AC Signal #1

Purpose
    The purpose of this lab is to explore the phase difference between AC sine signals at the same frequency in a simple RC circuit; besides, by adjust the frequency of the signal and resistance in the circuit, the phases will be different. Also, we can learn how to read the value of current or voltage on the scope display.


Knowledge
    If we can find that the amplitude of voltage is 6 divisions in 2V/Div scale from the scope, we can know the reading of voltage is 12V. Assumed that 6 divisions of horizontal time base set to 10 us/div, and we can find the period is 60 micro sec, and frequency is 1/T.  Between two peaks of two different channel of the signals, the phase difference between them would be t*f*360 degrees. If one peak of Ch1 is before Ch2, we call Ch1 signal is leading Ch2 signal.

Circuit

Figure 1. The Desired Circuit

Figure 2. Build-up Circuit connected to O-scope and Multimeter

Set-up

    First, We want to know the phase difference between CH1 and CH2, so we set the FG to produce 1 V peak-peak sine wave at 1k Hz. The RMS value for DMM is 0.318 V. The complex impedance of the 100 nF capacitor can be calculated as 1/(2*pi*C) = 1592 ohms. Next, we connect CH1 to FG and CH2 to the top of capacitor, and then Resistor box and C connected as shown above. After we assemble our circuit, we start measure our data.

Results

Part A

• peak-peak of CH2 = 0.852 V
• Vrms CH2 = 0.23 V
• tx =105.41 micro sec.
• Phi = tx * f * 360 = 37.9 degrees

==>CH1 is leading CH2 by 37.9 degrees

Part B

Next we adjust the frequency of input signal to 10 kHz rather than 1k Hz. 

• V peak-peak CH2=0.154V
• Vrms = 0.033 V
• tx=23.78 micro sec
• Phi = tx * f * 360 = 85.6 degrees

Part C

In order to see the impact of resistance, we adjust the resistor box to 10 Kohms, and we take the measurements.

• V peak-peak CH2 = 0.178 V
• Vrms = 0.049 V
• tx = 221.62 micro sec
• Phi = tx * f * 360 = 79.8 degrees

Figure 3. Time Difference between the Peaks of two Waveforms

Compare experimental phase difference to the theoretical value

Figure 4. Phase Relationship of VR and Vc in RC Circuit

Phi = arctan(VR/Vc)

      = arctan(R/Xc)

      = arctan(2*pi*f*R*C)

In Part A, the theoretical value is arctan(2*pi*1k*1k*102.6*10^-9) = 32.8 degrees

% error = (37.9-32.8)/32.8*100% = 15.5 %

In Part B, the theoretical value is arctan(2*pi*10k*1k*102.6*10^-9) = 81.2 degrees

% error = (85.6-81.2)/81.2*100% = 5.42 %

In Part C, the theoretical value is arctan(2*pi*10k*1k*102.6*10^-9) = 81.2 degrees

% error = (79.8-81.2)/81.2*100% = 1.72 %

Conclusion

    At low frequency range the capacitor voltage amplitude is the greatest; at high frequency range the capacitor voltage amplitude is the smallest. We also know that at very high frequency, the phase difference of two signals tend to be 90 degrees. So, we can treat this circuit as a lowpass filter.