Review

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Review of PhasorsGoal of phasor analysis is to simplify the analysis of constant frequency ac systems:

v(t) = Vmax cos(t + v),i(t) = Imax cos(t + I),

where:•v(t) and i(t) are the instantaneous voltage and current as a function of time t,• is the angular frequency (2πf, with f the frequency in Hertz), • Vmax and Imax are the magnitudes of voltage and current sinusoids,•v and I are angular offsets of the peaks of sinusoids from a reference waveform.

Root Mean Square (RMS) voltage of sinusoid:

2 maxmax

0

1( ) , so 2 .

2

T VV v t dt V V

T

2

Phasor Representation

The RMS, cosine-referenced voltage phasor is:

,

( ) Re 2 ,

cos sin ,

cos sin .

V

V

jV

jj t

V V

I I

V V e V

v t V e e

V V j V

I I j I

3

Phasor Analysis

0

2 2

Resistor ( ) ( )

( )Inductor ( )

1 1Capacitor ( ) ( ) (0)

C

= Impedance ,

= Resistance,

= Reactance,

Z = , =arctan .

t

v t Ri t V RI

di tv t L V j LI

dt

v t i t dt v V Ij C

Z R jX Z

R

X

XR X

R

Device Time Analysis Phasor

(Note: Z is a complex number but

not a phasor).

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Complex Power

max

max

max max

( ) ( ) ( ),

( ) = cos( ),

(t) = cos( ),

1cos cos [cos( ) cos( )],

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( ) [cos( )2cos(2 )].

V

I

V I

V I

p t v t i t

v t V t

i I t

p t V I

t

Instantaneous Power :

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Complex Power, cont’d

max max

0

max max

1( ) [cos( ) cos(2 )],

2

1( ) ,

1cos( ),

2cos( ),

= = .

V I V I

T

avg

V I

V I

V I

p t V I t

P p t dtT

V I

V I

Instantaneous Power is

Power F

sum

acto

of average

r Angl

and varying te

e

rms :

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Complex Power

*

cos( ) sin( ) ,

,

= Real Power (W, kW, MW),

= Reactive Power (VAr, kVAr, MVAr),

= magnitude of power into electric and magnetic fields,

= Complex power (VA, kVA, MVA),

Power Factor

,

(pf

V I V IS V I j

P jQ

V

P

Q

I

S

) = cos ,

If current leads voltage then pf is leading,

If current lags voltage then pf is lagging.

(Note: S is a complex number but not a phasor.)

7

Complex Power, cont’d

2

1

Relationships between real, reactive, and complex power:

cos ,

sin 1 pf ,

Example: A load draws 100 kW with a leading pf of 0.85.What are (power factor angle), and ?

cos 0.85 31.8 ,

10

P S

Q S S

Q S

S

0kW117.6 kVA,

0.85117.6sin( 31.8 ) 62.0 kVAr.Q

8

example

9

ZL=jwL=j*1000*1*10^-3 =j

10

ExampleEarlier we foundI = 20-6.9 amps

*

* *

2

* *

2

100 30 20 6.9 2000 36.9 VA,

36.9 pf = 0.8 lagging,

( ) 4 20 6.9 20 6.9 ,

1600W ( 0),

( ) 3 20 6.9 20 6.9 ,

1200VA r , ( 0).

R R

R R

L L

L L

S V I

S V I RI I

P I R Q

S V I jXI I j

Q I X P

= 1600W + j1200VAr

Power flowing from source to load at bus

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Power Consumption in Devices

2Resistor Resistor

2Inductor Inductor L

2Capacitor Capacitor C

Capaci

Resistors only consume real power:

,

Inductors only "consume" reactive power:

,

Capacitors only "generate" reactive power:

1.C

P I R

Q I X

Q I X XC

Q

2Capacitor

torC

(Note-some define negative.). CXV

X

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Example

*

40000 0400 0 Amps

100 0

40000 0 (5 40) 400 0

42000 16000 44.9 20.8 kV

44.9k 20.8 400 0

17.98 20.8 MVA 16.8 6.4 MVA

VI

V j

j

S V I

j

First solvebasic circuitI

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Example, cont’dNow add additionalreactive power loadand re-solve, assumingthat load voltage is maintained at 40 kV.

70.7 0.7 lagging

564 45 Amps

59.7 13.6 kV

33.7 58.6 MVA 17.6 28.8 MVA

LoadZ pf

I

V

S j

14

59.7 kV

17.6 MW

28.8 MVR

40.0 kV

16.0 MW

16.0 MVR

17.6 MW 16.0 MW-16.0 MVR 28.8 MVR

Power System NotationPower system components are usually shown as“one-line diagrams.” Previous circuit redrawn.

