Download 09-02-2010 Dept of Aeronautical Engineering SMM Rahman MIST

MIST
Direct Current
Limitations:
• Transmission Loss
• No Amplification
• Power Distribution Lim.
•Expensive
02-02-2010
Dept of Aeronautical Enggineering
S.M.M. Rahman
MIST
DC or AC ?
”A different type of Power System is
needed to overcome the limitations”
Solution?
Should we try with Alternating Current (AC) system?
02-02-2010
Dept of Aeronautical Enggineering
S.M.M. Rahman
MIST
Generation of Alternating Current
“the light bulb still lights but the electron
current is constantly reversing directions”
“The direction of current flowing in a circuit is
constantly being reversed back and forth”
90
02-02-2010
180
270
360
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Alternating Current
Oscillatory Current
An oscillatory current is a current which alternately
increases and decreases In magnitude with repect to
time according to some definite law
time
Periodic Current
A periodic current is a oscillating current the values
of which at equal interval
i = I0+ I1 sin (ωt+α1)+ I2 sin (2ωt+α2)+.........
Alternating Current
Alternating current is a periodic current, the average
of of which over a period is zero
i = I1 sin (ωt+α1)+ I2 sin (2ωt+α2)+.........
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Period and Frequency
Time(t)/angle(α)
Period(T)
The time for one complete cycle is defined as Period ( T ).
Freqency ( f )is the number of cycles per second
frequency,
f = 1/T
A complete cycle corresponded to 2π electrical radians or
360 degrees. Therefore the Angular Velocity,
ω = 2 π/T = 2 πf
We use 50 Hz AC sytem
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Alternating Current/Voltage - Sine Wave
Im
Triangular wave
t
ωt = π
Square wave
02-02-2010
ωt = 2π
”In practice, many AC waves approximate a sine wave
very closely and therefore its calculations are based
on sine waves”
i= Im sin ωt
or
i=Im sinα
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Phase
i= Im sin (ωt+Ө)
Ө
phase angle
”Phase is the fractional part of a period through which time or the
associated time angle ωt has advanced from an arbitrary reference”
i= Im sin (ωt+Ө) represents a sine wave of current with phase angle Ө
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Phase Difference
Applied voltage is,
v = Vm sin ωt
Due to nature of circuit parameters the current comes to a certain point before the
voltage wave by degrees to that point, then the current can be expressed as
i= Im sin (ωt+Ө)
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Phase Difference
To further describe the phase relationship between two sine waves, the terms
Lead and Lag are used. The amount by which one sine wave leads or lags another
sine wave is measured in degrees.
The positive maximum of the leading quantity occurs before the positive maximum
of the lagging quantity
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Impedance
“Electrical impedance extends the concept of resistance to AC circuits,
describing not only the relative amplitude of the voltage and current,
but also the relative phases”
Z
02-02-2010
Impedance is represented by, Z∟Ө
Z define the ratio of Vm to Im
Ө define their relative phase difference
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Impedance
C
R Circuit
RL Circuit
02-02-2010
L Circuit
C Circuit
RC Circuit
RLC Circuit
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Resistive Element
Applied voltage v= Vm sinωt
Current i= Im sinωt
Instantaneous Power is given by,
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Inductive Element
A sinusoidal voltage is applied to a pure inductor
Vm/Im= ωL and i lags v by 90 degree.
Therefore, the impedance of L branch is
Integrating both sides
The constant c1 will be considered to be zero,
then the expression for i reduces to
02-02-2010
Here ωL is called inductive reactance
XL= ωL = 2πfL
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Inductive Element
The instantaneous power delivered to the pure
inductance is
•The power variation is symmetrical and
The average power absorbed is equal to zero
• positive and negative power exist for a purely
Inductive circuit
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Inductive Element
The inductive element receives energy from source during one quarter of the
applied voltage And returns exactly the same amount of energy to the driving
source during the next one-quarter of a cycle.
The energy delivered to the circuit during a quarter of a cycle is
02-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Capacitive Element
A sinusoidal voltage is applied to a pure inductor
Differentiating the equation with respect to time
C
Vm/Im= 1/ωc and i leads v by 90 degree.
