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.

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. .