01 EE M EE Intro and Basic Concepts A

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Electric Equipment QP 2013-2014

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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


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




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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



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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


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 +



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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

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01 EE M EE Intro and Basic Concepts A