CHAPTER 4: THREE-PHASE CIRCUIT Balanced Three-Phase Voltages Balance Connections: Wye-Wye, Wye-Delta, Delta-Delta, Delta-Wye Power in a Balanced System

a) two-wire type b) three-wire type Introduction Single phase systems a) two-wire type b) three-wire type

Introduction Three Phase System

Balanced Three Phase Voltages Three-phase voltage sources a) wye-connected source b) delta-connected source

Balanced Three Phase Voltages Phase sequences a) abc or positive sequence b) acb or negative sequence

Balanced Three Phase Voltages If the voltage source have the same amplitude and frequency ω and are out of phase with each other by 120o, the voltage are said to be balanced. Balanced phase voltages are equal in magnitude and out of phase with each other by 120o

Balanced Three Phase Voltages abc sequence or positive sequence: is the effective or rms value acb sequence or negative sequence:

Balanced Three Phase Voltages Two possible three-phase load configurations: a) a wye-connected load b) a delta-connected load

Balanced Three Phase Voltages A balanced load is one in which the phase impedances are equal in magnitude and in phase. For a balanced wye connected load: For a balanced delta connected load:

Example 1 Determine the phase sequence of the set of voltages van = 200 cos(ωt + 10◦) vbn = 200 cos(ωt − 230◦), vcn = 200 cos(ωt − 110◦) Solution: The voltages can be expressed in phasor form as We notice that Van leads Vcn by 120◦ and Vcn in turn leads Vbn by 120◦. Hence, we have an acb sequence.

Exercise 1 Given that , find Van and Vcn, assuming a positive (abc) sequence. Answer:

Balanced Wye-Wye Connection A balanced Y-Y system is a three phase system with a balanced Y connected source and balanced Y connected load.

Balanced Wye-Wye Connection Source impedance Line impedance Load impedance Total impedance per phase

Balanced Wye-Wye Connection Line to line voltages or line voltages: Magnitude of line voltages:

Balanced Wye-Wye Connection Phasor diagram illustrating the relationship between line voltages and phase voltages.

Balanced Wye-Wye Connection Balanced Y-Y connection

Balanced Wye-Wye Connection Applying KVL to each phase:

Balanced Wye-Wye Connection A single phase equivalent circuit

Example 2 Calculate the line currents in the three wire Y-Y system of figure below. Reference : Alexander, Sadiku Chapter 12 - page 511

Exercise 2 A Y-connected balanced three-phase generator with an impedance of 0.4+j0.3 Ω per phase is connected to a Y-connected balanced load with an impedance of 24 + j19 Ω per phase. The line joining the generator and the load has an impedance of 0.6 + j0.7 Ω per phase. Assuming a positive sequence for the source voltages and that Find: (a) the line voltages (b) the line currents Reference : Alexander, Sadiku Chapter 12 – PP 12.2

Balanced Wye-Delta Connection A balanced Y- Δ system consists of balanced Y connected source feeding a balanced Δ connected load.

Balanced Wye-Delta Connection Line voltages: Phase currents: Line currents:

Balanced Wye-Delta Connection A single-phase equivalent circuit of a balanced Y- circuit

Balanced Wye-Delta Connection From phasor diagram Magnitude line currents:

Balanced Wye-Delta Connection A single phase equivalent circuit

Example 3 A balanced abc sequence Y-connected source with Is connected to a Δ-connected balanced load (8+j4)Ω per phase. Calculate the phase and line currents. Reference : Alexander, Sadiku Chapter 12 - page 513

Exercise 3 One line voltage of a balanced Y-connected source is If the source is connected to a Δ -connected load of , find the phase and line currents. Assume the abc sequence. Reference : Alexander, Sadiku Chapter 12 – PP 12.3

Balanced Delta-Delta Connection A balanced Δ - Δ system is one in which both balanced source and balanced load are Δ connected.

Balanced Delta-Delta Connection Phase currents: Line voltages: Line currents: Magnitude line currents: Total impedance:

Example 4 A balanced Δ connected load having an impedance 20-j15 Ω is connected to a Δ connected, positive sequence generator having Calculate the phase currents of the load and the line currents. Reference : Alexander, Sadiku Chapter 12 - page 515

Exercise 4 A positive-sequence, balanced -connected source supplies a balanced Δ-connected load. If the impedance per phase of the load is 18+j12 Ω and , find IAB and VAB. Reference : Alexander, Sadiku Chapter 12 – PP 12.4

Balanced Delta-Wye Connection A balanced Δ -Y system consists of balanced Δ connected source feeding a balanced Y connected load.

Balanced Delta-Wye Connection Applying KVL to loop aANBba: From: Line currents:

Balanced Delta-Wye Connection Replace Δ connected source to equivalent Y connected source. Phase voltages:

Balanced Delta-Wye Connection A single phase equivalent circuit

Example 5 A balanced Y connected load with a phase resistance of 40 Ω and a reactance of 25 Ω is supplies by a balanced, positive sequence Δ connected source with a line voltage of 210 V. Calculate the phase currents. Use Vab as reference. Reference : Alexander, Sadiku Chapter 12 - page 519

Exercise 5 In a balanced -Y circuit, and ZY = (12 + j15) Ω. Calculate the line currents. Reference : Alexander, Sadiku Chapter 12 – PP 12.5

POWER IN A BALANCED SYSTEM For Y connected load, the phase voltage:

POWER IN A BALANCED SYSTEM Phase current lag phase voltage by θ. If The phase current:

POWER IN A BALANCED SYSTEM Total instantaneous power: Average power per phase: Reactive power per phase: Apparent power per phase: Complex power per phase:

POWER IN A BALANCED SYSTEM Total average power: Total reactive power: Total complex power:

POWER IN A BALANCED SYSTEM Power loss in two wires: Power loss in three wires: PL : power absorbed by the load IL : magnitude of line current VL : line voltage R : line resistance

Example 6 A three-phase motor can be regarded as a balanced Y-load. A three-phase motor draws 5.6 kW when the line voltage is 220 V and the line current is 18.2 A. Determine the power factor of the motor. Reference : Alexander, Sadiku Chapter 12 - page 523

Exercise 6 Calculate the line current required for a 30-kW three-phase motor having a power factor of 0.85 lagging if it is connected to a balanced source with a line voltage of 440 V. Reference : Alexander, Sadiku Chapter 12 – PP 12.7

Exercise 7 For the Y-Y circuit in Exercise 2, calculate the complex power at the source and at the load. Reference : Alexander, Sadiku Chapter 12 – PP 12.6