Question:
The signal flow graph of a system is shown. The expression for \( Y(s)/X(s) \) is _________
The signal flow graph of a system is shown. The expression for \( Y(s)/X(s) \) is _________
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To calculate the transfer function using Mason's Gain Formula, identify the forward paths, loops, and feedback to apply the formula correctly.
Updated On: Nov 25, 2025
- ( \frac{2G_1(s)G_2(s) + 2G_1(s)G_3(s)}{1 + G_2(s) + G_3(s)} \)
- ( 2 + G_1(s) + G_3(s) + \frac{G_2(s)}{1 + G_2(s)} \)
- ( G_1(s) + G_3(s) - \frac{G_2(s)}{2 + G_2(s)} \)
- ( \frac{2G_1(s)G_2(s) + 2G_1(s)G_3(s) - G_1(s)}{1 + G_2(s) + G_3(s)} \)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the signal flow graph.
In a signal flow graph, the relationship between the input and output is determined using Mason's Gain Formula. For a system with multiple loops, we apply the formula by considering the forward paths and feedback loops. Step 2: Applying Mason's Gain Formula.
The gain of the system can be calculated by considering the forward paths and feedback loops. The general equation for the system with given components and the feedback loop is: \[ \frac{Y(s)}{X(s)} = \frac{2G_1(s)G_2(s) + 2G_1(s)G_3(s)}{1 + G_2(s) + G_3(s)}. \] Step 3: Analyzing the options.
- (A) Correct, this is the correct expression for the system based on the signal flow graph.
- (B) Incorrect, this expression does not follow from Mason’s Gain Formula.
- (C) Incorrect, this expression does not match the system structure.
- (D) Incorrect, this expression does not account for the correct feedback loop structure. Step 4: Conclusion.
Thus, the correct answer is (A) \( \frac{2G_1(s)G_2(s) + 2G_1(s)G_3(s)}{1 + G_2(s) + G_3(s)} \).
In a signal flow graph, the relationship between the input and output is determined using Mason's Gain Formula. For a system with multiple loops, we apply the formula by considering the forward paths and feedback loops. Step 2: Applying Mason's Gain Formula.
The gain of the system can be calculated by considering the forward paths and feedback loops. The general equation for the system with given components and the feedback loop is: \[ \frac{Y(s)}{X(s)} = \frac{2G_1(s)G_2(s) + 2G_1(s)G_3(s)}{1 + G_2(s) + G_3(s)}. \] Step 3: Analyzing the options.
- (A) Correct, this is the correct expression for the system based on the signal flow graph.
- (B) Incorrect, this expression does not follow from Mason’s Gain Formula.
- (C) Incorrect, this expression does not match the system structure.
- (D) Incorrect, this expression does not account for the correct feedback loop structure. Step 4: Conclusion.
Thus, the correct answer is (A) \( \frac{2G_1(s)G_2(s) + 2G_1(s)G_3(s)}{1 + G_2(s) + G_3(s)} \).
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