Juliusz B. Gajewski
Professor of Electrical Engineering
INSTITUTE OF HEAT ENGINEERING
AND FLUID MECHANICS
Electrostatics and Tribology Research Group
Fundamentals of Electrical
Engineering
WybrzeĪe S. WyspiaĔskiego 27
50-370 Wrocáaw, POLAND
1
Building A4 „Stara kotáownia”, Room 359
Tel.: +48 71 320 3201; Fax: +48 71 328 3218
E-mail: juliusz.b.gajewski@pwr.wroc.pl
Internet: www.itcmp.pwr.wroc.pl/elektra
2
Terms.
Fundamental Definitions and Units
Contents
Electrical Engineering
1.
2.
3.
4.
5.
6.
7.
E l e c t r i c a l e n g i n e e r i n g is engineering that deals with
practical applications of e l e c t r i c i t y; generally restricted to
applications involving current flow through conductors, as in motors
and generators.
E l e c t r i c a l e n g i n e e r i n g is an engineering discipline that
deals with the study and practical application of e l e c t r i c i t y and
e l e c t r o m a g n e t i s m.
Terms. Fundamental Definitions and Units.
Electrostatics. Electrostatic and Electric Fields.
Electrodynamics. DC Current.
Electromagnetism. Magnetic Field of DC Current.
Sinusoidal AC Voltage.
Electrical Measurements.
Three-Phase Circuits.
3
For electrical engineering the science of electricity is fundamental and
is the branch of physics. Physics studies, finds, and explains the principles of electrical phenomena, while electrical engineering explains
the applications of those phenomena to engineering and technology. 4
Terms.
Fundamental Definitions and Units
Terms.
Fundamental Definitions and Units
Electric Charge
Electric Charge
E l e c t r i c c h a r g e or c h a r g e is a basic property of
elementary particles of matter. One does not define charge but takes
it as a basic experimental quantity and defines other quantities in
terms of it.
Thales of Miletus (ca. 624–ca. 546 BC), a Greek, found that amber attracted different light
objects when rubbed with silk (fur). He is believed to be a discoverer of static electricity and
could be generally named a father of electricity. The Greek word for amber is ȒȜİțIJȡȠȞ
(élektron) ëelectron (English electron) from which one can get ëelectricity and ëelectronics.
The English word electric is based on the Greek amber. Both words derive from the electrostatic properties of amber. It is also said that “a first usage of the word e l e c t r i c i t y is
ascribed to Sir Thomas Browne in his 1646 work Pseudodoxia Epidemica”.
Ancient and medieval awareness of electrical effects includes lightning, electric fish, St.
Elmo’s fire, the amber effect, and, especially in early China, the lodestone (magnet). Little (or
even nothing) is known about the discoveries or inventions in the field of electricity between
ancient Greece and the Early Modern Times that is times after the development of printing —
Gutenberg’s moveable type printing machine — in 1452 and the increasing dispersion of
knowledge in the Renaissance and especially later in the Enlightenment. Those were the Dark
(Early) (AD 476–1000) and Middle Ages (AD 1000–1300).
The early Greek philosophers were aware that rubbing amber with
fur produced properties in each that were not possessed before the
rubbing. For example, the amber attracted the fur after rubbing, but
not before. These new properties were later said to be due to
“charge.” The amber was assigned a negative charge and the fur was
assigned a positive charge.
5
6
Terms.
Fundamental Definitions and Units
Terms.
Fundamental Definitions and Units
Electric Charge
Electric Charge — Laws and Principles
A charge can be p o s i t v e or n e g a t i v e, or z e r o. In nature there
occurs only an integral multiple of a universal basic charge of proton —
a positively charged particle that is the nucleus of the lightest chemical
element, hydrogen.
The term „charge” is a primitive notion and an independent quantity
(variable) in physics. Its unit is coulomb [C].
The charge of e l e c t r o n is conventionally n e g a t i v e, while that
of proton is p o s i t i v e. Both charges are the charged constituents of
ordinary matter and the smallest known particles (portions) of charge in
nature. They are referred to as e l e m e n t a r y and are marked as e i
e, where e 1.6021892 r 0.0000046×1019 C. They are exactly equal to
each other as to their absolute value and are the smallest undivided
„amount” of electricity. Each atom has an equal number of electrons and
protons, and therefore is electrically neutral as a whole.
Balance of electric charges is one of the most fundamental
laws of nature.
The electric charge can be neither c r e a t e d nor d e s t r o y e d.
One can only transfer some number of elementary charges, for example, electrons, from one body to another body which causes the
first body to be positively charged while the second body has a
negative charge of the same absolute value. This process is strictly
related to:
z
z Charge quantization — the principle that the electric charge of
an object must equal an integral multiple of a universal basic
charge.
z Conservation of charge — a law which states that the total
charge or the total algebraic sum of charges of an isolated system is
constant; no violation of this law has been discovered.
8
7
–2–
Terms.
Fundamental Definitions and Units
Terms.
Fundamental Definitions and Units
Electric Charge — Laws and Principles
Electric Charge — Laws and Principles
Isolated system is s u c h a system through which boundaries n o
charges can pass
Transfer of electrons from one body to the other causes the bodies
to be charged as a result of an e x c e s s or a d e f i c i e n c y of
charges.
or
Such a process is called e l e c t r i f i c a t i o n or c h a r g i n g
and is a physical proof of the law of charge conservation.
is a system which is s o i s o l a t e d that it c a n n o t exchange
charges with its surroundings and therefore the total charge inside
the system is p r e s e r v e d.
Therefore the charge is indestructible: never can be c r e a t e d or
d e s t r o y e d. The charges then can transfer from one place to
another one, but never come from nowhere. We therefore say that
the charge is p r e s e r v e d.
9
10
Terms.
Fundamental Definitions and Units
Terms.
Fundamental Definitions and Units
Charge Properties
z
There are negative charges as e l e c t r o n s or n e g a t i v e
i o n s and positive charges as i o n s which always are the
integral multiples of the smallest charge, that is an electron or a
proton.
z
Opposite charges a t t r a c t and like charges r e p e l.
z
Charges can be s t a t i c, i m m o b i l e and i n v a r i a b l e
or they can be in m o t i o n, or can v a r y with time.
