Power Electronics chapter02-2
Transcript of Power Electronics chapter02-2
-
7/26/2019 Power Electronics chapter02-2
1/12
1
Power
Electronics
CHAPTER 2
Power Computat ions
-
7/26/2019 Power Electronics chapter02-2
2/12
2
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
-
7/26/2019 Power Electronics chapter02-2
3/12
3
Power
Electronics
Apparent power and power factor
)()()()()()()()( 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
-
7/26/2019 Power Electronics chapter02-2
4/12
4
Power
Electronics
Apparent power and power factor
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
-
7/26/2019 Power Electronics chapter02-2
5/12
5
Power
Electronics
Apparent power and power factor
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
-
7/26/2019 Power Electronics chapter02-2
6/12
6
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)
-
7/26/2019 Power Electronics chapter02-2
7/12
7
Power
Electronics
Power computation (Nonsinusoidal periodic)
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
-
7/26/2019 Power Electronics chapter02-2
8/12
8
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
Power computation (Nonsinusoidal periodic)
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
-
7/26/2019 Power Electronics chapter02-2
9/12
9
Power
Electronics
2p
D1
D2
D3
D4
Power computation (Nonsinusoidal periodic)Sinusoidal source and nonlinear load
D1
D2
D3
D4
-
7/26/2019 Power Electronics chapter02-2
10/12
10
Power
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.
-
7/26/2019 Power Electronics chapter02-2
11/12
11
Power
Electronics
Power computation (Nonsinusoidal periodic)
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
-
7/26/2019 Power Electronics chapter02-2
12/12
12
Power
Electronics
Power computation (Nonsinusoidal periodic)
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