Asymptotic gain model

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

$Eout=\nu *Ec+\rho *Es$

$Ec=A*Ei$

$Ei=\beta *Ec+\xi *Es$

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

Here $Es$ is a source quantity and $Eout$ is an output quantity. In electronics, each of $Es$ and $Eout$ can be either a voltage or a current.

Since we are dealing with physical quantities, $\beta *A$ is dimensionless. $\rho$ has the same dimension as $\xi *A*\nu$.

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 $\xi$, $\beta$ and $\nu$ determines the overall gain, when $\rho$ 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

$A_{t}={\frac {A}{1-A*\beta }}+\rho$

but that is only true if $\xi *\nu =1$, in which case $A_{t}$ becomes dimensionless. The error they make is $\xi$ and $\nu$ cannot be disregarded unless they are dimensionless. Thet also claim that $\beta$ 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 $\beta$ is impedance (voltage/current), whereas the transfer function of the feedback network is dimensionless, and cannot be $\beta$.

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