FORM 6 PHYSICS: AC THEORY PART 3

INDUCTIVE REACTANCE

Inductive reactance is the opposition in which an inductor offers to current flow. It is denoted by X_L

 Inductance not only causes the current to lag behind the voltage but it also limits the magnitude of current in the circuit.

 We have seen above that,

                   I_o =

                   ωL =                       

Clearly the opposition of inductance to current flow is ωL.  This quantity ωL is called inductive reactance XL of the inductor.

            Inductive reactance X_L

a) From
X_L =

 But   =   and I_o =  

Then,

X_L =

                     
b) For d.c

       f = 0

so that,

         X_L = 2

         X_L = 2

Therefore a pure inductance offers zero opposition to d.c

c)  X_L = 2

Therefore,the greater  f,the greater is X_L and vice versa.

d) We can show that the units of X_L are that of ohm

X_L = ωL = x   Henry  =    and                                            

            X_L =  

III) Average Power consumed

E = sin ωt

     I = sin (ωt-          I = – cos ωt

Instantaneous power P

P = EI

P = (sin ωt) (-cos ωt)

               P = –  sin ωt cos ωt

              P =  Sin 2ωt 

Average power P is equal to average of power over one cycle.

P = Sin 2ωtdt  
          
            P = 0              

Hence average power absorbed by pure inductor is zero

During one quarter cycle of alternating source of e .m .f. energy is stored in the magnetic field of the inductor this energy is supplied by the source.

During the next quarter cycle the stored energy is returned to the source.   For this reason average power absorbed by a pure inductor over a complete cycle is zero.

NUMERICAL EXAMPLES

1. A pure inductive coil allows a current of 10A to flow from a voltage of 230V  and frequency 60Hz supply.

Find    

i) Inductive reactance

ii) Inductance of the coil

iii) Power consumed

Write down equations for voltage and current:

Solution

E_V = 230V

I_V = 10A

f =50Hz

      i)   Inductive reactance X_L

         X_L =  =

ii)  Inductance of the coil L

        From

                 X_L =2                                               

L = 0.073 H

iii)   Power absorbed = 0

Also

= 230 x                    = 10 x

= 325.27V              = 14.14A

   ω = 2

           ω = 314

Since in pure Inductive circuit current lags behind the applied voltage by    radians. The equation for voltage and current are,

E = 325 .27 sin 314t,      I = 14.14 sin (314t)

2. Calculate the frequency at which the inductive reactance of 0.7H inductor is 220Ω

Solution

  X_L = 220 Ω

  L = 0.7H

  f =?

  f =

 
  f = 50 Hz

3. A coil has self inductance of 1.4H.  The current through the coil varies sinusoidally with amplitude of 2A and frequency 50 Hz

Calculate

i) Potential difference across the coil

ii) r .m .s value of P.d across the coil.

Solution

(i) P.d across the coil

E = L

E = L

E = L

E = L cos t,

E = L 2 cos2

E = 2

E = 880 cos 100

ii)  r.m.s value of potential different across the coil

 =

=

 = 622.2V

4.  How much inductance should be connected to 200V, 50 Hz a.c supply so that a maximum current of 0.9A flows through it?

Solution

= 200V

I_o = 0.9A

f = 50 Hz

Peak value of voltage

= 

=  

Inductive reactance L
X_L =  

X_L =  

X_L = 314.27 Ω

Inductance L,
L =

L=

L = 1 H

5.  An Inductor of 2H and negligible resistance is connected to 12V, 50Hz supply.  Find the circuit current, what current flows when the inductance is changed to 6H?

Solution

* For the First case X_L

X_L = 2

X_L = 2

X_L = 628 Ω

Circuit current

 =

= 

 = 0.019A

* For the second case X_L^‘

X_L^‘ = 2

X_L^‘ = 2

X_L^‘ = 1884 Ω

Circuit current

=

= 0.0063A