Question

Determine the transfer function \(\dfrac{C(s)}{R(s)}\) of the system shown below.

• A. \[ \frac{G_{1}G_{2}G_{3}G_{4}} {(1+G_{1}G_{2}H_{1})(1+G_{3}G_{4}H_{2})+G_{2}G_{3}H_{3}} \]
• B. \[ \frac{G_{1}G_{4}} {(1+G_{1}G_{2}H_{1}H_{2})+(1+G_{2}G_{3}G_{4}H_{3})} \]
• C. \[ \frac{G_{1}G_{3}G_{4}} {(1+G_{1}G_{2}H_{1})(1+G_{3}G_{4})+(1+G_{2}G_{3}H_{2})} \]
• D. \[ \frac{G_{1}G_{2}G_{3}} {(1+G_{1}G_{2}H_{1})(1+G_{3}G_{4})+(G_{2}G_{3}H_{3}+G_{4})} \]

Hint:

For block diagrams containing multiple feedback loops: \[ T=\frac{\sum P_k\Delta_k}{\Delta} \] using Mason's Gain Formula is usually the fastest method. Always identify:
• Forward paths
• Individual loop gains
• Non-touching loops before calculating \(\Delta\).

Answer 1

The Correct Option is A