Question

Extra Credit: You have an undriven series RLC circuit in which the capacitor was given some initial charge at t=0. a. What is the natural frequency of the circuit in terms of R, L, and C? (2 pts) b. For what R value(s) will the circuit oscillate at the natural frequency? For other R values, will it oscillate at a higher or lower frequency? (2 pts) c. If we call the value of R that causes critical damping Re, then for all possible values of R, give a sketch of the voltage versus time, say what that particular type of response is called, and tell what range of R values correspond to each type of response. (6 pts)

Answer 1

ANSWER:
qo 0 R. L 10 J iR+ L di oblu 0 dt But we have ar dq dit Ldq dt² AR 44 dt + Z=0 2 d q R dq 1 t + dt L. dt Le q=0 2 d' q +28.dq ughq Here we = 1 / 2; 28 = & 2 dt dt Now, we have solution for of exists as Yer (t); By writing autially equation m²+28m tigh ao -23+ 1432442 2

زمع : m = -27 172 uz 2 =-81 i sugeda (a) Natural frequency of the system w=f-uz2782 (1) wyr te+ (4) (b) for WJ = Wo we - wg 28 2² - 2wj2 2. (2) 200 R Fwo 2L ल R 2L 2 tor) R = 282 VLC For the value of R=282 circuit oscillate with 하는 national frequency for other values of 'R' it will oscillate Dit kantly R'3R then we's W. 2 An both cases it is less than RcR then wil wd I natural frequency 'wo

8 (c) for critical damping P = woe (A = te R = 24 = R. for (2) 22 Swecon Re> rc called as over damping Respons under damping (on) له for (12.) <wg² tor) RL Re called small damping Response

< over damping Critical damping M t under - damping sesponse Los cillatery voltage)

: R=RL critical response RSRc.over over damping R ZRC under damping