Lesson 27: 3-phase circuit analysis 1Complex power (review) Determine the power (P, Q, and S) by load Z. Single-phase system (2-wire) A single-phase ac power system consists of a generator connected through a pair of wires (transmission line) to a load. - V P is the magnitude of the source voltage given in RMS (the subscript P is for phase, not peak). - | is the phase. - This is a _________-wire system. Single-phase system (3-wire) A three-wire single-phase system consists of two identical sources (equal __________ and ____________ ) connected to two loads by two outer wires and a neutral. This is typical of power supplied to houses. Three-phase system (4-wire) A three-phase system is produced by three sources with the same amplitude and frequency but separated in phase by ___________. Why three-phase? 1. Nearly all power generated and distributed is 3-phase, ___ Hz (or 50 Hz outside the U.S.). - When single phase is needed, it can simply be taken from one of the three phases. 2. The instantaneous power in 3-phase system can be ________________ (not pulsating). - Results in less _________________ in 3-phase machines. 3. Three phase systems are more economical that single. - Amount of ___________ needed is less than single phase. Balanced three-phase voltages Three-phase voltages are produced by a three-phase ac generator (or alternator). The stator contains 3 windings (a-a', b-b', c-c') physically separated by _____°. The voltages induced in the windings are equal in magnitude but out of phase by 120°. The voltages V an , V bn , and V cn are between lines a, b and c and the neutral line n. These are called ____________ voltages. Balanced three-phase voltages If the voltages are equal in magnitude but out of phase by 120° they are said to be balanced. This implies Positive phase sequence (abc) The ______ sequence or ___________ sequence. Note: V P is the rms value of the phase voltage. 0 an bn cn an bn cn + + = = = V V V V V V an bn cn = = = V V V Negative phase sequence (acb) The acb or negative sequence is produced when the generator rotates counterclockwise. Phase sequence The phase sequence is the time order in which the voltages pass through their respective maximum values. Phase sequence is important because it determines the ________________________ of a connected motor. Three-phase loads Three-phase loads can be either _____-connected or ______-connected. Loads are said to be balanced if phase impedances (Z 1 , Z 2 , Z 3 or Z a , Z b , Z c ) are equal in magnitude and phase. Wye-connected load Whether a ___________ line exists depends on if it is a three or four- wire system. For a balanced wye-connected load where Z Y is the load per phase. Delta-connected load A neutral line cannot exist in a A-connected load. For a balanced A-connected load (where Z A is the load per phase.) 1 2 3 = = = Z Z Z a b c = = = Z Z Z an cn bn = = = V V V Delta-wye transformation How can we equate the wye- and delta-connected (balanced) loads? For balanced loads, Delta-connected sources Sources may also be delta-connected. In practice, this is rarely done because slight imbalances in the voltages result in a circulating current in the delta-mesh Three-phase systems Given that both loads and sources can be Y-connected and A-connected, we have 4 possible combinations to consider - Y-Y connection - Y-A connection - A-A connection - A-Y connection In reality, the most common configuration is ________ . Balanced Y-Y connection We will consider a four-wire Y-Y system. Z Y = The impedance Z Y can be thought of as the sum of the ___________, __________ and _________ impedances. Balanced Y-Y connection Because Z S and Z l are often quite small compared to Z L , we will assume Z Y = Z L , simplifying our Y-Y system. Source-load labeling convention __________________ are labeled using lowercase letters. __________________ are labeled using uppercase letters. or Y A = = Z Z Line voltages How are the line voltages (V ab , V bc , V ca ) related to the phase voltages (V an , V bn , V an ) ? Line voltage are _____ times larger and lead their respective phase voltages by ____. Line currents What are the line currents (I a , I b , I c )? What is the neutral line current (I n )? Line currents are given Since I a + I b + I c = 0, if follows that I n = _____ . Single-phase equivalent In analyzing a balanced Y-Y system, it is sufficient to examine just one of the phases and apply the results to the other two phases (with appropriate phase delays). 