# Natural Oscillations - the RLC Circuit

Lab manual pp. 65-67

Overview:
The goal of this lab is to observe the effect of placing a resistor, capacitor, and inductor in the same circuit.  We will verify that the oscillations observed in the voltage has a natural frequency of approximately 1/(L*C)^1/2.  We will also verify the fact that the oscillations are damped over time.  From our theoretical equations for this behavior, we will calculate the value of the resistance of our circuit.

Notes:

In the last few labs, we have used a solenoid to create a magnetic field.  We always directly connected the solenoid to the power supply.  Now let's ask ourselves:  What happens if I put a solenoid in a circuit with other circuit elements (capacitors, resistors, frequency generators, etc.)?

First of all, a solenoid is a kind of inductor.

An inductor is a circuit element which stores energy in a magnetic field, in the same sense that a capacitor stores energy in an electric field.
Next you'll want to know: Why do we care about putting an inductor into a circuit?  Unfortunately (and computer guys, listen up!), it is very easy to make an inductor by accident.  All you have to do is get a wire and make it into a bunch of loops.  You could make an inductor by twisting the ends of a wire together.  If you are ever doing sensitive signal analysis of any sort, you always have to worry about inductors in your circuit.

If a magnetic field has energy, then we have to put energy into an inductor to create a magnetic field.  In the last lab, we saw that the magnetic field was proportional to I.  (For most of the solenoids we work with, you can assume that B is most nearly = (mu) * n * I.).  So as we increase the current, we increase the magnetic field, and we have to put in energy to do so.  This energy has to come from somewhere in the circuit (perhaps from a power supply like we used in the last lab).  And since this energy is taken out of our supply of energy we have, we can say that an inductor is a kind of resistor.  But of course, we don't call the resistance of an inductor "resistance"  we call it reactance.  Why?  There is only an energy loss when we are increasing the current (and, as we destroy the magnetic field by decreasing the current, we get an energy gain); but if we are holding the current constant (as in a DC circuit) there is no energy loss or gain because we are not building or destroying the field.  In a DC circuit, an inductor is just a wire; the energy gain/loss is zero because you are not changing the magnetic field.

That said, let's put a capacitor and an inductor together and see what happens.  Below, there is a circuit diagram of a capacitor and an inductor in a circuit.  The capacitor has been charged up (i.e. it has energy stored in its electric field.)

The positive charges on the top plate would reeeallly like to get to the bottom plate somehow (they hate their fellow positive charges and they love those negative charges on the bottom plate).  As soon as you connect the capacitor, those charges will try to zip through the inductor to get to the other plate.  But what happens when they do move?  That's right.  You get a current.  What happens when you get a current going through an inductor?  A magnetic field.  You've just increased the magnetic field from nothing to something.  Energy gets stored in the created magnetic field.  Some of the positive charges make it to the bottom plate.  Now the rest of the positive charges see less and less negative charge on the bottom plate, meaning there is less and less incentive to leave the top plate.  So the  current decreases.  But wait!  If you decrease the the current, you're destroying the field you created in the inductor!  Uh-oh.  The energy has to go somewhere.  So what happens?  The inductor pulls the rest of those positive charges from the top plate and puts them on the bottom plate.  Now there is an excess of positive charges on the bottom plate.  Then the positive charges are attracted to the top plate and... well you get the picture.  What happens is the charges oscillate back and forth forever.

I like to think of an inductor as a spring.  The increasing current compresses the spring, and when the current slows down, the spring releases and sends those charges back to where they came from.  This oscillation, by the way, has an angular frequency of w = 1/2*pi*(L*C)^1/2, where L is the inductance of the inductor (a measure of how much magnetic energy it can store).  This frequency has a special name: the natural frequency.

Now let's add a resistor to the mix.  The same thing will happen if you add a resistor, with one difference.  The energy of the circuit is not constant anymore.  Some of the energy is lost through the resistor (changed to heat and/or light).  That means that the inductor will still send those positive charges back... but not all of them.  Over time, less and less of the positive charges will be returned to the top plate, meaning that, over time, the positive charges will cancel out the charges on the bottom, and the oscillation will stop.  This case is called damped oscillation.

This is exactly analogous to the damped oscillation which you have seen in the mechanics lab.

The equation for damped oscillation is:

Q = Q0 * exp [- R * t / 2 * L] * cos(w' * t)
where w' is { (1/LC) - (R^2) / (4L^2) }^1/2
We will verify this in our experiment.

Procedure:
Part I:  Natural frequency.
Connect the RLC circuit, but do not attach the power supply.  Do not plug in the power supply.  (It has no "on" switch).
(Draw the circuit)

Attach the oscilloscope probes across the capacitor (one lead on each side).

Add the charging circuit across the capacitor.  You are not responsible for understanding how this works.  All it does is dump some charge on the capacitor every once in a while.

Connect the power supply to the charging circuit.

Now plug in the power supply.

Observe the oscillations on the oscilloscope.  Record the period of the oscillation.  Is the frequency approximately 1/(LC)^1/2 as we expect?

Unplug the power supply.

Connect the capacitors in series.  Repeat the above procedure, and compare your result with the predicted value for the period.

Repeat for capacitors in parallel.

Part II: Verifying the equation
Using the original RLC circuit from part I, we will verify the exponential decay of the amplitude of the voltage.

Use the multimeter to measure the resistance of the inductor.

Use the oscilloscope to measure the amplitude of consecutive oscillations.

Use the GA program to plot the natural log of (V/V0) versus the number of oscillations.

Data:

Part I:

L = ________________ +/- __________
R = ________________ +/- 10%
C1 = _______________ +/- __________
C2 = _______________ +/- __________

 number of oscillations number of divisions time/DIV setting period predicted period for C = C1 in the RLC circuit: for C = C1 series C2 for C = C1 parallel C2

Part II:

 number of oscillations number of divisions Volts/DIV setting Amplitude (V) ln (V/V0) 0 V0 = 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5