The signal flow graph of a system is shown below. U(s) is the input and C(s) is the output.

Assuming, hThis question was previously asked in

GATE EE 2014 Official Paper: Shift 3

Option 3 : \(G\left( s \right) = \frac{{{b_1}s + {b_0}}}{{{s^2} + {a_1}s + {a_0}}}\)

CT 1: Ratio and Proportion

2672

10 Questions
16 Marks
30 Mins

Concept:

According to Mason’s gain formula, the transfer function is given by

\(TF=\frac{\mathop{\sum }_{k-1}^{n}{{M}_{k}}{{\Delta }_{k}}}{\Delta }\)

Where,

n = no of forward paths

Mk = kth forward path gain

Δk = the value of Δ which is not touching the kth forward path

Δ = 1 – (sum of the loop gains) + (sum of the gain product of two non-touching loops) – (sum of the gain product of three non-touching loops)

Application:

Number of forward paths = 2

\({P_1} = \frac{{{h_0}}}{{{s^2}}},\;{P_2} = \frac{{{h_1}}}{s}\)

Number of loops = 2

\({L_1} = - \frac{{{a_1}}}{s},{L_2} = - \frac{{{a_0}}}{{{s^2}}}\)

\({\rm{\Delta }} = 1 + \frac{{{a_1}}}{s} + \frac{{{a_0}}}{{{s^2}}}\)

\({{\rm{\Delta }}_1} = 1,\;{{\rm{\Delta }}_2} = 1 + \frac{{{a_1}}}{s}\)

Transfer function \( = \frac{{\frac{{{h_0}}}{{{s^2}}}\left( 1 \right) + \frac{{{h_1}}}{s}\left( {1 + \frac{{{a_1}}}{s}} \right)}}{{1 + \frac{{{a_1}}}{s} + \frac{{{a_0}}}{{{s^2}}}}}\)

\( = \frac{{{h_0} + {h_1}\left( {s + {a_1}} \right)}}{{{s^2} + {a_1}s + {a_0}}}\)

\( = \frac{{{b_0} - {b_1}{a_1} + {b_1}\left( {s + {a_1}} \right)}}{{{s^2} + {a_1}s + {a_0}}}\)

\(G\left( s \right) = \frac{{{b_1}s + {b_0}}}{{{s^2} + {a_1}s + {a_0}}}\)