The overall transfer function C/R of the system shown in the figure below is
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In a signal flow graph, if two loops are non-touching, they appear as a product in the denominator. Here, the loops are separated by the gain $G$, making them independent in this specific structure.
1. Analyzing the Loops: Looking at the diagram, we see two individual loops:
• Loop 1 ($L_1$): The loop at the first node with gain $-H_1$.
• Loop 2 ($L_2$): The loop at the output node with gain $-H_2$.
2. Block Diagram Reduction: We can treat these as two separate feedback systems connected in series with a forward gain $G$.
The first node acts as a feedback system with transfer function $T_1 = \frac{1}{1 - (-H_1)} = \frac{1}{1 + H_1}$.
The forward path gain is $G$.
The final node acts as another feedback system with $T_2 = \frac{1}{1 - (-H_2)} = \frac{1}{1 + H_2}$.
3. Final Calculation: Since these components are in cascade (series), the total transfer function $T = \frac{C}{R}$ is the product of individual gains:
$$\frac{C}{R} = T_1 \times G \times T_2$$
$$\frac{C}{R} = \frac{1}{1 + H_1} \times G \times \frac{1}{1 + H_2}$$
$$\frac{C}{R} = \frac{G}{(1 + H_1)(1 + H_2)}$$