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Unformatted text preview: he LC circuit The LC circuit Suppose we put some charge Q on the capacitor, let some current i in the inductor and let this circuit go. g Since V L = V C , Ldi/dt=Q/C, or d 2 Q/dt2+(1/LC)Q=0. The solution that satisfies the initial conditions is (t)=Q os( +(i in( Q(t)=Q cos( t)+(i / )sin( t). The charge and current oscillate with a natural frequency =(1/LC) 1/2 . It is this natural oscillation that produces resonance in the driven circuit. scillation math Oscillation math It is convenient and possible to express the oscillation in the form Q(t)=Q max cos( t ), with Q max the amplitude, still the natural frequency, and the phase constant. /2 In terms of our previous solution, Q max =[Q 2 +i 2 / 2 ] 1/2 and tan = i /Q . e will see that quite generally that the naturals We will see that quite generally that the naturals frequency is always follows the form 2 =force/inertia, the amplitude is determined by the energy of the system, nd the phase constant by the initial conditions i e and the phase constant by the initial conditions , i.e., how the oscillations start. ass on a spring Mass on a spring Recall the spring force F=ky, where y is the displacement from the equilibrium position, and k is the spring constant (big for stiff springs) Newtons 2 nd Law gives md 2 y/dt 2 =ky. If y is the initial displacement and v is the itial velocity then we can clone the initial velocity, then we can clone the solution to this equation from the LC oscillator: y(t)=y cos( t)+(v / )sin( t). =(k/m) 1/2 . Alternatively y(t)=y max cos( t ), with ax 2 =y 2 +v 2 2 and =v y max y v / , and v / y . omparison of LC to mass on spring Comparison of LC to mass on spring For the spring, m comprises the inertia. It resists changes in velocity. In the LC circuit, L resists changes in current, acting like an electrical inertia. or the spring k measures the stiffness of the spring For the spring, k measures the stiffness of the spring. In the LC circuit, 1/C plays this roll of k....
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 Spring '10
 TimothyBolton
 Charge, Current

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