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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 Δ V L = Δ 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( ω t)+(i / ω )sin( ω t). • The charge and current oscillate with a natural frequency ω =(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( ω t φ ), with Q max the amplitude, ω still the natural frequency, and φ the phase constant. /2 • In terms of our previous solution, Q max =[Q 2 +i 2 / ω 2 ] 1/2 and tan φ = ω 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 ω 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”) • Newton’s 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( ω t)+(v / ω )sin( ω t). • ω =(k/m) 1/2 . • Alternatively y(t)=y max cos( ω t φ ), with ax 2 =y 2 +v 2 2 and =v y max y v / ω , and φ v / ω 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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This note was uploaded on 05/11/2010 for the course ＰＨＹ 214 taught by Professor Timothybolton during the Spring '10 term at Kansas State University.
 Spring '10
 TimothyBolton
 Charge, Current

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