Critically Damped Practice

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<\math> so we get: <math>\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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