Obtain the transfer function Y(s)/R(s) for the closed loop system shown in figure (6). Use signal flow graph (Masons gain formula) method.
The Correct Answer and Explanation is:
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Here is the step-by-step solution to obtain the transfer function for the given closed-loop system using Mason’s Gain Formula.
Final Answer
The transfer function C(s)/R(s) for the given system is:
C(s)/R(s) = [G₂(s)G₃(s)(G₁(s) + G₅(s)) + G₅(s)(G₃(s)G₆(s) + 1)] / [1 + G₂(s)G₆(s)(1 + G₃(s)G₄(s))]
Explanation
To find the overall transfer function C(s)/R(s) of the system, we will use Mason’s Gain Formula. This requires converting the block diagram into a signal flow graph and then identifying all forward paths and feedback loops.
The formula is given by:
T(s) = (Σ PₖΔₖ) / Δ
Where:
- T(s) is the overall transfer function.
- Pₖ is the gain of the k-th forward path from input R(s) to output C(s).
- Δ is the determinant of the graph, calculated as:
Δ = 1 – (Sum of all individual loop gains) + (Sum of gain products of all possible pairs of non-touching loops) – … - Δₖ is the cofactor for the k-th forward path, which is the value of Δ for the part of the graph that does not touch the k-th forward path.
Step 1: Identify Forward Paths
A forward path is a path from the input node R(s) to the output node C(s) that does not traverse any node more than once.
- P₁: R(s) → G₁(s) → G₂(s) → G₃(s) → C(s)
- Gain: P₁ = G₁G₂G₃
- P₂: R(s) → G₅(s) → G₂(s) → G₃(s) → C(s)
- Gain: P₂ = G₅G₂G₃
- P₃: R(s) → G₅(s) → G₆(s) [upper] → G₃(s) → C(s)
- Gain: P₃ = G₅G₆G₃
- P₄: R(s) → G₅(s) → C(s)
- Gain: P₄ = G₅
Step 2: Identify Individual Feedback Loops
A loop is a path that starts and ends at the same node without passing through any other node more than once. The diagram indicates that the output of G₄(s) sums with the output of G₂(s) at the input of the lower G₆(s) block.
- L₁: G₂(s) → G₆(s) [lower] → Summing Point 1 (negative feedback)
- Gain: L₁ = -G₂G₆
- L₂: G₂(s) → G₃(s) → C(s) → G₄(s) → G₆(s) [lower] → Summing Point 1 (negative feedback)
- Gain: L₂ = -G₂G₃G₄G₆
Step 3: Determine Non-Touching Loops
Two loops are non-touching if they do not share any common nodes. In this system, both loops L₁ and L₂ share the path through G₂(s) and G₆(s). Therefore, there are no non-touching loops.
Step 4: Calculate the Determinant (Δ) and Cofactors (Δₖ)
- Δ = 1 – (L₁ + L₂)
Δ = 1 – (-G₂G₆ – G₂G₃G₄G₆)
Δ = 1 + G₂G₆ + G₂G₃G₄G₆ - Since all forward paths (P₁, P₂, P₃, P₄) touch both loops (L₁, L₂), the graph remaining after removing any forward path will have no loops. Therefore, the cofactors for all paths are 1.
Δ₁ = 1, Δ₂ = 1, Δ₃ = 1, Δ₄ = 1
Step 5: Apply Mason’s Gain Formula
C(s)/R(s) = (P₁Δ₁ + P₂Δ₂ + P₃Δ₃ + P₄Δ₄) / Δ
C(s)/R(s) = (G₁G₂G₃ + G₅G₂G₃ + G₅G₆G₃ + G₅) / (1 + G₂G₆ + G₂G₃G₄G₆)
To simplify the expression, we can factor the numerator and denominator:
Numerator = G₂G₃(G₁ + G₅) + G₅(G₆G₃ + 1)
Denominator = 1 + G₂G₆(1 + G₃G₄)
Thus, the final transfer function is:
C(s)/R(s) = [G₂(s)G₃(s)(G₁(s) + G₅(s)) + G₅(s)(G₃(s)G₆(s) + 1)] / [1 + G₂(s)G₆(s)(1 + G₃(s)G₄(s))]
