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 ColoradoTechnical University. This lab report investigates the behavior of a simple AC circuit at various frequencies. Hand calculations areverified using P-Spice schematic calculations to determine viability of design prior to the physical build. P-Spice diagrams andcalculations are then verified by physically modeling the design on a bread board and taking measurements for observation. Theresults were then verified by the course instructor. The results illustrate the band passing behavior of a simple AC circuit due tovarious input frequencies. If you have any questions or concerns in regards to this laboratory assignment, this laboratory report, the process used in designingthe indicated circuitry, or the final conclusions and recommendations derived, please send an email to LSchwappach@yahoo.com. Allcomputer drawn figures and pictures used in this report are my own and of original and authentic content. I authorize the use of anyand 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 alongthe resistor or capacitor element). This is caused by thecapacitor’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 forcircuit 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 themagnitude and phase of sinusoidal voltages or currents. In this lab the following illustrated RC elements andPhasors do not contain any frequency information about AC source were converted into phasors to simplify circuitsinusoids and allow for the use of complex impedances for analysis.conducting AC circuit analysis. Through the use of Phasorsand complex impedances capacitor and resistor elements canbe interpreted by their impendence values allowing for simplecircuit analysis techniques. When using complex impedances for circuit analysis,impedance values are made of a real component and animaginary component. Resistors have no imaginarycomponent so they are simply represented without animaginary 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
CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit 2 Because we know this circuit is in series the currentis the same throughout the circuit and by using a Phasor to Circuit Current (Phasor Form) Arepresent the sinusoidal input by the relationship ������ = Frequency (Hz) Rectangular Polar Form1.5 cos ������������ + 0° → ������ = 1.5∠0° = 1.5 + ������0 and byconverting the resistor and capacitor into complex impedances 500 9.63 + j45.1 uA 46.1 ∟ 77.9º uAwe can discover the voltage at various frequencies along the 1000 34.1 + j79.7 uA 86.7 ∟ 66.8º uAresistor and capacitor. This is illustrated by the handcalculated 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 andcorresponding current Phasor ������ for each frequency (500Hz,1kHz, 1.5kHz, 2kHz, 4kHz, and 8kHz) we can then use thisknowledge to find the complex voltage Phasor across eachelement using Ohms law: ������ = ������ × ������ where ������ is the Figure 3: Hand calculations for voltage levels along theimpedance. 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.
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 atotal 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 wereis the resistor and capacitor are 90 degrees out of phase with connected across the second circuit elementeach 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 Resistorcircuit with the same RC values built using P-Spice. SeveralP-Spice simulations had to be run using a AC voltage source 500 1.45 ∟ -12º V 300 ∟ 80º mVset at specific frequencies. Circuit probes were attached 1000 1.35 ∟-23º V 600 ∟ 65º mVacross the source, resistor and capacitor and a simulation wasran to find the voltage and phase offsets at each frequency. 1500 1.2 ∟ -30º mV 800 ∟60º mVThe following table summarizes the P-Spice simulation 2000 1.1 ∟-41º mV 950 ∟ 50º mVresults. 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.
CTU: EE 331 - Circuit Analysis II: Lab 4: Simple AC Circuit 4 The next figure is a graph of the actual magnitudesmeasured of the source, resistor, and capacitor as a function offrequency. 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 outputvoltage across the capacitor the RC circuit acts as a low passfilter attenuating higher frequencies but when measuring thevoltage across the resistor the RC circuit acts as a band passfilter attenuating lower frequencies. The element phasors can be illustrated using vectorrepresentation on a graph of the complex plane (Polar graph).The following figures show the Voltage across each elementand the corresponding phase shit induced. Notice that theresistor 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 theFigure 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
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 indemonstrating the behavior of simple AC circuits at variousfrequencies. It also was beneficial in demonstrating the powerand beauty of using Phasors and complex numbers to simplifycircuit analysis. The ability for RC circuits to act as band passfilters is a powerful feature for engineers and will beinvaluable in work to come. Selective band pass filters arecritical in communication and digital systems and providesmost of the technology we have today. Using Phasor analysistechniques instead of dealing with differential equations is atremendous relief. Finally, the unexpected finding that only the secondcircuit element could be measured at the same time as Vs dueto grounding issues with the oscilloscope was frustrating butwill be invaluable in future laboratory work. REFERENCES 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.