What Happens to Average Power in an RLC Circuit at Resonance?

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SUMMARY

The average power in an RLC circuit at resonance is determined by the relationship between inductance (L) and capacitance (C). The resonance frequency, denoted as \( f_0 = \frac{1}{2\pi\sqrt{LC}} \), is where the circuit's impedance is minimized, resulting in maximum average power transfer. At this frequency, the average power is at its maximum due to the circuit's ability to efficiently transfer energy between the inductor and capacitor.

PREREQUISITES
  • Understanding of RLC circuit components (Resistor, Inductor, Capacitor)
  • Knowledge of resonance frequency calculations
  • Familiarity with AC circuit analysis
  • Basic principles of power in electrical circuits
NEXT STEPS
  • Study the derivation of the resonance frequency in RLC circuits
  • Learn about impedance in AC circuits
  • Explore the concept of power factor in RLC circuits
  • Investigate the effects of varying L and C on circuit behavior
USEFUL FOR

Electrical engineering students, circuit designers, and anyone interested in understanding RLC circuit dynamics and resonance phenomena.

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Homework Statement



4.) Determine the average power in an RLC circuit. Derive an expression for the resonance frequency in terms of L and C. Is the average power a maximum or a minimum at the resonance frequency?

Homework Equations





The Attempt at a Solution


I have circled in red the part that is giving me trouble, all the rest of the solution i can do, is that part some identity or rule i am missing?
EM6Q4-1.jpg
 
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this picture is probably a bit easier to see...

EM6Q4-3.jpg
 

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