Transcript ee221_12
Circuits II EE221 Unit 12 Instructor: Kevin D. Donohue Three Phase Circuits, Balanced Y-Y, Y-, and - Three-Phase Circuits Polyphase 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 3phase systems or phases manipulated to create additional phases. Balanced 3-Phase Voltages Balanced 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): Vˆan V p 0 Vˆbn V p 120 Vˆcn V p 240 V p 120 Negative sequence (Clockwise Rotation): Vˆan V p 0 Vˆbn V p 240 V p 120 Vˆcn V p 120 Show 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 amplitude Vˆp Vˆan Vˆbn Vˆcn Show that the line voltages are given by: Vˆab Vˆac Vˆbc 3 Vˆp In general: Vˆab 3V p 30 Vˆbc 3V p 90 Vˆca 3V p 210 0º Balanced 3-Phase Voltage Connections There are 2 ways to connect a Balanced set of sources: Y (wye)-Connected (delta)-Connected Balanced Loads Balanced loads are equal in magnitude and phase. There 2 ways to connect balanced loads Y (wye)-Connected A B N (delta)-Connected C A B C Show that for equivalent loads Z = 3ZY Load-Source Connections There 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 Connection The complete Y-Y connection is shown below with impedances listed separately for the source (subscript s), line (subscript l), and load (subscript L). V p Vˆan Vˆan Vˆan VL Vˆab Vˆbc Vˆca For a positive sequence with Vˆ an V p 0, it can be shown that Vˆab 3V p 30 Vˆbc 3V p 90 Vˆca 3V p 210 Balanced Y-Y Connection Show that the current in each phase can be expressed as: ˆ ˆI Van , Iˆb Iˆa 120, Iˆc Iˆa 240 , a ZY and that Iˆa Iˆb Iˆc Iˆn 0 Iˆa Ẑ Y Iˆn Ẑ Y Iˆc Ẑ Y Iˆb Because 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. Balanced Y- Connection In this case the line voltages are directly across each load. It can be shown that: 3Vˆp Vˆab VˆAB ˆ ˆ ˆ ˆ ˆ I AB Z Z Z , I BC I AB 120, I CA I AB 240 and the load currents and phase currents are related by: Iˆa IˆAB 3 - 30 Note the –connected load can be converted to a Yconnected load through: ˆ Z ZˆY 3 Balanced - Connection In this case the line voltages are the phase voltages and are directly across each load. It can be shown that: ˆ ˆ ˆI Vab VAB , AB Z Z IˆBC IˆAB 120, IˆCA IˆAB 240 The line currents can be obtained from the phase currents Iˆ Iˆ 3 - 30 a AB Balanced -Y Connection In this case the phase voltages are across the lines. It can be shown that: Vˆab V p 0, Vˆbc V p 120, Vˆca V p 120 the line current is related to the phase voltage by: V Iˆa ab - 30 3ZY Note the –connected source can be converted to a Y-connected source through: Vˆab ˆ Van 30 3 Power in Balanced System Show that the instantaneous power absorbed by a load in a balanced Y-Y system is a constant given by: p(t ) 3V p I p cos( ) where the impedance in a single phase is given by: ZˆY Z The complex power per phase is S V p I p exp( j ) Note that average power or real power is the same as the instantaneous power for the 3-phase system.