1. CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit 1
Colorado Technical University
EE 331 – Circuit Analysis II
Lab 4: Simple AC Circuit
December 2009
Loren Karl Schwappach
Student #06B7050651
This lab report was completed as a course requirement to obtain full course credit in EE331 Circuit Analysis II at Colorado
Technical University. This lab report investigates the behavior of a simple AC circuit at various frequencies. Hand calculations are
verified using P-Spice schematic calculations to determine viability of design prior to the physical build. P-Spice diagrams and
calculations are then verified by physically modeling the design on a bread board and taking measurements for observation. The
results were then verified by the course instructor. The results illustrate the band passing behavior of a simple AC circuit due to
various input frequencies.
If you have any questions or concerns in regards to this laboratory assignment, this laboratory report, the process used in designing
the indicated circuitry, or the final conclusions and recommendations derived, please send an email to LSchwappach@yahoo.com. All
computer drawn figures and pictures used in this report are my own and of original and authentic content. I authorize the use of any
and all content included in this report for academic use.
for all resistors. The imaginary number ������ = −1.
I. INTRODUCTION Capacitors and inductors have no real component and
only contain imaginary components and can be represented as
A simple R/C circuit driven by an alternating
current will act as a high or low pass filter complex impedances by using the following formulas where
������ = 2 × ������ ×f.
(dependent upon whether the voltage change is recorded along
the resistor or capacitor element). This is caused by the
capacitor’s ability to filter out (attenuate) higher frequencies Formula for complex impedance of a capacitor:
and pass lower frequencies (reactance). To demonstrate this 1
������������ =
behavior a simple RC circuit is designed with a variable AC ������ × ������ × ������
power source. Phasors are used in circuit analysis to allow for
circuit impendence and are used in this lab to simplify circuit Formula for complex impedance of an inductor:
analysis. ������������ = ������ × ������ × ������
Example of a capacitor as complex impedance (Zc):
II. PHASOR AND IMPEDANCE THEORY 1 100 ������
������ = 10������������ ⟹ ������������ = = 0 − ������( )
������ ×������ ×10������������ ������
Phasors use complex numbers to represent the
magnitude and phase of sinusoidal voltages or currents. In this lab the following illustrated RC elements and
Phasors do not contain any frequency information about AC source were converted into phasors to simplify circuit
sinusoids and allow for the use of complex impedances for analysis.
conducting AC circuit analysis. Through the use of Phasors
and complex impedances capacitor and resistor elements can
be interpreted by their impendence values allowing for simple
circuit analysis techniques.
When using complex impedances for circuit analysis,
impedance values are made of a real component and an
imaginary component. Resistors have no imaginary
component so they are simply represented without an
imaginary component or phase shift.
Example of a resistor as complex impedance (Zr):
������ = 6.8������ ⟹ ������������ = 6800 + 0������ = 6800∠0°
In the example above the 6800 represents the real Figure 1: Simple AC circuit with R=6.8k, C=10nF,
number resistance of the resistor. The 0������ represents the VA=1.5V, and freq = various (500Hz to 8000Hz)
imaginary component of the complex impedance which is zero

2. CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit 2
Because we know this circuit is in series the current
is the same throughout the circuit and by using a Phasor to Circuit Current (Phasor Form) A
represent the sinusoidal input by the relationship ������ = Frequency (Hz)
Rectangular Polar Form
1.5 cos ������������ + 0° → ������ = 1.5∠0° = 1.5 + ������0 and by
converting the resistor and capacitor into complex impedances 500 9.63 + j45.1 uA 46.1 ∟ 77.9º uA
we can discover the voltage at various frequencies along the 1000 34.1 + j79.7 uA 86.7 ∟ 66.8º uA
resistor and capacitor. This is illustrated by the hand
calculated formula using Ohms law below. 1500 64.2 + j100 uA 119 ∟ 57.3º uA
2000 93.1 + j109 uA 143 ∟ 49.5º uA
4000 164 + j96.2 uA 190 ∟ 30.4º uA
8000 203 + j59.4 uA 208 ∟ 12.8º uA
Table 2: Series current in Phasor form at various
frequencies.
With these results obtained Ohms law ������ = ������ × ������ is
utilized to find the respective voltage Phasors along each
element as identified below.
