Question

# The signal flow graph of a system is given below: The transfer function (C/R) of the system is

A
(G1G2+G1G3)/(1+G1G2H2)
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B
(G1G2+G1G3)/(1G1G2+G1H2H2)
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C
(G1G2+G1G3)/(1G1H1+G1G2H2+G1G3G2)
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D
(G1G2+G1G3)/(1G1H1+G1G2H2+G1G2G3H1)
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Solution

## The correct option is C (G1G2+G1G3)/(1−G1H1+G1G2H2+G1G3G2) Step 1 : There are tow forward paths between input and output P1=G1G2 P2=G1G3 Step 2 : G_{3} There are three loops, the loops gain are L1=G1H1 L2=G1,G3H2 L3=G1G2H1 Step 3 : There are no pairs of nontouching loops. Δ1=1 and Δ2=1 Step 4 : Δ=1−[L1+L2+L3] =1−G1H1+G1G3H2+G1G2H2 According to Mason's gain formula. T(s)=ΣPkΔkΔ =G1G2.1+G1G3.11−G1H1+G1G3H2+G1G2H2 ∴T(s)=G1G2+G1G31−G1H1+G1G3H2+G1G2H2

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