Arrows areused to show loads

Generators are shown as circles

Transmission lines are shown as a single line

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Reactive Compensation

44.94 kV

16.8 MW

6.4 MVR

40.0 kV

16.0 MW

16.0 MVR

16.8 MW 16.0 MW 0.0 MVR 6.4 MVR

16.0 MVR

Key idea of reactive compensation is to supply reactivepower locally. In the previous example this canbe done by adding a 16 MVAr capacitor at the load.

Compensated circuit is identical to first example with just real power load. Supply voltage magnitude and line current is lower with compensation.

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Reactive Compensation, cont’d Reactive compensation decreased the line flow

from 564 Amps to 400 Amps. This has advantages: – Lines losses, which are equal to I2 R, decrease,– Lower current allows use of smaller wires, or

alternatively, supply more load over the same wires,– Voltage drop on the line is less.

Reactive compensation is used extensively throughout transmission and distribution systems.

Capacitors can be used to “correct” a load’s power factor to an arbitrary value.

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Power Factor Correction Example

1

1desired

new cap

cap

Assume we have 100 kVA load with pf=0.8 lagging,

and would like to correct the pf to 0.95 lagging

80 60 kVA cos 0.8 36.9

PF of 0.95 requires cos 0.95 18.2

80 (60 )

60 -ta

80

S j

S j Q

Q

cap

cap

n18.2 60 26.3 kVAr

33.7 kVAr

Q

Q

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Distribution System Capacitors

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Balanced 3 Phase () SystemsA balanced 3 phase () system has:

– three voltage sources with equal magnitude, but with an angle shift of 120,

– equal loads on each phase,– equal impedance on the lines connecting the

generators to the loads. Bulk power systems are almost exclusively 3.Single phase is used primarily only in low

voltage, low power settings, such as residential and some commercial.

Single phase transmission used for electric trains in Europe.

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Balanced 3 -- Zero Neutral Current

* * * *

(1 0 1 1

3

Note: means voltage at point with respect to point .

n a b c

n

an a bn b cn c an a

xy

I I I I

VI

Z

S V I V I V I V I

V x y

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Advantages of 3 Power

Can transmit more power for same amount of wire (twice as much as single phase).

Total torque produced by 3 machines is constant, so less vibration.

Three phase machines start more easily than single phase machines.

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Three Phase - Wye ConnectionThere are two ways to connect 3 systems:

– Wye (Y), and– Delta ().

an

bn

cn

Wye Connection Voltages

V V

V V

V V

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Wye Connection Line Voltages

Van

Vcn

Vbn

VabVca

Vbc

-Vbn

(1 1 120

3 30

3 90

3 150

ab an bn

bc

ca

V V V V

V

V V

V V

Line to linevoltages arealso balanced.

(α = 0 in this case)

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Wye Connection, cont’dWe call the voltage across each element of a

wye connected device the “phase” voltage.We call the current through each element of a

wye connected device the “phase” current.Call the voltage across lines the “line-to-line” or

just the “line” voltage.Call the current through lines the “line” current.

6

*3

3 1 30 3

3

j

Line Phase Phase

Line Phase

Phase Phase

V V V e

I I

S V I

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Delta Connection

IcaIc

IabIbc

Ia

Ib

*3

For Delta connection,

voltages across elements

equals line voltages

For currents

3

3

a ab ca

ab

b bc ab

c ca bc

Phase Phase

I I I

I

I I I

I I I

S V I

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Three Phase Example

Assume a -connected load, with each leg Z = 10020 is supplied from a 3 13.8 kV (L-L) source

13.8 0 kV

13.8 0 kV

13.8 0 kV

ab

bc

ca

V

V

V

13.8 0 kV138 20 amps

138 140 amps 138 0 amps

ab

bc ca

I

I I

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Three Phase Example, cont’d

*

138 20 138 0

239 50 amps

239 170 amps 239 0 amps

3 3 13.8 0 kV 138 amps

5.7 MVA

5.37 1.95 MVA

pf cos 20 lagging

a ab ca

b c

ab ab

I I I

I I

S V I

j

28

Delta-Wye Transformation

Y

Linephase

To simplify analysis of balanced 3 systems:

1) Δ-connected loads can be replaced by 1

Y-connected loads with 3

2) Δ-connected sources can be replaced by

Y-connected sources with 3 30

Z Z

VV

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Delta-Wye Transformation Proof

For the

Suppose

side

the two sides have identical te

we get

rminal behavior

Hence

.

ab ca ab caa

ab ca

a

V V V VI

Z Z Z

V VZ

I

30

+

-

Delta-Wye Transformation, cont’dFor the side we get

( ) ( )

(2 )

Since 0

Hence 3

3

1Therefore

3

ab Y a b ca Y c a

ab ca Y a b c

a b c a b c

ab ca Y a

ab caY

a

Y

Y

V Z I I V Z I I

V V Z I I I

I I I I I I

V V Z I

V VZ Z

I

Z Z

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3 phase power calculation

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