The transfer function C/R of the system shown in the figure is

F1 U.B Madhu 20.06.20 D6

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  1. \(\frac{{{G_1}{G_2}}}{{1 + {G_1}{H_1} + {G_2}{G_2}}}\)
  2. \(\frac{{{G_1}{H_1}{G_2}{H_2}}}{{\left( {1 + {G_1}{H_1}} \right)\left( {1 + {G_2}{H_2}} \right)}}\)
  3. \(\frac{{{G_1}{G_2}}}{{1 - {G_1} - {G_2} + {G_1}{G_2}{H_1}{H_2}}}\)
  4. \(\frac{{{G_1}{G_2}}}{{1 + {G_1}{H_1} + {G_2}{H_2} + {G_1}{G_2}{H_1}{H_2}}}\)

Answer (Detailed Solution Below)

Option 4 : \(\frac{{{G_1}{G_2}}}{{1 + {G_1}{H_1} + {G_2}{H_2} + {G_1}{G_2}{H_1}{H_2}}}\)
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Detailed Solution

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The transfer function of the left side block \( = \frac{{{G_1}}}{{1 + {G_1}{H_1}}}\)

The transfer function of the right-side block \( = \frac{{{G_2}}}{{1 + {G_2}{H_2}}}\)

These two blocks are in cascade connection. Therefore, the overall transfer function will multiplication of their individual transfer functions.

\(TF = \frac{{{G_1}{G_2}}}{{\left( {1 + {G_1}{H_1}} \right)\left( {1 + {G_2}{H_2}} \right)}} = \frac{{{G_1}{G_2}}}{{1 + {G_1}{H_1} + {G_2}{H_2} + {G_1}{G_2}{H_1}{H_2}}}\)

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