Power Electronics chapter02-2

7/26/2019 Power Electronics chapter02-2

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Power

Electronics

CHAPTER 2

Power Computat ions


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Power

Electronics

Apparent power and power factor

R

L

v

i

)cos(2)(

)cos(2)(

tIti

tVtv

rms

rms

Decompose the current with two components

tItI

ttI

tIti

rmsrms

rms

rms

sin)sin2(cos)cos2(

)sinsincos(cos2

)cos(2)(

in-phasecurrent

component

tIti rmsp cos)cos2()(

out-of-phase

(quadrature)current

component

tIti rmsq sin)sin2()(

)()()( tititi qp

Sinusoidal source and linear load


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Power

Electronics


)()()()()()()()( tptptititvtitvtp qpqp

)2cos(1)cos(

cos)cos2()cos(2

)()()(

tIV

tItV

titvtp

rmsrms

rmsrms

pp

)2sin()sin(

sin)sin2()cos(2

)()()(

tIV

tItV

titvtp

rmsrms

rmsrms

qq

average power

cos)( rmsrmsp IVtp

average power

0)( tpq

only iPis

responsible for

the power

transfer

iqdoes nottransfer the

power.

Real power (active power)

][cos WIVP rmsrms

Reactive power

][sin VARIVQ rmsrms

The reactive power is commonly used in conjunction with voltage and current for

inductors and capacitors, which transfers no net power in steady-state.

Sinusoidal source and linear load


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Power

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Apparent power (S): the magnitude of the complex power

S =S ][22 VAIVQP rmsrms product of rms voltage magnitude and rmscurrent magnitude

Complex power:combining real and reactive power in a complex form

rmsrms

j

rmsrmsrmsrmsrmsrmsrmsrms IVeIVjIVIjVIV )sin(cossincosS=

P+jQ

Power factor (pf): the ratio of average power to apparent power

rmsrmsIV

P

S

Ppf

With sinusoidal voltage and current

cospf rmsrmsIV

P

S

P

angle between the

voltage and the current

Index to represent the effectiveness of power transfer

larger inductorQpf

0v& iin-phasepure resistive (R) loadpf =1

90ov& i90o-phase-shiftpure inductive (L) load pf =0

cosrmsrmsIVP


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Power

Electronics


Physical meanings

Apparent power (S): the cost of most electrical equipment such as generators,

transformers, ad transmission lines increases with S. i.e. specifying the rating of power

equipment, because the electrical insulation level depends on Vand the conductor size

depends onI.

Real power (P) : representing useful work

Reactive power (Q) : characterized by energy storage during one-half of the cycle andenergy retrieval during the other half.

In most situations, it is desirable to have the reactive power Qbe zero. i.e. desirable to

make pf = 1


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Power

Electronics

Power computation (Nonsinusoidal periodic)

Power electronics circuits typically generate nonsinusoidal periodic currents.

Nonsinusoidal periodic waveforms can be decomposed with series of sinusoids with

different frequencies by Fourier series.

Review of Fourier series

]sincos[)(1

00 tnbtnaatf nn

n 0

2/

2/ 0

2/

2/ 0

2/

2/0

sin)(2

cos)(2

)(1

T

Tn

T

Tn

T

T

dttntfT

b

dttntfT

a

dttfT

a

1

00 )cos()(n

nn tnCatf

n

nnnnn

a

bandbaC 122 tan

a0: average value of f(t) = dc value of f(t)


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rms value of nonsinusoids

1

00 )cos()(n

nn tnCatf

1

2

2

0

0

2

,2n

n

n

rmsnrmsCaFF

Average power of nonsinusoids

1

00

1

00

cos)(

cos)(

n

nn

n

nn

tnIIti

tnVVtv

nnn

nn

n n

nnrmsnrmsnn

IVIVP

or

IVIVPP

cos2

,

)cos(

1

max,max,

00

0 1

,,00

Tt

t

o

o

dttitvT

P )()(1

The average of voltage and current products at the same frequency is

cosrmsrmsIVP

The average of voltage and current products at the different frequency = 0


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Power

Electronics

Apparent power (S): the magnitude of the complex power

S =S ][VAIV rmsrms product of rms voltage magnitude and rmscurrent magnitude

Power factor (pf): the ratio of average power to apparent power

rmsrmsIV

P

S

Ppf Index to represent the effectiveness of power transfer


1

00

1

00

cos)(

cos)(

n

nn

n

nn

tnIIti

tnVVtv

1

2

2

0

0

2

,2n

n

n

rmsnrms

VVVV

1

2

2

0

0

2

,2n

n

n

rmsnrms

IIII


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2p

D1

D2

D3

D4

Power computation (Nonsinusoidal periodic)Sinusoidal source and nonlinear load

D1

D2

D3

D4


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Electronics

Power computation (Nonsinusoidal periodic)Sinusoidal source and nonlinear load

101 sin)( tVtv

1

00 )sin()(n

nn tnIIti

11,1,1111

2

max,

1111

0

1

max,max,

00

coscos

cos2

0cos

20

cos2

rmsrms1

nn

n

n

nn

n

nn

IV2IV

IIVI

IVIVP

11,1,1

11,1,1 coscos

pf

rms

rms

rmsrms

rmsrms

rmsrms I

I

IV

IV

IVP

SP

2

1

20

0

2,

2

n

n

n

rmsnrms IIII

v

iAverage power

Power factor

The only current component at the

frequency of the source voltage contributes

to average power.

In case of sinusoidal current (linear circuit)

1111

,1coscospf

rms

rms

I

Itermed as displacement power factor (DPF)

rmsrmsIVS

Apparent power

The current component at the all frequency contributes to apparent power.


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Power

Electronics


Sinusoidal source and nonlinear load

Distortion factor (DF) : quantifies how much the current includes nonsinusoidal harmonics termsrepresents the reduction inpfdue to the nonsinusoidal property of the current waveform

rms

rms

I

IDF

,1 11

,1cospf

rms

rms

I

I

Total Harmonic Distortion (THD) : another term to quantify the nonsinusoidal harmonics terms

rms

rmsrms

rms

n

rmsn

rms

n

rmsn

I

II

I

I

I

I

THD,1

2

,1

2

,1

1

2

,

2

,1

1

2

,

DF pf

THD pf

1

2

,

n

rmsnIAs highly as the current waveform is distorted

from sinusoidal waveformreduced

effectiveness of power transfer


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Electronics


Sinusoidal source and nonlinear load

Reactive power

11,1,1 sin rmsrmsIVQ

222

1

2

,

2

,111

2

11

22

,1

2

,1

1

2

,

2

,1

2

,1

2

,1

1

2

,

2

,1,1

sincos

DQP

IVIV

IVIVIIVIVS

n

rmsnrmsrmsrms

n

rmsnrmsrmsrms

n

rmsnrmsrmsrmsrms

1

2

,,1

n

rmsnrms IVD

Apparent power

Distortion volt-amps (D)

11,1,1 sin rmsrmsIVQ 11,1,1

cos rmsrms

IVP

222 DQPS Larger displacement angle

between the fundamental voltage

and current(1 - 1) Q

Increased amount of nonsinusoidal

harmonicsD

S pf

reduced

effectiveness of

power transfer

Power Electronics chapter02-2

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