University of North Carolina-Charlotte Department of Electrical and Computer Engineering ECGR 4143/5195 Electrical Machinery Fall 2009

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1 University of North Carolina-Charlotte Deartment of Electrical and Comuter Engineering ECG 4143/5195 Electrical Machinery Fall 9 Problem Set 5 Part Due: Friday October 3 Problem 3: Modeling the exerimental synchronous machine In the lab you will exeriment with the 6-ole machine described in the memo osted on the web site. Unlike the synchronous machines that we ve discussed in class, this machine does not have a field coil on the rotor. Instead, it has ermanent magnets that roduce DC magnetic fields in just the same way. The figure below on the left shows a schematic of how one hase of our exerimental machine is wound. It also shows the directions of the field roduced by a DC current flowing in this stator coil. The figure on the right shows how the fields roduced by the magnets would look from the to down. Just as stated in the memo, note that the rotor has two magnets with their north ole facing out, followed by two with their south ole facing out. This attern then reeats. With 6 such sets of magnets, the result is a 6-ole machine. Flux into/out of core a Assuming that the sace between individual magnets on the rotor is very small, sketch how the magnetic field density B varies as you make one comlete satial revolution around the air ga (i.e. as you rogress from θ = to θ = 36 degrees. Assume that each magnet roduces DC field strength B o or B o in the air ga. Make sure to label the x-axis in terms of mechanical degrees. b e-draw the grah from art a. Label the x-axis in electrical degrees. c If the rotor is turning, the magnetic field that you sketched in art a will rotate around the air ga. Clearly, this will roduce a voltage in each of the stator coils. Since our windings and our magnets are not sinusoidally distributed, the induced voltage should contain a fundamental sinusoid and a number of harmonics. If we examine hase A, we can aroximate the flux linkage from the magnets using only a fundamental comonent of the following form: λ a, rot = Λ Here, Λ is a constant, is the number of oles in the machine and θ is the mechanical angle of the rotor. Using the figure on the right above, briefly justify this. To aroach this roblem, you

2 may first want to consider how much flux should be assing through each coil as the figure is currently drawn. Then, consider what would haen if both rotor disks were to move to the right. d Stator coil has a self inductance L a and hase-to-hase mutual inductances L ab and L ac. Provide an exression for the total flux linkage λ a in coil A. Make sure to include the exression for the flux coming from the magnets. e Now, assume that the rotor is moving so that its osition is θ = Ωt + θ. Modify your exression from art d to reflect this. f Consider each coil to be concentrated, meaning that all of the turns of one coil are located in one sot rather than sread around some ortion of the stator. Note that the coils are located like so: What is the hysical angle between any one of the A coils and the nearest A coil (Note that A coils would contain a current flowing in the oosite direction? What is the electrical angle between then? What is the hysical angle between any one of the A coils and the nearest B coil? What is the electrical angle between them? What is the hysical angle between any one of the A coils and the nearest C coil? What is the electrical angle between them? g Using your results from art f, modify the exression λ a, rot = Λ to determine λ b, rot and λ c, rot. These are the flux linkages in coils b and c as a result of the flux from the magnets. h Use your exression from art e to determine the voltage in hase a. i Assuming that the machine is roducing a balanced three-hase set of currents and that L ab = L ac, simlify your exression from art h. j Modify your exression from art i to include winding resistance. k Use your exression from art j to rovide an equivalent circuit for one hase of the machine. l Your equivalent circuit should contain an AC voltage source. What is the eak value of this voltage source in terms of the variables given so far? By comarison with DC machines, this voltage source is often called a back-emf source and it written like so: KΩ, where K is a constant. Often, K is secified in data sheets for these machines. Physically, what determines the value of K? Why is this voltage source often called a seed voltage?

3 Problem 4: Lab Problem: Exerimenting validating the model In the lab you will exeriment with the 6-ole machine described above. The only things that need to be submitted from this roblem are the requested screenshots and/or sketches. a Effect of winding distributions: Go to the machine in the lab marked Voltage Exeriments. Aly 5V DC to the DC motor connected to the shaft of the synchronous machine. The machine will sin. Please do the following: a. Connect a scoe robe to the wire marked A. Connect the ground cli to the wire marked N. b. Connect a second scoe robe to the wire marked B. c. Measure the two voltages on the oscilloscoe and either make a careful sketch or take a d. ecord the hase shift between the voltages. e. ecord the frequency of the fundamental comonent of one of the voltages. f. ecord the frequency of the dominant harmonic comonent of one of the voltages. g. Move the second scoe robe to the wire marked C. h. Measure the two voltages on the oscilloscoe and either make a careful sketch or take a i. Disconnect both robes and connect one scoe robe to the wire marked A. Now connect the ground cli the wire marked B. j. Measure the voltage on the oscilloscoe and either make a careful sketch or take a b Effect of ole #: Go to the machine in the lab marked Voltage Exeriments. Aly a 5V DC voltage to the DC motor connected to the shaft of the synchronous machine. The machine will sin. Please do the following: a. Connect one scoe robe to the wire marked A. Now connect the ground cli to the wire marked B. b. Connect the other scoe robe to the outut of the tachometer. Connect the ground lead of this scoe robe to the ground in on the seed sensor board. You should see a square wave. ecord the frequency of the square wave. This frequency is related to the seed of the machine. c. Sto the machine and count the number of black and white ulses on the encoder disk. ecord this. d. Make sure that you have recorded the frequency of the sine wave measured between A and B on the motor. c Internal Voltage: As you know, there is a voltage generated in each hase of the machine as the rotor sins. Please do the following: a. Connect one scoe robe to the wire marked A. Now connect the ground cli to the wire marked B. b. Connect the other scoe robe to the outut of the tachometer. Connect the ground lead of this scoe robe to the ground in on the seed sensor board. You should see a square wave when the machine is moving. c. Aly 3V DC to the DC motor connected to the shaft of the synchronous machine. ecord the frequency of the square wave from the tachometer and record the amlitude of the sine wave measured between hase A and hase B. d. Increase the DC voltage alied to the motor in 1V stes from 3V u to 9V. At each voltage, record the frequency of the square wave from the tachometer and record the amlitude of the sine wave measured between hase A and hase B. d Torque: Now move to the machine labeled Torque. This machine has an arm on it with several holes. Use the following rocedure to measure the torque-angle relationshi:

4 a. Obtain the sring scale and rotractor from the TA. b. If necessary, adjust the sring scale so that it is reading N-m. c. Connect the hook on the sring scale through the to hole of the arm. d. ecord where the ower suly is connected to the machine. e. Turn on the ower suly. This will ass aroximately A DC into Phase A and out of Phase B. The arm should move to a vertical osition. f. Have one teammate hold the rotractor u to the arm. The other team mate should ull the sring scale until the arm is dislaced by about 5 degrees from vertical. ecord the force. g. TUN OFF THE POWE SUPPLY. DO NOT LEAVE THE SUPPLY ON FO MOE THAN SECONDS! h. Turn on the ower suly again. This will ass aroximately A DC into Phase A and out of Phase B. The arm should move to a vertical osition. i. Now dislace the arm by 1 degrees and record the force. Turn-off the ower suly. j. eeat stes h and i with the arm dislaced by 15 degrees, degrees, and 3 degrees. k. Disconnect the sring scale and turn the ower suly on again. Try to move the rotor (DO NOT PULL ON THE AM ast the 3 degree mark. What haens if you move the rotor far enough? Problem 5: Data Analysis Now, you are going to analyze the data that you recorded in the lab. a Effect of winding distributions: a. In the lab, you measured the line-to-neutral voltages v a, v b, and v c. What was the hase shift between each of these? Briefly exlain why this is so. b. Each of the hase voltages that you measured were not urely sinusoidal. Which harmonic was dominant? Why would harmonics aear in the measured voltages? c. When you measured the line-to-line voltage v ab it should have had retty close to a ure sine wave. Prove that this should haen. To do so, use the following aroach. Consider a more comlete model for the flux linking hase A from the rotor magnets: λa, rot = Λ + Λ1 cos(3 θ Note that this exression contains a third harmonic term. First, rovide an exression for the voltage induced in coil A if the rotor is sinning so that its osition is θ = Ωt + θ. Next, rovide a similar exression for the flux linking coil b. Next, rovide a similar exression for the voltage induced in coil b. Comute v ab by taking the difference between v a and v b. The third harmonic term should disaear. Why? d. (Grad students Your result from c has imortant imlications for ower systems. Consider a Y-connected load that draws the following currents from the system: = I cos( ω t + I cos(3ωt π π = I cos( ωt + I cos(3 ωt 3 3 π π = I cos( ωt + + I cos(3 ωt + 3 3

5 What is the neutral current + i b + i c? How is this result different than what you have exected for a balanced set of currents? b Effect of Pole #: a. At what seed was the motor sinning when you were in the laboratory? To determine this, consider the measured frequency of the square wave from the tachometer and the recorded number of black and white sots on the encoder. You ll have to determine how many times the square wave should ulse er revolution of the machine. b. Comare the frequency of the measured line-to-line voltage with the measured seed. By what factor are they related? Why? c Internal Voltage: a. Create a table showing the amlitude of the measured line-to-line voltage and the amlitude of the measured seed for each of the conditions you considered in the lab. Create a lot showing the amlitude of the measured voltage on the y-axis and the seed (in radians er second on the X-axis. The lot should be linear. Why? Think back to roblem 3. b. Using the LINEST function in Excel, determine the sloe of the line relating the seed of the machine to the amlitude of the measured line-to-line voltage. How can this value be related to the eak value of the flux linking coil A? Think back to your answer from art l in roblem 3. d Torque: a. Create a table showing the measured force versus angle. Create a lot showing how the torque varies with angle. What is the eak value? Call this T o. b. Plot T o sin(3θ between degrees and 3 degrees. Place this lot on to of your lot from a. How do the two lots comare? c. Even though you are lotting force versus angle, this force could be easily converted into a torque. How? d. Let s try to justify the torque exression that you just measured. To do so, use the following stes: i. Assuming that a DC current I is flowing into the coil on the machine, rovide an exression for the coenergy. Answer this by considering coil A and coil B to be one series connected coil with a total inductance L. The total flux linking this coil should be λ = L I + Λ a a Note that there is only one coil for which you have to comute an integral. ii. Use your co-energy exression to determine the torque. Comare this exression to your measured result, and comare this result to what we know for the more traditional three-hase synchronous machine with a balanced three-hase set of currents, i.e. 3 τ = I ami F sin( θ