Lesson 35Per Phase Analysis

Learning Objectives Derive the relationship between line to line voltages and line

to neutral voltages in a balanced Y-Y three phase circuit. For a balanced Y-Y three phase circuit convert the line to

line voltage phasor to the line to neutral voltage phasor and vice versa.

Plot the phasors of line to line voltages and line to neutral voltages for a Y-Y three phase circuit.

Derive the relationship between line current and phase load current in a balanced Y- Δ three phase circuit.

For a balanced Y- Δ three phase circuit convert the line current phasor to phase load current phasor and vice versa.

Plot the phasors of line current and phase load current in a balanced Y- Δ three phase circuit.

Y-Y system

Lines

Y LoadGenerator

Review

Line and phase voltages of the Y-connected three-phase generator.

Determining a line voltage for a three-phase generator.

PHASE/LINE VOLTAGES

Line/Phase Voltages for a Wye Circuit

Using the following relationship

we can also convert back…

Therefore, given any voltage at a point in a balanced, 3-phase Y system, you can determine the remaining 5 voltages by inspection.

303303

AB

ANab

an

VE

VV

303 anab VV

Law of Sines The ratio of the length of a side to the sine of its

corresponding opposite angle is constant

Applying the law of sines to the vector EAB:

sin sin sin

a b c

EAN

EBN

120

180 12030

2

sin 30 sin120sin120

3sin 30

B

B

AN A

A AN AN

E E

E E E

Nominal voltages If phase voltage Van = 120 V then line

voltages Vab = 208 V. This is referred to as 120/208-V system. Two other common nominal voltages are

220/380-V and 347/600-V.

Currents for a Wye Circuit

For Y loads, line current and phase current are the same.

anana ZVI /

Line current Phase current

Wye-Wye Single Phase Equivalent

Single-phase equivalent

Original circuit

Example Problem 1

EAB = 2080 V .

Find the phase voltages and line currents.

load definitions

Line currents are the currents in the line conductors. Phase currents are the currents through phases .

Y-Y

Review

For loads, phase voltage and line voltage are the same thing.

Line/Phase Currents for a Delta Circuit

Line current Phase current

Iab

Iab

I ca

Ibc

Line/Phase Currents for a Delta Circuit

Relationship between line and phase currents

For a balanced system, the magnitude of line current is 1.732 times the magnitude of the phase current and line current lags phase current by 30.

Therefore, given any current at a point in a balanced, 3-phase system, you can determine the remaining 5 currents by inspection.

303

/

aba

caaba

ababab

II

III

ZVI

Line/Phase Currents for a Delta Circuit

Phasor diagram

Equivalent Ckt to solve Wye-Delta Problem

This is just an equivalent circuit of the load side! What is missing from the equivalent circuit?

Ia , Eab , Ean

You can only use the equivalent circuit to solve for load side voltages and currents: Vab, Iab

Notice that this equivalent circuit uses LINE VOLTAGE #1 mistake with DELTA circuits is using the wrong kind of voltage!

Example Problem 2

Ia = 41.66.9 A .

Determine the phase currents in the load.

Determine the supply line voltage EAB and the supply phase voltage EAN.

Line Impedances in 3 Phase AC Impedances in transmission lines complicate

three phase analysis Voltage drop in the transmission line must be

accounted for: EAN will not be the same as Van

VAa

Wye-Wye Single Phase Equivalent

Single-phase equivalent

Original circuit

•Notice that the neutral line impedance drop is ignored on the single phase equivalent.•No voltage drop occurs across this neutral line since in reality, no current flows through it.

Wye-Wye Single Phase Equivalent

Single-phase equivalent

Original circuit

•Use KVL on the single phase equivalent to find unknowns:

EAN –VAa –Van = 0

+ VAa -

Example Problem 3EAB = 2080 V. Determine:

a. Line Currents

b. Load phase voltages

Delta To Wye (∆→Y) Conversion A Delta circuit with line impedances cannot be

solved without first performing a ∆→Y conversion. The Delta load can be converted to an equivalent Wye

load

3Y

ZZ

Equivalent Delta Ckt with LINE IMPEDANCES

WYE to DELTA CONVERSION

15 215 7

3 3Y

Z jZ j

Example Problem 4EAB = 2080 V. Determine:

a. Find load phase voltage Vab

b. Find the load phase current Iab .