Current
E l e c t r i c c u r r e n t is connected with the motion or timevariations of electric charges; it is strictly related to the classification (division) of bodies which is as follows:
11
–3–
z
c o n d u c t o r s: class I — metals and coal; class II —
electrolytes (water solutions of acids, salts and bases);
z
i n s u l a t o r s (dielectrics, or non-conductors) — gases,
insulating liquids (water without additives, distilled water),
insulating oil, glass, porcelain, paper, cotton, silk, isinglass,
plastics, etc.;
z
s e m i c o n d u c t o r s — germanium, silicon, oxides of
different metals and other bodies of complex structure.
12
Terms.
Fundamental Definitions and Units
Terms.
Fundamental Definitions and Units
Current
z
z
International System of Units SI
Conduction current in conductors — in a crystal lattice free
electrons are loosely bound with atomic nuclei (positive ions)
located in the lattice points and can move about in the space of a
lattice between at very high velocities of about 105 m/s at room
temperature and at almost as twice as great velocity at a
temperature of 1000 K.
SI base units
Displacement current in insulators — there are few or no free
electrons at all and hence the insulator (dielectric) ability to
carry electric current is minimal or it does not conduct the
current; electrons are strongly bound with the atomic nuclei and
can move only within a given atom. In an ideal (perfect)
dielectric charges can move in its interior without disturbing its
structure, and the so-called dielectric polarization occurs.
13
Symbol
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deca
Y
Z
E
P
T
G
M
k
h
da
1024
1021
1018
1015
1012
109
106
103
102
10
Name
Symbol
Factor
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
d
c
n
P
n
p
f
a
z
y
101
102
103
106
109
1012
1015
1018
1021
1024
angle
D, E, J
radian
rad
solid angle
Z, :
steradian
sr
14
Selected Quantities in Electrical Engineering
electric charge
potential
voltage, SEM
electric field strength
electric displacement
permittivity
capacitance
resistance
resistivity
conductance
conductivity
magnetic flux density
magnetic flux
magnetic field strength
Subdivisions
Factor
m
kg
s
A
K, deg
cd
Terms.
Fundamental Definitions and Units
Standard Prefixes for the SI Units of Measure
Name
metre
kilogram
second
amper
kelvin
candela
Derived units
Terms.
Fundamental Definitions and Units
Multiples
l, s
m
t, W
I, i
T
j
length
mass
time
current
thermodynamic temperature
luminous intensity
15
–4–
Q
V, I, )
U, E
E
D
H
C
R
U
G
J
B
)
H
coulomb
volt
volt
volt per metre
coulomb per square metre
farad per metre
farad
ohm
ohm metre
siemens
siemens per metre
tesla
weber
ampere per metre
C
V
V
V/m
C/m2
F/m
F
:
:·m
S
S/m
T
Wb
A/m 16
Terms.
Fundamental Definitions and Units
Selected Quantities in Electrical Engineering
magnetic permeability
inductance
magnetic resistance
frequency
angular velocity
work, energy
power
reactive power
apparent power
velocity
acceleration
force
torque, moment of force
other
P
L
RP
f
Z
A, W
P
Q
S
X
a
F
M
…
henry per metre
henry
turns per henry
hertz
radians per second
joule
watt
war
volt-ampere
metre per second
metre per second squared
newton
newton metre
…
H/m
H
1/H
Hz
rad/s
J
W
var
VA
m/s
m/s2
N
N·m
… 17
Electrostatics.
Electrostatic and Electric Fields
18
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Electrostatics
Electrostatics
E l e c t r o s t a t i c s — The class of phenomena recognized by
the presence of electrical charges, either stationary or moving,
and the interaction of these charges, this interaction being solely
by reason of the charges and their positions and not by reason of
their motion.
qs
19
–5–
qv
'Q
'V o0 'V
qv
'Q
of
h o 0 h 'S
'S o 0
lim
lim
lim hqv
ho 0
dQ
dV
'Q
'S o 0 'S
lim
dQ
dS
20
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Electrostatics
'Q
'l o 0 'l
ql
lim
Q
Electrostatic and Electric Fields
dQ
dl
E l e c t r i c f i e l d*) is space where positive and negative electric
charges are and interact with each other.
³ q v dV
E l e c t r o s t a t i c f i e l d is such an electric field which is
time-independent and in which stationary, not time-varying and
immobile with respect to the earth positive and negative electric
charges are and interact with each other.
³ qsdS
Both fields belong to vector fields.
V
Q
S
Q
³ q l dl
l
*) One of the fundamental fields in nature, causing a charged body to be a t t r a c t e d
to or r e p e l l e d by other charged bodies.
21
22
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Electric Field Strength*)
Coulomb’s Law (Coulomb’s Force)
F
F
q1q2 r
4SHH 0 r 2 r
q1q2
4SHH 0r 2
E
H0 — permittivity of empty (free) space ( 8.854×1012 F/m)
H
Qq r
4 SHH 0 r 3
F
F
q
Q r
4 SHH 0 r 3
— relative permittivity, dielectric constant [–]
*)
HH0 — absolute permittivity, permittivity [Fm1]
23
Also known as electric field intensity; electric field vector; electric vector.
24
–6–
Electrostatics.
Electrostatic and Electric Fields
Work in Electric Field
Electrostatics.
Electrostatic and Electric Fields
Work in Electric Field — Voltage
O(f)
dW
B
B
A
r
Q
U AB
rB
dr
³
4SHH 0 rA r 2
O
U AO
Q 4SHH 0 r 2 and dlcosD
Q
³ E ˜ dl
A
A
B
³ E ˜ dl ³ Edl cos D
A
A
U AB
electric voltage
26
Electrostatics.
Electrostatic and Electric Fields
Work in Electric Field — Potential
q ³ Edl cos D
25
Electrostatics.
Electrostatic and Electric Fields
For E
B
WAB
q
rA
A
B
³ qE ˜ dl
WAB
dl
dr
qE dl cos D
q E ˜ dl
scalar product
rB
D
E
F ˜ dl
Q
4SHH 0 rA
Electric Potential — Potential Difference
dr
O
B
³ E ˜ dl
³ E ˜ dl ³ E ˜ dl
A
Q
A
O
B
4SHH 0 rB
M A U AB M B
A
³ E ˜ dl M ( A )
U AB
O
electric potential
27
MA MB
28
–7–
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Electric Potential — Potential Difference
Work in Electric Field
M const
A
MB
WAA
MA
q ³ E ˜ dl
0
L
Conclusions:
E
Q
Q
³ qE ˜ dl
A
In an irrotational electric field
the voltage between two points is
equal to a difference of their
potentials:
UAB MA – MB.
z Surfaces in space with the same
electric potential [M = M(x, y, z) =
const] at every point are called
e q u i p o t e n t i a l surfaces.
z
E
0
29
30
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Work in Electric Field
Potential Gradient
From Stokes’ Theorem
An irrotational vector electric field E whose curl is identically zero: rotE = 0
is always the gradient*) of a scalar function, here the electric potential or
simply potential M. It is called a potential gradient.
