Self Inductance and Mutual Inductance Explained


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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.
So, 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. So, 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 number
4. 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)
So, 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. So, 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