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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

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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**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
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_
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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