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( ) ( ) D. Shaw
October 31, 2006

Resonance in an LCR Circuit


Equipment: Pasco RCL experiment board with iron cylinder, one Pasco voltage probe, two medium length wires with banana
plugs at each end and a short cylindrical aluminum rod.

Part A: Resonance in an Electric Circuit
Introduction: Faraday's Law is applied to a series circuit having a coil, capacitor and coil. This circuit is described by the same
differential equation as we used earlier to study the motion of a magnetically driven oscillator. The electrical circuit also shows the
property of "resonance" that we found for the oscillator. A few magnetic properties of materials are also examined.

Theory: Fig. 1 shows a circuit consisting of a signal generator in series with a coil, resistor and capacitor. The resistor shown is
not a separate resistor but represents the resistance of the wire in the coil and all, connecting wires.
The signal generator is producing a signal whose potential difference is a sine function of time. Suppose at some time the current is
increasing and flowing in the direction of the arrow. The high potential side of each element in the circuit is marked in Fig. 1.
Since the flux in the coil is increasing the induced current must flow in a direction opposite to the current caused by the signal
generator. For the coil to produce an induced current in the direction opposite to the arrow, the high potential side of the coil must
be as shown. The changing flux produces an induced Emf of -L(dI/dt). By applying Kirchoff's second law to the circuit we have at
any time:

.


The number of complete oscillations of the sine function per second is the frequency
"f". The parameter "
" is the angular frequency and is related to "f" by: = 2 f.
Differentiating each term with respect to the time produces the equation:




This second order differential equation has the same form as the equation found for the motion of the driven mechanical oscillator
studied in a previous experiment. The term R(dI/dt) which represents the energy loss in the form of heat in the circuit corresponds
to the air resistance term of the mechanical oscillator, the term I/C corresponds to the spring force and the term L(d
2
I/dt
2
) is the
mechanical acceleration term. Therefore the solution of Eq. 2 is the same as the solution found for the mechanical oscillator except
for different symbols. The steady solution of Eq. 2 is:



.

where the angle
is the "phase angle". As we found in the mechanical oscillator experiment, there is also a transient solution
which is unimportant in this experiment. Eq. (3) shows that the current in the circuit is not, in general, in phase with the applied
potential. The term that multiplies the sine function is the current amplitude I
0
. This amplitude depends on the angular frequency of
the applied potential and also shows the phenomenon of resonance found previously in the Driven Oscillator Experiment.
Electrical resonance is observed in this circuit since the current amplitude will have a maximum value when the denominator is a
minimum. This occurs, very nearly, when
L - 1/ C = 0. This occurs at the resonant frequency
r
:



The absolute value of the phase angle is found to be:



( )
( )
1
...
0
sin
0
= dt
dI
L
C
q
RI
t
V ( )
( )
2
...
0
cos
2
2
0
= dt
I
d
L
C
I
dt
dI
R
t
V
( )
(
) ( )
3
...
sin
1
)
2
2
0


+
=
t
C
L
R
V
t
I
( )
5
...
1
tan
1







= R
C
L ( )
( )
4
...
1
2
1
LC
r
= coil
resistor
Pasco signal
generator
RI
V = -L(dI/dt)
capacitor
q/c
h
h
h
+
0
Fig. 1
Due to the phase angle, the current will not reach its maximum at the same instant as the applied voltage reaches its maximum.
However, at the resonant frequency the tangent of the phase angle is zero and the phase angle is zero. At this frequency the circuit
acts as if the inductor and capacitor were removed and Ohm's Law (V = IR) is valid. However, at other frequencies a phase plot of
V versus I will not be linear.
Experimental:

As shown in Fig. 2, use a wire to connect the ground terminal of the 750 Pasco
Interface Box to terminal "C" of the board and a second wire from the signal
terminal of the box to terminal "D" of the circuit board. The output voltage of the
signal generator is applied across the coil and the 100
f capacitor in series to
create the circuit shown in Fig. 4. The resistor is the resistance of the coil.
The nominal values for the capacitance and inductance are 100
f and 8.2mH
respectively. The approximate resonant angular frequency computed from these
values and Eq. (3) is 1.10x10
3
hz and the corresponding resonant frequency is
176 hz.
In Data Studio select New Activity from the File Menu. In the setup window double click on the signal output icon button
to open a dialog box which controls the signal generator. Select the sine wave function. Set the amplitude for 1.0 volts
and the frequency for the nominal resonant frequency of 176 Hz. Click on the Measurements and Sample Rate button and
choose both the output voltage and current to measure. Finally select a sample rate of 5000
Hz and leave the generator on Auto. Minimize the generator window.
Use the Options button to select a data collection time of 0.1 seconds. Minimize the setup
window.

Measurement of the Resonant Frequency and the Coil Resistance:

The values for the inductance and capacitance given are only approximate and as a result the
calculated resonant frequency is not known exactly. Drag the voltage icon to the plot icon to
create a plot. Drag the current icon to this graph and drop it over the time axis after a box
appears surrounding the axis to produce a plot of the potential difference applied across the
circuit versus the current in the circuit.
Collect a data set.
At the resonant frequency the phase angle should be zero. The applied potential and current
should be exactly in phase leading to a linear plot. Click on the Settings icon (at right end of
the icon bar) and turn off the Data symbols and Data points to obtain the clearest view of the
phase relation. If you observe a slightly elliptical curve, change the frequency slightly and try
again until you have obtained the best possible linear plot. Record this frequency as the best
estimate of the resonant frequency. Find the % difference between this value and the nominal
calculated value of 176 hz. The capacitance of the nominal 100 microfarad capacitor could be in error by as much as 20%
according to the manufacturer. This estimate is intentionally large to cover manufacturing fluctuations, however the error
you are likely to find for your capacitor will likely be considerably less.
At the resonant frequency, the circuit acts like a resistor only. Do a linear fit of this graph and record the slope, which is
the resistance of the coil, and connecting wires. This value will be used in the next part of the experiment (The Resonance
Curve).
Place the iron cylinder inside the coil and find the new resonance frequency by using the phase diagram. The resonance
frequency with the iron core will be significantly smaller than the resonant frequency found for air or aluminum. Also
measure the slope of the potential difference versus current curve from a linear fit. This information will be used later in
Part B.
Repeat the previous step for an aluminum rod placed in the coil.

The Resonance Curve:
Keep your previous plot window open and in addition open a second plot window to display the current versus time. We
are going to apply sine voltages with several different frequencies. In each case we use the statistics option to display the
maximum value of the current in the current versus time plot. Record the maximum current which is the current
amplitude for that particular frequency. As the frequency changes we will have to adjust the data collection rate and the
Freq.
(hz)
Rate
(1/s)
Time
(s)
20 2500
.2
60 2500
.2
100 2500 .2
120 5000 .05
140 5000 .05
160 5000 .05
175 5000 .05
200 5000 .05
260 20000
.01
320 20000
.01
400 20000
.01
700 20000
.01
D
C
coil
Fig. 2
RLC board
ds
capacitor
data collection time. Use the frequencies, data collection rates and data collection times suggested in the following table
and for each frequency record the amplitude. Also observe how the phase plots change with the frequency.
Open an Excel worksheet and enter your approximate values for the four constants V
0
, L, R and C into cells B1, B2, B3
and B4. En