31. Alternating Current

The RC circuit shown in Fig. 30–39 is a low-pass filter because it passes low-frequency ac signals with less attenuation than high-frequency ac signals. (a) Show that the voltage gain is $A=V_{\text{out}}/V_{\text{in}}=1/(4{\pi }^2{f}^2{R}^2{C}^2+1{)}^{\frac{1}{2}}$ (b) Discuss the behavior of the gain A for f → 0 and f → ∞.

A circuit contains two elements, but it is not known if they are L, R, or C. The current in this circuit when connected to a 120-V rms 60.0-Hz source is 5.6 A and lags the voltage by 55°. What are the two elements and what are their values?

A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W. What is the motor's resistance?

The tuning circuit in an FM radio receiver is a series RLC circuit with a 0.200 μH inductor. FM radio stations are assigned frequencies every 0.2 MHz, but two nearby stations cannot use adjacent frequencies. What is the maximum resistance the tuning circuit can have if the peak current at a frequency of 103.9 MHz, the closest frequency that can be used by a nearby station, is to be no more than 0.10% of the peak current at 104.3 MHz? The radio is still tuned to 104.3 MHz, and you can assume the two stations have equal strength.

(II) An LRC series circuit with R = 120 Ω, L = 25 mH, and C = 2.0 μF is powered by an ac voltage source of peak voltage Vo = 340 V and frequency f = 660 Hz. Determine the peak voltage across C and its phase angle relative to the source voltage.

(II) An LRC series circuit with R = 120 Ω, L = 25 mH, and C = 2.0 μF is powered by an ac voltage source of peak voltage V₀ = 340 V and frequency f = 660 Hz. Determine the peak voltage across L and its phase angle relative to the source voltage.

A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the average power dissipated?

(II) An LRC series circuit with R = 120 Ω, L = 25 mH, and C = 2.0 μF is powered by an ac voltage source of peak voltage V₀ = 340 V and frequency f = 660 Hz. Determine the peak current that flows in this circuit.

A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W. How much series capacitance needs to be added to increase the power factor to 1.0?

(II) An LRC series circuit with R = 120 Ω, L = 25 mH, and C = 2.0 μF is powered by an ac voltage source of peak voltage V₀ = 340 V and frequency f = 660 Hz. Determine the phase angle of the source voltage relative to the current.

Show that the peak inductor voltage in a series RLC circuit is maximum at frequency ${\omega }_L={(\frac{1}{{\omega }_0^2}-\frac{1}{2}{R}^2{C}^2)}^{-1/2}$.

(II) An LRC series circuit with R = 120 Ω, L = 25 mH, and C = 2.0 μF is powered by an ac voltage source of peak voltage V₀ = 340 V and frequency f = 660 Hz. Determine the peak voltage across R and its phase angle relative to the source voltage.

Show that the fraction of electromagnetic energy lost (to thermal energy) per cycle in a lightly damped (R² ≪ 4L/C) LRC circuit is approximately $\frac{\Delta U}{U}=\frac{2\pi R}{L\omega }=\frac{2\pi }{Q}$. The quantity Q can be defined as Q = Lω/R, and is called the Q-value, or quality factor, of the circuit and is a measure of the damping present. A high Q-value means smaller damping and less energy input required to maintain oscillations.