³ E ˜ dl ³ rotE ˜ dS
L
S
rot E
Er gradM r 0
irrotational electric field
*) Potential gradient is the potential difference per unit length, as measured in the
direction in which it is a maximum at a point.
31
32
–8–
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Gauss’ Law — Gauss’ Flux Theorem*)
Gauss’ Law — Gauss’ Flux Theorem
)
dS
S
Dr
dS
³ E ˜ dS ³ EdS cos D
S
B
E
Q 4SHH 0 r 2 and dScosD
For E
E
S
d)
*)
)
E ˜ dS
Also known as the integral form of Gauss’ Law.
³ E ˜ dS
S
Q
dZ
4SHH ³
Z
dZr2
Q Z
HH 4S
33
34
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Gauss’ Law — Gauss’ Flux Theorem
Gauss’ Law — Gauss’ Flux Theorem
For Z
)
4S
³ E ˜ dS
S
³ HH E ˜ dS ³ D ˜ dS
Q
S
Q
S
HH D
HH 0E
electric induction*)
The electric flux ) through any closed surface S is proportional to the total electric charge Q
enclosed by S and divided by the absolute permittivity HH0.
*)
Also known as dielectric displacement; dielectric flux density; displacement;
electric displacement density; electric flux density.
35
36
–9–
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Divergence Theorem
Conductor in External Electric Field
M2!M1
From the Gauss-Ostrogradsky Theorem
³ D ˜ dS ³ divDdV
S
E
V
If Q
³ qv dV exists, then
V
divD
³ divDdV ³ qv dV
V
source
of electric field
Ee
Ei
E 0
E e Ei
E
qv
E 0
V
source electric field
37
Electrostatics.
Electrostatic and Electric Fields
Capacitance
Capacitance of Isolated Conductor
0
1. qv
HH 0 divE 0
2. E
gradM
M
C
0œM
const
M
E
En i Et
E
38
Electrostatics.
Electrostatic and Electric Fields
Conductor in External Electric Field
E
M1
0
kQ
Q
M
Q
4SHH 0 R
potential of sphere
qs
HH 0
39
– 10 –
C
4SHH 0 R
capacitance of sphere
40
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Capacitance
Capacitance
Mutual Capacitance of Two Isolated Conductors
Mutual Capacitance — Capacitor
M1 M 2
Q
1
E
2
C
C
1
Q
C
Q
M1 M 2
Q
M1 M 2
41
42
Electrostatics.
Electrostatic and Electric Fields
Capacitance
Capacitance
Capacitors in Series
Capacitors in Series
Q1
Electrostatics.
Electrostatic and Electric Fields
U1
Q2
C1
C2
U1
Q1
; U2
C1
U
U2
1
C
U
43
– 11 –
Q2
and U U1 U 2
C2
Q1 Q2 Q
Q1 Q2
C1 C2
1
1
or C
C1 C2
C1C2
C1 C2
1 ·
§ 1
Q¨ ¸
© C1 C2 ¹
1
C
Q1 Q2
C1 C2
Q
C
n
1
¦C
i 1
i
44
Electrostatics.
Electrostatic and Electric Fields
Electrostatics.
Electrostatic and Electric Fields
Capacitance
Capacitance
Capacitors in Parallel
Capacitors in Parallel
U
U1 U 2 and Q Q1 Q2
C1U1 C2U 2
Q1
C1
U
Q2
C2
U1
C
U2
Q
U
C1U1 C2U 2
U
C1 C2
n
C
¦C
i
i 1
45
46
Electrostatics.
Electrostatic and Electric Fields
Capacitance
Energy of Isolated Conductor; Energy of Electric Field
Q
dW
Q
Q
dQ Ÿ W ³ dQ
C
C
0
2
2
Q
CM
QM
W
2C
2
2
M dQ
Q2
2C
Elektrodynamika. PrĈd staãy
Energy of Capacitor
U
W
³ CU dU
0
CU 2
2
QU
2
Q2
2C
47
– 12 –
48
Electrodynamics.
DC Current
Electric Current
E l e c t r i c c u r r e n t or c u r r e n t — A net ordered (directed)
motion of electrically charged particles or charged macroscopic
bodies in space through the cross-section of a medium (solids, liquids,
gases, or free space) under an electric field — it is the phenomenon
arising from the presence of this field.
The electric field is given vectorially by the electric field strength E or
scalarly by the voltage U.
Elektrodynamika. PrĈd staãy
Thomas Alva Edison (1847–1931)
…promoted direct current
DC or d i r e c t c u r r e n t — Such an electric current which flows
in one direction only (the unidirectional flow of electric charge), as
opposed to alternating current.
Electrodynamics. DC Current
49
50
Electrodynamics.
DC Current
Electrodynamics.
DC Current
Schematic Circuit Diagram
Ohm’s Law
elements
E1
R1
General Ohm’s Law
C3
I1
I4
port
I2
U1
I
C2
A resultant electric field in a conductor is a vector sum of
Coulomb’s field EC and external forces Ee
I3
U2
II
R4
E EC Ee
III U4
B
loop
I³U
branches
A
node
z Loops I and II are c l o s e d ones, while Loop III is o p e n because I3
z Element E1 is an a c t i v e one, while other elements are p a s s i v e.
I4.
51
– 13 –
dl
S
B
B
³ EC ˜ dl ³ Ee ˜ dl
A
A
The conductor of a length
from A to B with the same
current intensity in all its
cross sections.
52
Electrodynamics.
DC Current
Electrodynamics.
DC Current
Ohm’s Law
Ohm’s Law
General Ohm’s Law
General Ohm’s Law
dM
For E·dl
B
³E
C
˜ dl M A M B
U AB
B
³ EC Ee ˜ dl ³ E ˜ dl
*
U AB
B
A
potential difference
between A and B
A
U AB EAB
A
const, U
For J
const and S
const
B
³E
e
˜ dl
EAB
electromotive force EMF
between A and B
EBA
B
B
I
³ UJ ˜ dl ³ U S dl
A
A
A
U
lAB
I
S
RAB I
53
54
Electrodynamics.
