AC Circuits 10. For the circuit shown, a. What is the resonance frequency in Hz? b. What is the circuit's peak current at resonance? (10 V) cos wot 10 Ω 10 mH 10 μF www-oooo
AC Circuits 10. For the circuit shown, a. What is the resonance frequency in Hz? b. What is the circuit's peak current at resonance? (10 V) cos wot 10 Ω 10 mH 10 μF www-oooo
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1. **Component values:**
- Resistor (\( R \)) = 10 Ω
- Inductor (\( L \)) = 10 mH
- Capacitor (\( C \)) = 10 μF
- Voltage source: \( (10 \text{ V}) \cos{\omega t} \)
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**Questions and Solutions:**
**a. What is the resonance frequency in Hz?**
The resonance frequency \( f_0 \) for an RLC circuit is given by:
\[ f_0 = \frac{1}{2 \pi \sqrt{LC}} \]
**b. What is the circuit's peak current at resonance?**
At resonance, the peak current \( I_{peak} \) is given by:
\[ I_{peak} = \frac{\epsilon_0}{R} \]
where \( \epsilon_0 \) is the peak voltage of the AC source.
**c. If the inductance is doubled, what is the new resonance frequency in Hz?**
If \( L' = 2L \), the new resonance frequency is:
\[ f_0' = \frac{1}{2 \pi \sqrt{L'C}} = \frac{1}{2 \pi \sqrt{2LC}} = \frac{f_0}{\sqrt{2}} \]
**d. If instead of (c), the resistance is tripled, what is the new resonance frequency?**
The resonance frequency \( f_0 \) is independent of the resistance \( R \). Hence, tripling the resistance does not affect the resonance frequency. The resonance frequency remains:
\[ f_0 = \frac{1}{2 \pi \sqrt{LC}} \]
**e. The peak current for this circuit at any frequency \( \omega \) is given by:**
\[ I = \frac{\epsilon_0}{\sqrt{R^2 + (X_L - X_C)^2}} \]
where \( X_L = \omega L \) is the inductive reactance and \( X_C = \frac{1}{\omega C} \) is the capacitive reactance.
Explain mathematically why this peak current is maxed out only when driven](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9a8543e-0e4d-4b59-a486-683fe3d30411%2F57d0e8ff-0df5-495f-affd-c039012c26e2%2F3ce2qib_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### AC Circuits
#### Problem 10: Analysis of an RLC Circuit
**Given:**
For the circuit shown,
1. **Component values:**
- Resistor (\( R \)) = 10 Ω
- Inductor (\( L \)) = 10 mH
- Capacitor (\( C \)) = 10 μF
- Voltage source: \( (10 \text{ V}) \cos{\omega t} \)
---
**Questions and Solutions:**
**a. What is the resonance frequency in Hz?**
The resonance frequency \( f_0 \) for an RLC circuit is given by:
\[ f_0 = \frac{1}{2 \pi \sqrt{LC}} \]
**b. What is the circuit's peak current at resonance?**
At resonance, the peak current \( I_{peak} \) is given by:
\[ I_{peak} = \frac{\epsilon_0}{R} \]
where \( \epsilon_0 \) is the peak voltage of the AC source.
**c. If the inductance is doubled, what is the new resonance frequency in Hz?**
If \( L' = 2L \), the new resonance frequency is:
\[ f_0' = \frac{1}{2 \pi \sqrt{L'C}} = \frac{1}{2 \pi \sqrt{2LC}} = \frac{f_0}{\sqrt{2}} \]
**d. If instead of (c), the resistance is tripled, what is the new resonance frequency?**
The resonance frequency \( f_0 \) is independent of the resistance \( R \). Hence, tripling the resistance does not affect the resonance frequency. The resonance frequency remains:
\[ f_0 = \frac{1}{2 \pi \sqrt{LC}} \]
**e. The peak current for this circuit at any frequency \( \omega \) is given by:**
\[ I = \frac{\epsilon_0}{\sqrt{R^2 + (X_L - X_C)^2}} \]
where \( X_L = \omega L \) is the inductive reactance and \( X_C = \frac{1}{\omega C} \) is the capacitive reactance.
Explain mathematically why this peak current is maxed out only when driven
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