Problem 1: Using DC Mchine University o North Crolin-Chrlotte Deprtment o Electricl nd Computer Engineering ECGR 4143/5195 Electricl Mchinery Fll 2009 Problem Set 4 Due: Thursdy October 8 Suggested Reding: Fitzgerld nd Kingsley Sections 3.6, 4.1-4.2 A schemtic o seprtely excited DC motor is shown below. Remember tht seprtely excited DC motor hs ield coil (the coil on the sttor) tht is driven with seprte DC source. A schemtic o the mchine is shown below. v BEMF = Gi ω = K ω m This mchine ws tested s genertor beore being put into use s motor. During those tests, we lerned the ollowing cts: The open-circuit voltge t the rmture terminls ws mesured t 200V while the rotor ws turning t 500 rottions per minute The ield current ws set to 1A (The ield current is the current in the ield winding). The rmture resistnce R is 1 Ohm. The ield resistnce R is 1 Ohm All inductnces re negligible. τ Plese nswer the ollowing questions. Until told otherwise, ssume tht the rmture is driven by 50V source nd tht the ield winding is driven by 1A current source. ) During testing s motor, n mmeter indictes tht the rmture current is 1A. Under these conditions, wht is the speed o the motor in RPM? b) Given tht the motor is operting in stedy-stte, wht must be the vlue o the torque pplied to the motor by the mechnicl lod?
c) Now, we double the mechnicl lod torque nd llow the motor to rech new stedy-stte. Wht is the speed o the motor in RPM t this new stedy-stte? d) Given your nswer to prt b, you could hve given pretty ccurte nswer to prt c without doing ny mth. Why? Hint: Think bout the key reltionships between the electricl nd mechnicl terminl vribles. e) The lod torque is now set to 10N-m. Wht is the new stedy-stte current? ) The lod torque is still set to 10N-m nd the ield current is reduced to 0.5A. In stedy-stte, wht is the new speed o the mchine? Problem 2: Torque Production in three-phse synchronous mchine Figure 1 shows cross section o simple, three-phse synchronous mchine. There is single winding on the rotor known s ield winding, nd three identicl windings tht re ech displced by 120 0 (or 2π/3 rdins) long the periphery o the sttor. The windings on the sttor re known s phses. The mchine extends length l. At the bck o the mchine, the windings lbeled A re connected to the windings lbeled A. Any current tht lows into the coil side lbeled A will thus return down the other side o the mchine in the coil sided lbeled A. The sme is lso true or the other two sttor windings nd the ield coil. This is simpliiction o the design used in most genertors. θ R ) Write n expression or the totl lux linkge in ech sttor coil (i.e. λ A, λ B, nd λ C ). Assume tht ech coil hs its own sel inductnce (i.e. L A, L B, nd L C ) tht includes the eects o both lekge nd mgnetiztion. Also include the eects o mutul inductnces nd use the ollowing nottion: L AB is the inductnce between coil A nd coil B, L BF is the inductnce between coil B nd coil F, etc. You do not need to provide explicit expressions or ny o the inductnces. b) I the sttor coils re completely identicl (i.e. they ech hve the sme number o turns), you cn simpliy your expressions rom prt. In prticulr, ech o the sel inductnces is equl to L A nd ech o the phse-to-phse mutul inductnces re equl to L AB (i.e. L AB = L BA = L AC = ). Explin. Use these results to simpliy your expressions rom prt. c) A resonble expression or the mutul inductnce between phse A nd the ield winding is L AF (θ R ) = Mcos(θ R ). Justiy this.
d) Modiy the expression or the mutul inductnce between phse A nd the ield winding to determine n expression or the mutul inductnce between phse B nd the ield winding. Similrly, determine the mutul inductnce between phse C nd the ield winding. e) Write n expression or the totl lux in the ield winding. Assume tht the ield coil hs its own sel inductnce (L F ) nd use the mutul inductnces rom the previous two prts. ) Write n expression or the totl coenergy in the system. g) Write n expression or the torque on the rotor. h) Now, ssume tht we drive the windings with blnced three-phse set o currents: i i b c ( ωt) i cos 2π cos ωt 3 2π cos ωt + 3 i Also, ssume tht the rotor is turning t speed ω nd tht t time t = 0 it is locted t ngle θ o (i.e. θ(t) = ωt + θ o Show tht i + i b + i c = 0. Recll tht i) Substitute the currents rom prt h into your torque expression rom prt g. The result should depend only upon I, I, M, nd θ o. The ollowing trigonometric identities my be helpul: j.) The result rom prt i, though slightly pinul to derive, is one o the most importnt in the history o the modern world. It succinctly summrizes one o the primry resons why we use blnced three-phse sets o currents. Given your result in prt i, speculte s to why blnced threephse set o currents is useul. (Don t drive yoursel crzy over this one It s not worth tht mny points).
