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Instrumentation and Process Control Question on Transfer functions and dynamic responses of various systems

For the block diagram shown in the figure, the correct expression for the transfer function Gd=y1(s)d(s)G_d = \frac{y_1(s)}{d(s)} is:
block diagram of transfer function

A

−Gp1Gc21+Gc2Gc2Gp1⋅(1+Gc2Gp2)-\frac{G_{p1} G_{c2}}{1 + G_{c2} G_{c2} G_{p1} \cdot (1 + G_{c2} G_{p2})}

B

−Gp1Gc21+Gc2Gp2+Gc1Gc2Gp1Gp2-\frac{G_{p1} G_{c2}}{1 + G_{c2} G_{p2} + G_{c1} G_{c2} G_{p1} G_{p2}}

C

−Gp1Gc21+Gc2Gp2+Gc1Gc2Gp1-\frac{G_{p1} G_{c2}}{1 + G_{c2} G_{p2} + G_{c1} G_{c2} G_{p1}}

D

11+Gc2Gp2+Gc1Gc2Gp1Gp2\frac{1}{1 + G_{c2} G_{p2} + G_{c1} G_{c2} G_{p1} G_{p2}}

Answer

−Gp1Gc21+Gc2Gp2+Gc1Gc2Gp1-\frac{G_{p1} G_{c2}}{1 + G_{c2} G_{p2} + G_{c1} G_{c2} G_{p1}}

Explanation

Solution

The correct Answers are (C) :−Gp1Gc21+Gc2Gp2+Gc1Gc2Gp1-\frac{G_{p1} G_{c2}}{1 + G_{c2} G_{p2} + G_{c1} G_{c2} G_{p1}}