Topic: Control Systems
For a negative feedback system shown in Figure, the equivalent transfer function is
- A. $\frac{s^{3}+5s^{2}+6s}{s^{3}+7s^{2}+12s+3}$
- B. $\frac{s^{3}+5s^{2}+6s}{s^{3}+5s^{2}+4s-3}$
- C. $\frac{s^{2}+5s+4}{s^{3}+7s^{2}+12s+3}$
- D. $\frac{s^{2}+5s+4}{s^{3}+5s^{2}+4s-3}$
Answer: C. $\frac{s^{2}+5s+4}{s^{3}+7s^{2}+12s+3}$
Explanation: $\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)H(s)} = \frac{\frac{s+1}{s(s+2)}}{1+\frac{s+1}{s(s+2)}\cdot\frac{s+3}{s+4}} = \frac{(s+1)(s+4)}{s(s^{2}+6s+8)+(s^{2}+4s+3)} = \frac{s^{2}+5s+4}{s^{3}+6s^{2}+8s+s^{2}+4s+3} = \frac{s^{2}+5s+4}{s^{3}+7s^{2}+12s+3}$.