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ELECTROMAGNETIC OSCILLATIONS

Introduction

The goal of this lab is to examine electromagnetic oscillations in an alternating current (AC)

circuit of the kind shown in Fig.1. The circuit differs from that in the RC circuit

experiment by including the extra component labeled L, a solenoid coil consisting of

numerous turns of wire. When the switch is moved from position 1 to position 2, the

charged capacitor discharges, but now the voltage across the capacitor does not simply

decay to zero. Instead, it oscillates between positive and negative values with an amplitude

that decreases as time passes, similar to the behavior of a pendulum.

There is in fact a strong analogy between mechanical oscillations and electromagnetic

oscillations. Physical characteristics of a mechanical oscillator (such as mass and friction)

correspond to specific electromagnetic characteristics of the AC circuit studied here, the

same equations that describe the oscillation of mechanical quantities also describe the

oscillation of electromagnetic quantities, and many results apply equally for mechanical

oscillators and electromagnetic oscillators. To explain the oscillatory behavior of the circuit

in Fig.1 and the close analogy between mechanical and electromagnetic oscillations, we

first discuss Faradays law, inductors, and alternating current circuits.

VR

LC

S

1

2

VCVLV

0

Figure 1 RLC circuit

VC

t

Figure 2 Time dependence of the

voltage VC across the capacitor i n

F ig . 1


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Every electric current generates a magnetic field B .

The magnetic field around the moving charge can be

visualized in terms of magnetic lines of force. At each point, the direction of the straight

line tangent to the magnetic line of force gives the direction of the magnetic field, in

complete analogy to the electric lines of force.

For example, the solenoid

used in this experiment as

an inductor (component L

of Fig.1) consists of a

long insulated wire wound

around a cylinder with a

constant number of turnsper unit length of the

cylinder. The current I

through the wire produces

a magnetic field, as

illustrated in Fig. 3.

To examine the role of the solenoid, consider more generally any area A enclosed by a

closed path P in space, as illustrated in Fig. 4. Then with B(r) the magnetic field at each

point in the region we can define the flux

= B daarea

(1)

as essentially the number of lines of the B field (or the amount of B field) passing

through the area. If the lines of the magnetic field are perpendicular to the area A, then the

flux is simply the product of the magnetic field times the area. The dimensions of the

magnetic flux are Webers (or Wb) with 1 Wb = 1 Volt-second.

Induction and Faradays law

S NI

I

Figure 3 Magnetic field lines near a solenoid


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as a back emf). By Lenzs law, the induced emf is in a direction that opposes the change

in flux, so that it acts to keep the current from dropping to zero even up to the time when

the capacitor is fully discharged and the voltage from the capacitor itself has reached zero.

In analogy to the mass of the pendulum bob that causes it to swing past equilibrium to the

opposite side, the back emf keeps the current flowing beyond the point where the charge onthe capacitor reverses its sign.

To examine this more quantitatively, note that for a coil ofN turns of wire each with flux

, the total flux N through the coil must be proportional to the current through the wire,

so that

N LI = . (4)

The proportionality constant in this relation is the inductanceL given by

LN

I=

. (5)

The inductanceL is a characteristic property of the inductor determined by its geometry (its

shape, size, number of windings, and arrangement of windings), just as the capacitance of

a capacitor depends on the geometry of its plates and on whatever separates them from each

other. The units ofL are volt-second/ampere with 1 Vs/A = 1 Henry (with the symbol for

Henry being H).

Because Eq. (3) relates the flux through the circuit to the current at each instant of time, the

time rate of change of the two sides of Eq. (4) must also be equal, and since the left side of

the equation is the total flux linkage N, its rate of change is the induced emf , while the

rate of change of the right hand side isL times the rate at which the current changes at each

instant. Therefore the back emf is equal toL multiplied by the rate of change of the current

or, equivalently, by simply differentiating both sides of Eq. (3),

Nd

dtL

dI

dt

= (6)

or

= LdI

dt. (7)


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As a charge dQ flows through the circuit, it gains energy

VdQ, where V=LdI/dt. Thus, the energy lost by the

charge (which is the energy given to the inductor) is

dU VdQ L dIdt

dQ l dI dQdt

L dI dQdt

LI dI= = = = = . (8)

If we start with zero current, and build up to a current I0, the energy stored in the inductor

is

U LI dI LI I

= = 012 0

20 . (9)

This energy is stored in the form of the magnetic field B.

