# Mutual induction.

Thankyou,B.Shahvir!

So what I understand is that my understanding of the coaxial cable as the core having a separate voltage source from the shield is indeed wrong,the source of power for both of them is the same i.e the alternating applied voltage source....that's sorted.

However,I should be more clear as to what exactly I'm looking for.
The thought that is presently nagging me is as what would happen if we indeed connected a separate voltage source to the secondary!

In that case,we would have to calculate the net flux shared between the two coils...but then....either of the two coils has a certain amount of its own generated flux(generated in response to the applied voltage)which it shares with the other coil...then how would we calculate the net flux in the transformer core?
(we can't just add them up.)

This is where the 'superposition principle' seems to comes in...but I don't really understand how we should go about using it.

Here's a sum that might illustrate what exactly I want .....

"2 coils A and B lie in parallel planes.Coil A has 150 turns and coil B has 120 turns.
55% of the flux produced by coil A links coil B. A current of 6A in coil A produces 0.05 mwb,while the same current in coil B produces 0.08mwb. Calculate the mutual inductance and coupling coefficient."

My doubts:
1. Firstly,as cabraham said in his last post,the coupling factors 'k1' and 'k2' for the two coils should not be the same..so why do they say " calculate the coupling coefficient"?

2. It says "55% of the flux produced by coil A links coil B" ...so would the coupling coefficient for A be 0.55?

3. As it says in the sum,both the coils are energised...so we need to calculate the net flux shared i.e the mutual flux (which would require a superposition principle) but in my book,the say:
"mutual flux=55% of 0.05mwb(flux produced by A)".....but this doesn't take into account the flux produced by B!!

(One important point to note: they want us to calculate the mutual inductance ...which means that they know that the net flux in the transformer core is not due to only A or B....but the question remains....how do we calculate the net flux in the transformer core when both the coils are energised,and taking into account their individual coupling factors)

Thankyou,B.Shahvir!

So what I understand is that my understanding of the coaxial cable as the core having a separate voltage source from the shield is indeed wrong,the source of power for both of them is the same i.e the alternating applied voltage source....that's sorted.

However,I should be more clear as to what exactly I'm looking for.
The thought that is presently nagging me is as what would happen if we indeed connected a separate voltage source to the secondary!

In that case,we would have to calculate the net flux shared between the two coils...but then....either of the two coils has a certain amount of its own generated flux(generated in response to the applied voltage)which it shares with the other coil...then how would we calculate the net flux in the transformer core?
(we can't just add them up.)

This is where the 'superposition principle' seems to comes in...but I don't really understand how we should go about using it.

The question is...whether the 2 voltage sources are out of phase. If the 2 voltage sources are in phase, then the 2 fluxes will cancel out in opposite direction (in case of core type Xmer) and the net flux within the core will add up to zero. However, in case of shell type Xmer, if both the windings are on the same limb, the both fluxes will add up in same direction (2 equivalent coils in parallel with addition of flux). Also, observe equivalent ckt. of Xmer, especially magnetizing reactance branch in parallel with the voltage sources responsible for flux creation.

An interesting case is that of induction motor during regenerative braking when rotor acts as a separate voltage source secondary. The primary (stator) voltage is in quadrature with secondary (rotor generated) voltage. Hence the fluxes are in quadrature with each other and do not directly add up by superposition principle. The rotor flux is immediately neutralized by reverse stator flux (primary current fed back to supply via stator i.e. reverse Xmer action). However, net flux in air gap is in same direction as before and is due to stator magnetizing current only.

The question is...whether the 2 voltage sources are out of phase. If the 2 voltage sources are in phase, then the 2 fluxes will cancel out in opposite direction (in case of core type Xmer) and the net flux within the core will add up to zero.

So what I understand is,suppose we have the two coils of a transformer,we apply separate voltages to the coils,then at any instant,the net flux in the core would be what we get by adding the flux by each of the coils as per superposition principle.

In that case,if the two applied voltages are not exaclty in phase,or exactly out of phase,we would have to perform this kind of addition for each instant separately,otherwise we couldn't know.

The thing is,NpIp=NsIs....here,Ip and Is are obviously not the same,as we have two different voltage sources attached...so in each coil,we would have to consider the current due to
1. the voltage source attached to it.
2.the mutually induced current.
3. the self induced current,

only then we could make NpIp=NsIs......hence,the transformer core would contain flux due to all of these factors....and so my simple reasoning of superposition principle fails!

So what I understand is,suppose we have the two coils of a transformer,we apply separate voltages to the coils,then at any instant,the net flux in the core would be what we get by adding the flux by each of the coils as per superposition principle.

