I. II. III. Electric Circuits Lab Series RC Circuits: Phase Angle, Phase Lag, and Capacitors as Integrators Part 1: 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. Parts List: 1. Resistor 1000, 1k, 6.8 k 2. Capacitors 0.1 µF, 0.01 µF Procedures: 1. Connect the following circuit. VS 1Vrms 1 kHz R1 ww 1.0k Figure 1: RC Circuit 1 C1 =0.1pF 2. Connect one DMM across the resistor and one DMM across the capacitor. Set both DMM sto read AC voltage. Measure the voltage drop across each component. Record the result in Table 1./n3. 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. 4. Calculate the capacitive reactance using Ohm's law. Record the result in Table 2. Vc Capacitive Reactance, Xc = I Total current, I = R 5. Now, calculate the capacitive reactance value using the equation below. Record the result in Table 1 under Computed Reactance, Xc. Capacitive Reactance, Xc = Frequency (in Hz) 300 1k Voltage across, R Voltage across, C Total Current, I Capacitive Reactance, Xc Computed Reactance, Xc 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. 1 2nfC Table 1: Calculated and measured values VR Vc (measured) (measured) Capacitor C₁ 2 VR 1 = R Xe= I (calculated) (calculated) Vc 1 (2nfc) (calculated) Xc=/n3k 5k 7k 9k 11k 13k 15k 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. 1.5Vpk 500Hz 0 Vs C1 HH 0.01 μF R1 6.8kQ Figure 2: Series RC Circuit 3 9. Set the source voltage amplitude to 1.5 V, and frequency to 500 Hz./n10. 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. VR I Total current = R Vc Xc Z₁ 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. Vc Capacitive Reactance Xc = 7 e Vs Total Impedance (ZT) = Ť 14. Now, calculate the total impedance (Z₁) value using the equation below. Record the result in Table 3. 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, 8 = -tan-¹ () 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./nMAAAA Vs VR 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) 5/nType of Angle Phase angle 8 Phase Lag Measured Period (T) Time difference Measured (At) Angle 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) Phase Lag 22. Spread the signals horizontally using the Timebase (Sec/Div) control until both signals are just visible across the screen as shown below. Calculated Angle 23. Measure the time duration between the two signals (At) 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, 8 = (At/T) 360°/nlag circuit will lag the input voltage. 1.5Vpk ~500Hz 0" Vs R2 6.8kQ C2 =0.01 μF Figure 7: RC Lag Circuit XSC2 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, ø = tan-¹) 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) 7 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 (At) 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, Ø = (At/T) * 360°/n33. 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./nru ru V1 10kHz 10V V1 10kHz 10V 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) R1 3kQ R1 mw 3kQ C1 5.6μF C1 5.6μF Figure 9a. Integrator Circuit with Oscilloscope Connections ET 36. Run the simulation. Your signals should look like the example in Figure 9b./nOscilloscope-XSC1 T2 T2-T1 VR Vc Time 164.481 ms 164.481 ms 0.000 s Timebase Scale: 20 us/Div X pos.(Div): 0 Y/T Add B/A A/B Channel A -4.999 V -4.999 V 0.000 V Channel B -1.675 mV -1.675 mV 0.000 V Channel A Scale: 2 V/Div Y pos.(Div): 0 AC O DC Figure 9b: Capacitor as an integrator waveforms Channel B Scale: 5 mV/Div Y pos.