Why is mutual inductance equal for both coils in a transformer?

[cabraham]

Regarding the latest questions, my response is verified in many textbooks, 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

 

[Urmi Roy]

Regarding the latest questions, my response is verified in many textbooks, 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!"

 

[b.shahvir]

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.

 

[Urmi Roy]

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