Experiment 12: Elements of AC Circuits - The Series RLC circuit
Purpose: To study the behavior of a resistor, capacitor and inductor in a series ac circuit.
Prior to Lab: Sketch a phasor diagram for an ideal LRC circuit consisting of an 33 μF
capacitor, a 70 mH inductor, and a 47 Ω resistor connected to a 5 v, 150 Hz source. Be sure
you calculate XL and XC. Show these calculations.
Theory:
For a series RLC circuit, Kirchhoff’s rules demand that the current be a constant throughout.
Furthermore, the potential differences across each element must combine to equal the source
voltage. However, each element also has a unique phase relation between the potential
difference and the current, so the potential differences must add according to the rules for
combining phasors for the circuit. If we look at each element in turn we can investigate the
individual behavior and then the collective behavior of the elements in the circuit.
For example, the capacitor’s potential difference is determined by the charge on it, while the
current is the rate of change of charge. This fact leads to a phase difference between current
and voltage of 90o with the potential difference lagging the current by that phase angle. The
effective potential difference across the capacitor VC equals the effective current I times the
capacitive reactance XC.
Figure 1. The apparatus wired and ready to read data. Note that the meter on the right is
connected to the DMM Output of the Function Generator and is indicating the frequency output
of the source. The meter on the left is set to read ac volts, and is reading the source voltage at
this time.
Procedure:
In order to completely analyze the behavior of the circuit, we will measure the resistance R and
five different potential differences at two different frequencies. We will also determine the
resonant frequency of the circuit.
Note: Before beginning the following procedure record the number of your circuit board
so you will be able to use the same circuit board for next week’s AC Resonance Lab.
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1. BEFORE connecting wires to the circuit, use a DMM set on the 200 Ω range to measure the
resistance between connection C and D in Fig. 2 or 3. The instrumental uncertainty for
measuring resistance is 0.5% of the reading plus 1 digit.
2. Connect one DMM to the DVM Output of the function generator. Set this DMM to measure
20VDC. Note that the frequency measured is the product of the reading on this meter and the
multiplier setting on the function generator. Connect the other DMM between A and D initially
and set it to measure 20 VAC.
3. Connect the ac source to the circuit board using the 10 Ω resistor and the 100 μF capacitor.
Set the function generator for 5.0 V at a frequency of 120 Hz. Note: Be certain the output is
set for the sine wave function.
D
R
C
A
Figure 2. Above is a diagram of the apparatus used in
the experiment.
L
r
B
C
Figure 3. Above is the schematic wiring
diagram representing the apparatus and its
source. Note that the inductor is not ideal, and
we have represented any (power) loss in the
circuit by a resistance, r, in series with the
inductor.
4. Complete the measurements listed below for the frequency set in step 3 by connecting the
probe wires of the DMM set to measure ACV to the connectors listed. Record this data in the
first line of a table in your data notebook which looks like the sample below. When
finished, change the frequency output of the function generator to 250 Hz and repeat the
measurements to fill in the second line of the table in your data notebook.
a. Vsource is measured by connecting the probes at A and D.
b. VC is measured by connecting the probes at A and B.
c. VLr is measured by connecting the probes at B and C.
d. VLR is measured by connecting the probes at B and D.
e. VR is measured by connecting the probes at C and D.
f
R
Vsource
VC
VLr
VLR
VR
5. To measure the resonant frequency of the circuit connect the DMM used to measure ACV
across points A and C to measure the combined voltage across the inductor and capacitor.
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Adjust the frequency output of the function generator until this measured voltage is a
minimum. For this frequency setting the combined reactance of the inductor and capacitor
is theoretically zero. This minimum voltage occurs at the resonant frequency of the circuit.
Record the measured resonant frequency, the voltage across R, and the voltage across the
capacitor and inductor.
Analysis:
1. Determine the current in the circuit by calculating the current in the resistor. Note: We will
use I (and VR since it has the same phase) for reference throughout our analysis.
VR
= I = ______ A
R
2. Determine the capacitance of the circuit from its reactance, the current calculated above,
and the frequency of the source.
VC
= X C = ______ Ω and
I
C=
1
= __________ F
2π fX C
3. Determine the inductance of the circuit from three different voltage measurements, the
current calculated in step 1, and the frequency of the source. Follow the instructions
carefully; this can be complicated.
a. Determine the phase of VLR with respect to VR using the law of cosines.
2
2
VLr
= VLR
+ VR2 − 2VLR VR cos α
VLR
VLr
V 2 + V 2 − VLr2
α = cos LR R
= ______
2VLRVR
α
−1
VR
Figure 5. A phasor diagram showing the relationship
between the measured voltages across the inductor,
resistor and combination.
b. Determine the fraction of VLR due VL alone.
VL = VLR sin α = ____ V
c. Determine the inductance of the circuit from its reactance using the current determined
in step 1 and the VL determined above.
VL
= X L = ______ Ω and
I
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L=
XL
= __________ H
2π f
3
d. Determine the loss in the inductor and write it as a resistance using the current found in
step 1.
Vr = VLR cos α − VR = _____ V and
r=
Vr
_____ Ω
I
4. Determine the phase between the source voltage and the current
φ = tan −1
VL − VC
= _____
VR + Vr
5. Calculate the resonant frequency of the circuit from the values you determined for L andC.
fo =
1
2π LC
= _____ Hz
Report:
1. For the first frequency that you collected data, and using R, r, XL and XC, draw to scale a
phasor diagram to determine Z and φ. Make a second drawing for the second frequency for
which you collected data. Assume the phase angle for the capacitor is 90o. Answer these
questions for each drawing: Does Vsource = IZ? Is the phase angle in the drawing close to
phase angle you calculated in step 4 of the analysis.
2. Are the values for C and L close to the manufacturer’s values? Explain.
3. What is the per cent difference between the measured resonant frequency and that
calculated in step 5 of the analysis?
4. Using the voltage across the resistor recorded in step 5of the procedure, calculate the
current in the circuit at resonance. If you now divide the current you just determined into
the voltage across the inductor and capacitor you also measured in step 5 of the procedure,
do you get the same value for the resistance of the inductor that you got in step 3d of the
analysis? Calculate a per cent difference if it exists? Try to explain any difference.
5. Discuss your results paying particular attention to instrumental error and other sources of
experimental error.
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