In some recent articles I posted here I showed that the formulas I
thought were OK simply
were not OK for use with shunt FB around an inverting amp.
After a night's sleep, I speny an hour untangling the mystery
of shunt NFB formulas that don'r work properly, and came up with this
one....
Gain with shunt feedback applied, or closed loop gain,
A' = ( A - Axß ) / ( Axß + 1 )
Where ß = R1 / ( R1 + R2 ),
A is open loop gain measured when FB network is connected.
A' is closed loop gain.
Let's see if it works, to prove this is right.
Example 1, using triode with A = 20.
Let us suppose this is a 1/2 x 6DJ8 Ia = 5mA, so Ra = 5k, µ =33, RL = 8k
approx.
R1 = R2 = 210k , so ß = R1 / ( R1 + R2 ) = 0.5.
This set up is an "anode follower".
A' = ( 20 - 20x0.5 ) / ( 20x0.5 +1 )
= 10 / ( 10 + 1 )
= 10 / 11, which is what we would measure,
ie, -11V input gives +10V output.
Example 2, using triode with gain = 20.
R1 = 70k, R2 = 210k, so ß = 70 / ( 70 + 210 ) = 0.25.
A' = ( 20 - 20x0.25 ) / ( 20x0.25 +1 )
= 15 / 6
= 2.5, which agrees with what we would measure,
because for -4V input, we would get +10V output.
How I derived the correct formula is my secret, and involves several
pages of trying to epress gain in terms of A, R1 and R2.
But since ß = R1 / ( R1 + R2 ), then if we let R1 = 1, then ß = 1 / 1 +
R2.
So finally by a process of schoolboy agebraic elimination of unwanted
terms I ended
up with an equation for A' that involved ONLY ß, A and 1.0, and without
having to know values of R1 or R2, but just knowing ß.
I am not sure of the equation for the output resistance, ie, effective
Ra, or Ra'
after NFB has been applied.
But Ra' depends on µ, so perhaps we might say
Ra' = Ra / ( 1 + µxß ).
Consider example 1, the classic anode follower.
This would mean Ra' = 5k / 1 + 33 x 0.5
= 5k / 17.5 = 284 ohms approx.
To check this you need to adjust the value of RL to
change the gain.
Rout = change in anode output voltage / change in load current.
The A for varying loads can be easily we worked out from A = µ x RL / (
RL + Ra ),
correct for all tubes.
Patrick Turner.
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