In that case,if the two applied voltages are not exaclty in phase,or exactly out of phase,we would have to perform this kind of addition for each instant separately,otherwise we couldn't know.

The thing is,NpIp=NsIs....here,Ip and Is are obviously not the same,as we have two different voltage sources attached...so in each coil,we would have to consider the current due to
1. the voltage source attached to it.
2.the mutually induced current.
3. the self induced current,

only then we could make NpIp=NsIs......hence,the transformer core would contain flux due to all of these factors....and so my simple reasoning of superposition principle fails!

Exactly!... for in-phase quantities superposition (arithmatic addition) is applicable, whilst for out of phase quantities vector addition is applicable. Hence AC quantities have more of vector mathematics than simple arithmatics as most of the AC quantities are out of phase with each other due to presence of reactive components in the AC system. This is applicable for both 3 phase & 1 phase AC system. Also, the type of magnetic circuit available affects magnetic flux distribution & measurements.

Hope this helps.

Regards,
Shahvir

Regarding the latest questions, my response is verified in many text books, which anyone can obtain. The "coupling coefficient" can be called "k". If k12 represents the coupling *from 1 to 2*, and k21 represents coupling from 2 to 1, then the overall coupling is given by:

k = sqrt (k12*k21).

As far as the coax cable goes, I don't think that the shield is a shorted turn. If we sketch an iron core xfmr, add a shorted secondary turn, and examine the direction of the core flux, we will see that the shorted turn is oriented normal to the flux. But the coax shield is along the flux of the center conductor, not normal.

The mutual inductance Lm, does equal the shield self inductance Ls, and Henry Ott of Bell Labs derives this relation in his highly acclaimed book "Noise Reduction Techniques In Electronic Systems". This is true for a general coaxial cable. When we say "in general" I presume that coax cable is under discussion. If "in general" refers to other configurations besides coax, then different relations are encountered.

In general, if 2 coils mutually interact, then k = sqrt (k12*k21), and Lm = k*sqrt(L1*L2). This can be derived but it is involved. An advanced fields text might have the derivation with illustrations. With grad school I have no time to derive it. Maybe in June when things slow down I might have time. Best regards.

Claude

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Regarding the latest questions, my response is verified in many text books, which anyone can obtain. The "coupling coefficient" can be called "k". If k12 represents the coupling *from 1 to 2*, and k21 represents coupling from 2 to 1, then the overall coupling is given by:

k = sqrt (k12*k21).

Actually it's me who seems to be getting mixed up.I had been thinking all this time that k depends exclusively on the individual coils themselves (no. of turns,length,etc.)...so we would have different k values for each...
also,as you said in an earlier post, the k values don't have to be the same for the coils,I thought that confirmed my idea,
However,you say,(like in my book )that k = sqrt (k12*k21).,which means there is only one value of k for a particular transformer.....I can't really explain to myself why there is a unique value of k!

(As 'The Electrician' knows,my book often contains mistakes,so I was wondering if the fact that k has a unique value as stated in my book is one of those mistakes,however,you've just confirmed that it's not.)

The mutual inductance Lm, does equal the shield self inductance Ls

I got that...then I tried to prove how the mutual inductances could be equal for both the core and the shield(and without success ofcourse)...
"Lm of core=Nc(k*phi_s)/Is Lm' of core =Ns(k'*phi_c)/Ic...and you said k' was less than unity and Ic=Is....since Nc>Ns (these are the magnetic heights,which is analogous to turns in coils,I suppose)....then (k*phi_s)<(k'*phi_c)....but k'<k.....so phi_s>phi_c....but that's the opposite!"

As far as the coax cable goes, I don't think that the shield is a shorted turn. If we sketch an iron core xfmr, add a shorted secondary turn, and examine the direction of the core flux, we will see that the shorted turn is oriented normal to the flux. But the coax shield is along the flux of the center conductor, not normal.

The mutual inductance Lm, does equal the shield self inductance Ls, and Henry Ott of Bell Labs derives this relation in his highly acclaimed book "Noise Reduction Techniques In Electronic Systems". This is true for a general coaxial cable. When we say "in general" I presume that coax cable is under discussion. If "in general" refers to other configurations besides coax, then different relations are encountered.

In general, if 2 coils mutually interact, then k = sqrt (k12*k21), and Lm = k*sqrt(L1*L2). This can be derived but it is involved. An advanced fields text might have the derivation with illustrations.
Claude

I agree. Also, mutual induction are compex quantities & depend on plane of co-incidence resulting in lesser net magnitude of flux linkage.

Could someone clear the confusion about the k values as I stated in post 32#, please.

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