A) LC circuits and Oscillation Frequency Physics 2112 Unit 19 Today’s Concepts: A) LC circuits and Oscillation Frequency B) Energy C) RLC circuits and Damping

LC Circuit I L C Q - + + - Circuit Equation: where Solution to this DE:

Just like in 2111 C L k x m F = -kx a Same thing if we notice that and

Also just like in 2111 k L C F = -kx a m x Total Energy constant (when no friction present) x C L Total Energy constant (when no resistor present)

Charge and Current “90o out of phase” Q and I Time Dependence C L I + + - - Charge and Current “90o out of phase”

CheckPoint 1 L C At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. Inductor does not need current to have voltage drop! It’s not like a resistor! What is the potential difference across the inductor at t = 0? A)  VL = 0 B)  VL = Qmax/C C)  VL = Qmax/2C since VL = VC

CheckPoint 2 At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C Q is max but dQ/dt is 0! What is the potential difference across the inductor at when the current is maximum? A)  VL = 0 B)  VL = Qmax/C C)  VL = Qmax/2C

Max current means all of U is in inductor! CheckPoint 3 L C At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. Max current means all of U is in inductor! How much energy is stored in the capacitor when the current is a maximum ? A)  U = Qmax2/(2C) B)  U = Qmax2/(4C) C)  U = 0

Example 19.1 (Charged LC circuit) After being left in position 1 for a long time, at t=0, the switch is flipped to position 2. All circuit elements are “ideal”. 1 2 V=12V C=0.01F L=0.82H R=10W What is the equation for the charge on the upper plate of the capacitor at any given time t? How long does it take the lower plate of the capacitor to fully discharge and recharge?

Example 19.1 (Charged LC circuit) What is the equation for the charge on the upper plate of the capacitor at any given time t? How long does it take the lower plate of the capacitor to fully discharge and recharge? Conceptual Analysis Fill in all the terms in Strategic Analysis Find initial current Use energy conservation to find Qmax Use L and C to find w Use initial conditions to final f

Prelecture Question Q(t) = Qmaxcos(ωt) Q(t) = Qmaxcos(ωt + π/4) The switch is closed for a long time, resulting in a steady current Vb/R through the inductor. At time t = 0, the switch is opened, leaving a simple LC circuit. Which formula best describes the charge on the capacitor as a function of time? Q(t) = Qmaxcos(ωt) Q(t) = Qmaxcos(ωt + π/4) Q(t) = Qmaxcos(ωt + π/2) Q(t) = Qmaxcos(ωt + π)

CheckPoint 4 The capacitor is charged such that the top plate has a charge +Q0 and the bottom plate -Q0. At time t = 0, the switch is closed and the circuit oscillates with frequency w = 500 radians/s. L C L = 4 x 10-3 H w = 500 rad/s + + - - What is the value of the capacitor C? A)  C = 1 x 10-3 F B)  C = 2 x 10-3 F C)  C = 4 x 10-3 F

CheckPoint 5 L C closed at t = 0 +Q0 -Q0 Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed?

CheckPoint 6 L C When the energy stored in the capacitor reaches its maximum again for the first time after t = 0, how much charge is stored on the top plate of the capacitor? closed at t = 0 +Q0 -Q0 A)  +Q0 B)  +Q0 /2 C)  0 D) -Q0/2 E) -Q0 Q is maximum when current goes to zero Current goes to zero twice during one cycle

Use “characteristic equation” Add Resistance 1 2 Switch is flipped to position 2. R Use “characteristic equation”

Natural oscillation frequency Damped oscillation frequency Damped Harmonic Motin Damping factor Natural oscillation frequency Damped oscillation frequency

Which one do you want for your shocks on your car? Remember from 2111? Overdamped Critically damped Under damped Which one do you want for your shocks on your car?

Example 19.2 (Charged LC circuit) After being left in position 1 for a long time, at t=0, the switch is flipped to position 2. The inductor now non-ideal has some internal resistance. RL=7W V=12V C=0.01F L=0.82H R=10W What is the equation for the charge on the capacitor at any given time t? How long does it take the lower plate of the capacitor to fully discharge and recharge?

V(t) across each elements The elements of a circuit are very simple: This is all we need to know to solve for anything. But these all depend on each other and vary with time!!

How would we actually do this? Start with some initial V, I, Q, VL Now take a tiny time step dt (1 ms) Repeat… What would this look like?

Example 19.3 In the circuit to the right R1 = 100 Ω, L1 = 300 mH, C = 120 μF and V = 12 V. The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive. What is the charge on the bottom plate of the capacitor at time t = 2.82 ms? Conceptual Analysis Find the equation of Q as a function of time The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Strategic Analysis Determine initial current Determine oscillation frequency w0 Find maximum charge on capacitor Determine phase angle

Example 19.4 In the circuit to the right R1 = 100 Ω, L1 = 300 mH, C = 120 μF and V = 12 V. The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive. What is the energy stored in the inductors at time t = 2.82 ms? Conceptual Analysis Use conservation of energy The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Strategic Analysis Find energy stored on capacitor using information from Ex 19.3 Subtract of the total energy calculated in the clicker question