Post on 26-Mar-2015
transcript
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-1
Chapter 3
Review of Basic Electrical and Magnetic Circuit Concepts
• Electric Circuits• Phasors• Power, Power Factor• Fourier Analysis• Inductors and Capacitors• Magnetic Circuits• Transformers
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-2
AVERAGE POWER AND RMS CURRENT
T
RMS
RMS
T
av
T
av
dtiT
I
IRdtiT
RP
iRv
dtivT
P
ivtp
0
2
2
0
2
0
1
1
, aresistênci uma apenas seja carga a caso
1
)(
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-3
Sinusoidal Steady State
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-4
Phasor Representation
0)(
fasorial notação
)0sin()0cos()(
lexponencia notação a utilizando
)0cos()(
0
)0(
VeVtv
tVjtVeVtv
tVtv
j
tj
IZ
V
ZeZ
eLRLjRj
RLarctgj
0
complexa carga uma de caso no
)/(22
Z
VI
Z
Z
R
jL
Z
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-5
Power, Reactive Power, Power Factor
II
0VVpI
qIj
Complex power S
jjj eIVeIeV 0*IVSReal power is
cosRe IVIVP p SReactive power is
22sinIm PSIVIVQ q S
Power Factor cos
cos
IV
IV
S
P
Ideally power factor should be 1.0
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-6
ExampleAn inductive load connected to a 120V, 60Hz ac source draws 1 kW at a power factor of 0.8. Calculate the capacitance required in parallel with the load in order to bring the power factor to 0.95.
)(
has one 0.95 offactor power aget order toIn
7501000
ispower complex the
750
1250
1000
load for the
22
CL
CL
QQjP
QjQjPS
j
QjPS
VAPSQ
VAPF
PS
WP
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-7
Example (end)
FV
QC
CVC
V
Z
VQ
VAQ
VAPP
PQQPS
C
CC
C
CL
CL
6.77602120
3.421
)/(1
is ecapacitanc ain power reactive but the
3.4217.328750
therefore
7.32895.0
)(
95.0)(
22
222
22
2
22
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-8
Non-sinusoidal waveforms in steady state
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-9
Non-sinusoidal waveforms in steady state (cont.)
-current drawn from power electronic equipment is highly distorted
-However
-In steady state waveforms repeat with period T=1/f
-f is the fundamental frequency (f1)
-The current signal has many harmonics (multiples) of the fundamental
-The harmonics can be calculated by Fourier Analysis
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-10
Fourier Analysis
-a non sinusoidal waveform f(t) repeating with angular frequency can be expressed as
)()sin()(1
)()cos()(1
)sin()cos(2
1)()(
1 1100
tdtntfb
tdtntfa
tnbtnaatfFtf
n
n
n nnn
nn
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-11
Fourier Analysis
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-12
Distortion in the Input Current
• Voltage is assumed to be sinusoidal
• Subscript “1” refers to the fundamental
• The phase 1 is between the voltage and the current fundamental
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-13
Line-Current Distortion(1)
11111
11
)sin()sin()(
)()()(
)sin()(
nnnsss
nsnss
ss
tnItIti
tititi
tVtv
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-14
Total Harmonic Distortion -THD
%100
%100100
as defined is distortion harmonic total
)()()(
zero, are rmsproduct te-cross theall of integrals thebecause
)(1
12
2
2/1
21
21
2
1
2/1
1
22/121
2
1
2/1
1
221
2/1
0
2
1
1
1
n
sn
s
ss
s
dist
nsnssdist
ssdist
nsnsRMS
T
ssRMS
sI
ITHD
I
II
I
ITHD
IIII
tititi
III
tiT
II
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-15
POWER FACTOR - PF
-starting with the definition of average power
DPFTHD
PF
DPFI
IPF
DPF
I
I
IV
IV
S
PPF
IVS
IVdttItVT
P
dttitvT
dttpT
P
s
s
s
s
ss
ss
ss
ss
T
ss
T
ss
T
2
1
1
1111
11
0
11111
0101
1
1
as expressed becan PF The
)cos(
is DPFfactor nt Displaceme The
)cos()cos(
asfactor power thedefine We
is Spower apparent the
)cos()sin()sin(1
zero are products cross allagain once
