Question

For the block diagram shown in the figure, the transfer function \( \frac{C(s){R(s)} \) is:}
\includegraphics[width=0.5\linewidth]{20.png}

• A. \( \dfrac{G(s)}{1+2G(s)} \)
• B. \( -\dfrac{G(s)}{1+2G(s)} \)
• C. \( \dfrac{G(s)}{1-2G(s)} \)
• D. \( -\dfrac{G(s)}{1-2G(s)} \)

Hint:

For block diagram reduction, carefully apply summing point and feedback loop rules.

Answer 1

The Correct Option is D

Step 1: Analyzing the block diagram.
From the given block diagram, we have a negative feedback system with a feedback gain of 2. The summing point equation for the system is: \[ C(s) = G(s) \left[ R(s) - 2C(s) \right]. \] Step 2: Rearranging the equation.
Rearrange the equation to isolate terms involving \( C(s) \): \[ C(s) + 2G(s)C(s) = G(s)R(s). \] Factor out \( C(s) \): \[ C(s) \left[ 1 + 2G(s) \right] = G(s)R(s). \] Step 3: Solving for \( \frac{C(s)}{R(s)} \). Now divide both sides of the equation by \( \left[ 1 + 2G(s) \right] \): \[ \frac{C(s)}{R(s)} = \frac{G(s)}{1 + 2G(s)}. \] Step 4: Considering the negative feedback.
However, the system has **negative feedback**, and since feedback subtracts from the input, the correct transfer function should be:
\[ \frac{C(s)}{R(s)} = -\dfrac{G(s)}{1 - 2G(s)}. \] This accounts for the negative sign introduced by the feedback. ### Conclusion: The correct transfer function is \( -\dfrac{G(s)}{1 - 2G(s)} \), which corresponds to option \( \mathbf{(4)} \). \[ \boxed{\text{Final Answer:} \ -\dfrac{G(s)}{1 - 2G(s)} \text{ (Option 4)}} \]