Equivalence of various forms of flux linkage

[lagrangman]

Hello all,

Apologies if this has already been asked before, but I tried researching this question for a while with no results.

I was reading Grainger's Power System Analysis' derivation of the inductance of a single wire and got confused by his definition of magnetic flux linkage.

He seems to state that magnetic flux linkage (λ) is ∫ I dΦ where dΦ is a differential increase in magnetic flux and I is the incremental current that causes the differential increase in magnetic flux.

I am familiar with the following definitions of flux linkage (λ):

(1) λ = N Φ where Φ is the flux that N turns of conductor "contain."
(2) λ = ∫ V dt where V is voltage caused by electromagnetic induction and t is time.
(3) λ = L I where L is the inductance of an inductor-like device and I is the current flowing through the inductor-like device.

My question is why is ∫ I dΦ equivalent anyone of the three expressions above?

Thanks in advance!

 

[Svein]

If the flux is constant and you have N turns you get (1).

 

[lagrangman]

Thanks for the response, however I am still confused. Where does the contributing current factor in then?

Hopefully this isn't a copyright violation, but Chang writes, "the flux linkages dλ per meter of length, which are caused by the flux in the tubular element, are the product of the flux per meter of length and the fraction of the current linked." (p. 132.)

This would correspond to a formula for flux linkage I found on the internet (∫ I dΦ as defined above), but I don't know how to derive it from the other definitions of magnetic flux.

I might be making some mistakes so feel free to correct me.

Thanks again.

 

[alan123hk]

formula for flux linkage I found on the internet (∫ I dΦ as defined above)

Could you please provide the web link ?

I really can't think of why flux linkage is equal to ∫ I dΦ .

 

[lagrangman]

I got it from here: https://www.physicsforums.com/threads/flux-linkage-in-inductance-calculation-for-single-wire.819129/

Sorry about the red herring and thank you for your time, but I think I figured it out. When people say fraction of current in this context, they mean I_enclosed/I_total. I thought fraction of current meant I_enclosed. So:

λ = ