At t=0 a switching operation occurs which produces the circuit shown in Figure 9. What is the resonant frequency, wo, for the circuit? 6 ΚΩ 25 nF + Vc 2 ΚΩ Figure 9 400 mH
At t=0 a switching operation occurs which produces the circuit shown in Figure 9. What is the resonant frequency, wo, for the circuit? 6 ΚΩ 25 nF + Vc 2 ΚΩ Figure 9 400 mH
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Author:Robert L. Boylestad
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Transcribed Image Text:**Question:** What type of response is displayed by the circuit in Figure 9 after switching?
**Options:**
- O Underdamped
- O Not damped (undamped)
- O Critically damped
- O Overdamped
- O Correct Answer Not Given
![### Understanding Resonant Frequency in RLC Circuits
At \( t = 0 \) a switching operation occurs which produces the circuit shown in **Figure 9**. What is the resonant frequency, \( \omega_0 \), for the circuit?
#### Circuit Diagram (Figure 9)
The given circuit consists of:
- A 6 kΩ resistor in series with
- A 400 mH inductor and a
- Parallel combination of a 2 kΩ resistor and a 25 nF capacitor.
The components are arranged as follows:
1. The 6 kΩ resistor is connected in series with the inductor (400 mH).
2. The junction of the 6 kΩ resistor and the inductor is connected to a parallel combination of a 2 kΩ resistor and a capacitor (25 nF).
### Explanation:
The circuit is typically analyzed by looking at the parallel RLC (Resistor, Inductor, Capacitor) components to find the resonant frequency. Resonant frequency (\( \omega_0 \)) is the frequency at which the reactive inductive and capacitive effects cancel each other out in an LC circuit.
### Formula to Determine Resonant Frequency:
For an LC circuit, the resonant angular frequency (in radians per second) is given by:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]
Where:
- \( L \) is the inductance in henries (H).
- \( C \) is the capacitance in farads (F).
Given:
- \( L = 400 \) mH \( = 400 \times 10^{-3} \) H
- \( C = 25 \) nF \( = 25 \times 10^{-9} \) F
Substituting these values into the formula:
\[ \omega_0 = \frac{1}{\sqrt{(400 \times 10^{-3}) \times (25 \times 10^{-9})}} \]
\[ \omega_0 = \frac{1}{\sqrt{10 \times 10^{-9}}} = \frac{1}{10^{-4.5}} \]
\[ \omega_0 = 10^4.5 \text{ rad/s} \]
\[ \omega_0 \approx 31622.78 \text{ rad/s} \]
Therefore, the resonant frequency \( \omega_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabbd02aa-e289-47d3-b3b4-87f7eeaaef64%2F8529e663-dc8d-4233-9a6d-69d58cbf5003%2Fx3396em_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Resonant Frequency in RLC Circuits
At \( t = 0 \) a switching operation occurs which produces the circuit shown in **Figure 9**. What is the resonant frequency, \( \omega_0 \), for the circuit?
#### Circuit Diagram (Figure 9)
The given circuit consists of:
- A 6 kΩ resistor in series with
- A 400 mH inductor and a
- Parallel combination of a 2 kΩ resistor and a 25 nF capacitor.
The components are arranged as follows:
1. The 6 kΩ resistor is connected in series with the inductor (400 mH).
2. The junction of the 6 kΩ resistor and the inductor is connected to a parallel combination of a 2 kΩ resistor and a capacitor (25 nF).
### Explanation:
The circuit is typically analyzed by looking at the parallel RLC (Resistor, Inductor, Capacitor) components to find the resonant frequency. Resonant frequency (\( \omega_0 \)) is the frequency at which the reactive inductive and capacitive effects cancel each other out in an LC circuit.
### Formula to Determine Resonant Frequency:
For an LC circuit, the resonant angular frequency (in radians per second) is given by:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]
Where:
- \( L \) is the inductance in henries (H).
- \( C \) is the capacitance in farads (F).
Given:
- \( L = 400 \) mH \( = 400 \times 10^{-3} \) H
- \( C = 25 \) nF \( = 25 \times 10^{-9} \) F
Substituting these values into the formula:
\[ \omega_0 = \frac{1}{\sqrt{(400 \times 10^{-3}) \times (25 \times 10^{-9})}} \]
\[ \omega_0 = \frac{1}{\sqrt{10 \times 10^{-9}}} = \frac{1}{10^{-4.5}} \]
\[ \omega_0 = 10^4.5 \text{ rad/s} \]
\[ \omega_0 \approx 31622.78 \text{ rad/s} \]
Therefore, the resonant frequency \( \omega_
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