Dept of Aeronautical Enggineering S.M.M. Rahman MIST Direct Current Limitations: Transmission Loss No Amplification Power Distribution Lim. Expensive

Dept of Aeronautical Enggineering S.M.M. Rahman MIST DC or AC ? ”A different type of Power System is needed to overcome the limitations” Should we try with Alternating Current (AC) system? Solution?

Dept of Aeronautical Engineering S.M.M. Rahman MIST Generation of Alternating Current “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”

Dept of Aeronautical Engineering S.M.M. Rahman MIST Alternating Current Oscillatory Current Periodic Current Alternating Current i = I 0 + I 1 sin (ωt+α 1 )+ I 2 sin (2ωt+α 2 ) i = I 1 sin (ωt+α 1 )+ I 2 sin (2ωt+α 2 ) 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

Dept of Aeronautical Engineering S.M.M. Rahman MIST 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)

Dept of Aeronautical Engineering S.M.M. Rahman MIST Alternating Current/Voltage - Sine Wave Triangular wave Square wave ImIm t ωt = π ωt = 2π i= I m sin ωt or i=I m sinα ”In practice, many AC waves approximate a sine wave very closely and therefore its calculations are based on sine waves”

Dept of Aeronautical Engineering S.M.M. Rahman MIST Phase i= I m 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= I m sin (ωt+Ө) represents a sine wave of current with phase angle Ө

Dept of Aeronautical Engineering S.M.M. Rahman MIST Phase Difference Applied voltage is, v = V m 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= I m sin (ωt+Ө)

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

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 ” Impedance is represented by, Z ∟ Ө Z define the ratio of V m to I m Ө define their relative phase difference Z

Dept of Aeronautical Engineering S.M.M. Rahman MIST Impedance C R CircuitL CircuitC Circuit RLC CircuitRC CircuitRL Circuit

Dept of Aeronautical Engineering S.M.M. Rahman MIST Resistive Element Applied voltage v= V m sinωt Current i= I m sinωt Instantaneous Power is given by,

Dept of Aeronautical Engineering S.M.M. Rahman MIST 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 V m /I m = ωL and i lags v by 90 degree. Therefore, the impedance of L branch is Here ωL is called inductive reactance X L = 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

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

Dept of Aeronautical Engineering S.M.M. Rahman MIST Capacitive Element C A sinusoidal voltage is applied to a pure inductor Differentiating the equation with respect to time V m /I m = 1/ωc and i leads v by 90 degree. Therefore, the impedance of C branch is Here 1/ωC is called capacitive reactance X c = 1/ωC = 1/2πfC

Dept of Aeronautical Engineering S.M.M. Rahman MIST Capacitive Element C The instantaneous power delivered to the pure capacitor 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

Dept of Aeronautical Engineering S.M.M. Rahman MIST 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

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 I m sin ωt flows in the circuit, then

Dept of Aeronautical Engineering S.M.M. Rahman MIST RL Branch

Dept of Aeronautical Engineering S.M.M. Rahman MIST RL Branch The instantaneous power of delivered to the RL circuit is Average Power

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

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

Dept of Aeronautical Engineering S.M.M. Rahman MIST RLC Branch

Dept of Aeronautical Engineering S.M.M. Rahman MIST 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

Dept of Aeronautical Engineering S.M.M. Rahman MIST RLC Branch i= I m sinωt and v=V m 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

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

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.

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

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

Dept of Aeronautical Engineering S.M.M. Rahman MIST Average Value of a Sinusoidal Wave Therefore, average value of a sinusoidal wave is of the peak value

Dept of Aeronautical Engineering S.M.M. Rahman MIST Form Factor Form Factor= RMS Value/ Average Value Form factor is the ratio of the effective voltage to the average value of the wave

Dept of Aeronautical Engineering S.M.M. Rahman MIST Peak Factor e = E m sin ωt + (5/12) E m sin 5ωt Form factor= 1.11 Pure sine wave Form factor=0.707 V m /0.636 V m = 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

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 Pure sine wave Peak factor= E m /0.707E m = 1.41 e = E m sin ωt + (5/12) E m sin 5ωt Peak factor= 1.85

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

Dept of Aeronautical Engineering S.M.M. Rahman MIST Form Factor and Crest Factor of Different Waves

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

Dept of Aeronautical Engineering S.M.M. Rahman MIST Vector Representation of a Sine Wave

Dept of Aeronautical Engineering S.M.M. Rahman MIST Vector Representation of a Sine Wave

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

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

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

Dept of Aeronautical Engineering S.M.M. Rahman MIST Vector Representation of a Sine Wave

Dept of Aeronautical Engineering S.M.M. Rahman MIST Vector Representation of a Sine Wave