Time Response of First-order Systems

$\frac{1}{sT}$

$\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)}$

$G(s)=\frac{1}{sT}$

$\frac{C(s)}{R(s)}=\frac{\frac{1}{sT}}{1+\frac{1}{sT}}=\frac{1}{sT+1}$

$C(s)=\left ( \frac{1}{sT+1} \right )R(s)$

$C(s)=\left ( \frac{1}{sT+1} \right )R(s)$

$R(s)$

$C(s)$

$C(s)$.

$r(t)=\delta (t)$

$R(s)=1$

$C(s)=\left ( \frac{1}{sT+1} \right )R(s)$

$R(s) = 1$

$C(s)=\left ( \frac{1}{sT+1} \right )(1)=\frac{1}{sT+1}$

$C(s)=\frac{1}{T\left (\ s+\frac{1}{T} \right )} \Rightarrow C(s)=\frac{1}{T}\left ( \frac{1}{s+\frac{1}{T}} \right )$

$c(t)=\frac{1}{T}e^\left ( {-\frac{t}{T}} \right )u(t)$

$r(t)=u(t)$

$R(s)=\frac{1}{s}$

$C(s)=\left ( \frac{1}{sT+1} \right )R(s)$

$R(s)=\frac{1}{s}$

$C(s)=\left ( \frac{1}{sT+1} \right )\left ( \frac{1}{s} \right )=\frac{1}{s\left ( sT+1 \right )}$

$C(s)=\frac{1}{s\left ( sT+1 \right )}=\frac{A}{s}+\frac{B}{sT+1}$

$\Rightarrow \frac{1}{s\left ( sT+1 \right )}=\frac{A\left ( sT+1 \right )+Bs}{s\left ( sT+1 \right )}$

$1=A\left ( sT+1 \right )+Bs$

$0=T+B \Rightarrow B=-T$

$C(s)$.

$C(s)=\frac{1}{s}-\frac{T}{sT+1}=\frac{1}{s}-\frac{T}{T\left ( s+\frac{1}{T} \right )}$

$\Rightarrow C(s)=\frac{1}{s}-\frac{1}{s+\frac{1}{T}}$

$c(t)=\left ( 1-e^{-\left ( \frac{t}{T} \right )} \right )u(t)$

$c_{tr}(t)=-e^{-\left ( \frac{t}{T} \right )}u(t)$

$c_{ss}(t)=u(t)$

$So, r(t)=tu(t)$

$R(s)=\frac{1}{s^2}$

$C(s)=\left ( \frac{1}{sT+1} \right )R(s)$

$R(s)=\frac{1}{s^2}$

$C(s)=\left ( \frac{1}{sT+1} \right )\left ( \frac{1}{s^2} \right )=\frac{1}{s^2(sT+1)}$

$C(s)$.

$C(s)=\frac{1}{s^2(sT+1)}=\frac{A}{s^2}+\frac{B}{s}+\frac{C}{sT+1}$

$\Rightarrow \frac{1}{s^2(sT+1)}=\frac{A(sT+1)+Bs(sT+1)+Cs^2}{s^2(sT+1)}$

$1=A(sT+1)+Bs(sT+1)+Cs^2$

$0=T+B \Rightarrow B=-T$

$s^2$ $C=T^2$.

$C = T^2$

$C(s)$.

$C(s)=\frac{1}{s^2}-\frac{T}{s}+\frac{T^2}{sT+1}=\frac{1}{s^2}-\frac{T}{s}+\frac{T^2}{T\left ( s+\frac{1}{T} \right )}$

$\Rightarrow C(s)=\frac{1}{s^2}-\frac{T}{s}+\frac{T}{s+\frac{1}{T}}$

$c(t)=\left ( t-T+Te^{-\left ( \frac{t}{T} \right )} \right )u(t)$

$c_{tr}(t)=Te^{-\left ( \frac{t}{T} \right )}u(t)$

$c_{ss}(t)=(t-T)u(t)$

$r(t)=\frac{t^2}{2}u(t)$

$R(s)=\frac{1}{s^3}$

$C(s)=\left ( \frac{1}{sT+1} \right )R(s)$

$R(s)=\frac{1}{s^3}$

$C(s)=\left ( \frac{1}{sT+1} \right )\left( \frac{1}{s^3} \right )=\frac{1}{s^3(sT+1)}$

$C(s)$.

$C(s)=\frac{1}{s^3(sT+1)}=\frac{A}{s^3}+\frac{B}{s^2}+\frac{C}{s}+\frac{D}{sT+1}$

$-T, \: T^2\: and \: −T^3$

$C(s)=\frac{1}{s^3}-\frac{T}{s^2}+\frac{T^2}{s}-\frac{T^3}{sT+1} \: \Rightarrow C(s)=\frac{1}{s^3}-\frac{T}{s^2}+\frac{T^2}{s}-\frac{T^2}{s+\frac{1}{T}}$

$c(t)=\left ( \frac{t^2}{2} -Tt+T^2-T^2e^{-\left ( \frac{t}{T} \right )} \right )u(t)$

$C_{tr}(t)=-T^2e^{-\left ( \frac{t}{T} \right )}u(t)$

$C_{ss}(t)=\left ( \frac{t^2}{2} -Tt+T^2 \right )u(t)$