Question

Please solve this as a differential equations problem not as a physics probelm, please.

3. (7pts) A crude tuning device can be created by an LRC-circuit forced by an external signal. An LRC-circuit is equipped with a variable capacitor, which can be dialed to different values to obtain the maximum response from an incoming radio signal. If I(t) is the current in the tuning device, which contains an inductor, L, a resistor, R, and a tunable capacitor, C, and receives an external signal, Vow cos(wt), the ODE describing this system satisfies: LÏ +Rİ + ¿I = Vow cos(wt), (1) where V, is the strength of the signal and w = 27 f and f is the frequency of the signal. a. Suppose the inductor is L = 30 mH, the resistor is R = 10 12, the maximum signal is Vo = 50, and the frequency f = 60 Hz. Solve Eqn. (1) for any C. Give the solution for t + o. Find the amplitude of this oscillatory solution. b. Find the value of C that gives the optimal response of this circuit to the external signal above, Cmar, and determine the amplitude of that response. With this value, Cmax, of tuning, what is the magnitude of the response to a f = 50 Hz signal.

Answer 1

page- Answer: Given data Li +Rit_I - Vow cos(wt), 1. An. LCR circuit with diffential Equation flï + RI + I - Yow coscoot)]-0 @ Griven data L = 30ml R=lor Vo=50 F: 60 By using given data . Q = Qлхбо [w=1200) We have to solve © for any e you know that for du d'u tay dy two = Acospt General solution can be writen as Sum of two act) - Afyt Sinçw'ta) + A, cos (Pt-O) Toansient) (Steady State Solution)

where A, & or are un knows and Calaclated from Pnitial values & san o = Qyp In our problem : DI + RdI + - Yow ) cos ut for us (wet, LY ), A - You) .: General Solution is I (t) = A, e-C At sin (w't +9) + Az Coscut-o) whore w'at (B5 w- aff Az = Now Ulte-we) <ul te xwe tan o - Ax xw - The-weiz

page! ® At f od Transient solution ie ist there will go away & we are left with Ar cos (wt-o) Amplitude ( t o ) Vow Vize-w2jkt u . At tya, curent & w in circuût ps decided by Vo cos wt ( forcing ostillatol) optimal responce mean when Az is manimum you can diffenetrate w.got Vize-603) 44 202 W& get resonant condition we will get, - [wt- woh : 242 So wh- wol-aya w- is - aveva can fj?. te - page 2 putting values we get

page: - un'x (60)2--L 30x1o Bc. - (10)2 2X (30x10 32 103 10x106 4x9.8 X 3600 - "148 x 3000 4x98x360 = 300 1000 - 2x900 10% X205 2x 900 300 141120 - 100 X 103 30 141120 + 5555 = 100 146675 X3 100 C = 0.000227 . Іс- ааз ие | and port putting f = 50 Hq find A = 9 put all valus & get A = 99 Vow 1. / (wow?) ?>4 ( 22 ) xw2