Circuits II EE221 Unit 12 Instructor: Kevin D. DonohuePolyphase CircuitsBalanced 3-Phase VoltagesSingle and 3-Phase Circuit ComparisonBalanced 3-Phase Voltage ConnectionsBalanced LoadsLoad-Source ConnectionsBalanced Y-Y ConnectionSlide 9Balanced Y- ConnectionBalanced - ConnectionBalanced -Y ConnectionPower in Balanced SystemCircuits IIEE221Unit 12Instructor: Kevin D. DonohueThree Phase Circuits, Balanced Y-Y, Y-, and - Three-Phase CircuitsPolyphase Circuits Polyphase circuits contain multiple sources at the same frequency but different phases. Power is distributed over the power grid in the form of three-phase sinusoids.Advantages of Three-Phase power distribution include:(Constant Power) Instantaneous power can be constant in a three phase system.(More Economical) For equivalent power, the 3-Phase systems are more economical than single-phase (can be driven with lower currents and voltages, and fewer wires required because of a common neutral connection between the phases).(Flexible) Single phase service can be extracted from the 3-phase systems or phases manipulated to create additional phases.Balanced 3-Phase VoltagesBalanced phase voltage are equal in magnitude and separate by 120 degrees in phase.Voltages generated from a 3-phase generator can have 2 phase sequence possibilities depending on direction of the rotor:Positive sequence (Counter Clockwise Rotation):Negative sequence (Clockwise Rotation):Show that the sum of all phase voltages in a balanced system is zero. 120240ˆ120ˆ0ˆppcnpbnpanVVVVVVV120ˆ120240ˆ0ˆpcnppbnpanVVVVVVVSingle and 3-Phase Circuit Comparison Consider the phase voltages of equal amplitudeShow that the line voltages are given by:In general: cnbnanpVVVVˆˆˆˆ0ºpbcacabVVVVˆ3ˆˆˆ2103ˆ903ˆ303ˆpcapbcpabVVVVVVBalanced 3-Phase Voltage ConnectionsThere are 2 ways to connect a Balanced set of sources:Y (wye)-Connected (delta)-ConnectedBalanced LoadsBalanced loads are equal in magnitude and phase.There 2 ways to connect balanced loadsY (wye)-Connected (delta)-ConnectedShow that for equivalent loads Z = 3ZYABNCABCLoad-Source ConnectionsThere are 4 possible ways balanced sources and loads can be connected:Y Source to Y Load (Y-Y) Source to Load (-)Y Source to Load (Y-) Source to Y Load (-Y)If not specified, the voltages on the sources will be assumed to be in RMS values.Balanced Y-Y ConnectionThe complete Y-Y connection is shown below with impedances listed separately for the source (subscript s), line (subscript l), and load (subscript L).cabcabLanananpVVVVVVVVˆˆˆˆˆˆFor a positive sequence with , it can be shown that2103ˆ903ˆ303ˆpcapbcpabVVVVVV 0ˆpanVVBalanced Y-Y ConnectionShow that the current in each phase can be expressed as: , and that 240ˆˆ ,120ˆˆ ,ˆˆacabYanaIIIIZVIBecause of the symmetry of a balanced 3 phase system, the neutral connection can be dropped and the system analyzed on a per phase basis. In a Y-Y connected system, the phase (source or load) and line currents are the same.aIˆcIˆbIˆnIˆYZˆYZˆYZˆ 0ˆˆˆ ˆncbaIIIIBalanced Y- ConnectionIn this case the line voltages are directly across each load. It can be shown that: and the load currents and phase currents are related by: 240ˆˆ ,120ˆˆ ,ˆˆˆ3ˆABCAABBCABabpABIIIIZVZVZVINote the –connected load can be converted to a Y-connected load through: 30-3ˆ ˆABaII3ˆˆZZYBalanced - ConnectionIn this case the line voltages are the phase voltages and are directly across each load. It can be shown that: The line currents can be obtained from the phase currents 240ˆˆ ,120ˆˆ ,ˆˆˆABCAABBCABabABIIIIZVZVI 30-3ˆ ˆABaIIBalanced -Y ConnectionIn this case the phase voltages are across the lines. It can be shown that: the line current is related to the phase voltage by: 120ˆ ,120ˆ ,0ˆpcapbcpabVVVVVVNote the –connected source can be converted to a Y-connected source through: 30-3ˆYabaZVI 303ˆˆabanVVPower in Balanced System Show that the instantaneous power absorbed by a load in a balanced Y-Y system is a constant given by:where the impedance in a single phase is given by: The complex power per phase is Note that average power or real power is the same as the instantaneous power for the 3-phase system. ZZYˆ)cos(3)(ppIVtp
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