Three Phase CircuitsTuesday, September 13th, 2005

Today

• Single-Phase Three-Wire Systems

• Three-Phase Wye-Wye Systems

• Delta Connection

• Wye-Delta Transformations

Single Phase Three-Wire Systems

a A

n

cB

N

V1

V2

100 0anV

100 120bnV

a

bn

Vbn

Vnb

Van

Vab

Z1

Z1

a A

n

bB

N

V1

V1

1

1 1

anaA

V VI

Z Z

1

1 1

bnbB aA

V VI I

Z Z

0nN aA bBI I I

Z1

Z1

aA

n

b B

N

V1

V1

Z3

Z2

Z2

Z4

IaA

IbB

I3

1 2 3 3 1 3 1aA bBZ Z Z I Z I Z I V

3 1 2 3 1 3 1aA bBZ I Z Z Z I Z I V + 1 2 3 3 0aA bB aA bBZ Z Z I I Z I I

0aA bBI I or

1150

1150

39 W

1 W

1 W

47 W

j W

j W

2 W

a A

n

b B

N

I1

I2

Example

Find I1, I2 and InN !

1

2

42 2 115 0

2 50 115 0

o

o

Ij

Ij

42 50 4 2097 2,51oj j

1

2

50 2 1 2,85 1,41115 0

2 42 1 2,41 1,21

oo

o

I j

I j

The determinant is

Solving by matrix inversion

Three-Phase Wye-Wye SystemsZl

Zl

Zl

ZL

ZL ZL

a

n

bc

A

B C

N

0

120

120

oan p

obn p

ocn p

V V

V V

V V

0

120

120

oan p

obn p

ocn p

V V

V V

V V

0an bn cnV V V

Positive or abc sequence Negative or acb sequence

V a n

V bn

V cn- V bn

V ab

V ab = V an - V b n

V ab = 3 V an 3 0 o

3 0 o

30||3

2

3

2

1||||

)120sin120(cos1||

120||0||

p

pp

p

pp

bnanab

V

jVV

jV

VV

VVV

210||3

90||3

pca

pbc

VV

VV

VoltageLine ||3 pL VV

Van

Vbn

Vcn

n N

o o

o

o

o

o

a A

b B

c C

ZP

ZL

ZL

ZL

ZP

ZP

ZN

120120

120120

anaA

P

oobn an

bB aAP P

oocn an

cC aAP P

VI

Z

V VI I

Z Z

V VI I

Z Z

0nN aA bB cCI I I I

120 120

120 120

aA L P

o obB L P

o ocC L P

I I I

I I I

I I I

cosp p pP V I 3 pP P

Delta ConnectionZl

Zl

Zl

a

bc

A

B CZL

ZLZ L

+

+

+

_

_

_

L PV V 3 L PI I

3 cosL LP V I

Wye-Delta Transformations

Y

RRR

RRR

RRR

RRR

RRR

RRR

c

b

a

321

13

321

32

321

21

Y

R

RRRRRRR

R

RRRRRRR

R

RRRRRRR

a

accbba

c

accbba

b

accbba

3

2

1

3 YZ Z

2300

2300

21 W

3 W

3 W

21 W

3 W

a A

n

b B

N

-j4 W

-j4 W

Find the currents IaA, IbB, and InN in this symetric single phase three wire systems. Sketch in phasor diagram !

Van

Vbn

Vcn

n

o

o

o

a A

bB

c

C

ZP

ZP

ZP

o

o

o

In the Wye-Delta system shown, the source is positive sequence with Van = 100 0 Vrms and the phase impedance is Zp = 3 – j4 W. Find the line voltage VL, the line current IL and the power delivered to the load !