Week 3 Lab 2 Series RC Circuits Lab Report
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ECPI University, Virginia Beach *
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Electrical Engineering
Date
Dec 6, 2023
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docx
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Electric Circuits Lab
Instructor: Cameron Ruddy
Series RC Circuits
Student Name(s):
Brandon Walker
Click or tap here to enter text.
Honor Pledge: 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:
0/19/2023
Contents
Abstract
.......................................................................................................................................................
3
I
ntroduction
................................................................................................................................................
3
Procedures
...................................................................................................................................................
3
Data Presentation & Analysis
.......................................................................................................................
4
Calculations
.............................................................................................................................................
4
Required Screenshots
..............................................................................................................................
4
Conclusion
...................................................................................................................................................
4
References
...................................................................................................................................................
5
2
Abstract
The lab we are completing will help us better understand how to measure the impedance of a RC
circuit. During the lab we will understand the effect of frequency on capacitive reactance while
using a oscilloscope. The use of the oscilloscope will also help us measure phase angles, phase
lag and better understand capacitor currents.
I
ntroduction
We find that impedance of a rc circuit is opposition of flow in a a/c circuit with the formula
Z=R+1/jwC. We find that phase angle represents the phase shift between voltage across the
resistor and voltage across the capacitor it is calculated using 0=arctan(-1/wRC). We
understand that phase lag is when the output voltage lags behind the input voltage using the
calculation 0=arctan(-1/wRC). Capacitors are able to integrate current because they store
electrical energy in a electric field between the plates with the equation Q=C*V.
Procedures
Part I:
1.
Connect
the following circuit.
VS
R1
1.0kΩ
C1
0.1µF
1Vrms
1 kHz
Figure 1: RC Circuit
3
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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 = V
R
R
4.
Calculate
the capacitive reactance using Ohm’s law. Record
the result in Table 2
.
Capacitive Reactance, X
C
= V
C
I
5.
Now, calculate
the capacitive reactance value using the equation below. Record
the result in Table 1
under Computed Reactance, X
C
.
Capacitive Reactance, X
C
=
1
2
πfC
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. V
C
.
Part II:
8.
Build
the circuit shown in Figure 2
.
4
Figure 2: Series RC Circuit
9.
Set
the source voltage amplitude to 1.5 V
p
and
frequency to
500 Hz. 10. Connect
Channel A of the oscilloscope across the resistor and measure
the peak
voltage drop (V
R
). 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 = V
R
R
12. Connect
Channel B of the oscilloscope across the capacitor and measure
the peak
voltage drop (V
C
). Record
the value in Table 3
.
13. Calculate
the capacitive reactance using Ohm’s law. Record
the result in Table 3
.
Capacitive Reactance X
C
= V
C
I
14.
Now, calculate
the total impedance (Z
T
) value using the equation below. Record
the
result in Table 3
.
Total Impedance (Z
T
) = V
S
I
5
15. Calculate
the phase angle between V
R
and V
S
using the formula below. Record
the
result in Table 3
. Also, record
this value in Table 4
under Phase Angle calculated value.
Phase angle, θ
=−
tan
−
1
(
X
C
R
)
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: V
S and V
R waveforms
18.
Obtain a stable display showing a couple of cycles for Channel B (which is showing V
S
)
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
.
6
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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
7
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 X
C
values from Table 3
.
Phase Lag, ϕ
=
tan
−
1
(
R
X
C
)
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.
8
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 V
R
and V
C
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)
9
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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
10
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.
i
=
C
dv
dt
v
(
t
)
=
1
C
∫
−
∞
t
i
(
τ
)
dτ
v
(
t
)
=
1
C
∫
−
∞
t
i
(
τ
)
dτ
=
1
C
∫
0
t
i
(
τ
)
dτ
+
1
C
∫
−
∞
0
i
(
τ
)
dτ
v
(
0
)
=
1
C
∫
−
∞
0
i
(
τ
)
dτ
v
(
t
)
=
1
C
∫
−
∞
t
i
(
τ
)
dτ
=
1
C
∫
0
t
i
(
τ
)
dτ
+
v
(
0
)
39.
Refer to Figure 10
to answer the following questions.
11
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, v
R
(t) is constant so the
current will be constant as well.
i
(
t
)
=
v
R
(
t
)
R
b.
Write
the equation for the voltage across the capacitor by solving the integral.
You will need to read the value v
C
(0) from Figure 10
.
12
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v
(
t
)
=
1
C
∫
0
50
µs
i
(
x
)
dx
+
v
(
0
)
c.
Confirm
your equation by predicting the value of v
C
(50 µs).
d.
Read
the value of v
C
(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, v
R
(t) is constant so
the current will be constant as well.
13
i
(
t
)
=
v
R
(
t
)
R
b.
Write
the equation for the voltage across the capacitor by solving the integral.
You will need to read the value v
C
(50) from Figure 11
.
v
(
t
)
=
1
C
∫
50
µs
t
i
(
τ
)
dτ
+
v
(
50
µs
)
c.
Confirm
your equation by predicting the value of v
C
(100 µs).
d.
Read
the value of v
C
(100 µs) from Figure 11
.
