Question

For the feedback system shown, the transfer function \(\dfrac{E(s)}{R(s)}\) is:

• A. \(\dfrac{G}{1+GH}\)
• B. \(\dfrac{GH}{1+GH}\)
• C. \(\dfrac{1}{1+GH}\)
• D. \(\dfrac{1}{1+G}\)

Hint:

Always use \(C = GE\) in feedback loops to derive relationships. The error signal ratio is always \(1/(1+GH)\).

Answer 1

The Correct Option is C

Step 1: Write the standard feedback equation.
For unity summing junction:
\[ E(s) = R(s) - H(s)C(s) \]

Step 2: Substitute forward path output.
Since \(C(s) = G(s)E(s)\):
\[ E(s) = R(s) - H G E(s) \]

Step 3: Solve for \(\dfrac{E(s)}{R(s)}\).
\[ E(s)(1 + GH) = R(s) \] \[ \frac{E(s)}{R(s)} = \frac{1}{1 + GH} \]

Step 4: Conclusion.
The correct option is (C).

Final Answer: \(\boxed{\dfrac{1}{1+GH}}\)