Experiment 11
The RLC Series Circuit, II
The phase difference between the total voltage applied to an RLC series circuit and the voltage across the capacitor in the circuit is to be examined at various frequencies, and from this information, the resonant frequency of the circuit is found.
Theory
When an AC signal is applied to the RLC series circuit, a phase difference occurs between the voltage across the circuit and the current through the circuit. This phase difference is
  tan

1
X
L

X
C
, (1)
R where the inductive reactance is X
L
= 27π fL , the capacitive reactance is X, = 1/27cjC, and the resistance is R. The expressions for the instantaneous current and voltage are i

I m sin 2 ft

, (2) and v

V m sin

2 ft
   
.
(3)
When the frequency of the applied signal is low, the circuit is capacitive and the current leads the voltage. At high frequencies, the circuit is inductive and the current lags the voltage. At some intermediate frequency, the circuit is purely resistive and the phase difference between the voltage and the current is zero. This frequency is the resonant frequency of the system and equals f
0

2

1
LC
.
(4)
In this experiment, the phase angle between the current and the Voltage is examined at various frequencies. This is done by finding the phase angle between the voltage across the resistor and the voltage across the entire circuit. Because the voltage across the resistor is in phase with the current through the circuit, the phase angle between the voltage across the resistor and the voltage across the entire circuit is, in fact, the phase angle between the current and the total voltage.
In order to determine the phase angle experimentally, the voltage across the resistor is connected to the horizontal input terminals of the oscilloscope and the total voltage is connected
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to the vertical input terminals of the oscilloscope.
(Refer to Figure 1.) If the amplitudes of the horizontal and vertical signals are adjusted to the same value V., then the voltage delivered to the oscilloscope can be written as
V x

V m sin

2 ft
 
(5) and
V y

V m sin

2 ft
   
.
(6)
When the time t is eliminated between (5) and (6) and the resulting expression rearranged, the result is
V x
2 
2 V x
V y cos
 
V y
2 
V m
2 sin 2

, (7)
Figure 1. The RLC series circuit. which is the equation of a rotated ellipse. Note from Figure 2 that when
V x
 a
2
, then V y
 a
2
, and when
V x
  b
2
, then V y
 b
2
, where a and b are the semi-major and semi-minor axes, respectively. When these values are substituted into (7), the result is tan

2
 b a
,
Figure 2. The rotated ellipse. or
 
2 tan

1 b a


.
(8)
The phase angle

can thus be determined by measuring the length of the semi-major axis, a, and the length of the semi-minor axis.
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o o
Apparatus o o o o o o inductor, approximately 20 mH decade resistance box set at approximately 100 ohms capacitance box set at approximately 0. 15

F oscilloscope with leads signal generator
2 leads
15 cm ruler (6 inch)
3-cycle semi-log graph paper
Procedure
1) Connect the circuit as shown in Figure 1. Adjust the oscilloscope so that it responds to both horizontal and vertical voltages. (Set TIME/DIV knob to X-Y.)
2) Turn on the equipment and set the frequency, of the of the signal generator to 100 Hz.
3) Adjust the voltage amplitude knob on the signal, generator and the voltage adjustment knobs on the oscilloscope until the line (or ellipse or circle) makes an angle of 45' with the vertical. The voltages in the horizontal and vertical direction sill then be the same.
4) Use the ruler and measure and record the lengths of the semi-major axis (a) and the semiminor axis (b) of the ellipse. (

is always the length of the figure leaning to the right at
45' and b is always the length of the figure leaning to the left at 45".
5) Change the frequency of the signal generator to 200 Hz and repeat steps (3) and (4).
Continue in this fashion for frequencies of 400 Hz, 700 Hz, 1000 Hz and multiples of 10 of the above frequencies up to I MHz.
Analysis
Calculate the phase angle

for each frequency. Graph

versus frequency on the three cycle semi-log graph paper with frequency plotted on the logarithmic scale. Draw a smooth curve through the points.
Determine the resonant frequency of the RLC series circuit from the graph and compare this value with the theoretical value of the resonant frequency. Find the percentage effort between these two values. Report the experimental and theoretical values of the resonant frequency in a table together with the percentage error.
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