# transfer function and error gain function 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 ?

• The stable output condition for an input of 1.0 is 0.5. Does that help at all? Regarding the rest of your question, I'm not entirely sure what you mean. Apr 10, 2014 at 18:50
• @Andyaka since 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.
– hbak
Apr 10, 2014 at 19:02

Let me label your circuit nodes for better understanding of your question simulate this circuit – Schematic created using CircuitLab

• 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:

$$0\leq TF<1$$

• the transfer function that i meant , that the function that you entered on it an input and give you an output , but the transfer function that you define , is the function that give it desired output 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 ....
– hbak
Apr 19, 2014 at 9:33
• Can you be clearer? 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. Apr 28, 2014 at 13:11