Data Presentation & Analysis
Table 1: Calculated and measured values
14
Capacitor C
1
Voltage across, R
533.245mV
Voltage across, C
845.958mV
Total Current, I
0.533mA
Capacitive Reactance, X
C
1.587 ohm
Computed Reactance, X
C
1.5925 ohm
Table 2: Calculated and measured values
Plot 1.
Frequency vs. Voltage,
V
C
15
Frequency
(in Hz)
V
R
(measured)
V
C
(measured)
I =
V
R
R
(calculated)
X
C =
V
C
I
(calculated)
X
C = 1
(
2
πfC
)
(calculated)
300
185.809mV
982.579mV
186mA
5.284 ohm
530.5 ohm
1k
533.245mV
845.958mV
533mA
1.588 ohm
1591.55 ohm
3k
883.997mV
467.468mV
884mA
0.529 ohm
530.52 ohm
5k
953.162mV
302.426mV
953mA
0.317 ohm
318.31 ohm
7k
975.266mV
221.028mV
975mA
0.277 ohm
227.51 ohm
9k
984.812mV
173.593mV
985mA
0.176 ohm
176.71 ohm
11k
989.752mV
142.743mV
990mA
0.144 ohm
144.6 ohm
13k
992.627mV
121.134mV
993mA
0.122 ohm
122.1 ohm
15k
994.443mV
105.175mV
994mA
0.106 ohm
212.13 ohm
Frequency
(in Hz)
V
R
(measured)
V
C
(measured)
300
185.809mV
982.579mV
1k
533.245mV
845.958mV
3k
883.997mV
467.468mV
5k
953.162mV
302.426mV
7k
975.266mV
221.028mV
984.812mV
173.593mV
11k
989.752mV
142.743mV
13k
992.627mV
121.134mV
15k
994.443mV
105.175mV
V
R
I
V
C
X
C
Z
T
Ө
312.957
mV
45.99mA
1.460V
31.748
ohm
23.08 ohm
77.81
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Table 3: Calculated and measured values Table 4: Phase angle and phase lag measurements
70
75
80
85
90
95
Chart Title
Measured Angle Period (T)
Calculated Angle Period (T)
Plot 2(a) Impedance Phasor Plot 2(b) Voltage Phasor
Calculations Part I step 3: I =0.533mA
Part I step 4: X
C
=1.587 ohm
Part I step 5: Xc =1.5925 ohm
Part II step 11: I =45.99mA
Part II step 13: X
C
=31.748 ohm
16
Type of Angle
Measured Period (T)
Time difference
(∆t)
Measured
Angle
Calculated
Angle
Phase angle θ
2.000 ms
432.812uS
77.81 77.81
Phase Lag Φ
2.000ms
432.812uS
77.81
89.9
Part II step 14: Z
T
=23.08 ohm
Part II step 15: θ
=
¿
77.81 deg
Part III step 24: θ
=
¿
77.81 deg
Part III step 26: ∅
=
¿
89.9
Part III step 32:
∅
=
¿
77.81
Part IV step 39 a: i(t) =i(t)=1.992v/3000 ohm=664uA
Part IV step 39 b: v
c
(t) =
Vc
(
t
)=
ωCI
0sin(
ωt
)−7.425×10−3
Part IV step 39 c: v
c
(50 µs) =
Vc
(50
μs
)=-7.425×10−3V
Part IV step 39 d: v
c
(50 µs) =7.425
Part IV step 40 a: i(t) =
i
(100
μs
)=1.992V/3000=0.000664A=664
μ
A
Part IV step 40 b: v
C
(t) =
vC
(
t
)=
C
1(−664.3×10−6×(
t
−50×10−6))−7.424×10−3
Part IV step 40 c: v
c
(100 µs) =
vC
(100
μ
s)=5.6×10−61(−664.3×10−6×(100×10−6−50×10−6))
−7.424×10−3
Part IV step 40 d: v
c
(100 µs) =7.457
Required Screenshots
Figure 12: Screenshot of Waveforms for Part 2 Step 10
17
Figure 13: Screenshot of Waveforms for Part 2 Step 12
Figure 14: Screenshot of Waveforms for Part 3 Step 19
18
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Figure 15: Screenshot of Waveforms for Part 3 Step 23
Figure 16: Screenshot of Waveforms for Part 3 Step 27
19
Figure 17: Screenshot of Waveforms for Part 3 Step 31
Figure 18: Screenshot of Waveforms for Part 3 Step 33
20
Conclusion
Yes my measured and calculated capacitive reactance values agreed. As frequency increases the overall capacitive reactance decreases. My measured and calculated phase angle did not perfectly agree but are
very close. In a RC circuit we find the voltage leading due to current lagging behind voltage. We find the phase angle decreases as frequency increases and phase angle increases as frequency decreases. My calculated and measured phase lag did not perfectly agree but are close. In the circuit we find the capacitor voltage lags behind the source voltage. Both phase angle and lag show the differences between the waves but lag is specific for one wave lagging behind the other while phase angle is the angular difference between the waves. As frequency increases the phase lag decreases and while frequency decreases the phase lag increases.
21
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References
Floyd, T. L., & Buchla, D. M. (2019).
Principles of Electric Circuits
(10th Edition). Pearson
Education (US).
https://bookshelf.vitalsource.com/books/9780134880068
22
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