Transcript Three phase system

Chapter 1 – Three phase circuit and
three phase power
Describe and explain the characteristic in three
phase circuits.
 Analyze the three phase circuit

General Format for the Sinusoidal
Voltage or Current
 The
basic mathematical
format for the sinusoidal
waveform is:
where:
Am is the peak value of the
waveform
 is the unit of measure for
the horizontal axis
General Format for the Sinusoidal
Voltage or Current
equation  = t states that the angle  through which the
rotating vector will pass is determined by the angular velocity of the
rotating vector and the length of time the vector rotates.
 For a particular angular velocity (fixed ), the longer the radius
vector is permitted to rotate (that is, the greater the value of t ), the
greater will be the number of degrees or radians through which the
vector will pass.
 The general format of a sine wave can also be as:
 The
General Format for the Sinusoidal
Voltage or Current
For electrical quantities such as current and voltage, the general
format is:

i  I m sin t  I m sin 
e  Em sin t  Em sin 
where: the capital letters with the subscript m
represent the amplitude, and the lower case letters i
and e represent the instantaneous value of current
and voltage, respectively, at any time t.
General Format for the Sinusoidal
Voltage or Current
Example 1.1
Given e = 5sin, determine e at  = 40 and  = 0.8.
Solution
For  = 40,
e  5 sin 40  3.21 V
For  = 0.8
 180 
  144
  0.8 
  
e  5 sin 144  2.94 V
General Format for the Sinusoidal
Voltage or Current
Example 1.2
(a) Determine the angle at which the magnitude of the
sinusoidal function v = 10 sin 377t is 4 V.
(b) Determine the time at which the magnitude is
attained.
General Format for the Sinusoidal
Voltage or Current
Example 1.2 - solution
Vm  10 V;   377 rad/s 
v  Vm sin t V
Hence,
v  10 sin 377t V
When v = 4 V,
4  10 sin 377t
Or;
  sin 1 0.4  23.58
4
sin 377t  sin  
 0.4
10
General Format for the Sinusoidal
Voltage or Current
Example 1.2 – solution
  377t  23.58  0.412 rad
0.412
t
 1.09 ms
377
Sinusoidal ac Voltage
Characteristics and Definitions
 Definitions
Waveform: The path traced by a quantity, such as voltage,
plotted as a function of some variable such as time, position,
degree, radius, temperature and so on.
 Instantaneous value: The magnitude of a waveform at any
instant of time; denoted by the lowercase letters (e1, e2).
Peak amplitude: The maximum value of the waveform as
measured from its average (or mean) value, denoted by the
uppercase letters Em (source of voltage) and Vm (voltage drop
across a load).

Sinusoidal ac Voltage
Characteristics and Definitions
 Definitions
Peak value: The maximum instantaneous value of a function
as measured from zero-volt level.
 Peak-to-peak value: Denoted by Ep-p or Vp-p, the full voltage
between positive and negative peaks of the waveform, that is,
the sum of the magnitude of the positive and negative peaks.
Periodic waveform: A waveform that continually repeats
itself after the same time interval.
Sinusoidal ac Voltage
Characteristics and Definitions

