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r 92 chapter 2 Transformers ciency is a maximum and calculate this efficiency if the power factor is 0.85 and load voltage is 230 V. 2.15 A 1 cp, 10 kVA, 2400/240 V, 60 Hz distribution transformer has the following characteristics: Core loss at full voltage= 100 W Copper loss at half load = 60 W (a) Determine the efficiency of the transformer when it delivers full load at 0.8 power factor lagging. (b) Determine the per-unit rating at which the transformer efficiency is a maximum. Determine this efficiency if the load power factor is 0.9. (c) The transformer has the following load cycle: No load for 6 hours 70% full load for 10 hours at 0.8 PF 90% full load for 8 hours at 0.9 PF Determine the ali-day efficiency of the transformer. 2.16 The transformer of Problem 2.15 is to be used as an autotransformer (a) Show the connection that will result in maximum kVA rating. (b) Determine the voltage ratings of the high-voltage and low-voltage sides. (c) Determine the kVA rating of the autotransformer. Calculate for both high-voltage and low-voltage sides. 2.17 A 1cp, 10 kVA, 460/120 V, 60Hz transformer has an efficiency of 96% when it delivers 9 kW at 0.9 power factor. This transformer is connected as an autotransformer to supply load to a 460 V circuit from a 580 V source. (a) Show the autotransformer connection. (b) Determine the maximum kVA the autotransformer ;::an supply to the 460 V circuit. (c) Determine the efficiency of the autotransformer for full load at 0.9 power factor. 2.18 Reconnect the windings of a 1¢, 3 kVA, 240/120 V, 60Hz transformer so that it can supply a load at 330 V from a 110 V supply. (a) Show the connection. (b) Determine the maximum kV A the reconnected transformer can deliver. 2.19 Three 1¢, 10 kVA, 460/120 V, 60Hz transformers are connected to form a 3¢, 460/208 V transformer bank. The equivalent impedance of each transformer referred to the high-voltage side is 1.0 + j2.0 n. The transformer delivers 20 kW at 0.8 power factor (leading). (a) Draw a schematic diagram showing the transformer connection. (b) Determine the transformer winding current. (c) Determine the primary voltage. (d) Determine the voltage regulation. 2.20 Three 1 cp, 100 kVA, 2300/460 V, 60 Hz transformers are connected to form a 3¢, 2300/460 V transformer bank. The equivalent impedance of each trans- I i I I l l Problems 93 former referred to its low-voltage side is 0.045 + j0.16 0. The transformer is connected to a 3¢ source through 3¢ feeders, the impedance of each feeder being 0.5 + j 1.5 0. The transformer delivers full load at 460 V and 0.85 power factor lagging. (a) Draw a schematic diagram showing the transformer connection. (b) Determine the single-phase equivalent circuit. (c) Determine the sending end voltage of the 3¢ source. (d) Determine the transformer winding currents. 2.21 Two identical250 kVA, 230/460 V transformers are connected in open delta to supply a balanced 3 c/J load at 460 V and a power factor of 0.8lagging. Determine (a) The maximum secondary line current without overloading the trans- formers. (b) The real power delivered by each transformer. (c) The primary line currents. (d) If a similar transformer is now added to complete the ~. find the percent- age increase in real power that can be supplied. Assume that the load voltage and power factor remain unchanged at 460 V and 0.8 lagging, re- spectively. 2.22 Three identical single-phase transformers, each of rating 20 kVA, 2300/230 V, 60Hz, are connected Y-Y to form a 3¢ transformer bank. The high-voltage side is connected to a 3¢, 4000 V, 60 Hz supply, and the secondary is left open. The neutral of the primary is not connected to the neutral of the supply. The voltage between the primary neutral and the supply neutral is measured to be 1200 V. (a) Describe the voltage waveform between primary neutral and supply neu- tral. Neglect harmonics higher than third. (b) Determine the ratio of (i) phase voltages of the two sides and (ii) line voltages of the two sides. (c) Determine the ratio of the rms line-to-line voltage to the rms line-to- neutral voltage on each side. 2.23 A 1 c/J, 200 kVA, 2100/210 V, 60 Hz transformer has the following characteris- tics. The impedance of the high-voltage winding is 0.25 + j 1.5 0 with the low- voltage winding short-circuited. The admittance (i.e., inverse of impedance) of the low-voltage winding is 0.025 - j0.075 mhos with the high-voltage winding open-circuited. (a) Taking the transformer rating as base, determine the base values of power, voltage, current, and impedance for both the high-voltage and low-voltage sides of the transformer. (b) Determine the per-unit value of the equivalent resistance and leakage reactance of the transformer. (c) Determine the per-unit value of the excitation current at rated voltage. (d) Determine the per-unit value of the total power loss in the transformer at full-load output condition. 2.24 A single-phase transformer has an equivalent leakage reactance of 0.04 per unit. The full-load copper loss is 0.015 per unit and the no-load power loss .., r I I I ~ l ... I l I • ··r 94 chapter 2 Transformers at rated voltage is 0.01 pu. The transformer supplies full-load power at rated voltage and 0.85 lagging power factor. (a) Determine the efficiency of the transformer. (h) Determine the voltage regulation. 2.25 A 1¢, 10 kVA, 7500/250 V, 60 Hz transformer has Ze4 = 0.015 + j0.06 pu, Rc = 60 pu, and Xm = 20 pu. (a) Determine the equivalent circuit in ohmic values referred to the low- voltage side. (h) The high-voltage winding is connected to a 7500 V supply, and a load of 5 L2.Q is connected to the low-voltage side. Determine the load voltage and load current. Determine the voltage regulation. 2.26 A 1¢, 10 kVA, 2200/220 V, 60Hz transformer has the following characteristics: No-load core loss = 100 W Full-load copper loss= 215 W Write a computer program to study the variation of efficiency with output kVA load and load power factor. The program should (a) Yield the results in a tabular form showing power factor, per-unit kVA load (i.e., X), and efficiency. (b) Produce a plot of efficiency versus percent kVA load for power factors of 1.0, 0.8, 0.6, 0.4, and 0.2. '+" I chapter three ELECTROMECHANICAL ENERGY CONVERSION Various devices can convert electrical energy to mechanical energy and vice versa. The structures of these devices may be different depending on the functions they perform. Some devices are used for continuous energy con- version, and these are known as motors and generators. Other devices are used to produce translational forces whenever necessary and are known as actuators, such as solenoids, relays, and electromagnets. The various converters may be different structurally, but they all operate on similar principles. This book deals with converters that use a magnetic field as the medium of energy conversion. In this chapter the basic principles of force production in electromagnetic energy conversion systems are discussed. Some general relationships are derived to tie together all conversion devices and to demon- strate that they all operate on the same basic principle. 3.1 ENERGY CONVERSION PROCESS There are various methods for calculating the force or torque developed in an energy conversion device. The method used here is based on the principle of conservation of energy, which states that energy can neither be created nor destroyed; it can only be changed from one form to another. An electro- mechanical converter system has three essential parts: ( 1) an electric system, (2) a mechanical system, and (3) a coupling field as shown in Fig. 3.1. The energy transfer equation is as follows: Electrical = mechanical + increase + energy energy input energy output in stored losses from source energy in coupling field (3.1) The electrical energy loss is the