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An RLC circuit such as that of Figure (a) has R = 6.99 2, C = 24.7 pF, L = 1.23 H, and Em = 44.2 V. (a) At what angular frequency wd will the current amplitude have its maximum value, as in the resonance curves of Figure (b)? (b) What is this maximum value? At what (c) lower angular frequency wdi and (d) higher angular frequency wd2 will the current amplitude be half this maximum value? (e) What is (wd2 - wd1)/wd, the fractional half-width of the resonance curve for this circuit?

An RLC circuit such as that of Figure (a) has R = 6.99 2, C = 24.7 pF, L = 1.23 H, and Em = 44.2 V. (a) At what angular frequency wd will the current amplitude have its maximum value, as in the resonance curves of Figure (b)? (b) What is this maximum value? At what (c) lower angular frequency wdi and (d) higher angular frequency wd2 will the current amplitude be half this maximum value? (e) What is (wd2 - wd1)/wd, the fractional half-width of the resonance curve for this circuit?

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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**RLC Circuit Resonance Analysis** An RLC circuit, as depicted in Figure (a), is analyzed with the following parameters: - Resistance \( R = 6.99 \, \Omega \) - Capacitance \( C = 24.7 \, \mu F \) - Inductance \( L = 1.23 \, H \) - Maximum emf \( \varepsilon_m = 44.2 \, V \) **Questions:** (a) At what angular frequency \( \omega_d \) will the current amplitude reach its maximum value, as shown in the resonance curves in Figure (b)? (b) What is the maximum current amplitude? (c) At which lower angular frequency \( \omega_{d1} \) will the current amplitude be half of this maximum value? (d) At which higher angular frequency \( \omega_{d2} \) will the current amplitude be half of the maximum value? (e) What is the fractional half-width \((\omega_{d2} - \omega_{d1})/\omega_d\) of the resonance curve for this circuit? **Figure (a):** This diagram shows an RLC circuit consisting of a resistor (R), inductor (L), and capacitor (C) connected in series with an AC source. The circuit illustrates how current (i) flows through each component. **Figure (b):** The graph displays the resonance curve of the RLC circuit, plotting current amplitude \( I \) against angular frequency. The graph highlights a peak where the current amplitude is maximum, and shows where \( X_C > X_L \) (capacitive reactance greater than inductive reactance) and \( X_L > X_C \) (inductive reactance greater than capacitive reactance). The curve includes points \( \omega_{d1} \) and \( \omega_{d2} \) where the current amplitude is half of the maximum.
Transcribed Image Text:**RLC Circuit Resonance Analysis** An RLC circuit, as depicted in Figure (a), is analyzed with the following parameters: - Resistance \( R = 6.99 \, \Omega \) - Capacitance \( C = 24.7 \, \mu F \) - Inductance \( L = 1.23 \, H \) - Maximum emf \( \varepsilon_m = 44.2 \, V \) **Questions:** (a) At what angular frequency \( \omega_d \) will the current amplitude reach its maximum value, as shown in the resonance curves in Figure (b)? (b) What is the maximum current amplitude? (c) At which lower angular frequency \( \omega_{d1} \) will the current amplitude be half of this maximum value? (d) At which higher angular frequency \( \omega_{d2} \) will the current amplitude be half of the maximum value? (e) What is the fractional half-width \((\omega_{d2} - \omega_{d1})/\omega_d\) of the resonance curve for this circuit? **Figure (a):** This diagram shows an RLC circuit consisting of a resistor (R), inductor (L), and capacitor (C) connected in series with an AC source. The circuit illustrates how current (i) flows through each component. **Figure (b):** The graph displays the resonance curve of the RLC circuit, plotting current amplitude \( I \) against angular frequency. The graph highlights a peak where the current amplitude is maximum, and shows where \( X_C > X_L \) (capacitive reactance greater than inductive reactance) and \( X_L > X_C \) (inductive reactance greater than capacitive reactance). The curve includes points \( \omega_{d1} \) and \( \omega_{d2} \) where the current amplitude is half of the maximum.
### Analysis of RLC Circuit Resonance #### Diagram Description The image contains a diagram labeled (a), displaying a graph of current amplitude \( I \) versus the normalized angular frequency \( \omega_d/\omega \). 1. **Graph Explanation**: - The graph shows three curves, each corresponding to different resistance values: \( R = 10 \, \Omega \), \( R = 30 \, \Omega \), and \( R = 100 \, \Omega \). - The vertical axis represents the current amplitude \( I \). - The horizontal axis represents the normalized angular frequency \( \omega_d/\omega \) ranging from 0.90 to 1.10. - The peak occurs at the resonant frequency, where \( X_C > X_L \) for frequencies lower than resonant frequency and \( X_L > X_C \) for higher frequencies. #### Input Fields Description The lower part of the image labeled (b) contains input fields for numerical values related to the graph. 1. **(a) Number**: Entered value is 181 with units in rad/s. 2. **(b) Number**: Entered value is 6.32 with units in Amps. 3. **(c) Number**: Entered value is 140 rad/s with a cross mark indicating an error. 4. **(d) Number**: Entered value is 0 rad/s. 5. **(e) Number**: No units needed, and no value entered. This setup typically depicts parameters involved in analyzing an RLC circuit, particularly focusing on resonance behavior by observing changes in current amplitude as resistance varies.
Transcribed Image Text:### Analysis of RLC Circuit Resonance #### Diagram Description The image contains a diagram labeled (a), displaying a graph of current amplitude \( I \) versus the normalized angular frequency \( \omega_d/\omega \). 1. **Graph Explanation**: - The graph shows three curves, each corresponding to different resistance values: \( R = 10 \, \Omega \), \( R = 30 \, \Omega \), and \( R = 100 \, \Omega \). - The vertical axis represents the current amplitude \( I \). - The horizontal axis represents the normalized angular frequency \( \omega_d/\omega \) ranging from 0.90 to 1.10. - The peak occurs at the resonant frequency, where \( X_C > X_L \) for frequencies lower than resonant frequency and \( X_L > X_C \) for higher frequencies. #### Input Fields Description The lower part of the image labeled (b) contains input fields for numerical values related to the graph. 1. **(a) Number**: Entered value is 181 with units in rad/s. 2. **(b) Number**: Entered value is 6.32 with units in Amps. 3. **(c) Number**: Entered value is 140 rad/s with a cross mark indicating an error. 4. **(d) Number**: Entered value is 0 rad/s. 5. **(e) Number**: No units needed, and no value entered. This setup typically depicts parameters involved in analyzing an RLC circuit, particularly focusing on resonance behavior by observing changes in current amplitude as resistance varies.
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