AC circuit containing inductive coil of zero ohmic resistance or non inductive ohmic resistance

The inductive reactance is different from the ohmic resistance where the inductive reactance for a coil of zero resistance does not cause loss in electric energy , because the current resistance is due to the reverse induced emf where the electric energy is stored in the coil in the form of a magnetic field , The non inductive ohmic resistance causes loss in electric energy in the form of heat energy .

AC circuits

The value of the self inductance coefficient ( L ) for a solenoid coil is determined by the relation : L = μAN²/l , The electric current intensity passing in an induction coil of no resistance is determined from the relation :

I = Voltage difference at coil terminals ( VL) / Inductive reactance of the coil ( XL)

In high frequencies , the inductive reactance ( XL ) becomes very big where ( XL∝ f ) , Consequently the electric current intensity decreases where ( I ∝ 1/XL ) and the circuit becomes as if it is opened .

AC circuit

AC circuit

AC circuit containing inductive coil of zero ohmic resistance 

When connecting inductive coil of zero ohmic resistance to AC source and a switch in series , then on closing the circuit , the electric current grows gradually from zero to maximum value at a rate ( ΔI/Δt ) .

The variation in the current intensity as time passes generates reverse induced emf by self induction equals ( − L ΔI/Δt ) , Where : L = self induction coefficient of the coil .

The frequency of the induced emf is the same as that of the AC supply but acting in the opposite direction to the emf of the supply , So , the instantaneous value of the potential difference ( V ) =  − L ΔI/Δt

The phase difference between the current and the potential difference

The electric current intensity ( I ) varies with the phase angle according to sine curve as in figure , the value ( ΔI/Δt ) represents the slope of the tangent drawn to the curve where :

  1. It reaches its peak value when the phase angle equals zero that means that the voltage ( V ) reaches maximum values .
  2. The slope decreases gradually to reach zero when ( I ) reaches its peak value and the voltage reaches zero .
  3. The slope ( ΔI/Δt ) becomes negative when the current intensity decreases and the voltage becomes negative value .

So , the voltage ( V ) leads the current ( I ) by ¼ cycle or by a phase angle 90° due to the self induction of the coil .

The inductive reactance in a coil ( XL

Since the reverse induced emf produced by self induction in a coil of zero resistance cause some kind of resistance to the flow of the original current , it is called inductive reactance , The inductive reactance in a coil ( XL) is the opposition to the flow of the AC current through the coil due to its self-inductance .

The inductive reactance is measured in Ohm ( Ω ) , The inductive reactance is determined from the relation : XL = 2πfL = ω L , Where : ( L ) coefficient of self induction , ( f ) frequency of the current passing in the coil , ( ω ) the angular velocity .

When the inductive reactance of a coil = 100 Ω , It means that the opposition to the flow of the AC current through the coil due to its self induction = 100 Ω .

Factors affecting the inductive reactance of a coil
  1. Current frequency ( f ) , ( directly proportional ) .
  2. Self induction coefficient ( L ) , directly proportional .
The inductive reactance of inductor network

When connecting many inductive coils together ( away from each other ) then , If conductors are connected in series .

L = L1 + L2 + L2

XL = ( XL )1 + ( XL )2 + ( XL )3

If the self induction coefficients for all coils are equal and number of coils ( n ) .

L = n L1   ,   XL = n ( XL )1

If conductors are connected in parallel .

1/L = 1/L1 + 1/L2 + 1/L2

1/XL =1/( XL )1 + 1/( XL )2 + 1/( XL )3

If the self induction coefficients for all coils are equal and number of coils ( n ) .

L = L1 / n    ,   XL = ( XL )1 / n

AC circuit containing non inductive ohmic resistance

When connecting non inductive ohmic resistance to AC source and a switch in series , then closing the circuit , the potential difference between the terminals of the resistance ( R ) :

V = Vmax sin θ = Vmax sin ωt

Where : V = instantaneous value of potential difference , Vmax = maximum value of potential difference , θ = phase angle ( θ = ωt ) , ω = the angular velocity ( ω = 2πf ) .

Based on Ohm’s law , the electric current is determined from the relation :

I = V/R = Vmax sin ωt / R = Imax sin ωt

The potential difference and the current intensity in an ohmic resistance are having the same phase , thus the current and potential grow together up to a maximum value and drop together to zero , The current and potential difference in a resistance without induction are represented by two vectors having the same direction .

Properties of the alternating current , Hot wire ammeter uses , cons and pros

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