An RLC circuit such as that of Figure ( a ) has R = 4.42 ?, C = 20.6 ?F, L = 1.0

Question

An RLC circuit such as that of Figure (a) has R = 4.42 ?, C = 20.6 ?F, L = 1.03 H, and ?m = 30.8 V. (a) At what angular frequency ?d will the current amplitude have its maximum value, as in the resonance curves of Figure (b)? (b) What is this maximum value? At what (c) lower angular frequency ?d1 and (d) higher angular frequency ?d2 will the current amplitude be half this maximum value? (e) What is (?d2 - ?d1)/?d, the fractional half-width of the resonance curve for this circuit?

An RLC circuit such as that of Figure (a) has R = 4.42 ?, C = 20.6 ?F, L = 1.03 H, and ?m = 30.8 V. (a) At what angular frequency ?d will the current amplitude have its maximum value, as in the resonance curves of Figure (b)? (b) What is this maximum value? At what (c) lower angular frequency ?d1 and (d) higher angular frequency ?d2 will the current amplitude be half this maximum value? (e) What is (?d2 - ?d1)/?d, the fractional half-width of the resonance curve for this circuit?

Explanation / Answer

a)

Resonant frequency

Wd=1/sqrt[LC) =1/sqrt[20.6*10-6*1.03]

Wd=217.1 rad/s

b)

at resonant frequency

XL=Xc

=>Impedance Z=R=4.42 ohms

so Current amplitude

I=V/Z=30.8/4,42

I=6.97 A

c)

Given

I=6.97/2 =3.48 A

Impedance

Z=V/I =30.8/3.48 =8.84 ohms

since

Z=sqrt[R2+(WL-1/WC)2]

8.84=sqrt[4.422+(1.03*W-1/(20.6*10-6)*W)2]

58.61=1.03W-48543.7/W

1.03W2-7.655W-48543.7=0

W=213.4 rad/s

W=220.8 rad/s

Lower angular frequency

Wd1=213.4 rad/s

d)

higher angular frequency

Wd2=220.8 rad/s

e)

Wd2-Wd1/Wd1 =220.8-213.4/213.4=0.03468

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