Math 2280-001 Mon Mar 2 , Use last Friday's notes to understand damped forced oscillations, m x##C c x#C k x = F0 cos w t i.e. with c O 0. As we discussed briefly on Friday, in this case the undetermined coefficients particular solution is called the "steady periodic" solution, because this sinusoidal solution persists as t/N, whereas the general homogeneous solution always decays exponentially to zero. And then do something completely different, and entirely the same: , 3.7 Electrical circuits. It turns out that the mathematics of RLC circuits is identical to that for damped mass spring configurations: http://cnx.org/content/m21475/latest/pic012.png Charge Q t coulombs accumulates on the capacitor, at a rate (current) I t amperes (coulombs/sec), i.e Q # t = I t . i t in the diagram above) Kirchoff's Law: The sum of the voltage drops around any closed circuit loop equals the applied voltage V t (volts). The units of voltage are energy units - Kirchoff's Law says that a test particle traversing any closed loop returns with the same potential energy level it started with: 1 For Q t : L Q ## t C R Q # t C Q t =V t C 1 For I t : L I## t C R I# t C I t = V# t C If we specify that the voltage is given by a periodic sine function then V# t will be a cosine function: 1 For Q t : L Q ## t C R Q # t C Q t = V t = E0 sin w t C 1 For I t : L I## t C R I# t C I t = V# t = E0 w cos w t . C 1 We can just transcribe all of our work from mass-spring systems. Just substitute L for m, R for c, for C k, and I t for x t . (There could be other situations where you want to study Q t instead, depending on the context.) 1 I t = V# t = E0 w cos w t C For example, for RLC circuits with R O 0 the general solution for I t is I t = Isp t C Itr t . L I## t C R I# t C p . 2 (One difference from the forced mass-spring configuration it makes sense to express Isp t as Isp t = I0 cos w t K a = I0 sin w t K g , g = a K I0 sin w t K g , since the applied voltage is E0 sin w t . Since 0 % a % p, it follows that p p %g% . Transcribing ... 2 2 K F0 C w = kKm w 2 2 C c2w 0 I0 w = The denominator 1 KL w Cw 2 2 E0 w 0 I0 w = 1 2 KL w C E0 1 KL w Cw 2 C R2 w 2 . 2 C R2 C R2 of I0 w is called the impedance Z w of the circuit (because the larger the impedance, the smaller the amplitude of the steady-periodic current that flows through the circuit). Notice that for fixed resistance, the impedance is minimized and the steady periodic current 1 amplitude is maximized when = L w , i.e. Cw 1 C= if L is fixed and C is adjustable (old radios). 2 Lw 1 L= if C is fixed and L is adjustable 2 Cw Both L and C are adjusted in this M.I.T. lab demonstration: http://www.youtube.com/watch?v=ZYgFuUl9_Vs.