Engineering Solve RC Circuit Using Laplace Transforms

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The discussion focuses on solving an RC circuit using Laplace transforms to find the voltage v(t) at t=800ms. The initial calculation yields a voltage of 802mV, but discrepancies arise regarding the partial fraction expansion, leading to a potential error in the final result. A participant suggests a correction in the expansion, indicating that an additional term is needed for accurate results. The correct approach involves identifying constants A, B, and C, with C being crucial for the accurate calculation. The conversation emphasizes the importance of careful algebraic manipulation in Laplace transform problems.
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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





 

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I haven't checked all the details, but you seem to have the right idea.

What is the question?
 
Well my answer is different from that given in the book. My answer is too high by 1V.
 
Your partial fraction expansion is wrong. You should have:

\frac{4}{s^{2}(s+2)}=\frac{A}{s^{2}}+\frac{C}{s}+\frac{B}{s+2}
Using the residues you have A = 2 and B = 1 as you have found.
Rewrite the second member with the common denominator and equate the numerator to 4.
You will find C = -1. This is what you need to find the correct answer.
 
Thanks for not giving up on my question.

Kevin
 

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