PES 0 Spring 04, Spendier Lecture 35/Page Today: chapter 3 - LC circuits We have explored the basic physics of electric and magnetic fields and how energy can be stored in capacitors and inductors. We next turn to the associated applied physics, in which the energy stored in one location can be transferred to another location so that it can be put to use. For example, energy produced at a power plant can show up at your home to run a computer. LC Circuit: Imagine we have just an inductor (inductance L) and a capacitor (Capacitance C) in a circuit in series. We will now investigate eight stages in a single cycle of oscillation of a resistanceless LC circuit. The bar graphs by each figure show the stored magnetic B and electrical E energies. The total energy of the system is E B Q E C LI B The magnetic field lines of the inductor and the electric field lines of the capacitor are shown. Assume that the wires have no resistance. (a) Capacitor with maximum charge, no current. (b) Capacitor discharging, current increasing, current flows ccw. Some of electric field energy is converted to magnetic field energy
PES 0 Spring 04, Spendier Lecture 35/Page (c) Capacitor fully discharged, current maximum. All of the energy is now in the magnetic field The inductor resists changes to the current/magnetic field so the current continues counter clockwise, even though Q = 0. (d) Capacitor charging but with polarity opposite that in (a), current decreasing. New eclectic field in opposite direction to original. The current continues to flow ccw but is decreasing. The capacitor "wants" to create cw current. The inductor resists the change. Eventually when enough charge is built up on the capacitor plates, the current can reverse. (e) Capacitor with maximum charge having polarity opposite that in (a), no current (for a short moment). (f) Capacitor discharging, current increasing with cw direction (opposite to direction in (b)).
PES 0 Spring 04, Spendier Lecture 35/Page 3 (g) Capacitor fully discharged, current maximum. (h) Capacitor charging, current decreasing. And we arrive back at the start, (a). So we see that there is an oscillation tin the magnetic and electric field going from (a) to (h). We can plot the potential difference across the capacitor of the circuit as a function of time. This quantity is proportional to the charge on the capacitor. The total energy of the system is conserved since there is no resistance to dissipate energy as heat. But the energies B and E oscillate. When B = 0, E is maximum, and vice-versa. This should look familiar to you. Physics : conservation of total energy for mass on a spring! Energy is transferred between kinetic and potential energy while the mass is oscillating back and forth about its equilibrium position. Generally, in a LC circuit, the current i(t), the electric field E(t), the magnetic field B(t), and the stored charge on the capacitor q(t) oscillates. Next we will show this mathematically.
PES 0 Spring 04, Spendier Lecture 35/Page 4 We start with a charged capacitor in series with a switch and an inductor. Let s apply Kirchhoff s loop rule to calculate the charge in an LC circuit q( t) di V L 0 C q( t) di L 0 C dq use i( t) d q q( t) L 0 C d q q( t) 0 LC What if I just changed q to x and renamed the constant? d x x 0 So the equation just describes oscillations with a frequency ω. You have seen this equation in Physics. So the general solution for d q q 0 LC is 0 q( t) Q cos t with LC The constant Q 0 (amplitude) and ϕ (phase shift) can be chosen to obey the boundary conditions: q(t=0) = Q 0 or q(t=0) = 0.5 (Q 0 )
PES 0 Spring 04, Spendier Lecture 35/Page 5 The natural frequency of a LC circuit is given by ω and is related to the value of the inductance L and the capacitance C. Period T of one oscillation: sin(0) sin T LC T LC T LC f So the larger the inductance or capacitance, the longer one period will take. Analogy to Spring-Mass Oscillations: Current and Charge Oscillations are out of phase! q( t) Q cos t 0 dq( t) d i( t) Q0 cos t Q0 sin t q(t) i(t)
PES 0 Spring 04, Spendier Lecture 35/Page 6 Energy oscillations q( t) Q0 E ( t) cos t C C B ( t) Li( t) L Q0 sin t 6 Example: In an LC circuit, L.0 H and C is unknown. It takes 0 μs for the circuit to go from all energy in the inductor, to all stored energy in the capacitor. What is C?