Three-Phase AC Circuits - Welcome to Boise State, College...

Three-Phase AC Circuits

Objectives:

1. To discuss balanced three-phase voltages.

2. To discuss ABC and ACB phase sequences.

3. To discuss the ∆-to-Y and Y-to-∆ trans-

formations.

4. To discuss the relationship between line-

to-line and line-to-neutral voltages in a Y

connection.

5. To discuss the relationship between line

and phase currents in a ∆ connection.

1

6. To discuss the complex power into three-

phase four-wire and three-phase three-wire

power systems.

7. To discuss the two-wattmeter method for

real power measurement in a balanced or

unbalanced three-phase three-wire power

system.

8. To discuss the two-wattmeter method and

the one-wattmeter method for reactive power

measurement in a balanced three-phase three-

wire system.

Balanced Three-Phase Voltages

anv (t)bn

v (t)cn

v (t)an

v (t)bnV

m

−Vm

t

v (t)

van(t) = Vm cos(ωt)

vbn(t) = Vm cos(ωt− 120o)

vcn(t) = Vm cos(ωt + 120o)

van(t) + vbn(t) + vcn(t) = 0 at any time t

2

ABC and ACB Phase Sequences

V~

an

Vcn

~

V~

bn

++

−120

+120+120

−120

o o

o o

~

Van

~

V

V~

bn

cn

Van = Vln 6 0o Van = Vln 6 0o

Vbn = Vln 6 − 120o Vbn = Vln 6 120o

Vcn = Vln 6 120o Vcn = Vln 6 − 120o

3

Line-to-Neutral and Line-to-Line Voltages

in a Y Connection

n

+

V

~Van

~

bn

Vcn

~

+

−

Vab

~

+

−

Vbc

~

+

−

Vca

~

−

+

−

+−

−

ZAB

−

ZCA

−

ZBC

Van = Vln 6 0o Vab = Vll 6 30o

Vbn = Vln 6 − 120o Vbc = Vll 6 − 90o

Vcn = Vln 6 120o Vca = Vll 6 150o

Vll =√

3Vln

4

Line-to-Neutral and Line-to-Line Voltages

in a Y Connection

V

~

V

~

V

~

V

~

30o

−120

120

o

o

bc

bn

ca

ca

ab

an

~

V

~

V

Vab = Van − Vbn =√

3Vanej30o

=√

3Vln 6 (0o + 30o) = Vll 6 30o

Vbc = Vbn − Vcn =√

3Vbnej30o

=√

3Vln 6 (−120o + 30o) = Vll 6 − 90o

Vca = Vcn − Van =√

3Vcnej30o

=√

3Vln 6 (120o + 30o) = Vll 6 150o

5

Line and Phase Currents

in a ∆ Connection

aI

ca

~

V

~

bcV

~

abV

−

+

−

+

~

bc

~

I

ca

~

I

ab

~

I

c

~

I

~

bI

−

+

−

BCZ

−

CAZ

−

ABZ

Iab = Ip 6 0o Ia = Il 6 − 30o

Ibc = Ip 6 − 120o Ib = Il 6 − 150o

Ica = Ip 6 120o Ic = Il 6 90o

Il =√

3Ip

6

Line and Phase Currents

in a ∆ Connection

I

I

I

−120o

I

~

I

~

I

~

120o

a

ab

bcb

ca

c

−30o

~

~

~

Ia = Iab − Ica =√

3Iae−j30o

=√

3Ip 6 (0o − 30o) = Il 6 − 30o

Ib = Ibc − Iab =√

3Ibce−j30o

=√

3Ip 6 (−120o − 30o) = Il 6 − 150o

Ic = Ica − Ibc =√

3Icae−j30o

=√

3Ip 6 (120o − 30o) = Il 6 90o

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∆-to-Y Transformation

A

B

C

ZAB

−

ZBC

−

ZCA

−

ZA

−

ZB

−

ZC

−

C

A

B

ZA =ZABZCA

ZAB + ZBC + ZCA

ZB =ZBCZAB

ZAB + ZBC + ZCA

ZC =ZCAZBC

ZAB + ZBC + ZCA

Balanced Load Case

ZY =Z∆

3

8

Y-to-∆ Transformation

ZAB

−

ZBC

−

ZCA

−

A

B

C

ZA

−

ZB

−

ZC

−

C

A

B

ZAB =ZAZB + ZBZC + ZCZA

ZC

ZBC =ZAZB + ZBZC + ZCZA

ZA

ZCA =ZAZB + ZBZC + ZCZA

ZB

Balanced Load Case

Z∆ = 3ZY

9

Balanced Y-Y Connection

−

lZ

−

lZ

−

sZ

−

sZ

−

sZ

+−

+

Z

nn

~

Van

cn

bn

~

V

~

V

−

−

−

l

−

Z

−

−

I

−

+

~

bI

~

cI

~

Ia

LZ

LZ

ZL

~

−Van + (Zs + Zl + ZL)Ia + ZnIn = 0

−Vbn + (Zs + Zl + ZL)Ib + ZnIn = 0

−Vcn + (Zs + Zl + ZL)Ic + ZnIn = 0

0 + (Zs + Zl + ZL)(Ia + Ib + Ic) + 3ZnIn = 0

(Zs + Zl + ZL + 3Zn)In = 0

=⇒ In = 0

10

Per-Phase Analysis

~

anV

−

sZ

−

+

−

LZ

−

lZ

a

~

I

Van = (Zs + Zl + ZL)Ia = ZIa

Ia =Van

Z=

Vln 6 0o

Z 6 γ= IL 6 − γ

Ib =Vbn

Z= IL 6 − γ − 120o