(Div): 0 OAC 0 DC- v(t) >= ²1 [₁ v(t) = ½ ju(t)dr {=C 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. dv dt i(t)dr Reverse = - 1/1(1) Save t(t)dt + + Ext. trigger Trigger Edge: A B Ext Level: 0 O Single Normal Auto None t(r)dr/nT2 T2-T1 39. Refer to Figure 10 to answer the following questions. Oscilloscope-XSC1 Time 164.100 ms 164.150 ms 50.001 us v(0) = 1 ju(r)dr v(t): 1) = ²½ [167)ar = √ [ 16 [1(r)dr Timebase Scale: 20 us/Div X pos.(Div): 0 Y/T Add B/A A/B Channel A -1.993 V 1.992 V 3.985 V Channel A Scale: 2 V/Div Y pos.(Div): 0 AC O DC Channel B -7.425 mV 14.882 mV Channel B Scale: 5 mV/Div Y pos.(Div): 0 O AC O DC Figure 10: Integrator values, 0 to 50 µs i(t): 11 1(T)dt + v(0) t=50μS = mVys a. The signal has a period of 100 us. Write the equation for the circuit current on the interval 0 to 50 us. On the interval of 0 to 50 µs, va(t) is constant so the current will be constant as well. VR(t) R Reverse Save Ext. trigger Trigger Edge: A B Ext Level: 0 V Single Normal Auto None/nb. 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. 50 μ v(t) = [1(x)dx+v(0) c. Confirm your equation by predicting the value of vc(50 µs). d. Read the value of vc(50 us) from Figure 10. 40. Refer to Figure 11 to answer the following questions. Oscilloscope-XSC1 T2 T2-T1 t=50μs Time 164.400 ms 164.450 ms 50.001 us Timebase Scale: 20 us/Div X pos.(Div): 0 Y/T Add B/A A/B Channel A -1.993 V 1.992 V 3.985 V Channel A Scale: 2 V/Div Y pos.(Div): 0 AC O DC t=100μs 12 Channel B -7.424 mV 7.457 mV 14.882 mV Channel B Scale: 5 mV/Div Y pos.(Div): 0 AC 0 DC Figure 11: Integrator values, 50 to 100 μs X Ext. trigger Reverse Save Trigger Edge: A B Ext Level: 0 V Single Normal Auto None/na. The signal has a period of 100 us. 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. i(t) = VR(t) R 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. v(t) = 11(x) + (50) 50 μs c. Confirm your equation by predicting the value of vc(100 µs). d. Read the value of vc(100 us) from Figure 11.

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I. II. III. Electric Circuits Lab Series RC Circuits: Phase Angle, Phase Lag, and Capacitors as Integrators Part 1: 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. Parts List: 1. Resistor 1000, 1k, 6.8 k 2. Capacitors 0.1 µF, 0.01 µF Procedures: 1. Connect the following circuit. VS 1Vrms 1 kHz R1 ww 1.0k Figure 1: RC Circuit 1 C1 =0.1pF 2. Connect one DMM across the resistor and one DMM across the capacitor. Set both DMM sto read AC voltage. Measure the voltage drop across each component. Record the result in Table 1./n3. 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. 4. Calculate the capacitive reactance using Ohm's law. Record the result in Table 2. Vc Capacitive Reactance, Xc = I Total current, I = R 5. Now, calculate the capacitive reactance value using the equation below. Record the result in Table 1 under Computed Reactance, Xc. Capacitive Reactance, Xc = Frequency (in Hz) 300 1k Voltage across, R Voltage across, C Total Current, I Capacitive Reactance, Xc Computed Reactance, Xc 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. 1 2nfC Table 1: Calculated and measured values VR Vc (measured) (measured) Capacitor C₁ 2 VR 1 = R Xe= I (calculated) (calculated) Vc 1 (2nfc) (calculated) Xc=/n3k 5k 7k 9k 11k 13k 15k 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. 1.5Vpk 500Hz 0 Vs C1 HH 0.01 μF R1 6.8kQ Figure 2: Series RC Circuit 3 9. Set the source voltage amplitude to 1.5 V, and frequency to 500 Hz./n10. 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. VR I Total current = R Vc Xc Z₁ 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. Vc Capacitive Reactance Xc = 7 e Vs Total Impedance (ZT) = Ť 14. Now, calculate the total impedance (Z₁) value using the equation below. Record the result in Table 3. 