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.