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Encyclopedia > Asymptotic gain model

A negative feedback system can be described by the following equations (a picture would be nice):

E o u t = ν E c + ρ E s

E c = A E i

E i = β E c + ξ E s

The model is maybe called "Black's model".

Here $E s$ is a source quantity and $E o u t$ is an output quantity. In electronics, each of $E s$ and $E o u t$ can be either a voltage or a current.

Since we are dealing with physical quantities, $β * A$ is dimensionless. $ρ$ has the same dimension as $ξ * A * ν$ .

Now, a feedback amplifier consist of a gain part and a feedback network. The (usually passive) feedback network makes the total gain precise.

The asymptotic behaviour of the model comes when $A$ goes towards infinity. Then $ξ$ , $β$ and $ν$ determines the overall gain, when $ρ$ is ignored (which is usually the case).

"Structured Electronic Design" (see references) makes a simplification, which perhaps is not correct. They say that the transfer function of the system is

but that is only true if $ξ * ν = 1$ , in which case $A t$ becomes dimensionless. The error they make is $ξ$ and $ν$ cannot be disregarded unless they are dimensionless. Thet also claim that $β$ is the transfer function of the feedback network. This is usually not the case.

Imagine a voltage amplifier where $A$ is taken as the transconductance of a transistor, and where the gain is set by a voltage divider. Then the dimension of $A$ is conductance (current/voltage), the dimension for $β$ is impedance (voltage/current), whereas the transfer function of the feedback network is dimensionless, and cannot be $β$ .

References

"Structured Electronic Design", by C.J.M. Verhoeven, A. van Staveren and G.L.E. Monna (draft Nov 1, 1996)
E.H. Nordholt, "Design of High-Performance Negative-Feedbank Amplifiers", PhD thesis, Delft University of Thechnology, The Netherlands, 1980

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