Figure 2: Simple AC circuit using a Phasor for Vs and
complex impedances for the resistor and capacitor.
III. HAND CALCULATIONS FOR VC AND VR
By finding the complex impedance values and
corresponding current Phasor ������ for each frequency (500Hz,
1kHz, 1.5kHz, 2kHz, 4kHz, and 8kHz) we can then use this
knowledge to find the complex voltage Phasor across each
element using Ohms law: ������ = ������ × ������ where ������ is the Figure 3: Hand calculations for voltage levels along the
impedance. Thus the following values were found using hand capacitor at various frequencies.
calculations.
Frequency (Hz) Impedance of Capacitor: Zc (Ohms)
500 0 - 31830j Ω
1000 0 - 12915j Ω
1500 0 - 10610j Ω
2000 0 - 7958j Ω
4000 0 - 3979j Ω
Figure 4: Hand calculations for voltage levels along the
8000 0 - 1989j Ω resistors at various frequencies.
Table 1: Impedance values of 10nF capacitor at various
frequencies.
Now using the series current can be found by:
������������ 1.5+������ 0
������ = = 100000000 so for the various frequencies the
������������ +������������ 6800 −������
������
following Phasor currents were found.

3. CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit 3
Voltage across capacitor (Phasor V. PHYSICAL MEASUREMENTS
Frequency (Hz) form) in Volts The circuit in Figure 1 was then constructed on a
breadboard using a sine wave generator for Vsource with VA =
Rectangular Polar Form
1.5V at various frequencies. Oscilloscope probes were
500 1.44V – j307mV 1.47 ∟ -12º V connected across the second circuit element (first the capacitor
1000 1.27V – j543mV 1.38 ∟ -23.1º V and then the resistor, switching there places each time) and
from +Vs to ground. This had to be done due to Oscilloscope
1500 1.06V – j681mV 1.26 ∟ -32.7º V grounding concerns and was discovered after two hours of
2000 867mV – j741mV 1.14 ∟ -40.5º V troubleshooting the circuit with the aid of instructors. The
oscilloscope was thus able to display the Voltage due to the
4000 383mV – j653mV 757 ∟ -59.6º mV
source in comparison to the voltage of the second element
8000 118mV – j404mV 421 ∟ -73.7º mV (either resistor or capacitor). This allowed measurements of
Table 3: Voltage in Phasor form across capacitor at the phase angle and voltage across the elements in comparison
various frequencies. to the source.
Voltage across resistor (Phasor form)
in Volts VI. COMPONENTS USED / REQUIRED
Frequency (Hz)
The following is a list of components that were used.
Rectangular Polar Form
 A digital multimeter for measuring circuit voltage,
500 65.5mV – j307mV 314 ∟ 78º mV circuit current, resistance, and capacitance.
1000 232mV – j542mV 590 ∟ 66.8º mV  A oscilloscope for viewing the input and output
waveforms of a simple RC circuit with a 1kHz
1500 437mV – j680mV 808 ∟ 57.3º mV square wave input.
2000 633mV – j741mV 975 ∟ 49.5º mV  A signal generator capable of delivering 1.5V
amplitude sine waves at various frequencies.
4000 1.12V – j654mV 1.3 ∟ 30.3º V
 6.8kΩ resistors
8000 1.38V – j404mV 1.44 ∟ 16.3º V  10nF capacitors.
Table 4: Voltage in Phasor form across resistor at various  Bread board with wires.
frequencies.
VII. RESULTS
These results were then confirmed by noting that the
The circuit in Figure 1 was then constructed on a
total Voltage across the resistor and capacitor equals the
breadboard using a sine wave generator for Vsource with VA =
voltage provided by the source (1.5V) per KVL. Also of note
1.5V at various frequencies. Oscilloscope probes were
is the resistor and capacitor are 90 degrees out of phase with
connected across the second circuit element
each other at each respective frequency.
The following table illustrates the measurements.