DC Current
Electrodynamics.
DC Current
Ohm’s Law
Ohm’s Law
General Ohm’s Law
General Ohm’s Law
RAB
U
lAB
S
resistance of conductor
E AB
MA
I
A
U
*
AB
IRAB
UAB
RAB
*
U AB
or
B
IRAB
U AB EAB
MB
55
– 14 –
56
Electrodynamics.
DC Current
Electrodynamics.
DC Current
Ohm’s Law
Energy, Power, Heat — Joule’s Law
General Ohm’s Law
For MA
dW
MB, RAB R, EAB E
IR
E
E
I
UIdt Ÿ W
P=
dW
dt
W
RI 2t [J ]
work of DC current
electric power of
DC current
UI [ W ]
electric energy
1J
R
UIt
Since
0.24 cal
Q # 0.24 RI 2t [cal]
heat
57
58
Electrodynamics.
DC Current
Electrodynamics.
DC Current
Kirchhoff’s Laws of Electric Circuits
Kirchhoff’s Laws of Electric Circuits
Kirchhoff’s Current or First Law
Kirchhoff’s Voltage or Second Law
w1
n
¦ Ik
0
E1
R1
I1 w 2
k 1
I5
I1
I4
w
I2
I2
R 1I 1
I1 I 2 I 3 I 4 I 5
0
U
R 3I3
or
I3
I1 I 4
E2
R2
R 2I 2
I3
I 2 I3 I5
59
– 15 –
w4
R3
w3
60
Electrodynamics.
DC Current
Electrodynamics.
DC Current
Kirchhoff’s Laws of Electric Circuits
Resistors
Kirchhoff’s Voltage or Second Law
Series Circuit
n
n
¦ Rk I k
k 1
R1
I
¦ Ek
k 1
U1
U E1 R1 I1 E2 R2 I 2 R3 I 3
0
U
U E1 E2
U3
R1 I1 R2 I 2 R3 I 3
R3
61
Electrodynamics.
DC Current
U1
U
62
Electrodynamics.
DC Current
Resistors
Resistors
Series Circuit
Parallel Circuit
R1 I ; U 2
U1 U 2 U 3
R2 I ; U 3
I
R3 I
R1 R2 R3 I
I1
I3
I2
Re I
U
R1 R2 R3
Re
R2
U2
U1
R1
U2
R2
U3
R3
n
Re
¦R
k
equivalent resistance
k 1
63
– 16 –
64
Electrodynamics.
DC Current
I1
I
Electrodynamics.
DC Current
Resistors
Resistors
Parallel Circuit
Parallel Circuit
U
; I2
R1
I1 I 2 I 3
1
Re
U
; I3
R2
U U U
R1 R2 R3
n
1
Re
U
R3
U
Re
Since G
Ge
1
1
1
R1 R2 R3
1
¦R
k 1
equivalent resistance
k
1R — conductance
G1 G2 G3
n
Ge
65
¦G
k
equivalent conductance
k 1
66
Electromagnetism.
Magnetic Field of DC Current
Magnetic Field
M a g n e t i c f i e l d is one of the elementary fields in nature; it
is found in the vicinity of a magnetic body or current-carrying
medium and, along with an electric field, in a light wave. It one of
the many field existing in nature in which electric charges are
affected by forces by magnets or by currents in conductors. This
field in turn acts on other magnets or conductors with currents
being in it.
Electromagnetism.
Magnetic Field of DC Current
A magnetic field is characterized by energy and inertia, and to
some extent is material and similar to an electric field. It possesses
two poles: positive (North — N) and negative (South — S) and the
opposite poles attract and the like poles repel.
67
– 17 –
68
Electromagnetism.
Magnetic Field of DC Current
Electromagnetism.
Magnetic Field of DC Current
Magnetic Field
Magnet(ostat)ic Field. Ampere’s Force
Each point in space around a current-carrying wire is described by
such a vector of magnetic induction and therefore a wire or current
circuit generates a magnetic field. The sources of such a field are
not only wires or circuits but also magnetic materials, the so-called
ferromagnetic materials or ferromagnetics, and strictly current
microcircuits in their atoms.
N
I
Between the electric and magnetic fields there is difference in the
interactions of both fields. It is a result of their character: an
electric field has a central, radial character and its lines of force
are open, while a magnetic field has a crosswise character — the
force acts on a charge in motion perpendicularly to its direction
and the lines of force are closed. Both fields are complementary in
69
the description of a general e l e c t r o m a g n e t i c f i e l d.
Electromagnetism.
Magnetic Field of DC Current
I l B sin D
F
I
F
B
B
S
S
70
Electromagnetism.
Magnetic Field of DC Current
Magnet(ostat)ic Field. Ampere’s Force
F
N
Magnet(ostat)ic Field. Oersted
I l B sin( l, B )
I
IluB
F
F
B
D
B
”left hand rule”
”right hand grip rule”
l
71
– 18 –
72
Electromagnetism.
Magnetic Field of DC Current
Electromagnetism.
Magnetic Field of DC Current
Biot–Savart–Laplace’s Law
Biot–Savart–Laplace’s Law
Magnetic Induction, Magnetic Flux Density
D
dB
PP0 dl u r
I 3
4S
r
PP0 I dl sin dl, r PP0 I dl sin D
4S
r2
4S
r2
dB
dl
P0 — permeability of free space (vacuum), magnetic constant
( 4S×107 H/m)
r
90°
dB
I
P — relative permeability [–]
PP0 — absolute permeability, permeability of a specific medium,
P
73
Electromagnetism.
Magnetic Field of DC Current
permeability [Hm1]
74
Electromagnetism.
Magnetic Field of DC Current
Biot–Savart–Laplace’s Law
Ampere’s Law
Magnetic Field Strength
dH
dH
I dl u r
4S r 3
I dl sin dl, r 4S
r2
B
r
I
PP0 H
L
75
– 19 –
dl
H
76
Electromagnetism.
Magnetic Field of DC Current
2 Sr
³ H ˜ dl ³
L
0
Ampere’s Law
1 2I
1 2I
cosH, dl dl =
4S r
4S r
1 2I
2Sr
4S r
Electromagnetism.
