31. Alternating Current

In a series $LRC$ circuit, what is the primary effect of resonance on the circuit's response?

Which of the following events is an example of resonance in a series $LRC$ circuit?

A series LRC circuit is formed with a power source operating at VRMS = 100 V, and is formed with a 15 Ω resistor, a 0.05 H inductor, and a 200 µF capacitor. What is the voltage across the inductor in resonance? The voltage across the capacitor?

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the resistor value is doubled?

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled?

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled and, at the same time, the inductor value is halved?

For the circuit of FIGURE EX32.32, What is the resonance frequency, in both rad/s and Hz?

For the circuit of FIGURE EX32.33, What is the resonance frequency, in both rad/s and Hz?

A television channel is assigned the frequency range from 54 MHz to 60 MHz. A series RLC tuning circuit in a TV receiver resonates in the middle of this frequency range. The circuit uses a 16 pF capacitor. What is the value of the inductor?

(II) A 25-mH coil whose resistance is 0.80 Ω is connected to a capacitor C and a 420-Hz source voltage. If the current and voltage are to be in phase, what value must C have?

What is the resonant frequency of the LRC circuit of Example 30–11? At what rate is energy taken from the generator, on the average, at this frequency?

The frequency of the ac voltage source (peak voltage Vo) in an LRC circuit is tuned to the circuit’s resonant frequency f₀ = 1 / (2π√LC). (a) Show that the peak voltage across the capacitor is Vco = VoTo/ (2πτ), where To ( =1/fo) is the period of the resonant frequency and τ = RC is the time constant for charging the capacitor C through a resistor R. (b) Define β = To/ (2πτ) so that Vco = βVo. Then β is the “amplification” of the source voltage across the capacitor. If a particular LRC circuit contains a 2.0-nF capacitor and has a resonant frequency of 5.0 kHz, what value of R will yield β = 125?

Determine a formula for the average power $\bar{P}$ dissipated in an LRC circuit in terms of L, R, C, ω and V₀.

Determine a formula for the average power $\bar{P}$ dissipated in an LRC circuit in terms of L, R, C, ω and V₀. At what frequency is the power a maximum?

(III) Determine a formula for the average power $\bar{P}$ dissipated in an LRC circuit in terms of L, R, C, ω and V₀. Find an approximate formula for the width of the resonance peak in average power, Δω, which is the difference in the two (angular) frequencies where $\bar{P}$ has half its maximum value. Assume a sharp peak.