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Find v(t) at t=800ms for the circuit in Figure 1.
Ans: 802mV
Writing a single node equation we have
\frac{v(t)-2tu(t)}{5}+0.1\frac{dv}{dt}=0.
Taking the Laplace transform we have
L\left\{ \frac{v(t)-2tu(t)}{5}+0.1\frac{dv}{dt}=0\right\} .
tu(t)\Rightarrow\frac{1}{s^{2}}
\frac{df}{dt}\Rightarrow sF(s)-f(0^{-})
\frac{V(s)}{5}-\frac{2}{5s^{2}}+0.1sV(s)-0.1v(0^{-})=0
Assume that v(0^{-})=0 we have
\frac{V(s)}{5}-\frac{2}{5s^{2}}+0.1sV(s)=0
multiplying through by 10 and combing like terms
V(s)\left(2+s\right)=\frac{4}{s^{2}}
Solving for V(s) we have
V(s)=\frac{4}{s^{2}(s+2)}
Applying the method of residues we have
\frac{4}{s^{2}(s+2)}=\frac{A}{s^{2}}+\frac{B}{s+2}
Multiplying throught by s^{2} we have
\frac{4}{s+2}=A+\frac{Bs^{2}}{s+2}
A=\frac{4-Bs^{2}}{s+2}\mid_{s=0}=2
Multiplying through by s+2 we have
\frac{4}{s^{2}}=\frac{A(s+2)}{s^{2}}+B
B=\frac{4-A(s+2)}{s^{2}}\mid_{s=-2}=1
Substituting A and B back into the equation we have
V(s)=\frac{2}{s^{2}}+\frac{1}{s+2}
Applying the known Laplace transform pairs we have
v(t)=2t+e^{-2t}
v(800ms)=2(0.8)+e^{-2(0.8s)}=1.802V
Ans: 802mV
Writing a single node equation we have
\frac{v(t)-2tu(t)}{5}+0.1\frac{dv}{dt}=0.
Taking the Laplace transform we have
L\left\{ \frac{v(t)-2tu(t)}{5}+0.1\frac{dv}{dt}=0\right\} .
tu(t)\Rightarrow\frac{1}{s^{2}}
\frac{df}{dt}\Rightarrow sF(s)-f(0^{-})
\frac{V(s)}{5}-\frac{2}{5s^{2}}+0.1sV(s)-0.1v(0^{-})=0
Assume that v(0^{-})=0 we have
\frac{V(s)}{5}-\frac{2}{5s^{2}}+0.1sV(s)=0
multiplying through by 10 and combing like terms
V(s)\left(2+s\right)=\frac{4}{s^{2}}
Solving for V(s) we have
V(s)=\frac{4}{s^{2}(s+2)}
Applying the method of residues we have
\frac{4}{s^{2}(s+2)}=\frac{A}{s^{2}}+\frac{B}{s+2}
Multiplying throught by s^{2} we have
\frac{4}{s+2}=A+\frac{Bs^{2}}{s+2}
A=\frac{4-Bs^{2}}{s+2}\mid_{s=0}=2
Multiplying through by s+2 we have
\frac{4}{s^{2}}=\frac{A(s+2)}{s^{2}}+B
B=\frac{4-A(s+2)}{s^{2}}\mid_{s=-2}=1
Substituting A and B back into the equation we have
V(s)=\frac{2}{s^{2}}+\frac{1}{s+2}
Applying the known Laplace transform pairs we have
v(t)=2t+e^{-2t}
v(800ms)=2(0.8)+e^{-2(0.8s)}=1.802V