1. Lesson 35 Per Phase Analysis

2. 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.

3. Review
Y-Y system
Y Load
Generator
Lines

4. PHASE/LINE VOLTAGES
Line and phase voltages of the Y-connected three-phase generator.
Determining a line voltage for a three-phase generator.

5. 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.

6. Applying the law of sines to the vector EAB:
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:
EAN
EBN

7. 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.

8. Currents for a Wye Circuit
For Y loads, line current and phase current are the same.
Line current
Phase current

9. Wye-Wye Single Phase Equivalent
Original circuit
Single-phase equivalent

10. Example Problem 1 EAB = 2080 V .
Find the phase voltages and line currents.

11. Review  load definitions
Y-Y
For  loads, phase voltage and line voltage are the same thing.
Line currents are the currents in the line conductors.
Phase currents are the currents through phases .

12. Line/Phase Currents for a Delta Circuit
Iab
Ica
Iab
Ibc
Line current
Phase current

13. Line/Phase Currents for a Delta Circuit
Relationship between line and phase currents
For a balanced  system, the magnitude of line current is 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.

14. Line/Phase Currents for a Delta Circuit
Phasor diagram

15. 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!

16. Example Problem 2 Ia = 41.66.9 A .
Determine the phase currents in the load.
Determine the supply line voltage EAB and the supply phase voltage EAN.

17. 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

18. Wye-Wye Single Phase Equivalent
Original circuit
Single-phase equivalent
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.

19. Wye-Wye Single Phase Equivalent
Original circuit
+ VAa -
Single-phase equivalent
Use KVL on the single phase equivalent to find unknowns:
EAN –VAa –Van = 0

20. Example Problem 3 EAB = 2080 V. Determine: a. Line Currents
b. Load phase voltages

21. 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

22. Equivalent Delta Ckt with LINE IMPEDANCES

23. Example Problem 4 EAB = 2080 V. Determine:
a. Find load phase voltage Vab
b. Find the load phase current Iab .