When the switch S in Fig. 1 is in position 1, the capacitor becomes charged. Eventually

the capacitor has the full voltage Vacross it and has energy 12CV2stored in its electric field

E between its plates. Once we set the switch to position 2 at t=0, current flows through

the inductor building up a magnetic field. As the voltage oscillates, the energy oscillates

between being magnetic energy of the inductor and electric field energy of the capacitor.

Some of the energy is lost to the environment as heat because of the ohmic resistance of the

circuit, decreasing the maximum of the energy 122CV stored in the capacitor in each

successive cycle. The amplitude as measured by the maximum of the voltageVC across the

capacitor therefore decreases from each cycle to the next.

We next need to examine the precise

time dependence of the voltage V(t) and

of the currentI(t) in a circuit such as that illustrated in Fig.1. Although the circuit includes

a capacitor, an inductor, and a resistor, the behavior of the circuit is determined by the total

resistance R=RL

+RC, total inductance L, and total capacitance C of the entire circuit,

rather than the value for each component. Each component of the circuit contributes to

these three physical aspects of the circuit; the resistor, for example, also has a slight

capacitance and inductance. Since the resistance of the wires is fairly negligible, the

resistance measured from one side of the capacitor to the other in Fig. 1 is seen to be

R =RL +R C. The relation between voltage and current for these three elements are

summarized in Table 1, together with the SI symbol and the expressions for the energy

associated with each physical quantity.

Energy considerations

Damped oscillations in an RLC circuit


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In any closed circuit the sum of the voltages across the components must be zero, so that

V V V L R C + + = 0 . (10)

Based on the expressions for the voltage differences in Table 1,

LdI

dtRI

Q

C+ + = 0. (11)

ButI=dQ/dt, so therefore

L

d Q

dt R

dQ

dt

Q

C

2

2 0+ + = . (12)

One possible solution to this equation is

Q t V C R

Lt t( ) exp cos '=

0 2

(13)

with

' = =

1

22

2

LC

R

Lf . (14)

You can convince yourself that Eqs.(13) and (14) give the correct solution by substituting

into Eq.(12).

Table 1 Symbols, voltages and energies for RLC circuit components

Symbol Voltage Energy

Resistor R VR

Inductor L VL =Ld I /d t U L =12 L I

2

Capacitor C VC=Q/C U C = 12 C V

2

Battery V

Pulse Generator V ( t )


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The voltage across the capacitor will be

VQ

CV tC

R

Lt

= =

02e

cos ' (15)

The function VC is that shown in Fig. 5. It is the product of an oscillatory term cost

and an exponential function exp

R

Lt

2that damps the amplitude of the oscillation as the

time increases. The

angular frequency and

period T in Eq.(14) are

related by

' = =22

fT

The damping factor does

not affect the period T of

the oscillatory term. The

transfer of energy back

and forth from the

capacitor to the inductor is

illustrated in Fig. 6. In

this figure the current

I=dQ/dt is also plotted.At the time marked 1 in

Fig. 6 all the energy is in

the electric field of the

fully charged capacitor. A

quarter cycle later at 2, the

capacitor is discharged and

nearly all this energy is

found in the magnetic field

of the coil. As the

oscillation continues, the

circuit resistance converts

electromagnetic energy

into thermal energy and

the amplitude decreases.

VC

t

Figure 5 Time dependence of Eq. (12).

3 4 1 2 3 4 1 2

V

I

4.

1.

2.

3._

+

B

_

+

E

Q

E

I

I B

Figure 6 Voltage, current, and energy oscillations in an

RLC circuit


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Experiment Observing damped oscillations in an RLC circuit

This experiment deals with a capacitor, a resistor, and an

inductor connected in series on a printed circuit board (PC) as

shown in Fig.7. The set-up differs from that of Fig.1 by including connections to the

computer-based oscilloscope to monitor the time dependence of the input and capacitor

voltages. Also, just as in the case of the RC circuit experiment, the switch in Fig.1 is

replaced by the square wave input from the pulser. If you have forgotten how to use the

computer-based oscilloscope, it would be a good idea to read the section of the second lab

write up in which its operation is described.

Table 2 below lists the units of the three main quantities characterizing the circuit.

