01 EE M EE Intro and Basic Concepts A
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Transcript of 01 EE M EE Intro and Basic Concepts A
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11
Voltage (or electrical potential difference):
Energy involved in the movement of the electrical charge
unit between two points.
Depending on the signs conventions the energy can be
transferred or absorbed.
Synonyms: potential, potential difference, voltage,electromotive force, back electromotive force, induced
voltage ...
= dd Joule
Coulomb= Volt (V)
BASIC CONCEPTS
12
Electric power:
Energy per unit time involved in passing a current
between two points that have a voltage difference of .
Voltage: Energy Availability
Current: Effectiveness of availability
=dd =dd dd= Joulesecond = Watt W
BASIC CONCEPTS
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Signs Conventions:
BASIC CONCEPTS
Matching
assessment arrows
Not matching
assessment arrows
i
u u
i
Pdelivered= -ui
Pconsumed = ui
Pdelivered= ui
Pconsumed = -ui
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Kirchhoffs laws:
Although always associated to electric circuits, they are
actually useful expressions of energy and mass conservation
laws.
An electrical knot is a connection point where converge more
than two electric currents.
BASIC CONCEPTS
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1st. Kirchhoffs Law or Kirchhoffs Current Law (KCL):
The sum of currents entering a knot must equal the sum of
the currents leaving the knot. Result in the electrical circuit
there is no possibility of storing mass and this must be
preserved (law of conservation of mass).
=
BASIC CONCEPTS
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2nd. Kirchhoffs Law or Kirchhoffs Voltage Law (KVL):
The addition of voltages in a closed path is zero.
Consequence of the electric field is conservative.
If you take a charge and moves it from one point to another
the voltage is the energy per unit charge moved, if theprocess is repeated by a closed path energy balance
should be zero because the point arrival is the same as the
output one (the law of conservation of energy).
= 0
BASIC CONCEPTS
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MODELING ELEMENTS
Lineal modelization of the electric system elements:
The elements presented are idealizations. Idealized
elements hardly correspond to reality; however, the
combined use of different models can describe reality quite
accurately.
They will be categorized into two groups: Actives
characterized by the fact that can generate and consume
energy (in average value in case of AC), and passives
characterized by the fact that they only consume energy
(on average value in case of AC).
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Active elements
Voltage source: Constant voltage in their terminals
(connection points) regardless of how much current is
delivering.
MODELING ELEMENTS
DC current AC current
u(t)
u(t)
t
u(t)
u(t)
t
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Resistor:
In the usual electro materials (Cu and Al) and the industrial
temperature range the resistor varies linearly with
temperature according to the expression
where the thermal coefficient is
= 1 + ( )
0,004 K
MODELING ELEMENTS
Passive elements
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Resistor:
In metals, in addition to temperature, the resistor depends
on the material and shape, for wire type conductors(slender), is directly proportional to length and inversely
proportional to cross section:
Where is the resistivity, characteristic of the material,
and, of course, a function of temperature:
=
= 1 + ( )
MODELING ELEMENTS
Passive elements
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Resistor:
The power consumed by a resistor is
The materials of very high resistivity are called insulators or
dielectrics, and are used to prevent the circulation of
currents.
= = =
0 as > 0
MODELING ELEMENTS
Passive elements
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Inductance: Linear relationship between the voltage and
the time derivative of the current valued in the same
direction. The ratio is the value of the self-inductioncoefficient or inductance L, measured in Henry (H).
= d()d
i(t)
u(t) L
MODELING ELEMENTS
Passive elements
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Inductance:
Inductance or self-inductance coefficient reflects the
presence of a magnetic field in a zone of the space. The
magnetic field is due (or linked) to the flow of electric
current. If the presence of the field was not desired it is said
that is a parasite inductance. The magnetic field stores
energy according to the expression
= 12
MODELING ELEMENTS
Passive elements
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Inductance:
The power consumed by an inductance is
At instantaneous value (in a given time) can have positive
or negative values, but in steady state (constant or variable
repetitive) the average value of the power consumed will
be null.
