THREE PHASE POWER

GenerationTransportationConsumption

THREE PHASE GENERATION

THREE PHASE GENERATION

A B C A𝑉 𝐴=𝑉 ′𝑝𝑐𝑜𝑠 (𝜔𝑡 )

𝑉 𝐵=𝑉 ′𝑝 𝑐𝑜𝑠 (𝜔𝑡−120𝑜 )

𝑉 𝐶=𝑉 ′𝑝 𝑐𝑜𝑠 (𝜔𝑡−240𝑜 )

Expressions relative to ground

3G

A

B

C

𝑉 𝐴𝐵=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡 )𝑉 𝐵𝐶=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡−120𝑜 )𝑉 𝐶𝐴=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡−240𝑜 )

Between terminals

THREE PHASE GENERATION

3G

A

B

C

𝑉 𝐴𝐵=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡 )𝑉 𝐵𝐶=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡−120𝑜 )𝑉 𝐶𝐴=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡−240𝑜 )

Between terminals

C

B

A

THREE PHASE GENERATION

Advantage of three phase Smooth flow of energy to load

Consider resistive load Instantaneous power P(t)

R

tV

R

tV

R

tVtP CABCAB )()()()(

222

𝑉 𝐴𝐵=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡 )𝑉 𝐵𝐶=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡−120𝑜 )𝑉 𝐶𝐴=𝑉 𝑝𝑐𝑜𝑠 (𝜔𝑡−240𝑜 )

))240(2cos())120(2cos()2cos(22

3)(

22oopp ttt

R

V

R

VtP

Add to zero

THREE PHASE GENERATION

))240(2cos())120(2cos()2cos(22

3)(

22oopp ttt

R

V

R

VtP

0

constant2

3)(

2

R

VtP p

THREE PHASE GENERATION

constant2

3)(

2

R

VtP p

Smooth transfer of powerSmooth operation of motor

Each phase carries 1/3 of the total power delivered to the load

3G

A

B

C

How to connect to transmission lines

A’

B’

C’

THREE PHASE GENERATION

3G

A

B

C3T

Three transmission lines

Three generators (ABC)

Terminal board

For the generators chose: ''' CCBBAA VVV Identical generators

and

THREE PHASE GENERATION

A’

B’

C’3G

A

B

C

Three generators (ABC) “DELTA” connected

A’

B’

C’

A

B

C 3T

THREE PHASE GENERATION

A’

B’

C’

A

B

C

Three generators (ABC) “DELTA” connected

GAB

C

G

GA’

B’

C’ 3T

C

A

B

THREE PHASE GENERATION

Phase diagram for DELTA connected generators

GAB

C

G

GA’

B’

C’

C

A

B

CAV~

ABV~

BCV

ABV~

BCV

CAV~

Voltages are across pairs of transmission lines

Simple

Voltage of generators

THREE PHASE GENERATION

A’

B’

C’3G

A

B

C

Three generators (ABC) “STAR” connected

A’

B’

C’

A

B

C 3T

GroundNeutral

THREE PHASE GENERATION

Three generators (ABC) “STAR” connected

A

B

C

GG

GA’ B’ C’

3T

C

A

B

A’

B’

C’

A

B

C

GroundNeutral

THREE PHASE GENERATION

Phase diagram for STAR connected generators

AB

C

GG

GA’ B’ C’

C

A

B

ACV~

BAV~

CBV

'

~AAV

'

~BBV

'

~CCV

'

~AAV

'

~BBV

'

~CCV

Recall the generator phase diagram

We will start with the voltage between lines A and B

THREE PHASE GENERATION

AB

A’ B’

A

BBAV~

'

~AAV

AB GG

A’ B’

'

~AAV'

~BBV

AABBBA VVV ''

~~~

''

~~~AABBBA VVV

Change order of indices

Change sign in front of supply

Phasor “Addition” (- = flip direction by 180o)

Back to phasor diagram for generators

THREE PHASE GENERATION

''

~~~AABBBA VVV

'

~AAV

'

~BBV

Original Diagram for AA’ and BB’

'

~AAV

'

~BBV

'

~AAV

'

~BBV

BAV~

30o120o

THREE PHASE GENERATION

For ACV~ For CBV

'

~BBV

'

~CCV

'

~AAV

'

~CCV

''

~~~CCAAAC VVV

'

~AAV '