Magnetic Field of DC Current
Lorentz’s Force
2 Sr
³ dl
0
F
B
I
rotational field
(nonpotential)
³ H ˜ dl
L
2 Sr
n
0
k 1
+
q!0
D
³ Hdl cosH, dl = ¦ I k
X
77
Electromagnetism.
Magnetic Field of DC Current
78
Electromagnetism.
Magnetic Field of DC Current
Lorentz’s Force and Electromagnetic Force
Faraday’s Law of Induction
I dl u B
dF
I
q X dn
I dl
dS
dF
q dn X u B
S
Ei
FL
F
dF
dn
Fe FL
qXuB
qE q X u B
k
d) m
dt
Lorentz’s force or magnetic force
d)
electromagnetic force
or Lorentz’s equation
B ˜ dS
B
Ei
R
79
– 20 –
80
Electromagnetism.
Magnetic Field of DC Current
Electromagnetism.
Magnetic Field of DC Current
Self-Induction*)
Faraday’s Law of Induction
Ei
k
d) m
dt
1
k

o
Ei
d) m
dt
Bi
Lenz’s Law
B
dS
I
I
S
Ii
90°
Es
S
Ii
d) m
Bi Ii
Ei
81
*)
B ˜ dS
P0
³ dS ³
4S S
l
P
r
3
dl u r n
E
82
Electromagnetism.
Magnetic Field of DC Current
Electromotive Force*) of Self-Induction
Self-Inductance or Inductance
I
a
The production of a voltage in a circuit by a varying current in that same circuit
Electromagnetism.
Magnetic Field of DC Current
)m
Bn B
Bn dS
Es
IL
d
LI dt
L z f(t)
L
P0
P
dS ³ dl u r ³
4S
r
3
S
Es
n
L dI
dt
l
83
– 21 –
*)
Also known as induced voltage; induced electromotive force
84
Sinusoidal AC Voltage
Sinusoidal AC Voltage
A l t e r n a t i n g v o l t a g e — Periodic voltage, the average
value of which over a period is zero.
The time variations of periodic voltages can be waves of different
shapes: square, rectangular, triangular, sine, and so forth. Their
distinctive feature is a cycle of changes repeated within the time T
called a period. Its reciprocal is the frequency of voltage f.
f
f
T
Nikola Tesla (1856–1943) with one of his early electrical generators…
Z
…advocated alternating current while Thomas A. Edison (1847–1931) promoted direct current
85
Z
1
T
2S
— voltage frequency [Hz]
— period [s]
— angular velocity of rotation of an electromotive force
(emf) vector Em [rad·s1] or else angular frequency [1/s] 86
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Voltage Generation
AC Voltage Generation
B = var
d
Z
Z
l
D
)m
e
Bl X
B
Bm sin D
e
BmlX sin D
D
B = const
)
E m sin D
e
E m sin Zt
B
87
– 22 –
i
e
R
Em
sin Zt
R
Bld cos D
Bld
) m cos Zt
d) m cos Zt dt
zZ) m sin Zt Em sin Zt
z
d)
dt
z
I m sin Zt
88
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Voltage
u(t)
AC Voltage
e(t) and hence Um
u
Um
u(t ) U m sin Zt
t
phase angle
In general
u
Z
\
89
0
\i
\u
90
Sinusoidal AC Voltage
AC Voltage
Rotating Vector*) — Phasor Diagram
I m sin Zt \ i u, i
u
u
M Zt \ u Zt \ i \ u \ i
i
i
i
M
angular
frequency
period
Conclusion: Any variable sinusoidal physical quantities
can be presented e x p l i c i t l y by means of three
quantities: amplitude, frequency and phase angle.
AC Voltage
u
t, Zt
T
— voltage angular frequency [1/s]
— phase angle [rad]
Phase Shift
u (t ) U m sin Zt \ u š i(t )
0
\
u (t ) U m sin Zt \ Sinusoidal AC Voltage
u, i
amplitude
u(t)
Em
0
M
Zt
\i
Zt
\u
91
– 23 –
*)
Also known as phasor.
92
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Voltage
RMS
AC Voltage
Value*)
dA i Rdt
T
AT
0
T
I
Im
| 0.707I m
2
1 2
I m sin 2Ztdt
T ³0
Also known as root-mean-square value, effective value.
93
U
Um
2 i E
Em
2
94
AC Power Circuit
AC Power Circuit
T
³ pdt
Resistance R
C
Pm
0
Pm
Ÿ P
2
P
instantaneous power
I m sin Zt œ u
p U m I m sin Zt
PT
i(t)
Sinusoidal AC Voltage
2
AT
T t
Sinusoidal AC Voltage
u U m sin Zt š i
0
T/2
0
Resistance R
Ideal resistor R const, L
p ui
AT
I m2
0
T
1 2
i dt
T ³0
*)
R ³ i 2 dt
I 2 RT
AT
i2 (t)
Im
I
T
³ Ri 2dt
RMS Value
i(t)
2
Ri š U m
Um Im
2 2
UI
[P]
W
active (real) power
RI m
p(t)
Pm sin Zt
2
P
i
UmIm
u
T
P
Um Im
2
1
Pm sin 2 Zt dt
T ³0
95
– 24 –
PŸR
0
T/2
i(t)
u(t)
UI
T
t
96
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Power Circuit
AC Power Circuit
Inductance L
Ideal inductor L const, R
u eL
i
eL
L
eL
C
eL
L
0 Ÿ u
eL
S·
§
E Lm sin ¨ Zt ¸
2¹
©
u eL
di
dt
I mZL cos Zt
u
S·
§
I mZL sin ¨ Zt ¸
297 ¹
©
0Ÿu
S·
S·
§
§
I mZL sin ¨ Zt ¸ U m sin ¨ Zt ¸
2¹
2¹
©
©
I mZL cos Zt
Conclusion: the phase of the current l a g s that of the voltage by ʌ/2.