= = d()d = () d
MODELING ELEMENTS
Passive elements
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Capacitance: Linear relationship between the current and
the time derivative of the voltage measured in the same
direction. The ratio is the value of the capacity C, measured
in Farads (F).
= d()
d
i(t)
u(t) C
MODELING ELEMENTS
Passive elements
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Capacitance: The capacitance reflects the presence of an
electric field in a zone of the space. The electric field is due
(or linked) to the voltage between two points separated by
a dielectric material. If the presence of the field was notdesired it is said that is a parasite capacitance; however if it
was desired then a device called capacitor is manufactured
specifically. The electric field stores energy according to
the expression
= 12
MODELING ELEMENTS
Passive elements
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Capacitance:
The power consumed by a capacitance is
At instantaneous value (in a given time) can have positive
or negative values, but in steady state (constant or variable
repetitive) the average value of the power consumed will
be null.
= = d()d = ()
d
MODELING ELEMENTS
Passive elements
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Magnetic coupling: Linear relationship between the
voltages and time derivatives of the currents from two
circuits, which are said to be magnetically coupled. The
coupling coefficients that are called self inductances (L
1and L2) and the mutual coupling coefficient (M ) aremeasured in Henry (H).
= d()d + d()
d = d()d +
d()d
Mi1(t
u1(t) L 1
i2(t)
u2(t)L2
MODELING ELEMENTS
Passive elements
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Magnetic coupling: The two inductances share part of the
magnetic field, which is represented by the mutual
induction coefficientM. Self inductances reflect the total
magnetic field seen by each inductance, which is formed by
the common part and the each self part. In order to be
magnetically coupled, two inductances must be relatively
close together. If the presence of the magnetic field,
especially the common part was not desired it is said that it
is parasite.
MODELING ELEMENTS
Passive elements
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General Response of a Circuit
General response of an electric circuit
Evolution of magnitudes in time:
In a circuit formed by the mentioned linear elements, its
behavior can be described by a system of linear differential
equations, which is called the system state equations whenwritten in the following format:
where Xis the vector of state variables, consisting of voltages on
capacitances and currents in inductances. The A matrix is a function of
the passive elements of the system (with constant coefficients) andB is
the excitations vector function of passive and active elements of the
system.
dd = + with the initial conditions 0 =
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From 1st. Kirchhoffs law:
From 2nd. Kirchhoffs law:
= + = =
=
dd= dd
+ = 0+ = 0
R1
R2
L
C uC uR2uR1 uL
iU iL
iC
U
General Response of a Circuit
System state equation example
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Substituting the previous equations into the last two result in:
And in matrix form:
dd =
d
d =
dd
=1
1
1
+ 0
with the initial condicions 0 0 =
R1
R2
L
C uC uR2uR1 uL
iU iL
iC
U
General Response of a Circuit
System state equation example
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Solving the differential equation we find the transient response
and the steady state response, but if we only want to find the
steady state response, we can go much faster, since we can
directly write: = +
= +
R1
R2
L
C uC uR2uR1 uL
iU iL
iC
U
iLiC
t
uC
t
General Response of a Circuit
System state equation example
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SINUSOIDAL MAGNITUDES
Sinusoidal Magnitudes
in rad/s and f in Hz (s-1)
Maximum value = Peak value = MpeakPeak to peak value = 2Mpeak
= 2 = cos +
t
m(t)
T = 1/f
Mpeak
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Mean value:
Rectified mean value:
= cos +
() = 12 cos () = 0
() =1 cos () =2 0,64
SINUSOIDAL MAGNITUDES
t
m(t)
T = 1/f
Mpeak
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Effective value (RMS, root mean square):
= = 12 ()
= 2 cos
()
=
2 cos
()
=
2= =
2 = =
2 0,707
= 2 cos + = 2 cos +
SINUSOIDAL MAGNITUDES
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Example: Sum of two sinusoidal magnitudes
= 2 cos + = = + = 2 cos + = = +
= + =+ == +
M1
M2
M3
PHASOR REPRESENTATION
Phasor Diagram
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Operations with phasors
Linear combination of sinusoids:
Derivative:
Triangle inequality (Schwarz):
()
2
dd 2
+ +
= 4 + 3 = 4 + 3 = 3 + 4
I1= 30 A
I2= 40 A
I3= ?
