CIRCUITS and SYSTEMS – part I Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 7 Three-phase circuits

Definition of 3-phase circuits It is the AC power system in which three AC sources operate at the same frequency but with diffeerent phases. The individual phases of the circuit will be denoted by the letters: A, B and C.

Phase voltages of 3-phase symmetric generator

The 3-phase generator is symmetric, when all 3 phase volatges are of the same value and are shifted by 120 o to each other. The system is of positive sequence (ABC), when phase B is delayed by 120 o to phase B,while phase C is delayed by 120 o to phase (it leads phase A by 120 o ). In other case the sequence is negative (ACB). 3-phase symmetric generator

Complex represenatation of 3-phase generator Complex phase voltages Phasor diagram for positive sequence

Rotation of phases Rotation of phases in complex plane is anti-clock wise direction.

Between the phase (line) voltages Line-to-line voltages Phasor diagram

Connections of generator and load Generator Load impedances

Types of 3-phase networks generator and load connected in Y generator and load connected in Δ Δ generator – Y load connection Y generator – Δ load

Y-connected 3-phase network Point 0 – common point of generator. Point N – common point of load Symmetrical generator

Analysis of Y-connected network Neutral voltage U N Phase voltages of the load Load currents

Phasor diagram of 3-phase network There are 2 visible stars of voltages: the generator phase voltages of common point 0 and load phase voltages of common point N.

Balanced (symmetric) load Neutral voltage equal 0 Load currents

Phasor diagram at balanced load

Non-balanced load at zero Z N Neutral voltage U N =0 Current of neutral wire I N =I A +I B +I C Phasor diagram

Short circuit of the load phase Neutral voltage equal to the phase voltage of the short circuit phase (Z=0), for example Z A =0 -> U N =E A. Current of the short circuit phase I A =-(I B +I C ) Phasor diagram

Example Calculate the currents and voltages of the 3-phase Y-connected circuit. Assume balanced generator of the rms phase voltage equal 400V. Assume: R=40 , X C =30 , X L =60 , X 12 =10 , X 23 =20 , X 31 =20 .

Solution Elimination of magnetic couplings

Complex values of parameters Complex representation of phase voltages Complex impedances of the load

Currents and voltages of the load Currents Voltages

Phasor diagram

Delta connection of 3-phase network Structure of the circuit

Currents of Δ connected circuit Voltages of the 3-phase load Currents of the load Line currents

Phasor diagram of Δ connected network.

Example Calculate the currents and voltages of the 3-phase delta-connected circuit. Assume balanced generator of the rms phase voltage equal 200V. Assume: R=X L =X C =10 .

Solution The voltages of the load Load currents Line currents

Phasor diagram

Power in Δ and Y connected balanced load Structures of the circuit Power in Y-connected circuit Power in delta-connected circuit