An RLC circuit such as that of Figure (a) has R = 7.22 Ohm, C = 23.9 mu F, L = 1

Question

An RLC circuit such as that of Figure (a) has R = 7.22 Ohm, C = 23.9 mu F, L = 1.41 H, and epsilon_m = 35.9 V. (a) At what angular frequency omega_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 omega_d1 and (d) higher angular frequency omega_d2 will the current amplitude be half this maximum value? (e) What is (omega_d2 - omega_d1)/omega_d, the fractional half-width of the resonance curve for this circuit?

Explanation / Answer

(A) it will be at resonate frequency,

at resonant frequency w0,

XL = XC

w0 L = 1 / w0 C

w0 = 1 / sqrt(LC) = 1 / sqrt(1.41 x 23.9 x 10^-6)

wo = 172.3 rad/s

(B) at this w0

Z = sqrt[ R^2 + (XL - Xc)^2] = R = 7.22 ohm

I = V/ Z = 35.9 / 7.22

I = 4.97 A

(C) I' = I/2 = 2.49 A

Z = V / I = 14.44 ohm

Z = sqrt[ R^2 + (XL - XC)^2]

XL - Xc = sqrt(14.44^2 - 7.22^2) = 12.5

w L - 1 / wC = 12.5

1.41 w^2 - 12.5w - 41841 =0

w = 176.75 rad/s Or 167.8 rad/s

w1 = 167.8 rad/s

(D) w2 = 176.75 rad/s

(E) (176.75 - 167.8) / 172.3 = 0.052

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