Unformatted text preview: Mass+spring system
x=0, v=Max K=Max, U=0 x=Max, v=0, K=0, U=Max x=0, v=Max K=Max, U=0 x=Max, v=0, K=0, U=Max x=0, v=Max K=Max, U=0 1 LC circuit oscillations Inductor energy proportional to i2. Capacitor energy proportional to q2. 2 LC circuit
Loop rule: Vcircuit dq Substitute, i = dt
2 di q = -L - = 0 dt C d q q 1 =- =- q 2 dt LC LC familiar? 3 Comparing equations
d x k =- x 2 dt m d 2q 1 =- q 2 dt LC
2 Mass on spring Inductor/Capacitor Solution to the LC circuit: q = Q cos( t + ) where 1 = LC and q is the charge on the capacitor. 4 LC circuit q Q VC = = cos(t + ) C C
What does the loop rule say about VL?
The phase constant depends on how we start the system out. What is the phase constant if we start the system with charge on the capacitor and no current flowing? 5 Energy in LC circuits q Q 2 UE = = cos (t + ) 2C 2C 1 Li 2 = 1 LI 2 sin 2 ( t + ) UB = 2 2
U B + U E = U = constant
max E 2 2 =U max B Q LI = = 2C 2 2 2 6 Energy oscillation in LC circuit
1.2 1 Ratio of UE to UB 0.8
B B E U=U +U 0.6 0.4 0.2 0 -0.2 0 0.2 0.4 0.6 Time in Periods 0.8 1 1.2 U UE Energy vs time when the current is a maximum at t=0. 7 Applications
LC circuits have practical applications for tuned systems: audio oscillators radio receivers An important concept for tuned systems: resonance Resonance occurs when a system is driven at the frequency at which it would naturally oscillate. 8 Natural resonance frequencies
k Mass on spring: = m Pendulum: = g L 1 LC circuit: = LC 9 Driven oscillations
When a harmonic system of natural frequency N is driven by a harmonic force of frequency F , 1) The system oscillates at F . 2) The response is out of phase with the driving force. For a force: F (t ) = F sin ( F t ) The response is: R(t ) F
2 2 F - N ) ( 2 F sin ( F- ) t (If you keep pushing a resonant system at the natural frequency, the amplitude keeps growing until something gives.) Most real systems have damping. 10 RLC circuit: a harmonic system with damping Add resistance R for damping and a source of energy to keep the oscillations going. 11 Resonant behavior of an RLC circuit
emf = VF sin F t = VR + VC + VR dq q d q VR = iR = R ; VC = ; VL = L 2 dt C dt Plugging in a guess of q(t)=sin ( Ft - ) we get: VF / L q(t) = sin ( t - ) 1/ 2 2 1 2 R 2 - F + F LC L 2 12 RLC oscillations We can vary the applied frequency to demonstrate a resonance in the current.
RLC link 13 ...
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