Series RC Circuits: Phase Angle, Phase Lag, and Capacitors as Integrators

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Electric Circuits Lab

Instructor: ———–

Series RC Circuits

Student Name(s):

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I pledge to support the Honor System of ECPI. I will refrain from any form of academic dishonesty or deception, such as cheating or plagiarism. I am aware that as a member of the academic community, it is my responsibility to turn in all suspected violators of the honor code. I understand that any failure on my part to support the Honor System will be turned over to a Judicial Review Board for determination. I will report to the Judicial Review Board hearing if summoned.

Date: 1/1/2018

Contents
Abstract 3
Introduction 3
Procedures 3
Data Presentation & Analysis 4
Calculations 4
Required Screenshots 4
Conclusion 4
References 5

Abstract

(This instruction box is to be deleted before submission of the Lab report)

What is an Abstract?

This should include a brief description of all parts of the lab. The abstract should be complete in itself. It should summarize the entire lab; what you did, why you did it, the results, and your conclusion. Think of it as a summary to include all work done. It needs to be succinct yet detailed enough for a person to know what this report deals with in its entirety.

Objectives of Week 3 Lab 2:

· Understand the effect of frequency on capacitive reactance.
· Measure the impedance of an RC circuit.
· Measure the phase angle and phase lag of an RC circuit using the oscilloscope.
· Draw the impedance and voltage phasor diagrams.
· Understand how a capacitor current.

Introduction

(This instruction box is to be deleted before submission of the Lab report)

What is an Introduction?

In your own words, explain the reason for performing the experiment and give a concise summary of the theory involved, including any mathematical detail relevant to later discussion in the report. State the objectives of the lab as well as the overall background of the relevant topic.

Address the following items in your introduction:
· What is Impedance for an RC circuit? (Give formula)
· What is phase angle for an RC circuit? How is it calculated?
· What is phase lag for an RC lag circuit? How is it calculated?
· How/why does a capacitor integrate current? Give formula.

Procedures

(This instruction box is to be deleted before submission of the Lab report)
This section should contain the procedures as outlined in the lab instructions.

Part I:

1.
Connect the following circuit.

Figure 1: RC Circuit

2.
Connect one DMM across the resistor and one DMM across the capacitor. Set both DMMs to read AC voltage.
Measure the voltage drop across each component.
Record the result in
Table 1.

3. Use Ohm’s law to calculate the current flowing through the resistor. Since the circuit in
Figure 1 is a series RC circuit, the same current will flow through the capacitor and the resistor.
Record the result in
Table 1.

Total current, I =

4.
Calculate the capacitive reactance using Ohm’s law.
Record the result in
Table 2.

Capacitive Reactance, XC =

5. Now,
calculate the capacitive reactance value using the equation below.
Record the result in
Table 1 under Computed Reactance, XC.

Capacitive Reactance,

6.
Adjust the function generator frequency following the steps in
Table 2. Use the DMM to
measure the voltage across the resistor and the capacitor.
Record your measurements below.

7.
Plot the graph for
Frequency vs. VC.

Part II:

8.
Build the circuit shown in
Figure 2.

Figure 2: Series RC Circuit

9.

Set the source voltage amplitude to
1.5 Vp and
frequency to
500 Hz.

10.
Connect Channel A of the oscilloscope across the resistor and
measure the peak voltage drop (VR).
Record the result in
Table 3.

11. Use Ohm’s law to
calculate the peak current flowing through the resistor. Because it is a series circuit, the same current will flow through the capacitor.
Record the result in
Table 3.

Total current I =

12.
Connect Channel B of the oscilloscope across the capacitor and
measure the peak voltage drop (VC).
Record the value in
Table 3.

13.
Calculate the capacitive reactance using Ohm’s law.
Record the result in
Table 3.

Capacitive Reactance XC =

14. Now,
calculate the total impedance (ZT) value using the equation below.
Record the result in
Table 3.

Total Impedance (ZT) =

15.
Calculate the phase angle between VR and VS using the formula below.
Record the result in
Table 3. Also,
record this value in
Table 4 under Phase Angle calculated value.

