Problem 24.94 An R L C circuit with R = 25.0 ? , L = 305 m H , and C = 45.9 ? F

Question

Problem 24.94       An RLC circuit with R = 25.0? , L = 305mH , and C = 45.9?F is connected to an ac generator with an rms voltage of 28V .

Determine the average power delivered to this circuit when the frequency of the generator is equal to the resonance frequency.         
Determine the average power delivered to this circuit when the frequency of the generator is twice the resonance frequency.
Determine the average power delivered to this circuit when the frequency of the generator is half the resonance frequency.         
Problem 24.94       An RLC circuit with R = 25.0? , L = 305mH , and C = 45.9?F is connected to an ac generator with an rms voltage of 28V .

Determine the average power delivered to this circuit when the frequency of the generator is equal to the resonance frequency.         
Determine the average power delivered to this circuit when the frequency of the generator is twice the resonance frequency.
Determine the average power delivered to this circuit when the frequency of the generator is half the resonance frequency.         

Explanation / Answer

Impedence of RLC circuit is given by Z^2 = R^2 + (XL-Xc)^2

at resonant frequency w^2=1/LC

w= 1/sqrt(0.305*45.9*10^-6)--> w = 267.26 rad/s

Where XL = WL = 2pifL = 267.26*0.305 =81.51 ohms

Xc= 1/wC = 1/2pifC = 1/267.26*45.9uF = 81.51 ohms

so Z = R = 25?

Current Irms = Vrms/Z= 28/25 = 1.12 A

a. avg power = Irms Vrms = 1.12* 28 = 31.36 watts

b. when w= 2* 267.26 =534.52 rad./s

Xl= 534.52 * 0.305 = 163.02 ohms

XC= 1/534.52*45.9*10^-6 = 40.7 ohms

thus Z^2 = 25^2 +(163.02-40.7)^2

Z = 124.82 ohms

i = 28/124.82 =0.224A

avg power = Irms Vrms =0.224 * 28 = 6.28 watts

c. when W= 267.26/2 = 133.63rad/s

Xc = 1/wC = 1/133.63*45.9uC =163 ohms

Xl=wL = 133.63* 0.305 =40.75 ohms

Z^2 = 25^2+ (163-40.75)^2

Z = 124.77 ohms

irms = 28/124.77 =0.2244 A

avg power = 0.2244 * 28 = 6.28 watts

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