transfer function and error gain function

Question

^{simulate this circuit – Schematic created using CircuitLab}

I know the difference between those two functions , but i want to see the relation between them , i think that the error gain function , is given by the transfer function , and the best transfer function which is

1. stable.
2. has an error gain function equal to 1.

if this is true , how can i derive the error gain function , from transfer function ?

Answer 1

Let me label your circuit nodes for better understanding of your question

^{simulate this circuit – Schematic created using CircuitLab}

To answer your question I must first clarify some definitions:

• Error gain function: a gain k that multiplies the error e = r-y
• Transfer function: the behavior of the closed-loop system y/r

Now,

$$ y=k\cdot e $$ $$ e=r-y $$ $$ y=k\cdot (r-y) $$

So, the error gain function follows as

$$ k=\frac{y}{r-y} $$

So, you don't even need the transfer function to establish the value of k. You just need y and r.

Anyway the transfer function is

$$ TF=\frac{y}{r}=\frac{k}{k+1} $$

So, if you're given the transfer function (note that this is just a constant for this theoretical system) you can find k as

$$ k=\frac{TF}{1-TF} $$

Note also that, with negative feedback, in stable system conditions, and with finite k:

$$ 0\leq TF<1 $$