Question

trouvez la fonction de transfert équivalente du système représenté sur la figure.
$w(p)=\frac{w_1(s)}{w_2(s)+w_3(s)}$
$w(p)=\frac{w_1(s)w_2(s)}{w_2(s)+w_3(s)}$
$w(p)=\frac{w_1(s)w_2(s)}{1 + w_2(s)w_3(s)}$
$w(p)=\frac{w_1(s)w_2(s)}{1 - w_2(s)w_3(s)}$

Explanation:

Step1: Analyze the system structure

This is a feedback - type system. Let's assume the input is $U(s)$ and the output is $Y(s)$. The forward - path transfer function is $W_1(s)W_2(s)$ and the feedback - path transfer function is $W_2(s)W_3(s)$.

Step2: Apply the formula for feedback system

The formula for the transfer function $W(p)$ of a feedback system is $W(p)=\frac{G(s)}{1 + H(s)}$, where $G(s)$ is the forward - path transfer function and $H(s)$ is the feedback - path transfer function. Here, $G(s)=W_1(s)W_2(s)$ and $H(s)=W_2(s)W_3(s)$.
So, $W(p)=\frac{W_1(s)W_2(s)}{1 + W_2(s)W_3(s)}$.

Answer:

$W(p)=\frac{W_1(s)W_2(s)}{1 + W_2(s)W_3(s)}$