Negative feedback (thing)

               ______
       +      |      |
  I -----0--->|  A   |-------+----> O
        +^    |______|       |
         |     ______        |
         |    |      |       |
         +----|  B   |<------+
              |______|

Open Loop Gain = A
Closed Loop Gain = O = Forward  =   A   
                   I   1 - Loop   1 - AB
                                           

  _      _
 |   A    |  /
 |_1 - AB_| /  A

 =    1  
   1 - AB

Negative Feedback is when:
  1    < 1
1 - AB

and positive feedback is when:
  1    > 1
1 - AB

On a nyquist diagram the region of positive feedback is the unit circle around the point (-1, j0); the rest is negative feedback.

Effects of Negative Feedback

1. It reduces errors in the system.
2. It reduces the effect of disturbances.
3. It reduces the effect of changes in A.

1. Reduced Errors
   What we normally want for the system above is for O = -I/B (for B = -1, O = I).
   Error can be defined: Output Error = Desired Output - Actual Output
   This is absolute error. A more useful measure would be error ratio:
   Error Ratio = Output Error / Desired Output
   It can be shown from the definition of closed loop gain or open loop gain (actual output) and the desired system gain (desired output) that the error ratio can be shown to be:

   For Closed Loop:
     ER = (-I/B) - AI/(1-AB) =   1   
               (-I/B)          (1-AB)
   
   For Open Loop:
     ER = (-I/B) - AI = 1 + AB
           (-I/B)
   

   Therefore error has been reduced if the error with feedback (closed loop) is less than the error without feedback (open loop). If this is plotted on a nyquist diagram the result looks a little like this:

                     Im
               __    |    __ <------ 0.5
         e-- _/  \_  |  _/  \_  e--
            /      \ | /      \
           /  e++   \|/   e++  \
        --|----------|----------|----Re
           \        /|\        /^
            \_    _/ | \_    _/ |
              \__/   |   \__/   |
                     |          |
                     |         sqrt(2)
   

   Where the line is where error is unchanged, inside error is increased and outside error is decreased.
   
   
2. Reduced Effect of Disturbances
   Disturbances are introduced into the system in the following way:

                             D 
                             |
                  ______     | 
          +      |      |    v
     I -----0--->|  A   |----0--+----> O
           +^    |______|       |
            |     ______        |
            |    |      |       |
            +----|  B   |<------+
                 |______|
   
   For simplicity's sake assume that I = 0
   
   Open Loop, O = D
   Closed Loop, O =   D  
                    (1-AB)    
   
   Thus disturbance is reduced if:
   D >   D    =>   1    < 1
       (1-AB)    (1-AB)
   
   

   This is the definition of negative feedback as above. Thus negative feedback reduces disturbances.

3. Reduced Effect of Changes in A
   Closed Loop Gain = G. Thus change in G from changing A, assuming B is constant, is:

   dG = (1-AB)*(dA/dA) - A * d(1-AB)/da =   1   
   dA              (1-AB)2                (1-AB)2
   

   This is not useful in itself but relative change dG/G is

   dG = dA/(1-AB)2 = dA   1   
   G =   A/(1-AB)    A  (1-AB)
   

   dA/A is the proportional change in A.
   The other term you may recognise as the effect of feedback term which will be less than one for negative feedback. Therefore if feedback is negative then the effect of the change in A will be reduced.