98
Sinusoidal AC Voltage
AC Power Circuit
AC Power Circuit
U
uL
Inductance L
ZLI
Um
2
eL
i
Im
ZL Ÿ U
2
XL
M S
S/2
S
2S Zt
U
IX L Ÿ I
XL
EL
99
– 25 –
[XL]
[Z] [L]
:
IZL
ZL
I
0
eL
Sinusoidal AC Voltage
Inductance L
u, i
0
ImsinZt
i
u
Inductance L
inductive reactance
U
XL
ZL 2SfL
(1 s)1·1 H (1 s)1·(:·s)
100
Sinusoidal AC Voltage
AC Power Circuit
AC Power Circuit
Inductance L
Inductance L
S·
§
ui U m I m sin Zt sin ¨ Zt ¸ UI sin 2Zt
2¹
©
p
u, i, p
Sinusoidal AC Voltage
T
AT
³ pdt
0
uL
i
0 S/2 S 0 Ÿ P
AT
p
T /4
2S Zt
T /4
³ uidt
AT / 4
³
L
0
0
di
idt
dt
AT
T
0
Im
LI m2
2
³ Lidi
0
Wm
101
102
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Power Circuit
AC Power Circuit
Inductance L
Capacitance C
Ideal capacitor C const, R
reactive power
§
Q UI ¨ ‘U, I
©
[Q]
S·
¸
2¹
u
reactive energy
d q i d t š d q C du
uC
u
Ab
Qt
[Ab]
var·s
C
i
i
103
– 26 –
0
UmsinZt
i
var
L
C
du
dt
CU m
d(sinZt )
dt
C
du
dt
ZCU m cos Zt
104
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Power Circuit
AC Power Circuit
Capacitance C
Capacitance C
u, i
uC
S·
S·
§
§
sin ¨ Zt ¸ sin ¨ Zt ¸
2¹
2¹
©
©
cos Zt
S·
§
i ZCU m sin ¨ Zt ¸
2¹
©
M S/2
i
S·
§
I m sin ¨ Zt ¸
2¹
©
0 S/2
S
2S Zt
Im
2
Conclusion: the phase of the voltage l a g s that of the current by ʌ/2.
105
I/ZC
Um
ZC Ÿ I
2
Sinusoidal AC Voltage
AC Power Circuit
AC Power Circuit
Capacitance C
Capacitance C
XC
U
Ÿ U
XC
I
XC
[XC]
U
Sinusoidal AC Voltage
[Z] 1 [C] 1
1
ZC
:
1
ZC
I
p
capacitive reactance
u, i, p
106
S·
§
ui U m I m sin Zt sin ¨ Zt ¸ UI sin 2Zt
2¹
©
uC
IX C
p
0 S/2
1
2SfC
ZCU
i
S
2S Zt
1 s:1 F 1s1F 1s:(1C:1V 1V:1A
107
– 27 –
108
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Power Circuit
AC Power Circuit
Capacitance C
Capacitance C
reactive power
T
0
0 Ÿ P
AT
T /4
AT / 4
³ uidt
0
T /4
³
0
uC
S·
§
Q UI ¨ ‘U, I
¸
2¹
©
Qc ZCU 2
³ pdt
AT
du
dt
dt
AT
T
0
reactive energy
Um
³
Cudu
0
CU m2
2
We
[Q]
var
Ab
Qc t
[Ab]
var·s
109
110
Sinusoidal AC Voltage
Sinusoidal AC Voltage
Resonance
Resonance
Series RLC Circuit
Series RLC Circuit
t
i
u
u
uR
R
uL
L
uC
C
UR
i
111
– 28 –
u R u L uC
RI ; U L
X LI
I m sin Zt
u U m sin Zt M ?
Ri L
di 1
i dt
dt C ³0
ZLI ; U C
U
XCI
1
I
ZC
Um
;I
2
Im
2
112
Sinusoidal AC Voltage
Sinusoidal AC Voltage
Resonance
Resonance
Series RLC Circuit
Series RLC Circuit
UL
UL
U
UC
1 ·
§
I R ¨ ZL ¸
ZC ¹
©
UR
U
1 ·
§
¨ ZLI I¸
ZC ¹
©
I R2 X L X C 2
114
Sinusoidal AC Voltage
Z
Z
XL
XC
X
Sinusoidal AC Voltage
Resonance
Resonance
Series RLC Circuit
Series RLC Circuit
—
—
—
—
IZ
U R UC U L
113
1 ·
§
R 2 ¨ ZL ¸
ZC ¹
©
2
2
2
I
UC
RI 2
Z
U
M
U U L U C 2
2
R
impedance triangle
2
R X L X C 2
2
R X
2
Z
2
X
M
R
impedance [:]
inductive reactance [:]
capacitive reactance [:]
reactance [:]
tgM
115
– 29 –
X
R
X L XC
R
1
ZC
R
ZL 116
Sinusoidal AC Voltage
Resonance
Resonance
Series RLC Circuit
Series RLC Circuit
X L XC
R
X
R
tgM
Sinusoidal AC Voltage
1
ZL ZC
R
UL
UL
UC
M 0
z
X ! 0 œ XL ! XC Ÿ M
\u \i ! 0 — inductive character;
z
X 0 œ XL XC Ÿ M
\u \i 0 — capacitive character;
z
X 0 œ XL XC Ÿ M \u \i 0 š UL UC — resistive
charakter o series (voltage) r e s o n a n c e.
ZL
Z
U
1
ZC
I
UR
Z0
f0
UC
1
2S LC
2S
117
118
Sinusoidal AC Voltage
Sinusoidal AC Voltage
Resonance
Resonance
Parallel RLC Circuit
Parallel RLC Circuit
t
i
i
iL
iR
u
R
L
iR iL iC
iC
IR
C
U
; IL
R
U
XL
u U m sin Zt
i
119
– 30 –
I m sin Zt M ?
u 1
du
³ ud t C
R L0
dt
U
XC
U
; IC
ZL
I
ZCU
Im
;U
2
Um
2
120
Sinusoidal AC Voltage
Sinusoidal AC Voltage
Resonance
Resonance
Parallel RLC Circuit
Parallel RLC Circuit
IC
2
Z
IR
I
I I C I L U
M
2
1 ·
§1· §
U ¨ ¸ ¨ ZC ¸
ZL ¹
©R¹ ©
I
IC
IL
U ·
§U · §
¨ ¸ ¨ ZCU ¸
ZL ¹
©R¹ ©
2
2
R
2
2
U G 2 BC BL 2
IL
I R IC I L
I
121
122
Sinusoidal AC Voltage
Sinusoidal AC Voltage
Resonance
Resonance
Parallel RLC Circuit
Parallel RLC Circuit
admittance triangle
2
Y
1 ·
§1· §
¸
¨ ¸ ¨ ZC ZL ¹
©R¹ ©
Y
BC
BL
B
UY
—
—
—
—
G
2
G 2 BC BL 2
G2 B2
M
B
Y
admittance [S]
capacitive susceptance [S]
inductive susceptance [S]
susceptance [S]
tg M 123
– 31 –
B
G
BC BL
G
ZC G
1
ZL Ÿ tgM
1
ZC
ZL
G
124
Sinusoidal AC Voltage
Sinusoidal AC Voltage
Resonance
Resonance
Parallel RLC Circuit
tg M B
G
BC BL
G
1
ZC ZL Ÿ tgM
G
Parallel RLC Circuit
1
ZC
ZL
G
z
B ! 0 œ BC ! BL Ÿ M
\u \i ! 0 — capacitive character;
z
B 0 œ BC BL Ÿ M
\u \i 0 — inductive character;
z
B 0 œ BC BL Ÿ M \u \i 0 š UL UC — resistive
charakter o parallel (current) r e s o n a n c e.
