31. Alternating Current

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?

(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?

(I) 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?

(II) 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/ (2π τ), 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/ (2π τ) 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?

(III) (a) Determine a formula for the average power P dissipated in an LRC circuit in terms of L, R, C, ω and Vo.

(III) Determine a formula for the average power P dissipated in an LRC circuit in terms of L, R, C, ω and Vo. (b) At what frequency is the power a maximum?

(III) Determine a formula for the average power P dissipated in an LRC circuit in terms of L, R, C, ω and Vo. (c) 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 P has half its maximum value. Assume a sharp peak.