Therefore, the impedance of C branch is
Here 1/ωC is called capacitive reactance
Xc= 1/ωC = 1/2πfC
03-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Capacitive Element
The instantaneous power delivered to the pure
capacitor is
C
•The power variation is symmetrical and
The average power absorbed is equal to zero
• positive and negative power exist for a purely
Inductive circuit
03-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Capacitive Element
The capacitive element receives energy from source during one quarter of
the applied voltage and returns exactly the same amount of energy to the
driving source during the next one-quarter of a cycle.
The energy received by the capacitor during a quarter of a cycle is
C
03-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RL Branch
Resistane R and Inductance L are connected in series and a sinusoidal sinusoidal
Current Imsin ωt flows in the circuit, then
03-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RL Branch
03-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RL Branch
The instantaneous power of delivered to the RL circuit is
Average Power
03-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RL Branch
Instantaneous power delivered to the RL branch
The equation has two components, real and reactive component
Instantaneous Real power
Real Power
Instantaneous Reactive power
Reactive Power
The real and reactive power may be combined to yield the volt-ampere of the circuit
03-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RLC Branch
A sinusoidal current flows to the RLC series circuit
The voltage accross R, L and C become
The sum of the three component voltages
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RLC Branch
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RLC Branch
• Inductive and capacitive branch cause exactly opposite phase
displacement of current with respect to voltage.
• ωL is positive quantity and 1/ωC is negative quantity
Impedance:
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RLC Branch
i= Im sinωt and v=Vm sin(ωt+Ө)
Instantanous power delivered to the RLC branch
Real power delivered to the RLC branch is
Reactive power delivered to the RLC branch is
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Effective Current and Voltage
DC
AC
”An alternating current which produces heat in a given resistance at the same
average rate as I amperes of direct current is said to have a value of I ”
The average rate of producing heat by direct current is
The average rate of producing heat by AC current during one cycle is
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Root Mean Square Value
The RMS value is the effective value of a varying voltage or current. It is the
equivalent steady DC (constant) value which gives the same effect.
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Average Value
Average value of a ac wave is zero,
however, average value of a ac wave
means the average of either the positive
or negative loop of the wave
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
RMS Value of a Sinusoidal Wave
Root mean square value of a sinusoidal wave is
0.7 of the peak value
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Average Value of a Sinusoidal Wave
Therefore, average value of a sinusoidal wave is 0.636 of the
peak value
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Form Factor
Form factor is the ratio of the effective voltage to the average value of the wave
Form Factor= RMS Value/ Average Value
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Peak Factor
e = Em sin ωt + (5/12) Em sin 5ωt
Form factor= 1.11
Pure sine wave
Form factor=0.707 Vm/0.636 Vm
= 1.11
•Form factor of both waves are same
•It gives no certain indication of wave shape or wave form
•Give some indication of relative hysteresis loss
• use in determining induced effective voltage when non sinusoidal flux
wave is present in the iron core
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Peak Factor
Peak factor is the ratio of the maximum value to the effective value of a wave
e = Em sin ωt + (5/12) Em sin 5ωt
Peak factor= 1.85
09-02-2010
Pure sine wave
Peak factor= Em/0.707Em = 1.41
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Peak Factor
•The purpose of the crest factor calculation is to give an analyst a
quick idea of how much impacting is occurring in a waveform
• For dielectric test a knowledge of crest factor is required
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Form Factor and Crest Factor of Different Waves
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
• AC computations are often based upon the assumption of sine waves
of voltage and current
• It is cumbersome to handle instantaneous values in the form of equations
of the waves
• Vector/ Phasor method could be used to represent sine waves
• It simplifies certain kinds of calculations
• The phasor/vector representations of sine functions may be manipulated
instead of the sine functions themselves to secure the desired result
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
• The lengths of two the vectors represent maximum values of
the waves respectively, the resultant vector will represent the
maximum value of the two waves
• Effective or rms value could be used instead of maximum values
• The vector can be considered to represent effective values
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
• A convanient reference axis should be established
• Counterclockwise is considered the positive direction of rotation of
axis
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
• For a pure resistance applied voltage is in phase
with the current
• For a pure inductance circuits, the voltage drop accross
the inductor leads the current by 90 degree
• For a pure capacitance circuits, the voltage drop accross
the capacitor lags the current by 90 degree
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman
MIST
Vector Representation of a Sine Wave
09-02-2010
Dept of Aeronautical Engineering
S.M.M. Rahman