Loren k. schwappach ee331 - lab 4

CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit

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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 [email protected]. 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.

I. INTRODUCTION

simple R/C circuit driven by an alternating

current will act as a high or low pass filter

(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

and pass lower frequencies (reactance). To demonstrate this

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

analysis.

II. PHASOR AND IMPEDANCE THEORY

Phasors use complex numbers to represent the

magnitude and phase of sinusoidal voltages or currents.

Phasors do not contain any frequency information about

sinusoids and allow for the use of complex impedances for

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):

𝑅 = 6.8𝑘 ⟹ 𝑍𝑟 = 6800 + 0𝑗 = 6800∠0°

In the example above the 6800 represents the real

number resistance of the resistor. The 0𝑗 represents the

imaginary component of the complex impedance which is zero

for all resistors. The imaginary number 𝑗 = −1.

Capacitors and inductors have no real component and

only contain imaginary components and can be represented as

complex impedances by using the following formulas where

𝜔 = 2 × 𝜋 ×f.

Formula for complex impedance of a capacitor:

𝑍𝑐 = 1

𝑗 × 𝜔 × 𝐶

Formula for complex impedance of an inductor:

𝑍𝑙 = 𝑗 × 𝜔 × 𝐿

Example of a capacitor as complex impedance (Zc):

𝐶 = 10𝑛𝐹 ⟹ 𝑍𝑐 =1

𝑗×𝜔×10𝑛𝐹= 0 − 𝑗(

100𝑀

𝜔)

In this lab the following illustrated RC elements and

AC source were converted into phasors to simplify circuit

analysis.

Figure 1: Simple AC circuit with R=6.8k, C=10nF,

VA=1.5V, and freq = various (500Hz to 8000Hz)

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Because we know this circuit is in series the current

is the same throughout the circuit and by using a Phasor to

represent the sinusoidal input by the relationship 𝑉 =

1.5 cos 𝜔𝑡 + 0° → 𝑉 = 1.5∠0° = 1.5 + 𝑗0 and by

converting the resistor and capacitor into complex impedances

we can discover the voltage at various frequencies along the

resistor and capacitor. This is illustrated by the hand

calculated formula using Ohms law 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 𝐼 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: 𝑉 = 𝐼 × 𝑍 where 𝑍 is the

impedance. Thus the following values were found using hand

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

8000 0 - 1989j Ω

Table 1: Impedance values of 10nF capacitor at various

frequencies.

Now using the series current can be found by:

𝐼 =𝑉𝑠

𝑍𝑟+𝑍𝑐=

1.5+𝑗0

6800−𝑗 100000000

𝜔

so for the various frequencies the

following Phasor currents were found.

Frequency (Hz) Circuit Current (Phasor Form) A

Rectangular Polar Form

500 9.63 + j45.1 uA 46.1 ∟ 77.9º uA

1000 34.1 + j79.7 uA 86.7 ∟ 66.8º uA

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 𝑉 = 𝐼 × 𝑍 is

utilized to find the respective voltage Phasors along each

element as identified below.

Figure 3: Hand calculations for voltage levels along the

capacitor at various frequencies.

Figure 4: Hand calculations for voltage levels along the

resistors at various frequencies.


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Frequency (Hz)

Voltage across capacitor (Phasor

form) in Volts


500 1.44V – j307mV 1.47 ∟ -12º V

1000 1.27V – j543mV 1.38 ∟ -23.1º V

1500 1.06V – j681mV 1.26 ∟ -32.7º V

2000 867mV – j741mV 1.14 ∟ -40.5º V

4000 383mV – j653mV 757 ∟ -59.6º mV

8000 118mV – j404mV 421 ∟ -73.7º mV

Table 3: Voltage in Phasor form across capacitor at

various frequencies.

Frequency (Hz)

Voltage across resistor (Phasor form)

in Volts


500 65.5mV – j307mV 314 ∟ 78º mV

1000 232mV – j542mV 590 ∟ 66.8º mV

1500 437mV – j680mV 808 ∟ 57.3º mV

2000 633mV – j741mV 975 ∟ 49.5º mV

4000 1.12V – j654mV 1.3 ∟ 30.3º V

8000 1.38V – j404mV 1.44 ∟ 16.3º V

Table 4: Voltage in Phasor form across resistor at various

frequencies.

These results were then confirmed by noting that the

total Voltage across the resistor and capacitor equals the

voltage provided by the source (1.5V) per KVL. Also of note

is the resistor and capacitor are 90 degrees out of phase with

each other at each respective frequency.

IV. P-SPICE SIMULATION

The hand calculated results were then compared against a

circuit with the same RC values built using P-Spice. Several

P-Spice simulations had to be run using a AC voltage source

set at specific frequencies. Circuit probes were attached

across the source, resistor and capacitor and a simulation was

ran to find the voltage and phase offsets at each frequency.

The following table summarizes the P-Spice simulation

results.

Frequency

Hz

Capacitor voltage

Phasor

Resistor voltage

Phasor

500 1.47 ∟ -12º V 313 ∟ 78º mV

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.

V. PHYSICAL MEASUREMENTS

The circuit in Figure 1 was then constructed on a

breadboard using a sine wave generator for Vsource with VA =

1.5V at various frequencies. Oscilloscope probes were

connected across the second circuit element (first the capacitor

and then the resistor, switching there places each time) and

from +Vs to ground. This had to be done due to Oscilloscope

grounding concerns and was discovered after two hours of

troubleshooting the circuit with the aid of instructors. The

oscilloscope was thus able to display the Voltage due to the

source in comparison to the voltage of the second element

(either resistor or capacitor). This allowed measurements of

the phase angle and voltage across the elements in comparison

to the source.

VI. COMPONENTS USED / REQUIRED

The following is a list of components that were used.

A digital multimeter for measuring circuit voltage,

circuit current, resistance, and capacitance.

A oscilloscope for viewing the input and output

waveforms of a simple RC circuit with a 1kHz

square wave input.

A signal generator capable of delivering 1.5V

amplitude sine waves at various frequencies.

6.8kΩ resistors

10nF capacitors.

Bread board with wires.

VII. RESULTS

The circuit in Figure 1 was then constructed on a

breadboard using a sine wave generator for Vsource with VA =

1.5V at various frequencies. Oscilloscope probes were

connected across the second circuit element

The following table illustrates the measurements.

Hz Measured Capacitor Measured Resistor

500 1.45 ∟ -12º V 300 ∟ 80º mV

1000 1.35 ∟-23º V 600 ∟ 65º mV

1500 1.2 ∟ -30º mV 800 ∟60º mV

2000 1.1 ∟-41º mV 950 ∟ 50º mV

4000 0.7∟ -60º mV 1.3 ∟ 30º V

8000 0.35 ∟ -74º mV 1.4 ∟ 17º V

Table 6: Measured voltages in Phasor form across resistor

and capacitor at various frequencies.


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The next figure is a graph of the actual magnitudes

measured of the source, resistor, and capacitor as a function of

frequency.

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 6: Source, resistor, and capacitor on complex plane

at 500 Hz


at 1000 Hz


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

large error induced by resistor / capacitor variances. These

results follow on the following table.

Note:

Percentage error = ((expected - measured) / expected) * 100


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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.

Technology

Loren k. schwappach ee331 - lab 4