1EENG 224

Chapter 11AC Power Analysis

Huseyin BilgekulEeng224 Circuit Theory II

Department of Electrical and Electronic Engineering Eastern Mediterranean University

Chapter Objectives: Know the difference between instantaneous power and average

power Learn the AC version of maximum power transfer theorem Learn about the concepts of effective or Rms value Learn about the complex power, apparent power and power

factor Understand the principle of conservation of AC power Learn about power factor correction

2EENG 224

An Electical Power Distribution Center

3EENG 224

Apparent Power and Power Factor

12 cos( ) cos( )m m v i Rms Rms v iP V I V I

1

2 m m Rms RmsS V I V I

Apparent Power

cos

Power Factor

( )v i

Ppf

S pS

P f

The Apparent Power is the product of the Rms value of voltage and current. It is measured in Volt amperes (VA).

The Power Factor (pf) is the cosine of the phase difference between voltage and current. It is also the cosine of the angle of load impedance. The power factor may also be regarded as the ratio of the real power dissipated to the apparent power of the load.

The Average Power depends on the Rms value of voltage and current and the phase angle between them.

4EENG 224

Apparent Power and Power Factor Not all the apparent power is consumed if the circuit is partly reactive.

Purely resistive load (R)

θv– θi = 0, Pf = 1 P/S = 1, all power are consumed

Purely reactive load (L or C)

θv– θi = ±90o,

pf = 0

P = 0, no real power consumption

Resistive and reactive load (R and L/C)

θv– θi > 0

θv– θi < 0

• Lagging - inductive load• Leading - capacitive loadP/S < 1, Part of the apparent

power is consumed

5EENG 224

6EENG 224

Power equipment are rated using their appparent power in KVA.

7EENG 224

Apparent Power and Power

Factor

Both have same P

Apparent Powers and pf’s are different

Generator of the second load is overloaded

8EENG 224

Apparent Power and Power Factor

Overloading of the generator of the second load is avoided by applying power factor correction.

9EENG 224

Complex Power The COMPLEX Power S contains all the information pertaining to the power absorbed by a given load.

221

2Rms

Rms

VI Z

Z

Rms RmsS VI V I

( )

cos( )

Re{ } Real Powe

sin( )

Im{ } Reactive Power+ r

Rms Rms v i

Rms Rms Rms Rms v i

Rms v Rm

v

s i

i jV I

V I

V I

P j

I

Q

V

j

Rms RmsI

S

S

S

V

10EENG 224

The REAL Power is the only useful power delivered to the load.

The REACTIVE Power represents the energy exchange between the source and reactive part of the load. It is being transferred back and forth between the load and the source

The unit of Q is volt-ampere reactive (VAR)

2 2 ( )Rms Rms jXI Z R P QI j S

Re{ }

=Real Power

Im{ }

+Reactive Power

jQP j S S S

2

2= cos( ) Re

= sin( )

{

Im

}

{ }

Rms Rms v i Rm

Rms

s

Rms v i RmsQ

P V I I

V I

R

I X

S

S

Complex Power

11EENG 224

Resistive Circuit and Real Power

1si

0 RESISTIVE

n( )sin(2 )2

sin( )sin(2 )

1cos( ) 1 cos(2 )

2cos( ) 1 cos(2 )

( ) sin( ) ( ) sin( )

( ) ( ) ( )

cos(2 )

m m

Rms Rms

Rms R

m m

Rms Rms

Rms Rm

m

ms

m

s

V I t

V I

v t V t i t I t

p V I tt v t

V I t

VV I

t

I

i t

t

p t( ) is always Positive

12EENG 224

1cos( ) 1 cos(

1sin( )sin(2 )

2 sin( )sin(2 )

sin(

( ) sin( ) ( ) sin( )

(

90 INDUCTIVE

2 )2

cos(

) (

) 1 cos(2 )

) ( )

2

)

m m

Rms

m m

Rms Rms

Rms Rms

ms

m

L

R

m

V

v t V t i t I t

p t v I t

V I t

V

t V I t

V I t

i t

I t

( ) is equally both positive and negative, power is circula ing tLp t

Inductive Circuit and Reactive Power

13EENG 224

If the average power is zero, and the energy supplied is returned within one cycle, why is a reactive power of any significance?

At every instant of time along the power curve that the curve is above the axis (positive), energy must be supplied to the inductor, even though it will be returned during the negative portion of the cycle. This power requirement during the positive portion of the cycle requires that the generating plant provide this energy during that interval, even though this power is not dissipated but simply “borrowed.”

The increased power demand during these intervals is a cost factor that must that must be passed on to the industrial consumer.

Most larger users of electrical energy pay for the apparent power demand rather than the watts dissipated since the volt-amperes used are sensitive to the reactive power requirement.

