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AC POWER CALCULATIONAC POWER CALCULATIONInstantaneous, average and reactive powerInstantaneous, average and reactive power

Apparent Power and Power FactorApparent Power and Power FactorComplex PowerComplex Power

Dr. Nik Rumzi Nik IdrisDr. Nik Rumzi Nik Idris

SEE 1023 Circuit TheorySEE 1023 Circuit Theory

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Instantaneous, Average and Reactive PowerInstantaneous, Average and Reactive Power

+v(t)

i(t)

Passive, linear network

Instantaneous power absorbed by the network is, p =v(t).i(t)

Let v(t) = Vm cos (t + v) and i(t) = Imcos(t + i)

Which can be written as

v(t) = Vm cos (t + v i) and i(t) = Imcos(t)

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v(t) = Vm cos (t + v i) and i(t) = Imcos(t)

p = Vm cos(t + v – i ) . Im cos(t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

-2

-1

0

1

2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-0.5

0

0.5

1

1.5

2

vi

Instantaneous Power (p)

Example when v i = 45o

positivepositive p p = power transferred from source to network

negativenegative p p = power transferred from network to source

45o

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v(t) = Vm cos (t + v i) and i(t) = Imcos(t)

p = Vm cos(t + v – i ) . Im cos(t)

t2sin)sin(2IVt2cos)cos(

2IV)cos(

2IV

ivmm

ivmm

ivmm p =

p = P + Pcos(2t) Qsin(2t)

Using trigonometry functions, it can be shown that:

)cos(2IVP ivmm = AVERAGE POWER (watt)

)sin(2IVQ ivmm = REACTIVE POWER (var)

Which can be written as

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t2sin)sin(2IVt2cos)cos(

2IV)cos(

2IV

ivmm

ivmm

ivmm p =

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t2sin)sin(2IVt2cos)cos(

2IV)cos(

2IV

ivmm

ivmm

ivmm p =

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

0

1

2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.5

1

1.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

-0.5

0

0.5

1

Example for v-i = 45o

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t2sin)sin(2IVt2cos)cos(

2IV)cos(

2IV

ivmm

ivmm

ivmm p =

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

0

1

2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.5

1

1.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

-0.5

0

0.5

1

P = average power

Q = reactive power

p = P + P cos(2t) Q sin(2t)

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P = AVERAGE POWER

Q = REACTIVE POWER

p = P + P cos(2t) Q sin(2t)

• Useful power – also known as ACTIVE POWER

• Converted to other useful form of energy – heat, light, sound, etc

• Power charged by TNB

• Power that is being transferred back and forth between load and source

• Associated with L or C – energy storage element – no losses

• Is not charged by TNB

• Inductive load: Q positive, Capacitive load: Q negative

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Power for a resistorPower for a resistor

t2sin)sin(2IVt2cos)cos(

2IV)cos(

2IV

ivmm

ivmm

ivmm p =

0)( iv Voltage and current are in phase,

t2cos0cos2IV0cos

2IV mmmm p = )t2cos1(

2IV mm p =

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2

-1

0

1

2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-0.5

0

0.5

1

1.5

2

2.5

P = average power = 2IV mm

Q = reactive power = 0

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Power for an inductorPower for an inductor

t2sin)sin(2IVt2cos)cos(

2IV)cos(

2IV

ivmm

ivmm

ivmm p =

oiv 90)( Voltage leads current by 90o,

P = average power = 0

t2sin)90sin(2IV Omm p = t2sin

2IV mm p =

Q = reactive power = 2IV mm

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2

-1

0

1

2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2

-1

0

1

2

v i

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Power for a capacitorPower for a capacitor

t2sin)sin(2IVt2cos)cos(

2IV)cos(

2IV

ivmm

ivmm

ivmm p =

oiv 90)( Voltage lags current by 90o,

P = average power = 0

t2sin)90sin(2IV Omm p = t2sin

2IV mm p =

Q = reactive power = 2IV mm

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2

-1

0

1

2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1

-0.5

0

0.5

1

1.5

v i

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Apparent Power and Power FactorApparent Power and Power Factor

Consider v(t) = Vm cos (t + v) and i(t) = Imcos(t + i)

