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Determine R, such that |Vol = 100|Vs] at the resonant frequency. Vs R 0.1 μF V₂ 1 mH

Determine R, such that |Vol = 100|Vs] at the resonant frequency. Vs R 0.1 μF V₂ 1 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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### Problem Statement: Determine the resistance \(R\) such that \(|V_o| = 100|V_s|\) at the resonant frequency. ### Description: The given problem requires us to find the value of resistance \(R\) in an electrical circuit. The circuit must be analyzed at its resonant frequency, and the condition is that the magnitude of the output voltage \(|V_o|\) is 100 times the magnitude of the source voltage \(|V_s|\). ### Circuit Explanation: The circuit diagram provided includes the following components connected in series: 1. **Voltage Source (\(V_s\))**: An AC voltage source. 2. **Resistor (R)**: A resistor with resistance \(R\). 3. **Capacitor**: A capacitor with capacitance \(0.1\,\mu F\). 4. **Inductor**: An inductor with inductance \(1\,mH\). These components create a series RLC circuit, where the current \(I\) flows through all the components in a single pathway. The output voltage \(V_o\) is measured across the capacitor and inductor in series. ### Concepts to Understand: 1. **Resonant Frequency**: The frequency at which the inductive reactance \(X_L\) and capacitive reactance \(X_C\) are equal in magnitude but opposite in phase. At resonance, the impedance of the LC combination is minimum, and the voltage across them can become significantly large. 2. **Magnitude Condition**: The relationship \(|V_o| = 100|V_s|\) sets a specific requirement on the voltage magnification factor at resonance. 3. **Circuit Resonance Analysis**: At resonance, the impedance of the LC circuit is purely resistive and given by the resistance \(R\). The resonant frequency \(f_r\) can be found using the formula: \[ f_r = \frac{1}{2\pi\sqrt{LC}} \] 4. **Impedance and Voltage Relationship**: At resonance, the voltage across the LC combination is maximized, and its relationship to the source voltage can be used to determine \(R\). ### Calculation Steps: 1. **Find the resonant frequency (\(f_r\))**: \[ f_r = \frac{1}{2\pi\sqrt{(0.1 \times 10^{-
Transcribed Image Text:### Problem Statement: Determine the resistance \(R\) such that \(|V_o| = 100|V_s|\) at the resonant frequency. ### Description: The given problem requires us to find the value of resistance \(R\) in an electrical circuit. The circuit must be analyzed at its resonant frequency, and the condition is that the magnitude of the output voltage \(|V_o|\) is 100 times the magnitude of the source voltage \(|V_s|\). ### Circuit Explanation: The circuit diagram provided includes the following components connected in series: 1. **Voltage Source (\(V_s\))**: An AC voltage source. 2. **Resistor (R)**: A resistor with resistance \(R\). 3. **Capacitor**: A capacitor with capacitance \(0.1\,\mu F\). 4. **Inductor**: An inductor with inductance \(1\,mH\). These components create a series RLC circuit, where the current \(I\) flows through all the components in a single pathway. The output voltage \(V_o\) is measured across the capacitor and inductor in series. ### Concepts to Understand: 1. **Resonant Frequency**: The frequency at which the inductive reactance \(X_L\) and capacitive reactance \(X_C\) are equal in magnitude but opposite in phase. At resonance, the impedance of the LC combination is minimum, and the voltage across them can become significantly large. 2. **Magnitude Condition**: The relationship \(|V_o| = 100|V_s|\) sets a specific requirement on the voltage magnification factor at resonance. 3. **Circuit Resonance Analysis**: At resonance, the impedance of the LC circuit is purely resistive and given by the resistance \(R\). The resonant frequency \(f_r\) can be found using the formula: \[ f_r = \frac{1}{2\pi\sqrt{LC}} \] 4. **Impedance and Voltage Relationship**: At resonance, the voltage across the LC combination is maximized, and its relationship to the source voltage can be used to determine \(R\). ### Calculation Steps: 1. **Find the resonant frequency (\(f_r\))**: \[ f_r = \frac{1}{2\pi\sqrt{(0.1 \times 10^{-
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