~CCV

ACV~ 30o

30o

''

~~~BBCCCB VVV

CBV

'

~BBV

'

~CCV

THREE PHASE GENERATION

Phase diagram for STAR connected generators

AB

C

GG

GA’ B’ C’

C

A

B

ACV~

BAV~

CBV

'

~AAV

'

~BBV

'

~CCV

'

~AAV

'

~BBV

'

~CCV

Recall the generator phase diagram

120o

120o

120o

BAV~

ACV~

CBV

Phase diagram

Line to line voltages have 120 degree phase difference

What about amplitude of the voltages

THREE PHASE GENERATION

'

~AAV

'

~BBV

BAV~

30o

''' CCBBAA VVV

BAV~ The length of the red

vector related to the length of the two black vectors

'

~AAV

'

~BBV

30o'2

3AAV

'2

3BBV

'3~

AABA VV

THREE PHASE GENERATION

Phase diagram for STAR connected generators

AB

C

GG

GA’ B’ C’

C

A

B

ACV~

BAV~

CBV

'

~AAV

'

~BBV

'

~CCV

'

~AAV

'

~BBV

'

~CCV

Recall the generator phase diagram

120o

120o

120o

BAV~

ACV~

CBV

Phase diagram

'3~

AABA VV

'3~

AAAC VV

'3~

AACB VV

''' CCBBAA VVV

THREE PHASE TERMINOLOGY

Phase Current: Current through a single generator

GI

~G

V~

Phase Voltage: Voltage across a single generator

THREE PHASE ROTATION CONVENTION

ABV~

BCV

CAV~

ABC ABC ABC …

'

~AAV

'BBV

'

~CCV

ABC ABC ABC …

ABC Motor turns clockwise

ACB Motor turns counter clockwise

THREE PHASE TERMINOLOGY

Line current: Current in a single line feeding a load

3G

A

B

C

3T

LI~

Line voltage: Voltage between the lines carrying the power

LV~

PHASE-LINE RELATIONSHIP

For the STAR connected generators

AB

C

GG

G

C

A

B

ACV~

BAV~

CBV

'

~AAV

'

~BBV

'

~CCV

VVL 3~

V~

I~

IIL~~

PHASE LINE RELATIONSHIP

GAB

C

G

GA’

B’

C’

C

A

B

CAV~

ABV~

V~

VVL ~I~

IIL~

3~

For the Delta connected generators

Phasor current done in class

THREE PHASE TRANSFORMER

NAp NAs

Ap

NBp NBs

NCp NCs

Ap’

Bp

Cp

Bp’

Cp’

As

As’

Bs

Bs’

Cs

Cs’

How do you connect the primary coils together and separately the secondary coils together

THREE PHASE TRANSFORMER

NAp NAs

Ap

NBp NBs

NCp NCs

Ap’

Bp

Cp

Bp’

Cp’

As

As’

Bs

Bs’

Cs

Cs’

Star ----- Star

THREE PHASE TRANSFORMER

NAp NAs

Ap

NBp NBs

NCp NCs

Ap’

Bp

Cp

Bp’

Cp’

As

As’

Bs

Bs’

Cs

Cs’

Star ----- Delta

THREE PHASE TRANSFORMER

NAp NAs

Ap

NBp NBs

NCp NCs

Ap’

Bp

Cp

Bp’

Cp’

As

As’

Bs

Bs’

Cs

Cs’

Delta ----- Star

THREE PHASE TRANSFORMER

NAp NAs

Ap

NBp NBs

NCp NCs

Ap’

Bp

Cp

Bp’

Cp’

As

As’

Bs

Bs’

Cs

Cs’

Delta ----- Delta

THREE PHASE LOADS

A

B

C

Three phase load

3 Transforme

r3G 3T 3Z

Step up or step down

3 Transformer

3Z What does the three phase load look like?Voltages, currents, power, …..

THREE PHASE LOADS

3ZMade up of three impedances

Usually three identical values

Can be connected in STAR or DELTA

Z

ZZ

DELTA

YZ

yZZ 3

YZ YZTransformation

expression

USE OF TRANSFORMATION EXPRESSION IN THREE PHASE

LOADS

Z

ZZ

DELTA

YZ

3Z

ZY

YZ YZLZ

Transmission line impedance

LZ

LZ

LZLZ

LZ

Now two impedances are in series

3

~~~

ZZZ L

THREE PHASE LOADS

3Z

Load can be connected in STAR or DELTA

A

B

C3G

Generators can be connected in STAR or DELTA

4 possible configurations

Delta --- DeltaDelta --- Wye (Star)