Phase angle,

Part III: Phase Angle and Phase Lag Measurement

Phase Angle

16.
Connect Channel A of the oscilloscope across the resistor and Channel B of the oscilloscope across the function generator and
run the simulation.

17. The waveforms should look like the ones shown in
Figure 4.

Figure 4: VS and VR waveforms

18. Obtain a stable display showing a couple of cycles for Channel B (which is showing VS) and disable Channel A by setting it to 0.
19.
Measure the time period (T) of the source voltage.
Record the result in
Table 4. (Use the cursors to measure the period (on the scope it will show as T2-T1). Remember that the period is the time taken to complete one cycle). See
Figure 5.

Figure 5: Measuring time period (T)

20. Now
set the oscilloscope to view both the channels.

21.
Adjust the amplitude of the signals using Channel A and Channel B V/Div scale until both channels appear to have the same amplitude as seen on the scope face. (as close as possible)

22. Spread the signals horizontally using the Timebase (Sec/Div) control until both signals are just visible across the screen as shown below.
23.
Measure the time duration between the two signals (∆t) and
record the result in
Table 4. (Use cursors as shown below in
Figure 6)

Figure 6: Measuring the time difference

24.
Calculate the phase angle using the formula below and
record the result in
Table 4.

Phase angle, θ = (∆t/T) * 360°

Phase Lag

25.
Connect your circuit as shown in
Figure 7. When the output of an RC circuit is taken across the capacitor, the circuit is called an RC lag circuit. The output voltage in an RC lag circuit will lag the input voltage.

Figure 7: RC Lag Circuit

26.
Calculate the phase lag using the equation below. Notice the similarity to the equation for the phase angle. The phase lag angle and phase angle of an RC circuit are complementary angles. (Their sum is 90°.) Use R and XC values from
Table 3.

Phase Lag,
27.
Measure the time period (T) of the source voltage (as in Step 19).
Record this value in
Table 4.

28. Now
set the oscilloscope to view both the channels.

29.
Adjust the amplitude of the signals using Channel A and Channel B V/Div scale until both channels appear to have the same amplitude as seen on the scope face. (as close as possible)

30. Spread the signals horizontally using the Timebase (Sec/Div) control until both signals are just visible across the screen as shown in
Figure 6.

31.
Measure the time duration between the two signals (∆t) and
record the result in
Table 4 above.

32.
Calculate the phase lag using the formula below and
record the result in
Table 4.

Phase lag, ∅ = (∆t/T) * 360°
33.
Plot the Voltage and Impedance Phasor Diagrams. Clearly indicate the phase angle and the phase lag.
Measure the peak voltages for VR and VC with the oscilloscope.

Part IV: The Capacitor Integrates Current

34.
Construct the following RC circuit in Multisim. Set the clock voltage source to 10 kHz, 10V, 50% duty cycle.

Figure 9. Integrator Circuit

35.
Connect Channel A across the resistor and Channel B across the capacitor. (Note: change one or both trace colors to better observe the two signals)

Figure 9a. Integrator Circuit with Oscilloscope Connections

36.
Run the simulation. Your signals should look like the example in
Figure 9b.

Figure 9b: Capacitor as an integrator waveforms

37. Channel A will show the voltage across the resistor. This signal can be used to find the circuit current using Ohm’s law.
38. Channel B shows the voltage across the capacitor. Show that this signal satisfies the following equation. We will do this in intervals in the following steps.

39. Refer to
Figure 10 to answer the following questions.

Figure 10: Integrator values, 0 to 50 µs

a. The signal has a period of 100 µs.
Write the equation for the circuit current on the interval 0 to 50 µs. On the interval of 0 to 50 µs, vR(t) is constant so the current will be constant as well.

b.
Write the equation for the voltage across the capacitor by solving the integral. You will need to read the value vC(0) from
Figure 10.

c.
Confirm your equation by predicting the value of vC (50 µs).

d.
Read the value of vC(50 µs) from
Figure 10.

40. Refer to
Figure 11 to answer the following questions.

Figure 11: Integrator values, 50 to 100 µs

a. The signal has a period of 100 µs.
Write the equation for the circuit current on the interval 50 µs to 100 µs. On the interval of 50 to 100 µs, vR(t) is constant so the current will be constant as well.

b.
Write the equation for the voltage across the capacitor by solving the integral. You will need to read the value vC(50) from
Figure 11.

c.
Confirm your equation by predicting the value of vC(100 µs).

d.
Read the value of vC(100 µs) from
Figure 11.

Data Presentation & Analysis

Capacitor C1

Voltage across, R

Voltage across, C

Total Current, I

Capacitive Reactance, XC

Computed Reactance, XC

Table 1: Calculated and measured values

Frequency (in Hz)

VR
(measured)

VC
(measured)

I =

(calculated)

XC =

(calculated)

XC =
(calculated)

300

1k

3k

5k

7k

9k

11k

13k

15k

Table 2: Calculated and measured values

(Use Excel or Word to Create the Plot)

Plot 1. Frequency vs. Voltage, VC

VR

I

VC

XC

ZT

Ө

Table 3: Calculated and measured values

Type of Angle

Measured
Period (T)

Time difference (∆t)

Measured Angle

Calculated Angle

Phase angle θ

Phase Lead Φ

Table 4: Phase angle and phase lag measurements

(Use Excel or Word to create diagrams)

Plot 2(a) Impedance Phasor Plot 2(b) Voltage Phasor

Calculations

Part I step 3: I =

Part I step 4: XC =

Part I step 5: Xc =

Part II step 11: I =

Part II step 13: XC =

Part II step 14: ZT =

Part II step 15:

Part III step 24:

Part III step 26:

Part III step 32:

Part IV step 39 a: i(t) =

Part IV step 39 b: vc(t) =

Part IV step 39 c: vc(50 µs) =

Part IV step 39 d: vc(50 µs) =

Part IV step 40 a: i(t) =

Part IV step 40 b: vC(t) =

Part IV step 40 c: vc(100 µs) =

Part IV step 40 d: vc(100 µs) =

Required Screenshots

Figure 12: Screenshot of Waveforms for Part 2 Step 10

Figure 13: Screenshot of Waveforms for Part 2 Step 12

Figure 14: Screenshot of Waveforms for Part 3 Step 19

Figure 15: Screenshot of Waveforms for Part 3 Step 23

Figure 16: Screenshot of Waveforms for Part 3 Step 27

Figure 17: Screenshot of Waveforms for Part 3 Step 31

Figure 18: Screenshot of Waveforms for Part 3 Step 33

Conclusion

(This instruction box is to be deleted before submission of the Lab report)

What is a Conclusion?

This section should reflect your understanding of the experiment conducted. Important points to include are a brief discussion of your results, and an interpretation of the actual experimental results as they apply to the objectives of the experiment set out in the introduction should be given. Also, discuss any problems encountered and how they were resolved.

Address the following in your conclusions:
· Were your measured and calculated capacitive reactance values in agreement?
· What happened to the inductance and the capacitive reactance as you increased the frequency of the voltage source?
· Were your measured and calculated phase angle values in agreement?
· Which quantity leads in a series RC circuit? (Current or voltage) How do you know?
· What happens to phase angle as the frequency increases? What happens to phase angle as the frequency decreases?
· Were your measured and calculated phase lag values in agreement?
· Which quantity lags in an RC lag circuit? (Source voltage or capacitor voltage)
· What is the relationship between phase angle and phase lag?
· What happens to the phase lag as the frequency increases? What happens to the phase lag as the frequency decreases?

References

Floyd, T. L., & Buchla, D. M. (2019). 
Principles of Electric Circuits (10th Edition). Pearson Education (US). 
https://bookshelf.vitalsource.com/books/9780134880068

(2017) National Instruments Multisim (V 14.1) [Windows]. Retrieved from http://www.ni.com/multisim/

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Electric Circuits Lab

Series RC Circuits: Phase Angle, Phase Lag, and Capacitors as Integrators

I.

Objectives:

After completing this lab experiment using, you should be able to:

1. Understand the effect of frequency on capacitive reactance.

2. Measure the impedance of an RC circuit.

3. Measure the phase angle and phase lag of an RC circuit using the oscilloscope.

4. Draw the impedance and voltage phasor diagrams.

5. Understand how a capacitor integrates current.

II.

