Let us examine this LC circuit mathematically. To do this let us examine the energy of the system. Conservation of Energy 2 nd order differential equation  is the natural frequency of oscillation for an LC circuit. The solution to this differential equation is: I max At t = 0 we know that I = 0 and Q = Q max Phase Constant This oscillation is similar to that observed for a mass spring system.

The final circuit configuration we will discuss is a combination of a resistor, capacitor and an inductor. This is more commonly called an RLC circuit. a b When the switch is at position a the capacitor is charged. When the switch is at position b the capacitor is discharged through the resistor and the inductor. This situation is very similar to the LC circuit. How does R change the effects observed in an LC circuit? The resistor damps the oscillations, causing the amplitude of the oscillations to decrease in time. Energy is dissipated by the resistor. The size of the resistor determines how quickly the oscillations are damped.

We can determine a mathematical expression by looking at the energy. Energy is lost to the environment. This is identical in form to the damped harmonic oscillator. R is the damping coefficient for this expression. If R = 0 we have the same expression as we did for the LC circuit. We have three different classes of solutions, depending on the relative size of R. Under damped: Critically damped: Over damped: Oscillations die off very quickly – no real oscillations occur. Oscillations die off even more quickly than critically damped case.

Under damped case Critically damped case Over damped case

Example: For the circuit containing C = 10  F and L = 50 mH shown determine the following (assuming an ideal circuit – no resistance and no radiation): a)The natural frequency of this circuit. b)The charge on the capacitor after 5 ms, if the voltage across the capacitor when fully charged is 10 V. c)The current in the circuit after 5 ms. d)The total energy stored by the circuit at 5 ms. e)If a resistor is added in series to the inductor and the capacitor, what value must it have for the circuit to be critically damped? f)If the resistance of the resistor is decreased to one tenth the critical resistance value, what is the charge on the capacitor after 5 ms? a) b)

c) d) or e) f)