02-02-2010 dept of aeronautical enggineering s.m.m. rahman mist direct current limitations:...

02-02-2010 Dept of Aeronautical Enggineering S.M.M. Rahman

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Direct Current

Limitations:• Transmission Loss• No Amplification• Power Distribution Lim.•Expensive

02-02-2010 Dept of Aeronautical Enggineering S.M.M. Rahman

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DC or AC ?

”A different type of Power System is needed to overcome the limitations”

Should we try with Alternating Current (AC) system?

Solution?

02-02-2010 Dept of Aeronautical Engineering S.M.M. Rahman

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Generation of Alternating Current

90 180 270 360

“The direction of current flowing in a circuit is constantly being reversed back and forth”

“the light bulb still lights but the electron current is constantly reversing directions”


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Alternating Current

Oscillatory Current

Periodic Current

Alternating Current

i = I0+ I1 sin (ωt+α1)+ I2 sin (2ωt+α2)+.........

i = I1 sin (ωt+α1)+ I2 sin (2ωt+α2)+.........

Alternating current is a periodic current, the average of of which over a period is zero

A periodic current is a oscillating current the values of which at equal interval

An oscillatory current is a current which alternately increases and decreases In magnitude with repect to time according to some definite law

time


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Period and Frequency

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

Period(T)

Time(t)/angle(α)


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Alternating Current/Voltage - Sine Wave

Triangular wave

Square wave

Im

t

ωt = π

ωt = 2π

i= Im sin ωtori=Im sinα

”In practice, many AC waves approximate a sine wave very closely and therefore its calculations are based on sine waves”


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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


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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)


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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 maximumof the lagging quantity


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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”

Impedance is represented by, Z∟Ө Z define the ratio of Vm to Im

Ө define their relative phase differenceZ


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Impedance

C

R Circuit L Circuit C Circuit

RLC CircuitRC CircuitRL Circuit


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Resistive Element

Applied voltage v= Vm sinωt

Current i= Im sinωt

Instantaneous Power is given by,


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Inductive Element

A sinusoidal voltage is applied to a pure inductor

Integrating both sides

The constant c1 will be considered to be zero, then the expression for i reduces to

Vm/Im= ωL and i lags v by 90 degree.Therefore, the impedance of L branch is

Here ωL is called inductive reactance

XL= ωL = 2πfL


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Inductive Element

The instantaneous power delivered to the pure inductance is

•The power variation is symmetrical andThe average power absorbed is equal to zero

• positive and negative power exist for a purelyInductive circuit


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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


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Capacitive Element

CA sinusoidal voltage is applied to a pure inductor

Differentiating the equation with respect to time

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


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Capacitive Element

C

The instantaneous power delivered to the pure capacitor is

•The power variation is symmetrical andThe average power absorbed is equal to zero

• positive and negative power exist for a purelyInductive circuit


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Capacitive Element

C

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


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RL Branch

Resistane R and Inductance L are connected in series and a sinusoidal sinusoidal Current Imsin ωt flows in the circuit, then


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RL Branch


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RL BranchThe instantaneous power of delivered to the RL circuit is

Average Power


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RL Branch

Instantaneous power delivered to the RL branch

The equation has two components, real and reactive component

Instantaneous Real power Instantaneous Reactive power

Real Power Reactive Power

The real and reactive power may be combined to yield the volt-ampere of the circuit


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RLC Branch

A sinusoidal current flows to the RLC series circuitThe voltage accross R, L and C become

The sum of the three component voltages


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RLC Branch


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RLC Branch

Impedance:

• 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


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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


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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 isThe average rate of producing heat by AC current during one cycle is


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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.


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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


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RMS Value of a Sinusoidal Wave

Root mean square value of a sinusoidal wave is 0.7 of the peak value


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Average Value of a Sinusoidal Wave

Therefore, average value of a sinusoidal wave is 0.636 of the peak value


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Form Factor

Form Factor= RMS Value/ Average Value

Form factor is the ratio of the effective voltage to the average value of the wave


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Peak Factor

e = Em sin ωt + (5/12) Em sin 5ωt Form factor= 1.11

Pure sine waveForm 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


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Peak Factor

Peak factor is the ratio of the maximum value to the effective value of a wave

Pure sine wavePeak factor= Em/0.707Em = 1.41

e = Em sin ωt + (5/12) Em sin 5ωt Peak factor= 1.85


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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


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Form Factor and Crest Factor of Different Waves


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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


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• 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


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• A convanient reference axis should be established• Counterclockwise is considered the positive direction of rotation of axis


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• 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


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02-02-2010 dept of aeronautical enggineering s.m.m. rahman mist direct current limitations:...

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