About Feedback Theory:

What's the feedback?

It is an electronic technique, known in the control theory, which consists in taking a piece of the output signal and bringing it back at the input of the system.

The advantage of this technique is:

- Check that the system operate correctly;

- Force the system to correct the output if required;

- Immunize the system agaist disturbs & noise;

- Increse the sensibility of the system;

- Increse or Decrese of Input/Output Immitance;

How study this technique?

There are several ways, methods or theorems to study, analyze and design a feedback network; the most famous is the Classical approach presented in a lot of engineering books (as Jacob Millmann's ones). A limitation of this method is often that it's required that the feedback network is Unidirectional network.

This is a strong way to force the study and several problems can rise-up, especially in HF or μwaves applications.

I have read a lot of papers in which several solutions are presented, but often, the formalism is very strict and weighty. One year ago I learned about prof. Bruno Pellegrini [14] and his papers... In this article i want to share with you a summary of these studies, I hope that this will be useful for you as it was for me.

The method is based on the Cut-Insertion Theorem that permits to find a transformation of a generic network into an equivalent one which is simpler. The generic network is transformed in a TTC (three terminal circuit) and this network permits to calculate easily the main properties (α, A, β).

We can consider the generic network N as showed in fig.1 and the trasformed network N' in fig.2

For the equivalence of two generic networks it's necessary to determinate the constraints of N', in particular its quantities (Wr & Wp), which guarantee the equivalence of TTC with N.

Such equivalence is assured if and only if, for any input:

Wr = Wp    &    Wr' = Wp'.

The notation is inspired to the original papers of prof. Pellegrini and W is intended as voltage or current and the " ' " as dual. In the pictures above we can see voltages Vr and Vp and currents Ir' and Ip'.

For the conditions above and the networks showed:

Vr=Vp   & Zp=Vp/Ip  (from Ir=Ip)

In this way we open the ring and we can find the network equations  defined, from the superposition theorem, for Vp=0 & Vin=0.

In particular for Vp=0: 

α =Vr/Vin,    γ= Vout/Vin,    ρ=Ip/Vr

And for Vin=0:

Zp=Ip/Vp,   A=Vout/Vp,     β=Vr/Vout

It is important to notice the arbitrariness of the cut, and also that the function ρ is a null function, otherwise the method doesn't simplify the analysis because the ring remains closed. For this reason it is convenient to verify this condition and eventually to find a new cut.

Founded the network's equation we can calculate the global tranfer function finding:

Vout/Vin= αA/(1-βA) + γ

The main difference with classical approach is the function gamma. It's assumed a non-null value if the feedback network is bidirectional (as in RF or μwaves applications). If the cut is a "good cut" the gamma function may be trascurable.

The βA is named (as you know) Ring-Gain and its value (in dB) is the Reaction Tax. If the ring-gain is negative the reaction is also negative, otherwise positive.

We can see some, classic, example for try the method:

The network's functions:

For Vin=0

Zp=Vp/Ip=hie;

A=Vout/Vp= -hfe*Rc/hie;

Vr=-hfe*ib*Re//(Rb+Zp)* Zp/(Zp+Rb)= -hfe*ib * hie*Re/(hie+Re+Rb)

Vout=hfe*ib*Rc

Therefore β=Vr/Vout= hie Re/(Re+Rb+hie)Rc

For Vp=0

γ = ρ = 0  [γ= Vout/Vin,    ρ=Ip/Vr]  ,

α=Vr/Vin= Zp/(Rb+Zp+Re)= hie/(Rb+hie+Re)

At the end the gain af amplifier is:

Vout/Vin= αA/(1-βA)= hie/(Rb+Re+hie)* (-hfe*Rc/hie)* 1/[1+(hfe*Rc/hie)* hie*Re/Rc(Re+Rb+hie)]

If 1 is minor of (hfe*Rc/hie)* hie*Re/Rc(Re+Rb+hie) we can write:

Vout/Vin = - Rc/Re

As you can see we refind the result of classical approach !!!

When I have a minute ( ... ), I will insert others examples.

[14] Prof. Bruno Pellegrini - Pisa's University - Improved feedback theory