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Phasor diagrams|Alternating Current|A.C through pure resistor






(4) Root Mean square value of AC

  • We know that time average value of AC over one cycle is zero and it can be proved easily
  • Instantaneous current I and time average of AC over half cycle could be positive for one half cycle and negative for another half cycle but quantity i2 would always remain positive
  • So time average of quantity i2 is

    Phasor diagrams|Alternating Current|A.C through pure resistor
    This is known as the mean square current
  • The square root of mean square current is called root mean square current or rms current.
    Thus,
    Phasor diagrams|Alternating Current|A.C through pure resistor
    thus ,the rms value of AC is .707i0 of the peak value of alternating current
  • Similarly rms value of alternating voltage or emf is

    Phasor diagrams|Alternating Current|A.C through pure resistor
  • If we allow the AC current represented by i=i0sin(ωt+φ) to pass through a resistor of resistance R,the power dissipated due to flow of current would be
    P=i2R
  • Since magnitude of current changes with time ,the power dissipation in circuit also changes
  • The average Power dissipated over one complete current cycle would be

    Phasor diagrams|Alternating Current|A.C through pure resistor
    If we pass direct current of magnitude irms through the resistor ,the power dissipate or rate of production of heat in this case would be
    P=(irms)2R
  • Thus rms value of AC is that value of steady current which would dissipate the same amount of power in a given resistance in a given tine as would gave been dissipated by alternating current
  • This is why rms value of AC is also known as virtual value of current

(5) Phasor diagram

  • Phasor diagrams are diagram representing alternating current and voltage of same frequency as vectors or phasors with the phase angle between them
  • Phasors are the arrows rotating in the anti-clockwise direction i.e. they are rotating vectors but they represents scalar quantities
  • Thus a sinusoidal alternating current and voltage can be represented by anticlockwise rotating vectors if they satisfy following conditions
  • Length of the vector must be equal to the peak value of alternating voltage or current
  • Vector representing alternating current and voltage would be at horizontal position at the instant when alternating quantity is zero
  • In certain circuits when current reaches its maximum value after emf becomes maximum then current is said to lag behind emf
  • When current reaches its maximum value before emf reaches its maximum then current is said to lead the emf
  • Figure below shows the current lagging behind the emf by 900


    alt=




(6) A.C through pure resistor

  • Figure below shows the circuit containing alternating voltage source V=V0sinω connected to a resistor of resistance R


    A.C through pure resistor

  • Let at any instant of time ,i is the current in the circuit ,then from Kirchhoff’s loop rule
    V0sinωtRi
    or
    i=(V0/R)sinωt
    =i0sinωt                                                ----(8)
    Where,
    i0=V0/R                                                ----(9)
  • From instantaneous values of alternating voltage and current ,we can conclude that in pure resistor ,the current is always in phase with applied voltage
  • Their relationship is graphically represented as


    Sinusodical represectation of alternating volatage and current in AC circuit




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