PURPOSE
This experiment is about the isolation of the three harmonics that is included in a non-sinusoidal waveform. The non-sinusoidal waveform here is a square wave, and the harmonics being isolated are the fundamental, third, and fifth harmonics. The experimental amplitudes and frequencies will be compared to the theoretical computations.
EQUIPMENT
Oscilloscope
Digital Multimeter
Function Generator
COMPONENTS
10 mH Inductor
10 kΩ Resistor
PROCEDURE AND RESULTS
Figure 1. RLC Circuit Diagram
PART A: Obtain the scaling factors
1) Build the circuit shown in Figure 1. Throughout the experiment, connect the input voltage to channel 1 of the oscilloscope. This input voltage is used as a trigger source for the oscilloscope. Therefore, channel 1 is the reference signal that will prevent the output voltage of the first, third, and fifth harmonics to be shifted in time.
2) Choose RS = 10kΩ and RL = 10kΩ.
3) Select a resonant frequency, such as 5kHz, and a value for the inductance, such as 10mH. Calculate the value of capacitance C1 using the formula for the resonant frequency:
foth=12πLC Hz
--Let L = 10mH, foth=5kHz. The capacitance is:
5k=12π10mC1→C1=0.1μF
4) Apply a sinusoidal input voltage of magnitude 10 Vp-p, and tune the function generator around 5kHz. At resonance the output is maximized, which can be seen on the oscilloscope. Note the actual resonant frequency, foexp, and the output voltage, V1.
--The actual resonant frequency, foexp, is significantly equal to 5kHz. The output voltage measured is:
V1=1.7V
5) Choose f3th = 3foth. Make the RLC circuit resonate at the third harmonic frequency. With the value of the inductance same as before, calculate the new value of capacitance, C3.
--The third harmonic frequency is set to f3th = 15kHz. The capacitance is:
15k=12π10mC3→C3=0.01μF
6) Replace C1 and C3 and with the same input voltage of 10Vp-p, obtain the experimental resonant frequency f3exp. Make sure the input voltage is still 10Vp-p. Measure the output voltage V3.
--The actual resonant frequency, f3exp, is close to 16kHz. The output voltage measured is:
V3=3.6V
7) Repeat steps (5) and (6) for the fifth harmonic, f5th=5foth. Calculate C5, and measure f5ex and V5.
--The fifth harmonic frequency is set to f5th = 25kHz. The capacitance is:
25k=12π10mC3→C3=4.7nF
--The actual resonant frequency, f5exp, is close to 24kHz. The output voltage measured is:
V5=3.68V
8) Calculate the scaling factor (SF) as follows:
SF1=V1V1 SF3=V1V3 SF5=V1V5
--The scaling factors are:
SF1=V1V1=1.71.7=1
SF3=V1V3=1.73.6=0.472
SF5=V1V5=1.73.68=0.462
PART B: Isolate the 1st, 3rd, and 5th Harmonics
9) Set the function generator to the actual fundamental frequency, foth. Keep the generator at this frequency for all of part B. Replace C5 with C1. View the square wave on channel 1, and the harmonics on channel 2.
10) Switch the input waveform to a square wave, maintaining the magnitude at 10Vp-p. Obtain the fundamental component. Note the frequency; draw the output waveform as seen on the oscilloscope.
11) Replace C1 with C3. Obtain the third harmonic. Draw the output waveform, measure the period between two peaks, and obtain its average amplitude V3. See NOTE. REMEMBER: do not change the frequency of the function generator; maintain the magnitude of the input voltage at 10Vp-p.
12) Replace C1 with C5. Obtain the fifth harmonic. Draw the output waveform, measure the period between two peaks, and obtain its average amplitude V5. See NOTE. REMEMBER: do not change the frequency of the function generator; maintain the magnitude of the input voltage at 10Vp-p.
NOTE: Waveform average amplitude is obtained as follows:
a. Count the number of maxima appearing within one period of the square wave.
b. Measure the peak amplitudes of the maxima.
c. Add the peak amplitudes.
d. Use the relation
V3=sum of maximanumber of maxima
Where V3 is the average amplitude of the third harmonic, and V5 is the average amplitude of the fifth harmonic.
--The waveform average amplitudes are:
V1=2.08V
V3=1.06+1.64+1.443=1.38V
V5=1+0.32+1.66+0.56+1.185=0.944V
13) Calculate the final adjusted outputs usingA1=V1×SF1
A3=V3×SF3
A5=V5×SF5
--The adjusted outputs are:
A1=V1×SF1=1×2.082=1.04
A3=V3×SF3=0.472×1.382=0.3257
A5=V5×SF5=0.462×0.9442=0.2181
14) A1 : A3 : A5 should be approximately 1:13:15. In reality, A3=13A1;A5=15A1. Obtain the %error.
--The expected values are:
A3=13A1=131.04=0.3467
A5=15A1=151.04=0.208
The %errors are:
%error3=0.3467-0.32570.3467×100%=6.06%
%error5=0.208-0.21810.208×100%=4.86%
It is certainly observable that the actual values are very significantly near the expected values. The error levels of 6% and 5% are very low, especially if the values are in the milli-Volts range.
CONCLUSION
The RLC circuit was tested and analyzed experimentally. In the isolation of the third and fifth harmonics, it was proven experimentally that the scaling factors can be achieved. The %errors of the adjusted outputs are very low (6% and 4.9%).
