A series RLC circuit has $R=20 \ \Omega$, $L=1\text{ H}$, $C=0.01\text{ F}$. Using $\zeta = R / (2\sqrt{L/C})$, the response is:
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For a series RLC circuit, another quick way is to compare \(R^2\) with \(\frac{4L}{C}\):
- If \(R^2 = \frac{4L}{C}\): Critically damped.
Here, \(R^2 = 20^2 = 400\) and \(\frac{4L}{C} = \frac{4 \times 1}{0.01} = 400\).
Since \(R^2 = \frac{4L}{C}\), it is immediately confirmed to be critically damped.
Step 1: Understanding the Problem:
The question asks to determine the damping nature (type of transient response) of a series RLC circuit with given values of resistance, inductance, and capacitance, using the damping ratio formula. Step 2: Key Formula or Approach:
The damping ratio (\(\zeta\)) of a series RLC circuit is given by:
\[ \zeta = \frac{R}{2} \sqrt{\frac{C}{L}} = \frac{R}{2\sqrt{L/C}} \]
The type of response is classified based on the value of \(\zeta\):
- If \(\zeta > 1\): Overdamped response
- If \(\zeta = 1\): Critically damped response
- If \(0 < \zeta < 1\): Underdamped response
- If \(\zeta = 0\): Undamped (sustained oscillations) response Step 3: Detailed Explanation:
• Compute the term \(\sqrt{L/C}\):
\[ \sqrt{\frac{L}{C}} = \sqrt{\frac{1}{0.01}} = \sqrt{100} = 10\ \Omega \]
• Substitute these values into the formula for damping ratio \(\zeta\):
\[ \zeta = \frac{R}{2\sqrt{L/C}} = \frac{20}{2 \times 10} \]
\[ \zeta = \frac{20}{20} = 1 \]
• Since \(\zeta = 1\), the system's response is exactly critically damped. Step 4: Final Answer:
The response is critically damped.