)()(1
)(1
1
11
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-16
Inductor and Capacitor Phasors (1)
2/1
11
)(
)(
)()()(
)(
)cos()(
Capacitor
2/
2/
C
eCC
j
Cjti
tvZ
etCvtCvjdt
tdvCti
eVvtVtv
jLC
jLL
L
tjLL
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-17
Inductor and Capacitor Phasors (2)
2/)(
)(
)()()(
)(
)cos()(
Inductor
2/
2/
LLeLjti
tvZ
etLitLijdt
tdiLtv
eIitIti
jLL
jL
tj
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-18
Phasor Representation
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-19
** Inductor and Capacitor Response
11
11
1
1
)()()(
Inductor
)()()(
Capacitor
ttdttvtiti
dt
diLv
ttdttitvtv
dt
dvCi
t
t
LLL
LL
t
t
CCC
CC
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-20
Response of L and C
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-21
Average vL and iC in steady state
-steady-state condition implies that voltage and current waveforms repeat with a time period T:
)()()()( tiTtiandtvTtv
zero bemust oltageinductor v average
01
inductor of casein
zero bemust current capacitor average
01
capacitor of casein
1
1
1
1
Tt
t
L
Tt
t
C
dtvT
dtiT
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-22
Inductor Voltage and Current in Steady State
• Volt-seconds over T equal zero.
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-23
Capacitor Voltage and Current in Steady State
• Amp-seconds over T equal zero.
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-24
Ampere’s Law
• Direction of magnetic field due to currents
• Ampere’s Law: Magnetic field along a path
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-25
Ampere’s Law
HB
iNlH
idlH
mmm
kkk
:Bflux magnetic ofdensity
circuits practicalmost For
:Hintensity of field magnetic
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-26
Direction of Magnetic Field
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-27
A
dAB
Flux Lines
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-28
Magnetic Flux
reluctance magnetic theis where
or
writtenbecan Equation
)(
cases practicalmost For
mmm
k kk
k
mmm
mmm
k kk
k
mmm
kkk
k k k kk
k
kk
kk
kk
kkk
k kk
kkkk
kkk
iN
A
l
iN
iNA
l
iNlH
A
l
A
l
A
lAB
A
lAHlH
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-29
Concept of Magnetic Reluctance
)/( Al
iN
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-30
Faraday’s Law
dt
dN
dt
Nde
)(
dt
d
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-31
Definition of Self-inductance L
dt
diLe
dt
Nde
iLN
writtenbecan )(
Equation
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-32
Inductance L
Al
NL
l
i
NL
2
A)/(
iN
coil toroidalafor
A l
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-33
Analysis of a Transformer (1)
dt
dNiRv
dt
dNiRv
22222
11111
Very simple case:
11
22
21
21 0
vN
Nv
RR
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-34
Analysis of a Transformer (2)
dt
di
Al
NN
dt
di
Al
NiRv
dt
di
Al
N
dt
di
Al
NiRv
vv
N
iNii
Al
iN
Al
iNiN
m
m
m
m
ll
l
l
)/()/(
)/()/(
writtenbecan and voltagesThe
with
)/(
)/(
core in theflux the and fluxes, leakage the, being
122
2
22
222
211
1
21
111
21
1
221
12211
21
22
11
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-35
Analysis of a Transformer (3)
11
22222
1
222222
11
1111
1111
21
2
2
22
21
2
1
simplified becan and voltagesThe
)/(;
)/(;
)/(
thatnoting
11
eN
N
dt
diLiR
dt
diL
N
N
dt
diLiRv
edt
diLiR
dt
diL
dt
diLiRv
vv
Al
NL
Al
NL
Al
NL
lm
ml
lm
ml
mll
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-36
Transformer Equivalent Circuit
Lm N2/N1 Lm
Electrónica de Potência© 2008 José Bastos
Chapter 3 Basic Electrical and Magnetic Circuit Concepts
3-37
Ideal Transformers
1. R1=R2=0
2. Ll1=Ll2=0
3. Core permeability =
1112
11
1
222
2
1
1
2
2211
11
21
1
222
11
0)/(
iviN
Nv
N
Niv
N
N
i
i
iNiNAl
vN
Ne
N
Nev
ev