Question

5. 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

Sol th oo L R + olu But we = ir + L di - 0 at have da TE L d²q + at2 at d²a + 28 dat ata we a at + R da a & +R dq tl q=0 a dt2 Į at LC 11 Here Wo²= LC 27=R L Now we have as solution for exists By writting auxillary equation q.(t). CF m2 + 2 zom two EO m 22 543² 4 wo? 2 m - I J 3² w2 = + is 2 22 a) the natural frequency of F c + (2 ² System 2 wd = +3 Wd = (b) For Wd = Wo wor 2 2 22 - wo 222 > awg J2 wo alog ( 1 / ² = 2 wo? 1 l = Uz ste R = for the Circuit osallate with natural 25g value of R= 252 frequency. Differently values of wa wd than natural frequency wó. wa' <wd for other it will oscillate R'>R then In Both the Cases it is w RICR then les

(C) for Critical damping 32= wo2 2 t QL - Wo 5 = R = 2 2 For ( 4 ) co R>R Called over damping called Response for ² ² Small -damping (2) ² < we? RL as مكيا under-damping Response. CN V over damping Critical damping Ar t under-darping Response. (Oscillatery voltage) R= Rc Critical Response R> Rc overdamping under damping Rc Rc