# Barkhausen's tubular formula

The Barkhausensche tube formula , named after the German physicist Heinrich Georg Barkhausen , summarizes the three variables characterizing an electron tube

in a relationship

${\ displaystyle S \ cdot D \ cdot R_ {i} = 1}$

together.

The discovery of the connections and the writing down as a formula is wrongly ascribed to Barkhausen, in fact it was recorded in 1914 by Hendrik van der Bijl , an employee of Western Electric , in the USA. Barkhausen made this formula accessible to a larger audience as part of his extensive research on the processes in electron tubes that began in 1918.

The steepness S is defined as

${\ displaystyle S = \ left [{\ frac {\ partial I_ {a}} {\ partial U_ {g}}} \ right] _ {U_ {a} = {\ rm {constant}}} \ approx \ left [{\ frac {\ Delta I_ {a}} {\ Delta U_ {g}}} \ right] _ {U_ {a} = {\ rm {constant}}}}$

which can be read as the slope of the graph I a plotted over U g .

The relationship applies to penetration D.

${\ displaystyle D = - \ left [{\ frac {\ partial U_ {g}} {\ partial U_ {a}}} \ right] _ {I_ {a} = {\ rm {constant}}} \ approx - \ left [{\ frac {\ Delta U_ {g}} {\ Delta U_ {a}}} \ right] _ {I_ {a} = {\ rm {constant}}}.}$

Finally, the internal resistance R i can be read off as the reciprocal slope of the graph I a plotted against U a . Expressed in a formula:

${\ displaystyle R_ {i} = \ left [{\ frac {\ partial U_ {a}} {\ partial I_ {a}}} \ right] _ {U_ {g} = {\ rm {constant}}} \ approx \ left [{\ frac {\ Delta U_ {a}} {\ Delta I_ {a}}} \ right] _ {U_ {g} = {\ rm {constant}}}.}$

In all three formulas I a denotes the anode current, U a the anode voltage and U g the voltage applied to the grid.

## Sign discrepancies

The sign of the penetration is always incorrectly stated in the specialist literature. Coupled with the fact that the result of the formula is +1, a single negative number within the formula will appear incorrect.

From the definition of the penetration it can be seen that the penetration is the ratio of the anode to grid voltage so that the anode current remains constant. This means that as the anode voltage increases, the grid bias must become more negative in order to keep the anode current constant. Conversely, when the anode voltage drops, the grid bias voltage must become more positive in order to keep the anode current constant. The sign is therefore negative.

On the right-hand side of the tube formula, after all the values ​​have been multiplied, there is still +1, because after the differences have been inserted into the tube formula, these differences must not be reduced. This is mathematically wrong because they hold under different conditions.

Barkhausen's tubular formula is an application of Euler's chain rule for partial derivatives , e.g. B. is often used in thermodynamics. The derivation from the properties of the total differential shows that either all derivatives must be taken positive and placed on the right-hand side −1, or, as here, one of the derivatives must be given a minus sign and +1 on the right-hand side.

The whole thing comes into its own better if one uses differentials instead of the differences. Exact is z. B. the following derivation:

• The behavior of a tube is described by its (non-linear) characteristic field: ${\ displaystyle I_ {a} = f (U_ {g}, U_ {a}). \,}$
• For the small-signal behavior (for which the tube formula only applies) the characteristic is linearized by forming the total differential :${\ displaystyle dI_ {a} = {\ frac {\ partial I_ {a}} {\ partial U_ {g}}} \ cdot dU_ {g} + {\ frac {\ partial I_ {a}} {\ partial U_ {a}}} \ cdot dU_ {a}.}$
• With the definitions for the steepness and the internal resistance one obtains${\ displaystyle S = \ left [{\ frac {\ partial I_ {a}} {\ partial U_ {g}}} \ right] _ {U_ {a} = {\ rm {constant}}}}$${\ displaystyle R_ {i} = \ left [{\ frac {\ partial U_ {a}} {\ partial I_ {a}}} \ right] _ {U_ {g} = {\ rm {constant}}}}$${\ displaystyle dI_ {a} = S \ cdot dU_ {g} + {\ frac {1} {R_ {i}}} \ cdot dU_ {a}.}$
• When “measuring” the penetration, I a is constant (ie dI a  = 0). Therefore applies${\ displaystyle S \ cdot dU_ {g} + {\ frac {1} {R_ {i}}} \ cdot dU_ {a} = 0.}$
• With the definition of the penetration you get exactly after conversion${\ displaystyle D = - \ left [{\ frac {\ partial U_ {g}} {\ partial U_ {a}}} \ right] _ {I_ {a} = {\ rm {constant}}}}$${\ displaystyle S \ cdot R_ {i} \ cdot D = 1.}$

## literature

• Heinrich Barkhausen: Electron tubes, Volume 1 . S. Hirzel, Leipzig 1924.
• Philippow: Basics of electrical engineering . Academic publishing company Geest & Portig K.-G., Leipzig 1967.
• Philippow: Taschenbuch Elektrotechnik - Volume 3 . Verlag Technik, Berlin 1969.
• Pfeifer: Electronics for Physicists - Volume II . Akademie-Verlag, Berlin 1966.
• Schröder: Electrical communications engineering - Volume II . Publishing house for Radio-Foto-Kinotechnik GmbH, Berlin-Borsigwalde 1966.
• Rint: Handbook for RF and electrical engineering - I. band . Publishing house for Radio-Foto-Kinotechnik GmbH, Berlin-Borsigwalde 1964.
• Lange: Signals and Systems - Volume 2 . Verlag Technik, Berlin 1968.
• Ernst Erb: Yesterday's radios . 4th edition. Funk Verlag Bernhard Hein e. K., Dessau-Roßlau 2009, ISBN 978-3-939197-49-2 .

## Individual evidence

1. For example in contemporary specialist literature such as: Friedrich Benz: Introduction to radio technology , Springer Verlag, 3rd edition, 1944, page 150 ff.