IV. P-SPICE SIMULATION
The hand calculated results were then compared against a
Hz Measured Capacitor Measured Resistor
circuit with the same RC values built using P-Spice. Several
P-Spice simulations had to be run using a AC voltage source 500 1.45 ∟ -12º V 300 ∟ 80º mV
set at specific frequencies. Circuit probes were attached 1000 1.35 ∟-23º V 600 ∟ 65º mV
across the source, resistor and capacitor and a simulation was
ran to find the voltage and phase offsets at each frequency. 1500 1.2 ∟ -30º mV 800 ∟60º mV
The following table summarizes the P-Spice simulation 2000 1.1 ∟-41º mV 950 ∟ 50º mV
results.
4000 0.7∟ -60º mV 1.3 ∟ 30º V
Frequency Capacitor voltage Resistor voltage
Hz Phasor Phasor 8000 0.35 ∟ -74º mV 1.4 ∟ 17º V
500 1.47 ∟ -12º V 313 ∟ 78º mV Table 6: Measured voltages in Phasor form across resistor
and capacitor at various frequencies.
1000 1.38 ∟-23º V 589 ∟ 67º mV
1500 1.22 ∟ -31º mV 809 ∟57º mV
2000 1.07 ∟-41º mV 974 ∟ 49º mV
4000 0.742 ∟ -60º mV 1.26 ∟ 30º V
8000 0.411 ∟ -74º mV 1.43 ∟ 17º V
Table 5: P-Spice element voltage results as Phasors.

4. CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit 4
The next figure is a graph of the actual magnitudes
measured of the source, resistor, and capacitor as a function of
frequency.
Figure 7: Source, resistor, and capacitor on complex plane
at 1000 Hz
Figure 5: Graph of source and element voltages as a
function of frequency.
As illustrated in Figure 5 when measuring the output
voltage across the capacitor the RC circuit acts as a low pass
filter attenuating higher frequencies but when measuring the
voltage across the resistor the RC circuit acts as a band pass
filter attenuating lower frequencies.
The element phasors can be illustrated using vector
representation on a graph of the complex plane (Polar graph).
The following figures show the Voltage across each element
and the corresponding phase shit induced. Notice that the
resistor and capacitor are 90 degrees out of phase each time.
Figure 8: Source, resistor, and capacitor on complex plane
at 4000 Hz
VIII. ANALYSIS
It can be observed from Figures 6, 7 and 8 that while
the Vr increases closer to Vs at higher frequencies Vs
increases closer to Vs at lower frequencies. It can also be
stated that at higher frequencies the phase offset at Vc
increases and the phase offset of Vr decreases, while at lower
frequencies the opposite is true. Also note the 90 degree
phase shift difference among the resistor and capacitor
elements.
Next a comparison of the measured voltages against
the predicted (P-Spice calculated) measurements was
completed to evaluate human errors induced as well as the
Figure 6: Source, resistor, and capacitor on complex plane large error induced by resistor / capacitor variances. These
at 500 Hz results follow on the following table.
Note:
Percentage error = ((expected - measured) / expected) * 100

5. CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit 5
Voltage Amplitudes Phase Angles
Resistor Capacitor Resistor Capacitor
-4.4% -1.36 2.5% 16.66%
1.69% 2.17% -1.6% 0.45%
-0.99% -4.7% 4.71% 8.25
2.5% -3.5% 1.01% 2.25%
0% -7.5% 0.99% 0.671%
0% -4.98% -8.15 0.407%
Table 7: Percentage error results
IX. CONCLUSION
This lab was a success and was effective in
demonstrating the behavior of simple AC circuits at various
frequencies. It also was beneficial in demonstrating the power
and beauty of using Phasors and complex numbers to simplify
circuit analysis. The ability for RC circuits to act as band pass
filters is a powerful feature for engineers and will be
invaluable in work to come. Selective band pass filters are
critical in communication and digital systems and provides
most of the technology we have today. Using Phasor analysis
techniques instead of dealing with differential equations is a
tremendous relief.
Finally, the unexpected finding that only the second
circuit element could be measured at the same time as Vs due
to grounding issues with the oscilloscope was frustrating but
will be invaluable in future laboratory work.
REFERENCES
[1] R. E. Thomas, A. J. Rosa, and G. J. Toussaint, “The Analysis & Design
of Linear Circuits, sixth edition” John Wiley & Sons, Inc. Hoboken, NJ,
pp. 309, 2009.