A series RLC circuit with L = 13 mH, C = 3.6 mu F, and R = 4 Ohm is driven by a

Question

A series RLC circuit with L = 13 mH, C = 3.6 mu F, and R = 4 Ohm is driven by a generator with a maximum emf of 120 V and a variable angular frequency omega. (a) Find the resonant (angular) frequency omega_0. rad/s 0 attempt(s) made (maximum allowed for credit = 10) (b) Find I_rms at resonance. A 0 attempt(s) made (maximum allowed for credit = 10) When the angular frequency omega = 7000 rad/s, (c) Find the capacitive reactance X_C in ohms. Ohm 0 attempt(s) made (maximum allowed for credit = 10) Find the inductive reactance X_L in ohms. Ohm 0 attempt(s) made (maximum allowed for credit = 10) (d) Find the impedance Z. (Give your answer in ohms.) Ohm 0 attempt(s) made (maximum allowed for credit = 10) Find I_rms. A 0 attempt(s) made (maximum allowed for credit = 10)

Explanation / Answer

Here, L = 13 x 10^-3 H, C = 3.6 x 10^-6 F, R = 4 ohm

(a) Resonant angular frequency w0 = 1/sqrt(LC) = 1 / sqrt(13 x 10^-3x3.6 x 10^-6) = sqrt(10^8 / 4.68)

= 4622.50 rad/s.

(b) At w = w0, XC = 1/(wC) = XL = wL = (±)60.0925 . Since XL+XC = 0, Z = R = 4.0 .
Now, maximum emf = 120 V

So, rms emf = 120/root2 = 84.86 V.
So, Irms at resonance = 84.86/4 = 21.21 A

(c) Now, w = 7000 rad/s

So, Capacitive reactance Xc = 1/(wC) = 1/(7000x3.6 x 10^-6) = 39.68 Ohm.

Inductive reactance XL = wL = 7000x13 x 10^-3 = 91 ohm.

(d) Impedence, Z = R + i (w L - 1 / wC) = 4 + i(91 - 39.68) = (4 + i51.32) ohm

Magnitude of impedence = sqrt(4^2 + 51.32^2) = 51.47 Ohm

Irms = 84.86/51.47 = 1.65 A

(e) Phase angle, theta = arctan (imaginary / real) = arctan (51.32/4) = 85.54 deg.

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