode are mainly affected by the
control grid g1 voltage. The screen grid which is at a fixed voltage prevents the
anode voltage from having any major effect on Ia. Screen grid wires are aligned
in the shadows of the g1 wires so that electrons traveling to the anode don't
have to change direction. But some electrons are pulled to the positive screen
so screens always have dc current, and in typical small signal pentodes it is
roughly 1/3 of anode dc current. Once 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 some 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.

In Beam Tetrodes, there is no actual suppressor grid but the necessary suppressor
action was duplicated by using 2 beam forming plates located between screen and
anode and each side of the assembly to concentrate the electron stream into 2
equal 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 both offer similar stable operation in signal circuits.
There are very few small signal beam tetrodes, and most small "multigrids" are
pentodes. The 6AU6 has a fairly high screen current percentage of total tube current
but it does not matter 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 favored in audio circuits because
they gave more circuit gain than triodes and this allowed more external loop 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 number
of tubes. Most beam tetrodes are power output tubes where they usually have
screen currents less than 10% of anode currents, eg, 6L6GC, and less screen
current of power pentodes, eg EL34. The pentode or beam tetrode is best set
up as a class A single tube with Eg2 at about 2/3 of the Ea value. Having said that
it 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 signal
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 anode 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. But where an RL is used,
we would also find that that gain would change almost proportionately to the RL
value; A = pentode gm x RL. 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 very high Ra of pentodes and lack of internal
allows amplification of radio frequencies without the Miller effect which limits
most triode operation to about 1MHz. A 6AU6 is happy at 10MHz, and many
pentodes will work to well over 100MHz. The high Ra means that the use of tuned
LC circuits may be used to take advantage of the frequency selectivity available
with tuned circuits and hence radio and television became easily possible
with pentodes.

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 virtually no effect
on the electron stream. The spectra of harmonics produced in pentodes and
beam tetrodes is usually a lot more complex with a higher % of odd numbered
than even numbered H mostly produced by triodes.

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.

Pentodes may be made to work like triodes.
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 virtual
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 6AU6 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 favorable. 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. The gain reduction factor by triode connection is 26/94,
and is -11.1dB of NFB.

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 change at the anode, there is +1V change 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 Vg1 to 0V = Vin = + ( 2.6V + 1V ) = +3.6V.
overall gain is thus 94 / 3.6 = 26, the same as the triode produces. ß, the
fraction of output voltage 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 without NFB,
µ = 2,000,000 x 0.0015 = 3,000.

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. But theory is one thing, and
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 exploiting 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, 6DJ8, 6EJ7, 6BX6 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 Fig4, the Cgk is the data figure and usually less than 4pF.
But where there is gain, the Cga measured at the grid without signal becomes
Cga x tube gain when signal is present. 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.

To Tube operation 3

To tube operation 4

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