I1I2
I3= 10 A
I1 I2
I3 = 70 A
I1I2
10 A I3 70 A
j 90
PHASOR REPRESENTATION
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Capacitances (C ):
Taking into account phasors:
= 2 cos + = 2 cos + + 2
= = 1
=
i(t
u(t) Ct
u(t)
i(t)
XC
LINEAR ELEMENTS IN AC
Linear elements in sinusoidalsteady state
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Magnetic coupling (M ):
= + = +
Mi1(t
u1(t) L 1
i2(t)
u2(t)L2
= + = +
= d()d + d()
d = d()d + d()d
LINEAR ELEMENTS IN AC
Linear elements in sinusoidal
steady state
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IMPEDANCE
Impedance
Complex numberZ is named (complex) impedance whose
real part is the resistive part and the imaginary part is the
reactance (inductive or capacitive) part. This definition
allows to express the generalized Ohm's Law:
Where Z, in a generic form, it can be expressed as:
=
= +
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= = + =
= = 2 cos +
= 2 cos +
R
XL
XC
Z
X
u(t)
i(t)
t
Z
IMPEDANCE
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Active power P, energy consumed by time unit, measured
in Watts (W)
Reactive power Q, measured in volt-ampere-reactive (var),is defined (for convenience) as
Apparent power S, measured in volt-ampere (VA), isdefined as
= 1 ()
= cos
= sin
=
POWER IN AC
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The three AC powers can be grouped under the named
complex apparent power:
The power factor fdpis defined as
We are interested in unity power factor. Why?
= = cos + sin = +
== cos
POWER IN AC
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Power in each element
Generic Z:
R :
L :
C :
= =
=
=
= =
= + 0
= =
=
= 0 + = =
=
= 0
POWER IN AC
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Magnetic coupling M :
= + = + + +
= + + + = + + 2 cos
= 0 +
POWER IN AC
Power in each element
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THREE-PHASE SYSTEMS
Three-phase systems
Usually, a three-phase system is obtained with a three-phase
generator and not with three single-phase generators.
= + + n
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Three-phase generator
Direct sequence (abc) and inverse sequence (acb).
t
uan(t ) ucn(t )ubn(t )
THREE-PHASE SYSTEMS
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The generator provides a balanced (phase to neutral voltages
are equal) and symmetric (phase to phase voltages are equal)
three-phase system.
= = 2 cos 30 = 2 32 = 3 Line to line voltage = 3 Line to neutral voltage
120
30
THREE-PHASE SYSTEMS
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The rated voltage of a three-phase system is the line to line
voltage!
120 120
t
uan ucnubn
uab ubc uca 120 120
30
THREE-PHASE SYSTEMS
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Three-phase systems: a operator
= = 120
= = =
= = =
THREE-PHASE SYSTEMS
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Wye connection (generators and load)
n
N
N
THREE-PHASE SYSTEMS
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Why not to supply loads with line to line voltages?
n
THREE-PHASE SYSTEMS
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A three-phase system of line to line voltages can be
created by connecting generator windings in the named
delta connection
a
b
c
THREE-PHASE SYSTEMS
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PowerIf the three-phase load is symmetric (quite common) the
total power is as follows
= 3 3 cos = 3 c os
= 3 sin
= +
= + = 3 cos + 3 s in = 3 cos + sin
= 3
THREE-PHASE SYSTEMS