Ic =Vcn

Z= IL 6 − γ + 120o

11

Three-Phase Complex Power

Three-Phase Four-Wire System

+

−

+

−

V

~

I

~

C

IB

~

I

~

A

VBN

~

VAN

~

CN

N

−

ZB

−

ZC

−

ZA

−

+

S3ph = VAN I∗A + VBN I∗B + VCN I∗C= (VLN 6 0o)(IL 6 − φ)∗

+(VLN 6 − 120o)(IL 6 − φ− 120o)∗

+(VLN 6 120o)(IL 6 − φ + 120o)∗

= 3VLNIL 6 φ = S3ph 6 φ

= 3VLNIL cosφ︸︷︷︸ +j 3VLNIL sinφ︸︷︷︸= P3ph + j Q3ph

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Three-Phase Power Equations

S3ph = 3VLNIL

P3ph = 3VLNIL cosφ

Q3ph = 3VLNIL sinφ

S3ph =√

3VLLIL

P3ph =√

3VLLIL cosφ

Q3ph =√

3VLLIL sinφ

S3ph =

√P23ph + Q2

3ph

pf = cosφ =P3ph

S3ph

IL =S3ph√3VLL

=P3ph√

3VLL cosφ

13

Three-Phase Complex PowerThree-Phase Three-Wire System

~

A

IC

~

IB

~

−

+

−+

−

VAB

~

VBC

~

VCA

~

A

I

ZB

−

ZC

−

Z−

+

N

Equivalent Y Load

S3ph = VAN I∗A + VBN I∗B + VCN I∗C= VAN I∗A + VBN(−IA − IC)∗ + VCN I∗C= (VAN − VBN)I∗A + (VCN − VBN)I∗C= VABI∗A + VCBI∗C= |VAB||IA| cos(6 VAB − 6 IA)

+|VCB||IC| cos(6 VCB − 6 IC)

+j|VAB||IA| sin(6 VAB − 6 IA)

+j|VCB||IC| sin(6 VCB − 6 IC)

= P3ph + jQ3ph

14

Wattmeter Measurement

VL

~

+

VS

~

IS

~IL

~

+−

W

−

+−

−

LOAD

+

W = |VL||IS| cos(6 VL − 6 IS)

W ∼= |VL||IL| cos(6 VL − 6 IL)

W ∼= VLIL cosφ

15

Two-Wattmeter Method for Power

Measurement in Unbalanced/Balanced

Three-Phase Three-Wire Systems

−−

+

−+

−+

−+−

+

W−

CZ

−

B

+

Equivalent Y−Load

A

~IA

AW

C

~

CI

~

CBV

~

ABV

Z

−Z

WA = |VAB||IA| cos(6 VAB − 6 IA)

WC = |VCB||IC| cos(6 VCB − 6 IC)

Balanced/Unbalanced Load

P3ph = WA + WC

Balanced Load

Q3ph =√

3(WC −WA)

16

Two-Wattmeter Method for Real

Power Measurement in Balanced


AB

~V

BC

~V

CB

~V

A

~I

~

CNV

o

o

−φ

−φ

−φ

30

90

~

CI

B

~I

~

CA

V

~

BNV

~

ANV

P3ph = WA + WC


= VLLIL cos(30o − (−φ))

= VLLIL cos(φ + 30o)


= VLLIL cos(−90o − (120o − φ))

= VLLIL cos(φ− 30o)

17

Two-Wattmeter Method for Real



Check:

WA + WC = VLLIL [cos(φ + 30o) + cos(φ− 30o)]

= VLLIL [cosφ cos 30o − sinφ sin 30o

+cosφ cos 30o + sinφ sin 30o)]

= 2VLLIL cosφ cos 30o

= 2VLLIL cosφ

(√3

2

)

=√

3VLLIL cosφ

=√

3(√

3VLN

)IL cosφ

= 3VLNIL cosφ

= P3ph

18

Two-Wattmeter Method for Reactive



AB

~

V

BC

~

V

CB

~

V

A

~

I

~

CNV

o

o

−φ

−φ

−φ

30

90

~

CI

B

~

I

~

CA

V

~

BNV

~

ANV


= VLLIL cos(30o − (−φ))

= VLLIL cos(φ + 30o)


= VLLIL cos(−90o − (120o − φ))

= VLLIL cos(φ− 30o)

Q3ph =√

3 (WC −WA)

19

Two-Wattmeter Method for Reactive



Check:

WC −WA = VLLIL [cos(φ− 30o)− cos(φ + 30o)]

= VLLIL [cosφ cos 30o + sinφ sin 30o

− cosφ cos 30o + sinφ sin 30o]

= 2VLLIL sinφ sin 30o

= 2VLLIL sinφ

(1

2

)

= VLLIL sinφ

=Q3ph√

3

Q3ph =√

3 (WC −WA)

20

One-Wattmeter Method for Reactive



~

BCV

~

ABV

−

+

−

+

−+

−

C

I

~

BI

−+

Equivalent Y−Load

A

~

IA

AW

~

C

Z

−

B

−

Z

Z

WA = |VBC||IA| cos(6 VBC − 6 IA)

Q3ph =√

3WA

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One-Wattmeter Method for ReactivePower Measurement in BalancedThree-Phase Three-Wire Systems

VCN

~

I

~

A

V

~

BC

V

~

ABVCA

o−90

o

−φ

−φ

−φ

30

~

CI

B

~

I

~

~

BNV

~

ANV

WA = |VBC||IA| cos(6 VBC − 6 IA)

= VLLIL cos(−90o − (−φ))

= VLLIL cos(−90o + φ)

= VLLIL cos(90o − φ)

= VLLIL sinφ

Q3ph =√

3WA

Q3ph =√

3VLLIL sinφ

Q3ph = 3VLNIL sinφ

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Three-Phase AC Circuits - Welcome to Boise State, College...

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