Discussion Notes and Class Agenda
Physics 240
March 29, 2006
Problem Assignments
Problem Groups
30.66 1, 3
30.75 5, 7
30.28 2, 4
30.62 6, 8
Discussion: The LC and LRC Circuits
So far we have studied RC and LR circuits in complete detail, first deriving and then solving the differential equations resulting from the application of the Kirchoff loop rule by integration. The next combination of circuit elements we will consider is the LC circuit, which combines an inductor and a capacitor. The LC circuit is the zero-resistance limiting case of the LRC circuit, which includes damping due to resistance in the circuit. We’ll study the LC circuit first because it is simple to solve and then use the insight we have gained to understand the more general LRC circuit.
The LC Circuit
Figure 30.15 illustrates the LC circuit and defines the convention we will use for the current, which is chosen to be positive when flowing onto the capacitor so that .
Applying the loop rule clockwise as shown from the bold point indicated at the lower right hand corner of the loop
Substituting for the current yields
Recall from mechanics that any system described by a differential equation of the form
is a harmonic oscillator with a general solution of the form
where the amplitude of the oscillation A and the phase angle j are determined from the initial conditions imposed on the system at t=0. The solution for the charge on the capacitor can therefore be expressed as
The current in the circuit at time t is readily found as
If we impose the specific initial conditions , which corresponds to initially charging the capacitor and then connecting it across the inductor, the solution takes the form
As promised, the solution has indicated that the current is initially flowing in the opposite sense (counter clockwise) of the defined current, as evidenced by the minus sign. This indicates that the capacitor is initially discharging across the inductor, as expected with only the charge initially present on the capacitor and no other EMFs in the circuit. The condition is also satisfied, indicating that no current flows in the inductor when the capacitor is first connected.
Finally, we can consider the energy present in the circuit. At any time the energy will be shared between the capacitor and inductor
For the initial conditions we’ve chosen
so that initially all the energy is stored on the capacitor, as expected. We also see that no energy is dissipated in the oscillating circuit and therefore that the total energy in the system is constant with value
.
Obviously, all real circuits have some resistance and therefore the LC circuit as we have described it is an idealization. The next step is to add resistance to the LC circuit.
The LRC Circuit
Now consider an LRC circuit
The capacitor is first connected across the EMF and charged to an initial charge . The switch is then closed to connect points a and d. Applying the loop rule clockwise beginning at point c
Again substitute for the current to obtain the differential equation governing the charge q(t) present on the capacitor
At this point we simply take advantage of the fact that the solution for this differential equation is well known
It is the task of Problem 30.38 to verify that this is a solution to the LRC differential equation, and that if the capacitor is initially charged to and then connected so that the constants A and j are
The form of the solution makes sense physically. The angular frequency of the oscillation is shifted from the basic LC value by the presence of the resistance, but the shift is small for small values of R, and the effect is to lower the frequency of oscillation as expected for a friction-like effect such as resistance. The exponential decay factor causes the oscillation to die out at long times as expected, since the resistance in the circuit dissipates energy so that the oscillation must eventually die away to zero. And the time scale for the exponential decay of the oscillation is an exact multiple of the LR time constant L/R. Finally, note that if R=0 is substituted into the differential equation and the solution the result is
The LC circuit solution is therefore the R=0 limit of the LRC solution. Remembering the general solution for the LRC circuit covers the LC circuit as well. Simply substitute R=0 into the LRC solution to recover the LC case.
6