iiCircuits IIEE221EE221Unit 12Instructor: Kevin D DonohueInstructor: 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 hidihffhhover 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)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. pBalanced 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):0ˆVV120240ˆ120ˆ0pbnpanVVVVVVVNegative sequence (Clockwise Rotation):120240ppcnVVVˆ0ˆpanVVShow that the sum of all phase voltages in a balanced system is zero120ˆ120240ˆpcnppbnVVVVVShow that the sum of all phase voltages in a balanced system is zero.Single and 3-Phase Circuit Comparison Consider the phase voltages of equal amplitudecnbnanpVVVVˆˆˆˆShow that the line voltages are given by:p0ºVVVVˆ3ˆˆˆIn general:pbcacabVVVV3303ˆpabVV2103ˆ903ˆpcapbcVVVVBalanced 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( )CdY (wye)-ConnectedABNNC (delta)-ConnectedABShow that for equivalent loads Z= 3ZYCLoad-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-)SYLd(Y)Source to YLoad (-Y)If not specified the voltages on the sources will be assumedIf not specified, the voltages on the sources will be assumed to be in RMS values.Balanced Y-Y ConnectionBalanced YY ConnectionThe complete Y-Y connection is shown below with impedances listed separately for the source (subscript s), line (subscript l), and load (subscript L)(subscript L).cabcabLanananpVVVVVVVVˆˆˆˆˆˆFor a positive sequence with it b h th t0ˆVV, it can be shown that303ˆpabVV0panVV2103ˆ903ˆpcapbcVVVVBalanced Y-Y ConnectionShow that the current in each phase can be expressed as:, 240ˆˆ ,120ˆˆ ,ˆˆacabYanaIIIIZVIand that aIˆ 0ˆˆˆ ˆncbaIIIIBecause of the symmetry of a balanced 3 phase system, the neutral IˆYZˆyconnection can be dropped and the system analyzed on a per phase basis. In a nIYZˆYZˆY-Y connected system, the phase (source or load) and line currents are the same.cIˆbIˆBalanced Y-ConnectionBalanced YConnectionIn this case the line voltages are directly across each load. It can be shown that: 240ˆˆ ,120ˆˆ ,ˆˆˆ303ˆABCAABBCABabanABIIIIVVVIand the load currents and phase currents are related by:,,ABCAABBCABZZZ303ˆˆIINote the –connected load can be converted to a Y-connected load through: 30-3 ABaIIg3ˆˆZZYBalanced - ConnectionIn this case the line voltages are the phase voltages and are directly across each load. It can be shown that: ˆˆThe line currents can be obtained from the phase currents 240ˆˆ ,120ˆˆ ,ˆABCAABBCABabABIIIIZVZVIp 30-3ˆ ˆABaIIBalanced -Y ConnectionIn this case the phase voltages are across the lines. It can be shown that: 120ˆ,120ˆ,0ˆbbVVVVVVthe line current is related to the phase voltage by: 120 ,120 ,0pcapbcpabVVVVVVˆVNote the –connected source can be converted 30-3ˆYabaZVIto a Y-connected source through:30ˆˆabVV303anVPower in Balanced System Show that the instantaneous power absorbed by a load in a balanced Y-Y system is a constant given by:)(3)(IVtwhere the impedance in a single phase is given by:ˆ)cos(3)(ppIVtpThe complex power per phase is ZZYˆ)exp(jIVSppNote that average power or real power is the same as the instantaneous power for the 3-phase
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