Block diagrams can be very useful for visualizing feedback and avoiding errors in reducing a model a polynomial expression for the transfer function. (Really, they're not that tough. You just need to get the hang of them.)
If you want to use block diagrams (and thus block diagram 'algebra'), here is "Black's Formula" for simplifying negative feedback:
Let us define an input equal to 'x' and output
of 'y'. Then:
y = G*(x - (H*y))
y = G*(x - G*(H*x)
y*(1 + G*H) = G*x
We wish to find the
transfer function
, which is defined as (OUTPUT)/(INPUT), or (for us) (y/x):
(y/x) = G/(1 + G*H).
Thus, these two block diagram representations are
equivalent. (The righthand reduction just puts the lefthand block diagram into a neater format...)