3 3 ab ab bn bc P ca P V V = ÷ = = Z = Z V V V V V 120 240 a bn an b Y Y cn an c Y Y = Z ÷ ° = = = Z÷ ° = = = I V V I Z Z V V I Z Z Example Problem 1 Determine the phase sequence of the set of voltages. Example Problem 2 Calculate the line currents in the three-wire Y-Y system. 200cos( 10 ) 200cos( 230 ) 200cos( 110 ) an bn cn v t v t v t e e e = + ° = ÷ ° = ÷ ° When single phase is needed. Balanced three-phase voltages Three-phase voltages are produced by a three-phase ac generator (or alternator). b and c and the neutral line n. Vbn. Balanced three-phase voltages If the voltages are equal in magnitude but out of phase by 120° they are said to be balanced. The instantaneous power in 3-phase system can be ________________ (not pulsating). ___ Hz (or 50 Hz outside the U.). Van Vbn Vcn . and Vcn are between lines a. Results in less _________________ in 3-phase machines. Three phase systems are more economical that single.Why three-phase? 1. 2. The stator contains 3 windings (a-a'. Nearly all power generated and distributed is 3-phase. Amount of ___________ needed is less than single phase. c-c') physically separated by _____°. it can simply be taken from one of the three phases. Note: VP is the rms value of the phase voltage. This implies Van Vbn Vcn 0 Van Vbn Vcn Positive phase sequence (abc) The ______ sequence or ___________ sequence.S. These are called ____________ voltages. b-b'. 3. The voltages Van. The voltages induced in the windings are equal in magnitude but out of phase by 120°. Zc ) are equal in magnitude and phase. Van Vcn Vbn Phase sequence The phase sequence is the time order in which the voltages pass through their respective maximum values. Delta-connected load A neutral line cannot exist in a -connected load.) Z a Zb Z c . Phase sequence is important because it determines the ________________________ of a connected motor. For a balanced wye-connected load Z1 Z 2 Z3 where ZY is the load per phase. Three-phase loads Three-phase loads can be either _____-connected or ______-connected. Z3 or Za. Zb. Loads are said to be balanced if phase impedances (Z1. Wye-connected load Whether a ___________ line exists depends on if it is a three or fourwire system.Negative phase sequence (acb) The acb or negative sequence is produced when the generator rotates counterclockwise. Z2. For a balanced -connected load (where Z is the load per phase. Balanced Y-Y connection We will consider a four-wire Y-Y system. Z or ZY Delta-connected sources Sources may also be delta-connected.Delta-wye transformation How can we equate the wye. we have 4 possible combinations to consider Y-Y connection Y- connection - connection -Y connection In reality.and delta-connected (balanced) loads? For balanced loads. this is rarely done because slight imbalances in the voltages result in a circulating current in the delta-mesh Three-phase systems Given that both loads and sources can be Y-connected and -connected. __________ and _________ impedances. . Source-load labeling convention __________________ are labeled using lowercase letters. In practice. the most common configuration is ________ . we will assume ZY = ZL. ZY = The impedance ZY can be thought of as the sum of the ___________. Balanced Y-Y connection Because ZS and Zl are often quite small compared to ZL. simplifying our Y-Y system. __________________ are labeled using uppercase letters. Vbc. Vca ) related to the phase voltages (Van. Vbn. it is sufficient to examine just one of the phases and apply the results to the other two phases (with appropriate phase delays).Line voltages How are the line voltages (Vab. Line currents What are the line currents (Ia. Single-phase equivalent In analyzing a balanced Y-Y system. Ib. if follows that In = _____ . Van) ? Vab Vab Vbn Vbc Vca 3VP 3VP Line voltage are _____ times larger and lead their respective phase voltages by ____. Ic)? What is the neutral line current (In)? Line currents are given Ia Ib Ic Vbn ZY Vcn ZY Van 120 ZY Van 240 ZY Since Ia + Ib + Ic = 0. . . van 200cos(t 10) vbn 200cos(t 230) vcn 200cos(t 110) Example Problem 2 Calculate the line currents in the three-wire Y-Y system.Example Problem 1 Determine the phase sequence of the set of voltages.