# 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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