An R L C circuit with R=20.0 Ω, L=295 mH, and C=49.7 μF is connected to an ac generator with an

An R L C circuit with R=20.0 Ω, L=295 mH, and C=49.7 μF is...

An $R L C$ circuit with $R=20.0 \Omega, L=295 \mathrm{mH},$ and $C=49.7 \mu \mathrm{F}$ is connected to an ac generator with an rms voltage of 21.5 $\mathrm{V}$ . Determine the average power delivered to this circuit when the frequency of the generator is (a) equal to the resonance frequency, (b) twice the resonance frequency, and (c) half the resonance frequency.

Key Concepts

Frequency Dependence of Reactance

Reactance in an AC circuit, which includes both inductive and capacitive components, depends directly on the frequency of the applied voltage. Inductive reactance increases with frequency while capacitive reactance decreases. This frequency dependence is key in determining the overall impedance of the circuit and thereby impacts the current, phase relationships, and power consumption.

Average Power in AC Circuits

The average power delivered to an AC circuit is determined by the rms (root mean square) voltage and the impedance of the circuit. Typically, it is calculated using the formula P_avg = (V_rms^2 * R) / (R^2 + (X_L - X_C)^2), where X_L and X_C are the inductive and capacitive reactances, respectively. This concept is essential for evaluating how much energy is effectively converted into work or heat in the circuit.

Impedance in AC Circuits

Impedance is the total opposition a circuit presents to the flow of alternating current and is a combination of resistance and reactance (both inductive and capacitive). Unlike resistance, impedance is frequency dependent, meaning that as the signal frequency changes, the net impedance of the circuit changes. This is crucial when analyzing circuit response at different frequencies.

Series RLC Circuit

A series RLC circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in a single loop. The behavior of this circuit depends on the combined effects of resistance and the frequency?dependent reactances of the inductor and capacitor. Understanding series RLC circuits is fundamental in analyzing how alternating current (AC) signals are affected by these components together.

Resonance in RLC Circuits

Resonance occurs when the inductive reactance and capacitive reactance in an RLC circuit cancel each other out. At this frequency, known as the resonance frequency, the circuit’s impedance is minimized to the resistance value alone, which maximizes the current for a given voltage. This concept is critical in many applications, such as tuning circuits and filters.

Transcript

00:01 The average power that is delivered to the circuit is equal to the rms voltage times the rms current times the cosine of the phase.

00:16 But now remember that the rms voltage is equal to the rms current times the impedance.

00:26 Then the rms current is equal to the voltage divided by the impedance.

00:34 And therefore, the average power can be written as v rms squared divided by the impedance times the cosine of the phase.

00:46 But the cosine of the phase is equal to the resistance divided by the impedance.

00:51 And the average power is equal to the rmass voltage squared divided by the impedance squared times the resistance.

00:59 In the first item, the circuit is driven at resonance.

01:05 Then, remember that at resonance, both reactances are equal, and therefore the impedance is equal to the resistance.

01:20 Then, in this case, the average power is equal to the rms voltage squared divided by the resistance.

01:31 And this is equal to 21 .5 squared divided by 20, which is approximately 23 .1 watts.

01:46 On the next item, the frequency of the circuit is equal to two times the resonant frequency, and the resonant frequency is equal to 1 divided by the square root of l times c.

02:01 Then this is the frequency of the circuit.

02:05 The impedance is equal to the square root of r squared plus the difference between both reactances squared.

02:18 Before plugging anything in this equation, let us take a look at the reactances.

02:24 The capacity reactance is equal to 1 divided by the frequency times the capacitance...

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