Critically Damped Practice

Practice question for ECE201: "Linear circuit analysis I"

By: Chinar Dhamija

Topic: Critically Damped Second Order Equation

Question

Find the value for C that will make the zero input response critically damped with roots at -4.

Answer

For a response to be critically damped we know that:
$b^2 - 4c = 0$
The next step would be to simplify the circuit as shown in the image below. Once simplified it becomes a parallel RLC circuit where we know:
$b = \frac{1}{RC}$ and $c = \frac{1}{LC}$

Since the root was given to be -4 we can find b.
$\frac{-b}{2} = s$ so we get: $\frac{-b}{2} = -4$ therefore b = 8.
Once we know b we can use the critically damped equation to solve for C.
\begin{align} 8^2 - \frac{4}{2C} = 0\\ 64 = \frac{2}{C}\\ C = \frac{1}{32}\\ \end{align}

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