Three Phase Power Circuits, Slides of Power Electronics

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EEE 360

Lecture 2:

Three-Phase Circuits

1

Three-Phase Circuits

▪ Practically all electrical energy generation and

transmission systems use a three-phase circuit.

▪ The three-phase energy is transported through

three or four conductors to large customers.

▪ Only the small household and light commercial

loads are supplied by a single phase.

▪ The major advantage of the three-phase system

is the efficiency of power transmission.

2

Three-Phase Circuits

Three-Phase Quantities

Basic Definitions

Basic Definitions

▪ In this chapter only the balanced three-phase

systems are presented.

▪ A three-phase system has three sinusoidal

voltages sources.

▪ In a balanced system, each voltage source has

the same magnitude ( VM ) and frequency ( ω ),

and each is 120° out-of-phase with the other two

= −  =  − 

= =  

( ) cos 240 240

( ) cos 120 120

( ) cos 0

cn M M

bn M M

an M M

v t V t V

v t V t V

v t V t V

cn

bn

an

V

V

V

5

Basic Definitions

▪ A balanced three-phase circuit is one in which

the loads are such that the currents produced by

the voltages are also balanced.

▪ The balanced load currents are described by the

equations below

where the voltage of each phase leads its

corresponding current by an angle of θ

= − −  = − − 

= − = −

( ) cos 240 240

( ) cos 120 120

( ) cos

c M M

b M M

a M M

i t I t I

i t I t I

i t I t I

c

b

a

I

I

I

(^) 7

Basic Definitions

▪ The sum of the balanced voltages and the sum

of the balanced currents equal zero

▪ The instantaneous power in each phase is the

product of voltage and current. The total

instantaneous power is

▪ Hence, the instantaneous three-phase power is

constant over time.

i t i t i t

v t v t v t

a b c

an bn cn

cos( ) 3 cos( )

2

( ) ( ) ( ) ( ) 3  rms rms 

M M T a b c V I

V I p t = p t + p t + p t = =

(^8)

Basic Definitions

▪ Similarly, the total three-phase complex power

is three times the complex power of any of the

phases: S T = S A + S B + S C = 3 S 1

▪ Figure 4.2 shows the sinusoidal voltage and

currents as well as the instantaneous power

variation in time.

▪ The frequency of the voltage and current is 60

Hz, but the power of each phase varies at a

frequency of 120 Hz.

▪ The sum of the three instantaneous powers is

constant power.

10

Basic Definitions

Figure 4.2 Time-varying voltage, current and power

in a balanced three-phase circuit.

0 5 10 15 20 25 30 35 40

0

500

Voltage (kV)

Three-Phase Balanced System at 60 Hz, and V Leads I by 45°

0 5 10 15 20 25 30 35 40

0

1

Current (kA)

0 5 10 15 20 25 30 35 40

0

100

200

300

Time (ms)

Power (MW)

Va Vb Vc

Ia Ib Ic

Pa Pb Pc Ptotal

11

Delta-Wye Connections

▪ A balanced three-phase load can be connected

in a delta (Δ) or in a wye (Y).

▪ Figure 4.3 shows wye-connected loads.

ZY

ZY

ZY

a

c

b

n

Load

ZY

Z ZY Y

a

b

c

Load

n

Figure 4.3 Wye-connected loads. 13

Delta-Wye Connections

Figure 4.4 shows delta-connected loads; the series

and parallel impedance combination techniques

can not be used for a delta connected load.

Z 

Z 

Z 

a

c

b

Load

Z



Z



Z 

a

b

c

Load

Figure 4.4 Delta-connected loads. 14

Delta-Wye Connections

The equations for the delta to wye transformation

are

1 2 3

2 3

1 2 3

1 3

1 2 3

1 2

Z Z Z

Z Z Z

Z Z Z

Z Z Z

Z Z Z

Z Z Z

=

=

=

c

b

a

These relations can

be read as the

impedance next to a

particular Y node is

the product of the two

delta impedances

connected to that

node divided by the

sum of the three delta

impedances. 16

Delta-Wye Connections

▪ The reverse transformation (Y to Δ) can be

performed by the equations below

▪ For balanced load, when the impedances are

equal

a

a b b c c a

b

a b b c c a

c

a b b c c a

Z

Z Z Z Z Z Z

Z

Z

Z Z Z Z Z Z

Z

Z

Z Z Z Z Z Z

Z

3

2

1

Z (^) Δ = 3 ZY 17

Wye Connected Generator

Figure 4.6 Three-phase wye-connected generator

n

a

Vab

Ia

Van

Vcn Vbn

b

Ib

Ic c

Vbc

Vca

▪ Figure 4.6 exhibits a three-phase generator connected

in a wye (Y) configuration.

▪ The generator is represented by three voltage sources.

▪ The magnitudes of the ac

source voltages are the

same.

▪ The phase shift between the

voltages is 120º.

▪ This system has three phase

conductors ( a , b and c ) and

a grounded neutral ( n ).

19

Wye Connected Generator

▪ The voltage of phase a ( Van ) is selected as the

reference with a phase angle of .

▪ The phasor voltage expressions are

where VP is the phase voltage magnitude between

the phase conductors and neutral and is called the

line-to-neutral voltage****.

▪ An important property of this balanced voltage set is

P

P

P

V

V

V

cn

bn

an

V

V

V

Van + Vbn + Vcn = 0

VP is the rms value

of the voltage

20