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, 8 = -tan-¹ (*) 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./nMAAAA Vs VR 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) 5/nType of Angle Phase angle 8 Phase Lag Measured Period (T) Time difference Measured (At) Angle 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) Phase Lag 22. Spread the signals horizontally using the Timebase (Sec/Div) control until both signals are just visible across the screen as shown below. Calculated Angle 23. Measure the time duration between the two signals (At) 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, 8 = (At/T)* 360°/nlag circuit will lag the input voltage. 1.5Vpk ~500Hz 0" Vs R2 6.8kQ C2 =0.01 μF Figure 7: RC Lag Circuit XSC2 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, ø = tan-¹) 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) 7 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 (At) 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, Ø = (At/T) * 360°/n33. 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./nru ru V1 10kHz 10V V1 10kHz 10V 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) R1 3kQ R1 mw 3kQ C1 5.6μF C1 5.6μF Figure 9a. Integrator Circuit with Oscilloscope Connections ET 36. Run the simulation. Your signals should look like the example in Figure 9b./nOscilloscope-XSC1 T2 T2-T1 VR Vc Time 164.481 ms 164.481 ms 0.000 s Timebase Scale: 20 us/Div X pos.(Div): 0 Y/T Add B/A A/B Channel A -4.999 V -4.999 V 0.000 V Channel B -1.675 mV -1.675 mV 0.000 V Channel A Scale: 2 V/Div Y pos.(Div): 0 AC O DC Figure 9b: Capacitor as an integrator waveforms Channel B Scale: 5 mV/Div Y pos.(Div): 0 OAC 0 DC- v(t) >= ²1 [₁ v(t) = ½ ju(t)dr {=C 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. dv dt i(t)dr Reverse = - 1/1(1) Save t(t)dt + + Ext. trigger Trigger Edge: A B Ext Level: 0 O Single Normal Auto None t(r)dr/nT2 T2-T1 39. Refer to Figure 10 to answer the following questions. Oscilloscope-XSC1 Time 164.100 ms 164.150 ms 50.001 us v(0) = 1 ju(r)dr v(t): 1) = ²½ [167)ar = √ [ 16 [1(r)dr Timebase Scale: 20 us/Div X pos.(Div): 0 Y/T Add B/A A/B Channel A -1.993 V 1.992 V 3.985 V Channel A Scale: 2 V/Div Y pos.(Div): 0 AC O DC Channel B -7.425 mV 14.882 mV Channel B Scale: 5 mV/Div Y pos.(Div): 0 O AC O DC Figure 10: Integrator values, 0 to 50 µs i(t): 11 1(T)dt + v(0) t=50μS = mVys a. The signal has a period of 100 us. Write the equation for the circuit current on the interval 0 to 50 us. On the interval of 0 to 50 µs, va(t) is constant so the current will be constant as well. VR(t) R Reverse Save Ext. trigger Trigger Edge: A B Ext Level: 0 V Single Normal Auto None/nb. 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. 50 μ v(t) = [1(x)dx+v(0) c. Confirm your equation by predicting the value of vc(50 µs). d. Read the value of vc(50 us) from Figure 10. 40. Refer to Figure 11 to answer the following questions. Oscilloscope-XSC1 T2 T2-T1 t=50μs Time 164.400 ms 164.450 ms 50.001 us Timebase Scale: 20 us/Div X pos.(Div): 0 Y/T Add B/A A/B Channel A -1.993 V 1.992 V 3.985 V Channel A Scale: 2 V/Div Y pos.(Div): 0 AC O DC t=100μs 12 Channel B -7.424 mV 7.457 mV 14.882 mV Channel B Scale: 5 mV/Div Y pos.(Div): 0 AC 0 DC Figure 11: Integrator values, 50 to 100 μs X Ext. trigger Reverse Save Trigger Edge: A B Ext Level: 0 V Single Normal Auto None/na. The signal has a period of 100 us. 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. i(t) = VR(t) R 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. v(t) = 11(x) + (50) 50 μs c. Confirm your equation by predicting the value of vc(100 µs). d. Read the value of vc(100 us) from Figure 11.

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