Question

The transfer function of a first order system is given by $G\left(s\right)=\frac{10}{10s+1}$. It is used in a unity feedback system configuration with a proportional controller of proportional band $50\%,$ the time constant of the closed loop system is _________ (sec)

• A. $\frac{20}{21}$ sec
• B. $\frac{10}{21}$ sec
• C. $\frac{100}{21}$ sec
• D. $\frac{50}{21}$ sec

Answer 1

The correct option is B $\frac{10}{21}$ sec

PB = 50%

Proportional controller $K_P=\frac{100}{PB}=\frac{100}{50}=2$

the closed loop transfer function $C\left(S\right)=\frac{k_{P}G\left(s\right)}{1+K_{P}G\left(s\right)}$

$=\frac{2\times \frac{10}{10s+1}}{1+\frac{2\times 10}{10s+1}}=\frac{20}{10s+21}=\frac{20/21}{\frac{10}{21}s+1}$

time constant $\left(\tau \right)=\frac{10}{21}sec$