Experimental set-up

Coaxialcables

Printed circuit

board

C

R VR

VC

L

Pusler

A B C

Computer-based

Oscilloscope

Figure 7 Experimental set-up

SI unit symbol in terms of other SI units

Resistance [ R ] ohm = volt/ampere = V/A

Capacitance [C] farad F = coulomb/volt = C/V

Inductance [ L ] henry H = volt-second/ampere = Vs/A


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Insert in the PC board of your setup:

(a) a 0.10 F capacitor. (It is marked .10 MF),

(b) the large inductor,(c) the plug with the bypassing wire at the location of the resistor.

Set the pulser at a frequency between 20 Hz and 200 Hz, with the output selected to

rectangular pulses and the amplitude set to a few volts. Connect the pulser to the PC board

(see Fig. 7).

Observe the shape of the input pulses through one channel of the computer-based

oscilloscope display, while simultaneously observing with the voltage VC

across the

capacitor with the other channel. As seen in Fig. 5 of the second lab, the braid of the

coaxial cable you are using is connected to the ground of the BNC connector (the one that

goes to the oscilloscope). One of the two ends of the dual banana plug (at the end of the

cable going to PC board) must also be connected at ground potential to avoid short

circuiting the signal across the capacitor C. Two small capacitors (mounted underneath the

PC board) allow you to observe the AC signal across the capacitor, ignoring the DC level.

The two signals that you observe on the oscilloscope are shown in Fig. 8.

Experimental procedure

Vpulser

Tpulser

t

T

t

Figure 8 Input and capacitor voltage in Experiment 1


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Sketch in your lab notebook the voltage VCfor a single damped oscillation that you observe

with the oscilloscope.

Indicate on this sketch, using small squares as markers, the points where the energy of the

system is all in the electric field of the capacitor. Indicate with a circle where it is all in theform of magnetic energy.

Use the oscilloscope to measure the period Tof oscillation. It may be more accurate to

measure the time ofn periods, rather than just one.

Calculate the angular frequency

' = =22

fT

.

When the amplitude has dropped from V0 at t=0 to Vnat time t=nT, then the relation

V V Vn

R

LnT

R

Lt

= = ( )

02

02e e

(16)

holds. By taking the logarithm of both sides leads to

ln lnV VR

LnTn = ( )0

2. (17)

The damping constantR/(2L) should then be given by the slope of the lnVn vs. tplot.

Take five or more data points and use the GA program to plot Eq. (17) and evaluate the

damping constantR/2L from the slope. Be sure to include the standard deviation and the

correct units in your stated result.

Equation (14) gives the angular frequency of the oscillation. If the resistanceR in the RLC

circuit is zero (R=0) it will resonate at the resonant frequency

01

=LC

. (18)


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You have measured from the period Ttogether with the damping constant. As you can

see from Eq.(14), ifR L/ 2( ) >> then ' = 0 . CalculateL of your circuit, indicating

the correct units. This is possible if the RLC system oscillates at a frequency close to the

resonant frequency

01

=/ LC.

Determine the inductance of the solenoid directly from the number of turns of wire

indicated on it and from its physical dimensions. Compare it with the value ofL from your

data to confirm Eq.(18). (Note that this frequency depends only on the CandL of your

circuit and has nothing to do with the frequency of your pulse generator.)

Calculate the resistanceR of the circuit. You can do this using the measured values of the

damping factor2L

R

and the inductance.

Compare the measured value ofR with that determined from known resistances of parts of

the circuit.


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The following list of questions is intended to help you prepare for this laboratory session.

If you have read and understood this write-up, you should be able to answer most of these

questions. Some of these questions may be asked in the quiz preceding the lab.

In terms of energy, when can a system oscillate??

In what forms can electromagnetic energy be stored?

A battery stores electromagnetic energy. In which form?

What is magnetic flux?

What is Faradays law? Who first discovered it?

What is the energy in an inductor in terms of quantities such as charge, current, voltage?

What physical features determine the inductance of a solenoid?

What physical quantity in an AC circuit plays the same role that frictional forces play in a

mechanical oscillator? Why?

What quantity has the same role in the mechanical oscillation of a pendulum that the

inductance has for an RLC circuit? Why?

For a swinging pendulum, what is the resonant frequency?

Which of these circuits RC, RL, LC, RLC can produce oscillations? Explain.

QUESTIONS

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