Self Inductance and Mutual Inductance Explained

 

    Hey, friends, welcome to the new post on ALL ABOUT ELECTRONICS, and today we will see  what is Self Inductance and Mutual Inductance in the electrical circuit.  

    So, first, we will understand the concept of self-inductance and mutual inductance and  then we will derive the expression for the self and the mutual inductance.  So, let's first see the inductance. So, it is a property of the electric conductor  by which the rate of change of current produces the electromotive force or emf.  So, let's say we have one coil. And the current I is flowing through this  coil. So, the rate of change of current through  this coil produces emf or voltage. 

    Now, if this electromotive force or voltage  is produced within that same coil then that is called self-inductance.  And if the rate of change of current produces the emf or voltage in nearby coil then that  is called mutual inductance. So, let's understand first the self-inductance.  So as we have said earlier, the voltage or emf that is generated is proportional to the  rate of change of current. So, we can write, V= - L* (di/dt)  Where L is the self-inductance or simply inductance. And the unit of the self-inductance or inductance  is Henry. 

    So, you will observe negative sign.  And this negative sign is because of the Lenz's law. So, according to the Lenz's law, the generated emf or voltage opposes the rate of change  of current through which it is been generated. So, here this negative sign implies the voltage  that generated is opposes the rate of change of current.  So, let's say this is equation number 1. So, now let's find the expression for this  inductance in terms of magnetic flux. So, let's say we have one coil.  

    And in this coil the current I is flowing. So, because of flow of current, there will  be a generation of magnetic flux phi. And if the current that is flowing through  this coil is varying with time, or we can say that the current flowing through  this coil is time varying then the magnetic flux that will be generated will also be time  varying. So, this time varying magnetic flux produces  the emf or voltage in this coil. 

    According to the Faraday's law, the voltage  that is induced in this coil can be given as,  V= -N*(dɸ/dt) Where N is the number of turns in this coil. 
 So, let's say this equation number 2. So, here we got two equations, so now let's  compare this two equations. So, we can write, 
 -L*(di/dt) = -N* (dɸ/dt) 
Or we can write,  
L*di = N*dɸ
 That means  L*i = N* ɸ Now here this N*ɸ also known as the flux  linkage or magnetic flux linkage. And sometimes it is also denoted by symbol  ci. 

    So, we can write,  L = N* (phi)/i or  (Ci)/i So, this is the expression for inductance  in terms of magnetic flux and the current. So, now let's see the mutual inductance.  So, let's here we have two coils one and two. And the number of turns in this coils are  N1 and N2 respectively. Now let's say the current i1 is flowing through  this coil number 1. And because of the flow of current, there  is a generation of magnetic flux. Let's say that is ɸ1.  So, now this magnetic flux will link with this coil number 1 and 2.  

    Let's say the  ɸ11 is the flux that is linked with the coil number 1.   And ɸ12 is the flux that is linked with this coil number 2.  So, here the current that is flowing through this coil number 1 is time varying.  That is, it is varying with time. So, because of that, the flux that is linked  with this coil number 2 will also be a time-varying. So, this time varying magnetic field will  generate a voltage in this coil number 2. 

    And let's say that is V2.  So according to the Faraday's law, we can write,  
V2= - N2* (dɸ12)/dt) Where N2 is a number of turns in this coil  number 2.

     Let's say this is the equation number 3.  Now, the voltage that is generated in this coil number 2 is proportional to the rate  of change of current in the coil number 1.so, we can write V2 is proportional to the  rate of change of current in the coil number 1.  

    That means, V2= -M*(di1/dt)  Where M is nothing but mutual inductance between  this two coils. And here the negative sign implies the voltage  that is generated in the coil number 2 opposes the rate of change of current. The unit of this mutual inductance is same  as the self-inductance. That is Henry.  So, let's say this is the equation number4.  So, now here we have got two equations 3 and 4.  

    So, now let's find the value of this mutual  inductance in terms of the magnetic flux. So, we can write 
-M* (di1/dt) = -N2* (dɸ12/dt)  
or we can write,
  M*(di1) = N2* d(ɸ12)  
That means, 
M*i1 = N2*ɸ12  
 So we can write
M= N2*ɸ12/i1 
 So this is the expression for mutual inductance when the current is flowing in the coil number   1. Similarly, if the current is flowing in the  coil number 2, and because of that if the voltage is generated  in the coil number 1 then the mutual inductance M can be given as 
 M= N1*ɸ21/i2 
Where ɸ21 is the flux that is linked to the  coil number 1 because of the current that is flowing in the coil number 2. 

     So, now we can write mutual inductance M= (N2*ɸ12/i1) =(N1*ɸ21/i2)  So, now here the coupling between the two coil defines the how well the flux that is  linked to the another coil. If the coupling between the two coil is very  good, then the flux that is linked to the another coil will be the good.  

    Similarly, if the coupling between the two coil is bad then the flux that is linked to  the another coil will be poor. So, to define this coupling between the two  coils, we use a term Coefficient of coupling.  That is the fraction of total flux that is linked to the another coil.  And it is denoted by symbol K. So, let's say ɸ1 is the total flux that is  generated because of the current that is flowing in the coil number 1.  

    And out of this ɸ1, ɸ12 is the flux that  is linked to the coil number 2 And the ratio of flux (ɸ12/ɸ1) is known  as the coefficient of coupling. Similarly, if ɸ2 is the total flux, that is generated because of the current flowing   in the coil number 2,And out of the total flux if the flux ɸ21  that is linked to the coil number 1, then the ratio of this ɸ21/ɸ2 is known as the  coefficient of coupling. So, the value of this k is between the 0 and  1.

    If the value of k is 1, that means the coupling between the two coil is 100%. If the value of k is 0, then there is no coupling  between the two coils. So, now let's just find the relation between  the coefficient of coupling and the mutual inductance.  So, earlier we have found this expression for mutual inductance.  So, now let's just multiply this two equations. So, we can write, 
 M^2 = (N1*ɸ21/i2)*(N2*ɸ12/i1) 
So, let's just multiply and divide this term  by ɸ1*ɸ2. And by rearranging the terms we can write,  
(N1*ɸ1/i1)*(N2*ɸ2/i2)*(ɸ21/ɸ2)*(ɸ12/ɸ1).

    As we have seen earlier, the first two  So, as we have seen earlier, the first two terms are nothing but the self inductance of the coil 1 and 2.  That is L1 and L2. And if we observe the last two terms,  they are nothing but the coefficient of coupling. That is k.  So, we can write them as k*k. That means,  
M^2= K^2*L1*L2 That means,
 M=k*sqrt(L1*L2)
  So, this is the expression between the mutual  inductance and the coefficient of coupling. So, now let's just take one simple example  and find the value of this coefficient of coupling k.  

    Here we have given the values of L1, L2, and M.  And we need to find the value of this coefficient of coupling.  So, we can write K= M/(sqrt(L1*L2)) That is nothing but,  0.05/(sqrt(0.1*0.1)) That is equal to 0.05/0.1  So, the  value of coefficient of coupling k = 0.5

So, in this way, if we have given the value of self-inductance and mutual inductance then  we can find the value of the coefficient of coupling.  Or in another way, we have given the value of self-inductance and the coefficient of  coupling, then we can find the value of this mutual inductance.  So, I hope you understood what is self-inductance and the mutual inductance in the electrical circuit.

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