Problem 3: Chrcterizing DC mchine (Two prts Lbortory nd home) In the lb, there re two DC mchines tht hve been linked vi their shts. One mchine is intended to operte s genertor nd the other is intended to operte s motor. In this cse, you re going to chrcterize the motor nd the genertor. Section 1: Dt collection in the lb ) First, we re going to determine the motor constnt K. To do so, use the ollowing procedure: Apply voltge to the genertor. Note tht the genertor terminls re the yellow nd green terminls. This will cuse the motor to spin. Obtin tchometer rom the TA so tht you cn mesure the speed. Use the tch s shown in clss. Apply the ollowing voltges to the genertor terminls: 3V, 4V, 5V, 6V, 7V. At ech voltge, record both the sht speed nd the open-circuit voltge t the motor terminls. Note tht the speed dt is in revolutions per minute (RPM). b) Mke sure tht both the motor nd the genertor re open-circuited. Using the multimeter, mesure the resistnce t the motor terminls. Then, mesure the resistnce t genertor terminls. c) Now, we re going to drive the motor nd mesure the current nd speed. Use the ollowing procedure: Connect the red led o the meter mrked Amps to the positive terminl o the power supply. Connect the blck led o the meter mrked Amps to the positive terminl (the red led) o the motor. Connect the negtive terminl o the motor (the blck led) to the negtive terminl o the power supply. Connect the red led o the meter mrked Volts to the positive motor terminl (the red led) Connect the blck led o the meter mrked Volts to the negtive motor terminl (the blck led) Set the meter mrked Amps to Amps mode. Set the meter mrked Volts to Volts mode Obtin the three lod resistors rom the TA Adjust the power supply voltge until the meter mrked Volts reds pproximtely 6V Record the ctul voltge With the genertor open-circuited, record the current nd speed Connect the 9.4 Ohm resistor cross the genertor terminls. Record the current nd speed. Connect the 4.7 Ohm resistor cross the genertor terminls. Record the current nd speed. Connect the 2.35 Ohm resistor cross the genertor terminls. Record the current nd speed. Short circuit the genertor. Record the current nd speed. At this point, you re inished your lbortory work. Go hed nd return your equipment to the TA. Section 2: Dt nlysis d) Once you re home, enter your recorded motor voltge nd speed mesurements rom prt into Excel. Convert your speed dt rom RPM to rd/sec. Crete plot showing speed in rdins per second on the x-xis nd motor voltge on the y-xis. The reltionship is pproximtely liner. Why should you expect this? Think bck to wht we lerned in clss.
e) Use the LINEST unction in Excel to it line to your voltge nd speed dt. You must provide LINEST with set o known x-vlues (in this cse, the speed vlues) nd set o known y-vlues (in this cse, the voltges). LINEST will return the slope o the line tht best its the dt. In terms o the DC mchine model, wht is this vlue? ) In prt b, you mesured the resistnce o the motor nd genertor. This mesurement is not necessrily ccurte becuse it ws mde while the motor ws cold. When the motor is operting, the resistnce will increse. To determine the hot resistnce, use the current nd speed vlues tht you recorded t ech o the vrious genertor loding conditions in prt c. Using your motor constnt K, determine the vlue o the bck-em speed voltge t ech recorded speed. Determine the voltge cross the motor resistnce t ech speed. Use this vlue nd the mesured currents to clculte the resistnce t ech speed. Averge ll o your resistnce redings. Is this resistnce dierent thn the resistnce you mesured in prt b? Why or why not. g) Using your motor constnt rom prt e nd your resistnce clcultion rom prt, provide detiled schemtic or your motor nd genertor. Clerly show the vlues o the resistnces s well s the vlues o the constnts K. Since the motor nd genertor re identicl, ssume tht K is the sme or both mchines. Consider inductnce to be negligible. h) Typiclly, motor mnucturers provide you with dtsheet showing the torque-speed reltionship or prticulr motor. Using your model rom prt, determine the idel ormul or the torquespeed curve. With no mechnicl lod on the mchine, wht should be the speed? This is sometimes clled the no-lod speed. When the mchine is stlled (i.e. the speed is zero), wht is the torque? This is vlue is sometimes clled the stll torque. i) Now exmine your current nd speed dt rom prt c. When the genertor is open-circuited, there is idelly no mechnicl lod on the sht. Idelly, the motor should spin t constnt speed without producing ny torque. This mens tht no current should be drwn rom the power supply. In relity, you should hve mesured some current low when the genertor ws open-circuited. The reson or this current low is tht there is riction torque coming rom the combintion o the comuttors, berings, nd ir low tht cts to slow the motor down. Using Newton s second lw, write n expression or the net torque. This net torque should hve two components, nd their sum should be zero. Why should their sum be zero? Use your expression to clculte the riction torque. As we will see in more detil lter in the semester, riction is typiclly proportionl to speed. Oten, we sy tht τ ric = βω, where β is constnt. Using the current nd speed dt tht you clculted when the genertor ws open-circuited, determine the vlue o β. j) Once the genertor is drwing some current, the motor should pply some torque to the sht. Why? Provide physicl explntion using the irst lw o thermodynmics. The torque pplied to the sht should be relted to the current provided by the genertor. Explin. I the generted current increses, the mchine speed will decrese. Explin. k) Using your results rom prt c, determine the totl torque produced by the motor t ech mesured speed. Plot the torque (in N-m) versus speed (in rd/sec). The plot should be nerly liner. On top o this, plot predicted torque-speed curve using the eqution you ound in prt h. The curves should be very similr. Determine the percent error between the mesured torque nd the predicted torque t ech speed. Give some potentil cuses or the errors.