The closer the power factor of an industrial consumer is to 1, the more efficient is the plant’s operation since it is limiting its use of “borrowed” power.

Inductive Circuit and Reactive Power

14EENG 224

1cos( ) 1 cos(2 )

1sin( )sin(2 )

2 sin( )sin(2 )

si 9

( ) sin( ) ( ) sin( )

(

n(

) ( ) ( )

0 CAPACITIVE

2cos(

) 1 cos(2 )

2

)

m m

Rms

m m

Rms Rms

Rms Rms

Rms

m m

C V I t

V I t

V I t

v t V t i t I t

p t v t V I t

V I t

i t

( ) is equally both positive and negative, power is circulatingCp t

Capacitive Circuit and Reactive Power

15EENG 224

Complex Power The COMPLEX Power contains all the information pertaining to the power absorbed by a given load.

2 2

Complex Power

Apparent Power

Real Pow

1

er

Reactive

= ( )2

=

= Re{ } cos( )

=Q Im{ } s Power

Power Fa

in( )

= =cctor os( )

Rms Rms v i

Rms Rms

v i

v i

v i

P jQ V I

S V I P Q

P S

S

P

S

S VI

S

S

S

• Real Power is the actual power dissipated by the load.

• Reactive Power is a measure of the energy exchange between source and reactive part of the load.

16EENG 224

Power Triangle

a) Power Triangle b) Impedance Triangle Power Triangle

The COMPLEX Power is represented by the POWER TRIANGLE similar to IMPEDANCE TRIANGLE. Power triangle has four items: P, Q, S and θ.

0 Resistive Loads (Unity )

0 Capacitive Loads (Leading )

0 Inductive Loads (Lagging )

Q Pf

Q Pf

Q Pf

17EENG 224

T

100 200 300 600 Watt

0 700 1500 800Var

S 600 800 1000 53.13

T

T

P

Q

j

Power Triangle Finding the total COMPLEX Power of the three loads.

18EENG 224

1 2 1 2 1 2( ) ( )S P jQ S S P P j Q Q

Power Triangle

19EENG 224

Real and Reactive Power Formulation

20EENG 224

Real and Reactive Power Formulation

21EENG 224

Real and Reactive Power Formulation

22EENG 224

P is the REAL AVERAGE POWER

Q is the maximum value of the circulating power flowing back and forward

cos sinrms rms rms rmsP V I Q V I

Real and Reactive Power Formulation

( ) cos( ) ( ) cos( )

( ) cos( ) sin( )

=

=Real Power R eactive

sin 2( )

si

1 cos 2(

n

P

)

1 cos 2(

owe

2 )

r

()

m v m i

Rms Rms v i Rms Rms v i

v

v

v

v

v t V t i t I t

p t V I V I

P Q

t

t

t

t

23EENG 224

Real and Reactive Powers

CIRCULATING POWER

REAL POWER

24EENG 224

Real and Reactive Powers

• Vrms =100 V Irms =1 A Apparent power = Vrms Irms =100 VA• From p(t) curve, check that power flows from the supply into the load for the entire duration of the cycle!• Also, the average power delivered to the load is 100 W. No Reactive power.

25EENG 224

Real and Reactive Powers

• Vrms =100 V Irms =1 A Apparent power = Vrms Irms =100 VA• From p(t) curve, power flows from the supply into the load for only a part of the cycle! For a portion of the cycle, power actually flows back to the source from the load!• Also, the average power delivered to the load is 50 W! So, the useful power is less than in Case 1! There is reactive power in the circuit.

Power Flowing Back

26EENG 224

Practice Problem 11.13: The 60 resistor absorbs 240 Watt of average power. Calculate V and the complex power of each branch. What is the total complex power?

27EENG 224

Practice Problem 11.13: The 60 resistor absorbs 240 Watt of average power. Calculate V and the complex power of each branch. What is the total complex power?

28EENG 224

Practice Problem 11.14: Two loads are connected in parallel. Load 1 has 2 kW, pf=0.75 leading and Load 2 has 4 kW, pf=0.95 lagging. Calculate the pf of two loads and the complex power supplied by the source.

LOAD 12 kWPf=0.75Leading

LOAD 24 kWPf=0.95Lagging

29EENG 224

30EENG 224

Conservation of AC Power The complex, real and reactive power of the sources equal the respective sum of the complex, real and reactive power of the individual loads.

a) Loads in Parallel b) Loads in Series

For parallel connection:

* * * * *1 2 1 2 1 2

1 1 1 1S V I V (I I ) V I V I S S

2 2 2 2

Same results can be obtained for a series connection.

31EENG 224

32EENG 224

33EENG 224

Complex power is Conserved