Parts List:

1. Resistor 100 Ω, 1 kΩ, 6.8 kΩ

2. Capacitors 0.1 µF, 0.01 µF

III.

Procedures:

Part I:

1.
Connect the following circuit.

Figure 1: RC Circuit

2.
Connect one DMM across the resistor and one DMM across the capacitor. Set both DMMs to read AC voltage.
Measure the voltage drop across each component. Record the result in
Table 1.

3. Use Ohm’s law to
calculate the current flowing through the resistor. Since the circuit in Figure 1 is a series RC circuit, the same current will flow through the capacitor and the resistor.
Record the result in
Table 1.

Total current, I =

4.
Calculate the capacitive reactance using Ohm’s law. Record the result in
Table 2.

Capacitive Reactance, XC =

5. Now,
calculate the capacitive reactance value using the equation below.
Record the result in
Table 1 under Computed Reactance, XC.

Capacitive Reactance,

Capacitor C1

Voltage across, R

846 mV

Voltage across, C

583 mV

Total Current, I

0.846 mA

Capacitive Reactance, XC

686 Ω

Computed Reactance, XC

Table 1: Calculated and measured values

6.
Adjust the function generator frequency following the steps in
Table 2. Use the DMM to
measure the voltage across the resistor and the capacitor.
Record your measurements below.

Frequency (in Hz)

VR
(measured)

VC
(measured)

I =

(calculated)

XC =

(calculated)

XC =
(calculated)

300

983 mV

186 mV

0.983 mA

189 Ω

1k

846 mV

583 mV

0.846 mA

686 Ω

3k

884 mV

468 mV

0.884 mA

529 Ω

5k

953 mV

302 mV

0.953 mA

317 Ω

7k

975 mV

221 mV

0.975 mA

227 Ω

9k

985 mV

174 mV

0.985 mA

177 Ω

11k

990 mV

145 mV

0.990 mA

147Ω

13k

993 mV

121 mV

0.993 mA

122 Ω

15k

994 mV

105 mV

0.994 mA

106 Ω

Table 2: Calculated and measured values

7.
Plot the graph for
Frequency vs. VC.

(Use Excel or Word to Create the Plot)

Plot 1: Frequency vs. VC

Part II:

8.
Build the circuit shown in Figure 2.

Figure 2: Series RC Circuit

9.
Set the source voltage amplitude to
1.5 Vp and
frequency to
500 Hz.

10.

Connect Channel A of the oscilloscope across the resistor and measure the peak voltage drop (VR). Record the result in
Table 3.

11. Use Ohm’s law to
calculate the peak current flowing through the resistor. Because it is a series circuit, the same current will flow through the capacitor.
Record the result in
Table 3.

Total current I =

VR

I

VC

XC

ZT

Ө

313 mV

46 mA

1.46 V

32 mΩ

32.6 mΩ

Table 3: Calculated and measured values

12.
Connect Channel B of the oscilloscope across the capacitor and
measure the peak voltage drop (VC). Record the value in
Table 3.

13.
Calculate the capacitive reactance using Ohm’s law.
Record the result in
Table 3.

Capacitive Reactance XC =

14. Now,
calculate the total impedance (ZT) value using the equation below.
Record the result in
Table 3.

Total Impedance (ZT) =

15.
Calculate the phase angle between VR and VS using the formula below.
Record the result in
Table 3. Also,
record this value in
Table 4 under Phase Angle calculated value.

Phase angle,

Part III: Phase Angle and Phase Lag Measurement

Phase Angle

16.
Connect Channel A of the oscilloscope across the resistor and Channel B of the oscilloscope across the function generator and run the simulation.

17. The waveforms should look like the ones shown in Figure 4.

Figure 4: VS and VR waveforms

18. Obtain a stable display showing a couple of cycles for Channel B (which is showing VS) and disable Channel A by setting it to 0.
19.
Measure the time period (T) of the source voltage.
Record the result in
Table 4 below. (Use the cursors to measure the period (on the scope it will show as T2-T1). Remember that the period is the time taken to complete one cycle). See Figure 5.

