AC Circuits: An AC circuit involves an inductor, a capacitor and a resistor arra

Question

AC Circuits: An AC circuit involves an inductor, a capacitor and a resistor arranged as an LRC circuit as shown. The AC function generator has 20 volts maximum potential through the circuit at a frequency of 1000 Hz. The resistor is 100 ohms, the inductor 0.05 H and the capacitor is 0.008 Farads. a. What is the V_rms for this circuit? b. What is the reactive capacitance of the circuit? c. What is the reactive impedance of the circuit? d. What is the impedance (Z) of the circuit? e. What is the phase angle of this circuit? Does it lead or trail the current (sign)? f What is the resonance frequency of this circuit? g. What is the current in this circuit? h. What is the power being delivered in this circuit?

Explanation / Answer

Given

   LCR circuit with

   V0 = 20 v, L = 0.05H, C = 0.008 F, R = 100 ohm , f= 1000 Hz, W = 2pif = 2pi*1000 = 6283.2 rad/s

a) Vrms = V0/(sqrt(2)) = 20/sqrt2 v = 14.142 v

b) reactive capacitance XC = 1/W*C = 1/(6283.2*0.008) ohm = 0.019894 ohm

c) for reactive impedance , first calculate the reactive inductance Xl = W*L = (6283.2*0.05)ohm = 314.16 ohm

   so reactive impedance is = XL- Xc= 314.16 - 0.019894 ohm = 314.140106 ohm

   XL > XC

d)impedance of the circuit Z = sqrt(R^2+(XL-XC)^2) = sqrt(100^2+(314.140106)^2) ohm = 329.673 ohm

e) phase angle is Tan phi = (XL-Xc)/R = 314.140106 /100 ==> alpha = arc tan (314.140106 /100) = 72.34 degrees

f) resonance frequency of the circuit is f = 1/(2pisqrt(L*C))

       f = 1/(2pi(sqrt(0.05*0.008))) HZ =7.96 Hz
g) curernt in the circuit is

   i = V/Z = 20/329.673 A = 0.0606662 A

h) power delivered avereage power is P = vrms^2*cos alpha /Z

                   = 14.142^2cos(72.34) /(329.673) w

                   = 0.1840381742587 W

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