Lecture 21 - he LC circuit The LC circuit Suppose we put...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 18

Lecture 21 - he LC circuit The LC circuit Suppose we put...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online