Figure 5: Measuring time period (T)

Type of Angle

Measured
Period (T)

Time difference (∆t)

Measured Angle

Calculated Angle

Phase angle θ

2 ms

432.812 us

77.9

77.9

Phase Lead Φ

Table 4: Phase angle and phase lag measurements

20. Now
set the oscilloscope to view both the channels.

21.
Adjust the amplitude of the signals using Channel A and Channel B V/Div scale until both channels appear to have the same amplitude as seen on the scope face. (as close as possible)

22. Spread the signals horizontally using the Timebase (Sec/Div) control until both signals are just visible across the screen as shown below.
23.
Measure the time duration between the two signals (∆t) and record the result in
Table 4 above. (Use cursors as shown below in Figure 6)

Figure 6: Measuring the time difference

24.
Calculate the phase angle using the formula below and record the result in
Table 4.

Phase angle, θ = (∆t/T) * 360°

Phase Lag

25.
Connect your circuit as shown in
Figure 7. When the output of an RC circuit is taken across the capacitor, the circuit is called an RC lag circuit. The output voltage in an RC lag circuit will lag the input voltage.

Figure 7: RC Lag Circuit

26.
Calculate the phase lag using the equation below. Notice the similarity to the equation for the phase angle. The phase lag angle and phase angle of an RC circuit are complementary angles. (Their sum is 90°.) Use R and XC values from
Table 3.

Phase Lag,
27.
Measure the time period (T) of the source voltage (as in Step 19). Record this value in
Table 4.

28. Now
set the oscilloscope to view both the channels.

29.
Adjust the amplitude of the signals using Channel A and Channel B V/Div scale until both channels appear to have the same amplitude as seen on the scope face. (as close as possible)

30. Spread the signals horizontally using the Timebase (Sec/Div) control until both signals are just visible across the screen as shown in Figure 6.
31.
Measure the time duration between the two signals (∆t) and record the result in
Table 4 above.

32.
Calculate the phase lag using the formula below and
record the result in
Table 4.

Phase lag, ∅ = (∆t/T) * 360°
33.
Plot the Voltage and Impedance Phasor Diagrams. Clearly indicate the phase angle and the phase lag.
Measure the peak voltages for VR and VC with the oscilloscope.

(Use Excel or Word to create diagrams)

Plot 2(a) Impedance Phasor Plot 2(b) Voltage Phasor

Part IV: The Capacitor Integrates Current

34.
Construct the following RC circuit in Multisim. Set the clock voltage source to 10 kHz, 10V, 50% duty cycle.

Figure 9. Integrator Circuit

35.
Connect Channel A across the resistor and Channel B across the capacitor. (Note: change one or both trace colors to better observe the two signals)

Figure 9a. Integrator Circuit with Oscilloscope Connections

36.
Run the simulation. Your signals should look like the example in Figure 9b.

Figure 9b: Capacitor as an integrator waveforms

37. Channel A will show the voltage across the resistor. This signal can be used to find the circuit current using Ohm’s law.
38. Channel B shows the voltage across the capacitor.
Show that this signal satisfies the following equation. We will do this in intervals in the following steps.

39. Refer to Figure 10 to answer the following questions.

Figure 10: Integrator values, 0 to 50 µs

a. The signal has a period of 100 µs.
Write the equation for the circuit current on the interval 0 to 50 µs. On the interval of 0 to 50 µs, vR(t) is constant so the current will be constant as well.

b.
Write the equation for the voltage across the capacitor by solving the integral. You will need to read the value vC(0) from Figure 10.

c.
Confirm your equation by predicting the value of vC(50 µs).

d.
Read the value of vC(50 µs) from Figure 10.

40. Refer to Figure 11 to answer the following questions.

Figure 11: Integrator values, 50 to 100 µs

a. The signal has a period of 100 µs.
Write the equation for the circuit current on the interval 50 µs to 100 µs. On the interval of 50 to 100 µs, vR(t) is constant so the current will be constant as well.

b.
Write the equation for the voltage across the capacitor by solving the integral. You will need to read the value vC(50) from Figure 11.

c.
Confirm your equation by predicting the value of vC(100 µs).

d.
Read the value of vC(100 µs) from Figure 11.

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