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

In summary, an RLC circuit is an electrical circuit that contains a resistor, inductor, and capacitor, and its interaction creates a complex impedance that affects current flow. The average power of an RLC circuit is the average amount of power dissipated or consumed, calculated using the integral of instantaneous power over one period. It can also be calculated using specific formulas depending on the type of circuit. The average power is affected by the resistance, inductance, and capacitance of components, as well as the frequency and phase difference between voltage and current. To optimize the average power, components with appropriate values should be chosen, and a higher frequency input signal can be used to reduce the phase difference.
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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?
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this picture is probably a bit easier to see...

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

1. What is an RLC circuit?

An RLC circuit is an electrical circuit that contains a resistor (R), an inductor (L), and a capacitor (C). These components are connected in series or parallel, and their interaction creates a complex impedance that affects the flow of current in the circuit.

2. What is the average power of an RLC circuit?

The average power of an RLC circuit is the average amount of power dissipated or consumed by the circuit. It is calculated by taking the integral of the instantaneous power over one period of the input voltage or current.

3. How is the average power calculated in an RLC circuit?

The average power in an RLC circuit can be calculated using the formula P=VIcosφ, where V is the rms voltage, I is the rms current, and φ is the phase difference between the voltage and current. Alternatively, it can also be calculated using P=I²R or P=V²/R, depending on the type of circuit.

4. What factors affect the average power in an RLC circuit?

The average power in an RLC circuit is affected by the resistance, inductance, and capacitance of the components, as well as the frequency of the input signal. The phase difference between the voltage and current also plays a role in determining the average power.

5. How can the average power be optimized in an RLC circuit?

The average power in an RLC circuit can be optimized by choosing components with the appropriate values of resistance, inductance, and capacitance. Additionally, using a higher frequency input signal can reduce the phase difference between voltage and current, leading to a more efficient use of power in the circuit.

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