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Department of Electrical Engineering and Computer Science EEL6208 Advanced Electric Machinery Homework Set No. 1 Problem 1. (1) Please study “Energy Sources” from http://www.doe.gov/science-innovation/energy-sources and write a summary on your study. (2) Please study “Energy Efficiency” from http://www.doe.gov/science-innovation/energy-efficiency and write a summary on your study. Problem 2. A ferromagnetic core with relative permeability of 1500 is shown in the following figure. The dimensions are shown in the diagram. Consider airgap fringing using the permeance formula in my course notes. The coil has 300 turns, and the current in the coil is 1 A. (1) Find the flux values for the left, righ and center legs of the core. (2) Find the flux density in each airgap. (3) Find the inductance of the coil. Problem 3. The following figure shows the magnetic circuit for a permanent-magnet loudspeaker. The voice coil (not shown) is in the form of a circular cylindrical coil which fits in the air gap. A SmCo magnet is used to create the air-gap DC magnetic field which interacts with the voice coil currents to produce the motion of the voice coil. The designer has determined that the air gap must have radius R = 1.8 cm, length g = 0.1 cm, and height h = 0.9 cm. Assuming that the yoke and pole piece are of infinite magnetic permeability ( ), find the magnet height hm and the magnet radius R m that will result in an air-gap magnetic flux density of 1.2 T and require the smallest magnet volume. (Hint: Refer to the table of Hard materials in my class notes. For SmCo-28, find the point for maximum energy product at first.) Please review undergraduate materials on three phase circuits (EEL4205 notes and videos on my website) before doing the following Problem 5. Problem 4. The following figure shows a three phase power system with two loads. The connected generator is producing a line voltage of 480 V, and the line impedance is 0.09+j0.16 . Load 1 is Y connected with a phase impedance of 2.536.87 o . Load 2 is connected with a phase impedance of 5 20o . (1) What is the line to line voltage of two loads? (2) What is the voltage drop on the transmission line? (3) Find the real and reactive powers supplied to each load. (4) Find the real and reactive power loss in the transmission line. (5) Find the real power, reactive power and power factor supplied by the generator. Problem 5. For the linear DC machine shown in the following figure, the parameters are: B 0.1T, l 10m, VB 120V, R 0.3, M 1kg, Fapply 40 N, Fload 20 N Write a Simulink code and plot: (1) velocity versus time from 0 to 2s; (2) current versus time from 0 to 2s. Problem 6. For a 6 pole, 36 slot, 3 phase, 5/6 fractional pitch machine, calculate k ph , k dh ksh and the winding factor k wh for the first 30 odd harmonics ( h 1,3,...59 ) using MatLab. Assuming the stator has 1 slot skew. Please list your results in a table. Also calculate the total harmonic distortion (THD) defined as: THD F h 3 2 ah Fa1 Problem 7. Please work out detailed double layer lap winding schemes for the following two three phase machines. Please plot detailed winding diagrams: (1) 12 slots, 8 poles (2) 18 slots, 4 poles Problem 8. For a 3 phase, 4 pole, 48 slot synchronous generator with 5/6 fractional pitch double layer lap winding on stator, the prime mover supplies 15 N.m torque to its load at steady state speed of 6000 rpm. The stator has 1 slot skew. The stator inner diameter D 30 mm and the length of the machine l 30 mm . The airgap g 1.05 mm . The Carter’s coefficient for the stator and rotor are is 1.03 and 1.07, respectively. There are 2 turns per coil on stator. The number of parallel circuits C=2 on stator. The three phase stator windings are connected in Y. The rotor has 36 slots with concentric winding. Assume N1 3 , N 2 4 , N 3 2 , N 4 2 , and N 5 1 . The rotor slot pitch r 8 . The rotor coils are all connected in series. When the stator windings are open circuit, the peak magnetic field intensity in the airgap is 0.7 T. Find: (1) the electrical frequency f e at steady state; (2) the mechanical power supplied to the machine; (3) k p , k d , ks and the winding factor k w for the fundamental harmonic of stator winding; (4) the effective number of series turns Nˆ a on stator; (5) the stator self inductance LA and mutual inductance M s ; (6) the winding factor k w for the fundamental harmonic of rotor winding; (7) the effective number of series turns Nˆ f on rotor; (8) the rotor self inductance Lmf and the mutual inductance Laf between stator and rotor windings. (9) the open circuit rms phase voltage and terminal voltage of the machine; (10) the rotor field current i f . Problem 12. Design a 3 phase round rotor turboalternator to satisfy the following specs: 150 MW (real power), Y connected, 12.6 kV (terminal voltage), 60 Hz 4 pole , 0.85 pf lagging Maximum allowable rotor peripheral speed 250 m/s for 20% over speed run. Directly cooled stator (water) Directly cooled rotor (hydrogen) Vfmax = 500 V 60 slots, 14/15 pitch, skew 1 slot Number of parallel circuits C = 12 Working temperature 25oC (1) List your geometry details. (2) Calculate the self and mutual inductances based on the formulas derived in class; (3) Find the synchronous reactance X s at steady state; (4) Estimate the armature winding resistance Rs and field winding resistance R f ; (5) Calculate and plot the OCC curve (only the air-gap line) Problem 13. A 12.6 kV (line to line or terminal voltage), 150 MW (real power), 60 Hz, Y connected, 0.85 power factor lagging, four pole synchronous generator has the OCC curve (ET vs If) and SCC curve (IL,sc vs If) shown in the figure. Armature and field resistances is are listed in the figure. (1) Find the rated current. (2) Find the mechanical speed of the generator. (3) Input the OCC and SCC curves into MatLab and then use them to calculate the approximate synchronous reactance X s . Plot X s versus I f . What’s the unsaturated synchronous reactance X s ,u ? (4) Find short circuit current ratio (SCR). Use the X s ,u for X s in the following calculations: (5) If the generator is now operating at full load, how much field current and field voltage is required to keep the terminal voltage VT equal to 12.6 kV? (6) Suppose that the generator is connected to a 50MW load at 0.8 PF leading, how much field current and field voltage would be required to keep VT at 12.6 kV? (7) How much field current is needed to keep open circuit terminal voltage at 12.6 kV? Keep the field current and operate the machine at 150 MW, 0.85 PF lagging load. What’s the terminal voltage? Calculate voltage regulation. Problem 14. Design a 5 kW, 24 slot, 6 pole, 3 phase air cooled surface permanent magnet motor. The motor needs to use 30 V (rms) three phase AC power source to drive. The steady state speed is 2000 rpm. Illustrate your design procedure together with all the geometry. You need to pick up your own PM material, soft material, etc. After the design is done, estimate the self and mutual inductances of stator windings, moment of inertia of shaft, winding resistance and magnitude of flux linkage from rotor, etc.