Definitions
Period
(T): The time interval between successive
repetitions of a periodic waveform (the period T1 = T2 = T3), as
long as successive similar points of the periodic waveform
are used in determining T
 Cycle: The portion of a waveform contained in one period
of time
Frequency: (Hertz) the number of cycles that occur in 1 s
1
hertz, Hz 
f 
T
Sinusoidal ac Voltage
Characteristics and Definitions
Sinusoidal ac Voltage
Characteristics and Definitions
Sinusoidal ac Voltage
Characteristics and Definitions
Example 1.3
Determine:
(a) peak value
(b) instantaneous value at 0.3 s and 0.6 s
(c) peak-to-peak value
(d) period
(e) how many cycles are shown
(f) frequency
Sinusoidal ac Voltage
Characteristics and Definitions
Example 1.3 – solution
(a) 8 V; (b) -8 V at 3 s and 0 V at 0.6 s; (c) 16 V;
(d) 0.4 s; (e) 3.5 cycles; (f) 2.5 Hz
Three Phase System
An ac generator designed to develop a single
sinusoidal voltage for each rotation of the shaft (rotor) is
referred to as a single-phase ac generator
If the number of coils on the rotor is increased in a
specified manner, the result is a polyphase ac
generator, which develops more than one ac phase
voltage per rotation of the rotor
In general, three-phase systems are preferred over
single-phase systems for the transmission of power for
many reasons, including the following:
Three Phase System
1.
2.
3.
4.
Thinner conductors can be used to transmit the same kVA at the
same voltage, which reduces the amount of copper required
(typically about 25% less) and in turn reduces construction and
maintenance costs.
The lighter lines are easier to install, and the supporting structures
can be less massive and farther apart.
Three-phase equipment and motors have preferred running and
starting characteristics compared to single-phase systems
because of a more even flow of power to the transducer than can
be delivered with a single-phase supply.
In general, most larger motors are three phase because they are
essentially self-starting and do not require a special design or
additional starting circuitry.
Three Phase System
The frequency generated is determined by the number
of poles on the rotor (the rotating part of the generator)
and the speed with which the shaft is turned.
Throughout the United States the line frequency is 60
Hz, whereas in Europe the chosen standard is 50 Hz.
On aircraft and ships the demand levels permit the use
of a 400 Hz line frequency.
The three-phase system is used by almost all
commercial electric generators.
Three Phase System
Most small emergency generators, such as the gasoline type,
are one-phased generating systems.
The two-phase system is commonly used in
servomechanisms, which are self-correcting control systems
capable of detecting and adjusting their own operation.
Servomechanisms are used in ships and aircraft to keep them
on course automatically, or, in simpler devices such as a
thermostatic circuit, to regulate heat output.
The number of phase voltages that can be produced by a
polyphase generator is not limited to three. Any number of
phases can be obtained by spacing the windings for each
phase at the proper angular position around the stator.
Three-Phase Generator
The three-phase generator has three induction coils
placed 120° apart on the stator.
The three coils have an equal number of turns, the
voltage induced across each coil will have the same
peak value, shape and frequency.
Three-Phase Generator
At any instant of time, the algebraic sum of the three
phase voltages of a three-phase generator is zero.
Three-Phase Generator
The sinusoidal expression for each of the induced
voltage is:
Y-Connected Generator
If the three terminals denoted N are connected
together, the generator is referred to as a Y-connected
three-phase generator.
Y-Connected Generator
The point at which all the terminals are connected is
called the neutral point.
If a conductor is not attached from this point to the
load, the system is called a Y-connected, three-phase,
three-wire generator.
If the neutral is connected, the system is a Yconnected three-phase, four-wire generator.
The three conductors connected from A, B and C to
the load are called lines.
Y-Connected Generator
The voltage from one line to another is called a line
voltage
The magnitude of the line voltage of a Y-connected
generator is:
Phase Sequence (Y-Connected
Generator)
The phase sequence can
be determined by the order
in which the phasors
representing the phase
voltages pass through a
fixed point on the phasor
diagram if the phasors are
rotated in a counterclockwise
direction.
Y-Connected Generator with a YConnected Load
Loads connected with three-phase supplies are of two
types: the Y and the ∆.
If a Y-connected load is connected to a Y-connected
generator, the system is symbolically represented by YY.
Y-Connected Generator with a YConnected Load
If the load is balanced, the neutral connection can be
removed without affecting the circuit in any manner; that is, if
Z 1 = Z2 = Z3
then IN will be zero
Since IL = V / Z the magnitude of the current in each phase
will be equal for a balanced load and unequal for an unbalanced
load. In either case, the line voltage is
Y-∆ System
There is no neutral connection for the Y-∆ system
shown below.
Any variation in the impedance of a phase that
produces an unbalanced system will simply vary the line
and phase currents of the system.
Y-∆ System
For a balanced load, Z1 = Z2 = Z3.
The voltage across each phase of the load is equal to
the line voltage of the generator for a balanced or an
unbalanced load: V = EL.
Kirchhoff’s current law is employed instead of
Kirchhoff’s voltage law.
The results obtained are:
Example
For the system in figure shown below
a.
Find the phase angle delta2,3 for phase sequence ABC
b.
Find the current in each phase of load
c.
Find the magnetude of the line current
∆-Connected Generator
In the figure below, if we rearrange the coils of the
generator in (a) as shown in (b), the system is referred to
as a three-phase, three-wire.
∆-Connected Generator
∆-connected ac generator
In this system, the phase and line voltages are equivalent
and equal to the voltage induced across each coil of the
generator:
E AB  E AN and e AN  2 E AN sin t
EBC  E BN and eBN  2 EBN sin( t  120)
or
ECA  ECN and eCN  2 ECN sin( t  120)
EL = Eg
Only one voltage (magnitude) is available instead of the two
in the Y-Connected system.
∆-Connected Generator
Unlike the line current for the Y-connected generator,
the line current for the ∆-connected system is not equal
to the phase current. The relationship between the two
can be found by applying Kirchhoff’s current law at one
of the nodes and solving for the line current in terms of
the phase current; that is, at node A,
IBA = IAa + IAC
or
IAa = IBA - IAC = IBA + ICA
∆-Connected Generator
The phasor diagram is shown below for a balanced
load.
In general, line current is:
Phase Sequence (∆- Connected
Generator)
Even though the line and phase voltages of a ∆ connected system are the same, it is standard practice to
describe the phase sequence in terms of the line voltages
In drawing such a diagram, one must take care to have the
sequence of the first and second subscripts the same
In phasor notation,
EAB = EAB 0
EBC = EBC 120
ECA = ECA 120
Example
For the system shown below :
a.
Find the voltage across each phase of load.
b.
Find the magnetude of the line voltage