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
3-1
AVERAGE POWER AND RMS CURRENT
p (t )  v i
T
1
Pav   v i dt
T 0
caso a carga seja apenasuma resist ência v  R i,
T
1 2
2
Pav  R  i dt  R I RMS
T 0
T
I RMS
1 2

i dt

T 0
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-2
Sinusoidal Steady State
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-3
Phasor Representation
v(t )  V cos( t  0)
utilizandoa notaçãoexponencial
v(t )  V e j ( t  0 )  V cos( t  0)  j V sin( t  0)
notaçãofasorial
v(t )  V e j 0  V0
no caso de uma carga complexa
Z  R  j  L  R 2   L  e j arctg ( L / R )
2
Z  Z e j   Z 
V V 0
I 
 I  
Z Z 
Electrónica de Potência
© 2008 José Bastos
Z
jL

R
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-4
Power, Reactive Power, Power Factor
Ip
Complex power S
S  V I*  V e j 0 I e j  V I e j
 j Iq
V  V 0
I  I  
Real power is
P  ReS  V I p  V I cos
Reactive power is
Q  ImS  V I q  V I sin   S  P
2
Power Factor
2
P V I cos
 
 cos
S
VI
Ideally power factor should be 1.0
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-5
Example
An 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.
for t heload
P  1000W
S
Q
P
 1250VA
PF
S 2  P 2  750VA
t hecomplexpower is
S  P jQ
 1000 j 750
In order t oget a power fact orof 0.95one has
S  P  j QL  j QC
 P  j (QL  QC )
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-6
Example (end)
P
S  P  (QL  QC ) 
0.95
2
(QL  QC ) 
2
P2
2

P
 328.7 VA
2
0.95
t herefore
QC  750 328.7  421.3VA
but t hereact ivepower in a capacit ance is
V2
V2
QC 

V 2 C
Z C 1 /( C )
QC
421.3
C 2 
 77.6 F
2
V  120  2    60
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-7
Non-sinusoidal waveforms in steady state
-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
3-8
Non-sinusoidal waveforms in steady state (cont.)
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-9
Fourier Analysis
-a non sinusoidal waveform f(t) repeating with angular frequency  can be
expressed as

f (t )  F0  
n 1
an 
bn 
1

1



1
f n (t )  a0  an cos(nt )   bn sin(nt )
2
n 1
n 1

 f (t ) cos(nt )d (t )


 f (t ) sin(nt )d (t )

Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-10
Fourier Analysis
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-11
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
3-12
Line-Current Distortion
vs (t )  Vs sin(t )
is (t )  is1 (t )   isn (t )
n 1
is (t )  I s1 sin(1t  1 )   I s n sin(n1t   n )
n 1
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-13
Total Harmonic Distortion -THD
1/ 2
 1 T1 2 
I RMS  I s    is (t ) 
T

 10

because theintegralsof all thecross- product terms are zero,
1/ 2


I RMS   I s21   I sn2 
n 1


idist (t )  is (t )  is1 (t )


1/ 2


I dist  I s2  I
   I sn2 
 n 1 
totalharmonicdistortionis defined as
2 1/ 2
s1
THD  100
I  I 
I dist
 100  2 
I s1
 I s1 
2
s
2
s1
1/ 2
%
I sn2
THD  100  2 %
n 1 I s 1
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-14
POWER FACTOR - PF
-starting with the definition of average power
1
P
T1
T1
T
1 1
0 p(t )dt  T1 0 vs (t )is (t )dt
onceagain all cross product sare zero
T
1 1
P   Vs sin(1t ) I s1 sin(1t  1 )dt  Vs I s1 cos(1 )
T1 0
t heapparentpower S is
S  Vs I s
W e define t hepower fact oras
PF 
P Vs I s1 cos(1 ) I s1 cos(1 )


