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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
if this is true , how can i derive the error gain function , from transfer function ?
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:
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:
$$ 0\leq TF<1 $$
r , and give you an output y , in another woed , you define the system above as transfer function , and i want the transfer function that described by this error gain function ....
\$\endgroup\$
r is an input, not an output. Anyway the transfer function described by the error gain function (assuming you agree that this error gain function is just k, otherwise I don't understand your question) is what I defined as TF.
\$\endgroup\$
E(s) = Y(s) - X(s)G(S), where X : is the desired output(input to system) , and G is the error gain function , then X(s)G(S) = X(s) . which is equal to Y(s) , then E(s) = 0. \$\endgroup\$