Let's solve the problem step by step: ## **Given Data** - \( R = 100 \ \Omega \) - \( L = 20 \ \text{mH} = 0.02 \ \text{H} \) - Resonant frequency: \( \omega_0 = 20,000 \ \text{rad/s} \) - Find one root \( s_1 \) of the characteristic equation. --- ## **Step 1: Find Capacitance \( C \)** The resonant frequency for a parallel RLC circuit is: \[ \omega_0 = \frac{1}{\sqrt{LC}} \] \[ C = \frac{1}{L \omega_0^2} \] Plug in the values: \[ C = \frac{1}{0.02 \times (20,000)^2} \] \[ C = \frac{1}{0.02 \times 400,000,000} \] \[ C = \frac{1}{8,000,000} \] \[ C = 1.25 \times 10^{-7} \ \text{F} = 125 \ \text{nF} \] --- ## **Step 2: Characteristic Equation for Parallel RLC** The standard characteristic equation is: \[ s^2 + \frac{1}{RC}s + \frac{1}{LC} = 0 \] --- ## **Step 3: Substitute Values** - \( \frac{1}{RC} = \frac{1}{100 \times 1.25 \times 10^{-7}} = \frac{1}{1.25 \times 10^{-5}} = 80,000 \) - \( \frac{1}{LC} = \frac{1}{0.02 \times 1.25 \times 10^{-7}} = \frac{1}{2.5 \times 10^{-9}} = 400,000,000 \) So, the equation: \[ s^2 + 80,000 s + 400,000,000 = 0 \] --- ## **Step 4: Find the Roots** The roots are: \[ s_{1,2} = \frac{-\frac{1}{RC} \pm \sqrt{\left(\frac{1}{RC}\right)^2 - 4 \frac{1}{LC}}}{2} \] \[ s_{1,2} = \frac{-80,000 \pm \sqrt{(80,000)^2 - 4 \times 400,000,000}}{2} \] \[ = \frac{-80,000 \pm \sqrt{6,400,000,000 - 1,600,000,000}}{2} \] \[ = \frac{-80,000 \pm \sqrt{4,800,000,000}}{2} \] \[ \sqrt{4,800,000,000} \approx 69,282.03 \] So, \[ s_1 = \frac{-80,000 + 69,282.03}{2} = \frac{-10,717.97}{2} \approx -5,359 \ \text{rad/s} \] \[ s_2 = \frac{-80,000 - 69,282.03}{2} = \frac{-149,282.03}{2} \approx -74,641 \ \text{rad/s} \] --- ## **Step 5: Final Answer** **The value of one root \( s_1 \) is:** \[ \boxed{s_1 = -5359 \ \text{rad/s}} \] So, **the correct option is:** > s1 = -5359 rad/s --- ### **Summary Table** | Step | Calculation | Result | |-----------|-----------------------------------------------|-------------------------| | Capacitance \(C\) | \(C = 1/(L\omega_0^2)\) | \(1.25 \times 10^{-7}\) F | | \(1/RC\) | \(1/(100 \times 1.25 \times 10^{-7})\) | 80,000 | | \(1/LC\) | \(1/(0.02 \times 1.25 \times 10^{-7})\) | 400,000,000 | | Roots | \(\frac{-80,000 \pm \sqrt{4,800,000,000}}{2}\) | \(-5359, -74641\) | | **Answer**| | **-5359 rad/s** |

## **Explanation of Incorrect Options** ### **Option Analysis** 1. **s1 = -5000 rad/s** - This value does not match our calculated root of -5359 rad/s. It is arbitrary and does not arise from the characteristic equation. 2. **s1 = +5359 rad/s** - The root value should be negative due to the damping factor in the characteristic equation. A positive root indicates an unstable system, which is not applicable here. 3. **None of the above** - This option would imply that all provided options are incorrect. However, we have found a valid root value of -5359 rad/s, making this statement false. 4. **s1 = +5935 rad/s** - Similar to the previous positive root option, this value does not correspond to any calculations made from the circuit parameters and is also indicative of an unstable system. ### **Conclusion** The only valid calculation yields a single root of: \[ s_1 = -5359 \ \text{rad/s} \] This corresponds with the system's behavior, as negative roots indicate a stable, damped response in a parallel RLC circuit. Hence, the correct option is: > **s1 = -5359 rad/s**