We have seen, )cos(2IVP ivmm

2I

2V mm

rmsrms IV = Is known as the APPARENT POWERAPPARENT POWER

rmsrms IVS VA

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Apparent Power and Power FactorApparent Power and Power Factor

rmsrms IVS

We can now write,

)cos(SP iv

The term )cos( iv is known as the POWER FACTORPOWER FACTOR

)cos(SPpfFACTORPOWER iv

For inductive load, (v i) is positive current lags voltage lagging pflagging pf

For capacitive load, (v i) is negative current leads voltage leading pfleading pf

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Apparent Power and Power FactorApparent Power and Power Factor

)cos(SPpfFACTORPOWER iv

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Apparent Power and Power FactorApparent Power and Power Factor

)cos(SPpfFACTORPOWER iv

Irms = 5- 40o

Vrms = 25010o

Load+Source

+

VL

Active power absorbed by the load is 250(5) cos (50o)= 1250(0.6428) = 803.5 watt

Power factor of the load = cos (10-(-40)) = cos (50o) = 0.6428

Apparent power, S = 1250 VA

Reactive power absorbed by load is 250(5) sin (50o)= 1250(0.6428) = 957.56 var

(lagging)

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Complex PowerComplex Power

Defined as:

2*IVS

(VA)

Where, vmV V imI I imI I*and

If we let vrmsvm V2

VrmsV irmsi

m I2

IrmsIand

rmsrms *IVS (VA)

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Complex PowerComplex Power

2*IVS

(VA)

Where, imvm IV21

S

)(IV21

ivmm

)(IV ivrmsrms

)sin(IjV)cos(IV ivrmsrmsivrmsrms

jQP

)-(S) iv cosIV(P rmsrmsRe )-(S) iv sinIV(Q rmsrmsIm

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Complex PowerComplex Power

jQP SThe complex power contains all information about the load

Irms = 5- 40o

Vrms = 25010o

Load+Source

+

VL

We have seen before:

Active power, P = 803.5 watt

Apparent power, S = 1250 VA

Reactive power, Q = 957.56 var

803.5 watt

957.56 varS S = (803.5 + j957.56) VA

S = 1250 50o VA

|S| = S = Apparent power

S = 25010o (5-40o) VA

= 1250 VA

With complex power,

50o

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Complex PowerComplex Power

rmsrms *IVS

Other useful forms of complex powerOther useful forms of complex power

rmsrms ZIV We know that

rmsrms *IZIS

2rmsIZS

)jXR(2 rmsIS

)XjR( 22rmsrms IIS

PP QQ

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Complex PowerComplex Power

rmsrms *IVS

Other useful forms of complex powerOther useful forms of complex power

Zrms

rmsVI We know that

*

Z

rms

rmsVVS

Z

2rmsVS

For a pure resistive element, R

2rmsVP

For a pure reactive element, X

2rmsVQ

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Conservation of AC PowerConservation of AC Power

Complex, real, and reactive powers of the sources equal the respective sums of the complex, real and reactive powers of the individual loads

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Conservation of AC PowerConservation of AC Power

Complex, real, and reactive powers of the sources equal the respective sums of the complex, real and reactive powers of the individual loads

Ss = Ps +jQs = (P1 + P2 + P3) + j (Q1 + Q2 + Q3)

But

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Maximum Average Power TransferMaximum Average Power Transfer

Max power transfer in DC circuit can be applied to AC circuit analysis

ZL

+

V

IZTh

VTh +

AC linear circuit

What is the value of ZL so that maximum averageaverage power is transferred to it?

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Maximum Average Power TransferMaximum Average Power Transfer

ZL

+

V

IZTh

VTh +

What is the value of ZL so that maximum averageaverage power is transferred to it?

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Maximum Average Power TransferMaximum Average Power Transfer

ZL

+

V

IZTh

VTh +

What is the value of ZL so that maximum averageaverage power is transferred to it?

ZTh= RTh + jXTh

ZL= RL + jXL

L2R

21P I P max when 0

RP

L

0

XP

L

and

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Maximum Average Power TransferMaximum Average Power Transfer

ZL

+

V

IZTh

VTh +

What is the value of ZL so that maximum averageaverage power is transferred to it?

P max when 0RP

L

0

XP

L

and

)jXR()jXR( LLThTh

Th

VI

L2R

21P I

2R

)XX()RR(L

2ThL

2LTh

2

ThVP

XL = XTh , RL= RTh