IC
IR
M 0
IL
Z0
2S
1
2S LC
126
AC Network Analysis — Complex Numbers
Voltage and Current Relationships
in The Time and Frequency Domains
1
u (t ) Ri (t )
i (t )
u (t ) Gu (t )
R
di (t )
u (t ) L
1
i (t )
u (t )dt
dt
L³
1
u (t )
i (t )dt
du (t )
i (t ) C
C³
dt
I m e j Zt
f0
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
2U e jZt š i (t )
U
125
Sinusoidal AC Voltage
u (t ) U m e jZt
1
ZC
IC
IL
Comparison of series and parallel circuits
X!0ŸB0šX0ŸB!0
ZL
Z
2 I e jZt
Voltage and Current Relationships
in The Time and Frequency Domains
U
RI
I
GU
U
jZLI
I
U
1 I
jZC
1 U
jZL
I
jZCU
U
127
– 32 –
ZI › I
YU
128
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Network Analysis — Complex Numbers
Voltage and Current Relationships
in The Time and Frequency Domains
1 ·
§
Z R j ¨ ZL ¸ R jX Ze jM
Z
C
©
¹
Voltage and Current Relationships
in The Time and Frequency Domains
1 ·
§
Y G j ¨ ZC ¸ G jB Ye jM c
L
Z
©
¹
G2 B2
Z
R2 X 2
— modulus of the complex impedance
Y
M
arctg X R — argument of the impedance (phase
shift)
M c arctgB G Re Y
G Y cos M c
— conductance of a circuit
Im Y
B Y sin M c
— susceptance of a circuit
Re Z
R
Z cosM
— resistance of a circuit
Im Z
X
Z sin M
— reactance of a circuit
129
— modulus of the complex admittance
Sinusoidal AC Voltage
— argument of the admittance (phase
shift)
130
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Network Analysis — Complex Numbers
Voltage and Current Relationships
in The Time and Frequency Domains
Voltage and Current Relationships
in The Time and Frequency Domains
Z
R jX
R
G jB
G2 B2
B
š X
G2 B2
1
G jB
G
G2 B2
1
š M
Y
G jB
G
Z
M c
U
I
Ue j\ u
Ie j\ i
U j \ u \ i e
I
1
R jX
R
R X2
2
R jX
R2 X 2
X
š B
2
R X2
Z e jM
Z
131
– 33 –
U
I
Um
š M \ u \ i
Im
132
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Network Analysis — Complex Numbers
Voltage and Current Relationships
in The Time and Frequency Domains
U m U m e j\ u š I m I m e j\ i
Ohm’s Law
U e j\ u š I
U
u (t ) U m e jZt
i (t )
I me
j Zt
Ie j\ i
U m e j Zt e j \ u
j Zt
I me e
U
j\ i
I me
1 ·
§
R j ¨ ZL ¸
ZC ¹
©
Z
U m e j (Zt \ u )
j
j (Zt \ i )
133
1
1
, since
ZC
jZC
Sinusoidal AC Voltage
ZI
R jX
j
j
j ZC
1 ZC
2
j
2
ZC
134
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Network Analysis — Complex Numbers
Ohm’s Law — Series Circuit
Ohm’s Law — Series Circuit
I
U
Z
U
I
Z1
Z2
Z
U1
U2
n
Z1 Z 2
Z
¦Z
i 1
n
i
1
¦
i 1 Yi
1
Y
135
– 34 –
n
n
i 1
i 1
¦ Ri j ¦ X i
Z
Z
M
arctg
R1 R2 j X 1 X 2 R1 R2 2 X1 X 2 2
X1 X 2 R1 R2 136
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Network Analysis — Complex Numbers
Ohm’s Law — Parallel Circuit
Ohm’s Law — Parallel Circuit
I
I1
Z1
U
Y
I
U
I2
Z2
n
Y
Y1 Y 2
Y
¦Y
i 1
n
i
1
¦
i 1 Zi
1
Z
n
n
i 1
i 1
¦ Gi j ¦ Bi
G1 G2 j B1 B2 G1 G2 2 B1 B2 2
Y
Y
M
arctg
B1 B2 G1 G2 137
138
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Network Analysis — Complex Numbers
Ohm’s Law
Ohm’s Law
I
Ie j 0
ZL ! (1/ZC) œ X ! 0, UX ! 0 š M
\u \i ! 0 — inductive
z
ZL < (1/ZC) œ X < 0, UX < 0 š M
character;
\u \i < 0 — capacitive
z
ZL = (1/ZC) œ X = 0, UX = 0 š M
\u \i = 0 — resistive
z
character;
U
ZI
U
M
R jX I
U R2
U X2
RI jXI
U R jU X
Ue jM
character o v o l t a g e r e s o n a n c e.
