Class 12 Physics Alternating Current Inductive AC Circuit

Inductive AC Circuit

In Inductive AC circuit only circuit element which is present is inductor.

AC voltage supplied to a inductor:-

  • The source of voltage is alternating as is represented as V = Vm sinωt.
  • In the circuit there is source voltage(V) and an inductor with inductance = L.
  • In this circuit there are no resistors. There is one source EMF i.e. is the source voltage and another emf is self-induced.
    • As current is changing therefore the magnetic flux associated with the current also changes.
    • According to Faraday’s Lenz’s law whenever there is change in the flux a current is induced or an EMF is induced in the inductor.
    • As a result there will be self-induced EMF in the inductor which will oppose the change which is causing it.
  • Therefore V + e = 0.
    • Where V = source voltage and e = self- induced emf in the inductor L.
  • => V - L(dI/dt) = 0 . Using e = -L (dI/dt)
  • => Vm sinωt - L(dI/dt) = 0.
  • =>dI = (Vm sinωt dt /L)
  • Integrating both sides , therefore 0I∫dI = ∫ (Vm sinωt dt /L)
  • After simplifying, I = (Vm/L) [ -cosωt/ω] + constant
  • I = - (Vm/ ω L) cosωt + 0
    • (constant = 0 because as source voltage oscillate symmetrically about 0, therefore current should also oscillate about 0.)
  • I = - Im cosωt  
    • where Im = (Vm/ ω L)
  • I = Im sin(ωt –(∏/2)) .  This is the current which will flow through the circuit.

Conclusion:-

The current and voltage are not in phase with each other. They are out of phase by (∏/2).

(Circuit diagram containing a voltage source and an inductor).

 

Inductive Reactance

  • Current amplitude Im = (Vm / ω L) .
  • In an inductance circuit  (ω L) acts as resistance, when compared with I = (V/R). Therefore the resistance of inductive circuit is  known as inductive reactance.
  • Inductive reactance  is  the resistance  associated with  a  pure inductive AC circuit.
  • It is denoted by XL.
  • S.I. unit: ohm(Ω).
  • It limits the current flowing through an inductive circuit.
  • XL = ω L . =>  XL  ∝ ω  and XL  ∝ L.

Problem:- A 44 mH inductor is connected to 220 V, 50 Hz ac supply. Determine the RMS value of the current in the circuit.

Answer:-

Inductance of inductor, L = 44 mH = 44 × 10−3 H

Supply voltage, V = 220 V

Frequency, ν = 50 Hz

Angular frequency, ω = 2πν

Inductive reactance, XL = ω L = 2πν L = (2π × 50 × 44 × 10−3) Ω

RMS value of current is given as:

I = (V/XL)

= (220)/(2π × 50 × 44 × 10−3) = 15.92 A

Hence, the rms value of current in the circuit is 15.92 A.

 

Graphical representation of Voltage & Current

  • Voltage and current are represented as:- V = Vm sinωt  and I = Im sin(ωt –(∏/2)) respectively. Voltage and current are out of phase by (∏/2).
  • Current lags voltage by (∏/2).Current will reach its maximum value after(∏/2).
  •  Average current over a complete cycle is 0.
  • Average voltage over a complete cycle is 0.

 

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