Wye (Star) --- DeltaWye (Star) --- Wye (Star)

THREE PHASE LOADS

3Z Delta Connected

DELTA

A

B

C

A

BC

AI~

AI~

CI~

BI~

BI~

CI~

Line currents

ABI~

R CAI~

R

R

BCI~

ABV~

CAV~

BCV~

Line to line voltages

Phase currents

Balanced three phase load

CABCAB VVV~~~

o0 o120 o240

THREE PHASE LOADS

3ZDelta Connected: Consider node “A”

A

B

C

A

BC

AI~

AI~

BI~

CI~

ABI~

R CAI~

R

ABV~

CAV~

BCV~

Line to line voltages

CABCAB VVV~~~

o0 o120 o240

ACABA III~~~

CAABA III~~~

Flip

Since load is only resistive

ABAB IV~

with phasein ~

THREE PHASE LOADS

Delta Connected: Consider node “A”

A

BC

AI~

ABI~

R CAI~

R CAABA III~~~

ABAB IV~

with phasein ~

ABV~

ABI~

CAI~

BCI~

ABV~

CAI~

AI~

30o

3

~~ A

AB

II

Result

Leads by 30 degrees AI~

R

VI ABAB

~~

THREE PHASE LOADSDelta Connected: Consider node “A”

Per phase power

A

BC

AI~

CI~

BI~

ABI~

R CAI~

R

R

BCI~

)cos( IVP

Power Factor (PF)

For purely resistive load Power Factor = 1.0

Total power: )cos(333 IVPP

Note: is the phase angle of the load impedance. It may or may not be the phase angle between the line voltage and line current.

For complex load: jXRZL ~Phase angle )(

R

Xatn

THREE PHASE LOADSDelta Connected: Consider node “A”

Per phase power

A

BC

AI~

CI~

BI~

ABI~

R CAI~

R

R

BCI~

)cos( IVP

Total power:

)cos(333 IVPP

Note: Phase voltage and phase current may be difficult to measure. For example inside a motor. However line voltage and line current can always be measured

)cos(3

333 L

L

IVPP

)cos(333 LLIVPP

Phase angle of load (Can be measured)

THREE PHASE LOADS

3Z Wye (Star) Connected

STAR

A

B

C

A

B C

AI~

AI~

CI~

BI~

BI~

CI~

jXRZL

LZ

ABV~

CAV~

BCV~

Balanced three phase load

CABCAB VVV~~~

o0 o120 o240

LZLZ

NABV

~

CAV~

BCV~

ANV~

BNV~

CNV~

THREE PHASE LOADS

STARA

B C

AI~

CI~

BI~

LZ

LZLZ

NABV

~

CAV~

BCV~

ANV~

BNV~

CNV~

NBANAB VVV~~~

Analysis

BNANAB VVV~~~

flip

CNV~

BNV~

ANV~

BNV~

ANV~

BNV~

ABV~

THREE PHASE LOADS

CNV~

BNV~

ANV~

BNV~

ANV~

BNV~

ABV~

3

~~ AB

AN

VV

Lags by 30 degrees ABV~

ANV~

ABV~

30o

STAR

THREE PHASE LOADS

Star Connected:Per phase power

)cos( IVP

Power Factor (PF)

For purely resistive load Power Factor = 1.0

Total power: )cos(333 IVPP

Note: is the phase angle of the load impedance. It may or may not be the phase angle between the line voltage and line current.

For complex load: jXRZL ~Phase angle )(

R

Xatn

A

B C

AI~

CI~

BI~

LZ

LZLZ

N

THREE PHASE LOADS

Star Connected:Per phase power

)cos( IVP

Power factor (PF)

A

B C

AI~

CI~

BI~

LZ

LZLZ

N

ANVV~~

Consider phase for point A and neutral

LII~~ and

3

~~ AB

AN

VV and

)cos(3

LL IV

P Per phase power

)cos(333 LLIVPP Total power: Same as for Delta load

THREE PHASE LOADS SUMMARY

3Z

Load can be connected in STAR or DELTA

A

B

C3G

Generators can be connected in STAR or DELTA

Real Power Reactive Power Apparent Power

)cos(33 IVP

)cos(33 LLIVP

)sin(33 IVQ

)sin(33 LLIVQ

22 QPS

IVS 3

LLIVS 3