— modulus of voltage
arctgU X U R — argument of voltage (phase
shift)
139
– 35 –
140
Sinusoidal AC Voltage
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Network Analysis — Complex Numbers
AC Power
AC Power
Remark: I *
j\ u
U
Ue
I
Ie j\ i š I *
S
Ie j\ i
S
P2 Q2
UI
Ie j\ i is the conjugate of the complex current
S
S
P
Q
S U I Ue j\ u Ie j\ i UIe j \ u \ i UIe jM
UI (cos M j sin M ) P jQ
P UI cos M
Q UI sin M
—
—
—
—
apparent power [VA]
complex power (absolute value of complex power) [VA]
active (real, true) power [W]
reactive power [var]
141
142
Electrical Measurements
Sinusoidal AC Voltage
AC Network Analysis — Complex Numbers
AC Power — Power Triangle
Im S
j
S
jQ
M
P
Re S
Remark: cosM is called power (phase) factor
143
– 36 –
144
Electrical Measurements
Pomiar oporu czynnego (rezystancji)
Electrical Measurements
Pomiar oporu czynnego (rezystancji)
Technical Method of Resistance Measurement Technical Method of Resistance Measurement
Accurate Measurement of Voltage
I
Iv
A
U
Iv
I Iv
V
Rx
Rv
Rx
Rv o f
Rx*
Rx*
G
Rx
Rx
Rx*
Rv
U
Rv
U
I Iv
U
I
Accurate Measurement of Current
I
U
U
I
Rv
U
G
145
Rx* Rx
Rx
R1
A
R2
D
E
I1Rx
I 2 R1
I1Rn
I 2 R2
B
W2
Rx 0.1 (1.0) :— Thomson (Kelvin) bridge
Rx
Ra
Ra
U
I
Rx !! Ra
Rx ! 1.0 :
146
Active Power and Resistance
U AC U AD i U CB U DB
V
Ra o 0
Measurement of Active and Apparent Powers
and Power Factor
Wheatstone Bridge
Rn
Rx*
U
Ra
I
Electrical Measurements
Resistance Measurement
W1
Rx
Rx
V
Electrical Measurements
Pomiar oporu czynnego (rezystancji)
Rx
Ra
Rx*
Rx Rv Rx 1.0 :
C
A
I
W
U
R
R
Rn 1
R2
P UI
0.1 (1.0) : Rx 106 :
147
– 37 –
148
Electrical Measurements
Electrical Measurements
Measurement of Active and Apparent Powers Measurement of Active and Apparent Powers
and Power Factor
and Power Factor
Active Power and Impedance
I
Apparent Power, Power Factor and Impedance
I
W
U
U
Z
P UI cos M , since D
cwUI cos M
A
W
V
P UI cos M , S UI š cos M
cw P
149
Z
P
S
150
Three-Phase Circuits
Three-Phase Voltage and Current
u L1 U m sin Zt
uL2
U m sin Zt 2S
uL3
U m sin Zt 4S U m sin Zt 2S
UL1
Three-Phase Circuits
151
– 38 –
ZL1 ZL2 ZL3
UL2 UL3 š IL1 IL2
iL1
I m sin Zt M iL 2
I m sin Zt 2S / 3 M iL 3
I m sin Zt 4S / 3 M IL3
152
Three-Phase Circuits
Three-Phase Circuits
Three-Phase Star (Y) Configuration
Three-Phase Delta (') Configuration
L1
L1
U
L1
U12
X
UL1
U31
Y
Z
UL2
V
W
L2
L3
N
U12
30°
UL2
L3
UL1
W
X
Y
L2
U23
U
UL1
U12
UL3
L2
V
UL2
3
2
3U p
U p I p cos M Ÿ P 3Pp
U
I
L2
U L2
U 31 U L 3
U p and I
154
Power in Delta Configuration
3U p I p cos M
Pp
U p I p cos M Ÿ P 3Pp
Ip
I
L1
Zp
3I p
Three-Phase Circuits
L1
—3 p
U 23
153
Power in Star Configuration
U
U L1
L3
Three-Phase Circuits
Pp
U12
U 23 U 31 U
2U p cos 30q 2U p
U
Z
UL3
60°
3U p I p cos M
—3 I f
Up
If
U
Uf
Zf
L2
star
delta
L3
L3
155
– 39 –
156
Three-Phase Circuits
Three-Phase Circuits
Power of Symmetric Three-Phase System
Pp
U p I p cos M Ÿ P 3Pp
3U p I p cos M
star
Up
Measurement of Power and Energy
in Three-Phase System
P1
L1
delta
U
, Ip I
3
P
3UI cos M
Q
S
Up
U , Ip
ZL1
W
P2
L2
I
3
P3
L3
[ var]
3UI
[VA]
ZL3
W
[W]
3UI sin M
ZL2
W
N
P
P1 P2 P3
P1
P2
Asymmetric load
157
Three-Phase Circuits
P3
Pp Ÿ P 3Pp
Symmetric load
Three-Phase Circuits
Measurement of Power and Energy
in Three-Phase System
Measurement of Power and Energy
in Three-Phase System
Aaron’s System
Aaron’s System
L1
IL1
W
Asymmetric load
PE
P
PD PE
U12
PD
U32
L3
IL3
U 32 I L 3 cos D U12 I L1 cos E
Symmetric load
Load
L2
158
W
159
P
– 40 –
PD PE
PD
UI cosM 30q
PE
UI cosM 30q
UI >cos M 30q cos M 30q @
3UI cos M160
Three-Phase Circuits
Pomiary mocy i energii prądu trójfazowego
Three-Phase Circuits
Pomiary mocy i energii prądu trójfazowego
Measurement of Power and Energy
in Three-Phase System
Measurement of Power and Energy
in Three-Phase System
Aaron’s System
Aaron’s System
200
P
When M
150
P [%]
100
PD
PD
50
UI cos 30q
PE
PD PE
P
PE
0 š cosM
1
3
UI
2
3UI
0
capacitive
character
-50
-100
90
60
30
When M
inductive
character
0
30
60
PE
90
161
M [°]
60° š cosM
0; P
PD
0,5
3
UI
2
162
Three-Phase Circuits
Pomiary mocy i energii prądu trójfazowego
Three-Phase Circuits
Pomiary mocy i energii prądu trójfazowego
Measurement of Power and Energy
in Three-Phase System
Measurement of Power and Energy
in Three-Phase System
Power and Reactive Energy in Symmetric System
Power and Reactive Energy in Symmetric System
L1
IL1
ZL1
Wattmeter or watt-hour
meter
UL1
M
W
90° M
0
ZL2
L2
U23
IL1
UL3
ZL3
U23
UL2
QL1 U L1 I L1 sin M
L3
163
P U 23 I L1 cos90q M U 23 I L1 sin M
– 41 –
reactive power by definition
active power measured
164
Thank you forTerms.
your attention!
Three-Phase Circuits
Pomiary mocy i energii prądu trójfazowego
Fundamental Definitions and Units
Measurement of Power and Energy
in Three-Phase System
Power and Reactive Energy in Symmetric System
U 23
QL1
3U L1
P
š Q 3QL1
3
Ab Qt
3P
reactive
power
reactive
energy165
© 2010 Juliusz B. Gajewski
– 42 –
166