S
Vs I s
Is
T heDisplacement fact orDP Fis
DPF  cos(1 )
PF 
I s1
DPF
Is
T heP F can be expressedas
PF 
Electrónica de Potência
© 2008 José Bastos
1
1  THD2
DPF
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-15
Inductor and Capacitor Phasors (1)
Capacitor
vL (t )  V cos(t )  vL  V e
j t
dvL (t )
j / 2
i (t )  C
 jCv L (t )  Cv L (t )e
dt
vL (t )
1
j
1  j / 2
ZC 



e
i (t )
jC C C
1

  / 2
C
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-16
Inductor and Capacitor Phasors (2)
Inductor
i (t )  I cos(t )  i  I e
jt
di(t )
j / 2
vL (t )  L
 jLi(t )  Li(t )e
dt
vL (t )
j / 2
ZL 
 jL  Le
 L   / 2
i (t )
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-17
Phasor Representation
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-18
** Inductor and Capacitor Response
Capacitor
dvC
iC  C
dt
t
1
vC (t )  vC (t1 )   iC (t )dt t  t1
C t1
Induct or
diL
vL  L
dt
t
1
iL (t )  iL (t1 )   vL (t )dt t  t1
L t1
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-19
Response of L and C
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-20
Average vL and iC in steady state
-steady-state condition implies that voltage and current waveforms
repeat with a time period T:
v(t  T )  v(t )
and
i(t  T )  i(t )
in case of capacitor
1
T
t1 T
i
C
dt  0
t1
 averagecapacitorcurrentmust be zero
in case of inductor
1
T
t1 T
 v dt  0
L
t1
 averageinductor voltagemust be zero
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-21
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
3-22
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
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
3-24
Ampere’s Law
magnet icfield of int ensit yH :
 H dl   i
For most pract icalcircuit s
H l  N
k k
k
i
m m
m
densit y of magnet icflux B :
B H
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-25
Direction of Magnetic Field
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-26
Flux Lines
   B dA
A
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-27
Magnetic Flux 
For most practicalcases
 H k lk   H k ( k Ak )
k
k
lk
 k Ak
  Bk Ak
k
lk
 k Ak
  k
Equation H k lk   N mim can be written
k


k
lk
k
Ak
k
lk
 k Ak


k
lk
k
Ak
m
  N mim
m
or
 
N
i
m m
m

k
lk
k

N
i
m m
m

Ak
where  is themagneticreluctance
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-28
Concept of Magnetic
Reluctance
Ni

l /( A)
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-29
Faraday’s Law
d
dt
v
Electrónica de Potência
© 2008 José Bastos
d ( N )
d
v
N
dt
dt
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-30
Definition of Self-inductance L
N Li
d ( N )
Equationv 
can be written
dt
di
vL
dt
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-31
Inductance L
A
l
N
L
i
for a toroidalcoil
Ni

l/(  A)
2
N
L
A
l
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-32
Analysis of a Transformer (1)
d1
v1  R1 i1  N1
dt
d 2
v2   R2 i2  N 2
dt
Very simple case:
R1  R2  0
1  2
N2
v2  
v1
N1
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-33
Analysis of a Transformer (2)
1    l1
2    l 2
being l1 , l 2 theleakagefluxes, and  theflux in t hecore
N1im
N1i1  N 2i2

l /(  A)
l /(  A)
with
N 2i2
im  i1 
N1

T he voltagesv1 and v2 can be written
N12 di1
N12 dim
v1  R1 i1 

l1 /(  A) dt l /(  A) dt
N 22 di2
N 2 N1 dim
v2   R2 i2 

l2 /(  A) dt l /(  A) dt
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-34
Analysis of a Transformer (3)
notingthat
N 12
2
2
N 12
N
Ll1 
; Ll 2 
; Lm 
l1 /(  A)
l2 /(  A)
l /(  A)
T he voltagesv1 and v2 can be simplified
dim
di1
di1
v1  R1 i1  Ll1
 Lm
 R1 i1  Ll1
 e1
dt
dt
dt
dim
di2 N 2
di2 N 2
v2   R2 i2  Ll 2

Lm
  R2 i2  Ll 2

e1
dt N1
dt
dt N1
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-35
Transformer Equivalent Circuit
Lm
Electrónica de Potência
© 2008 José Bastos
N2/N1 Lm
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
3-36
Ideal Transformers
v1  e1
1. R1=R2=0
N2
N2
v2  e2  
e1  
v1
N1
N1
2. Ll1=Ll2=0
3. Core permeability =
 l /(  A)  N1i1  N 2i2  0
i2 N1

i1 N 2
v2i2  
Electrónica de Potência
© 2008 José Bastos
Chapter 3 Basic Electrical and
Magnetic Circuit Concepts
N 2 N1
v1
i1  v1i1
N1 N 2
3-37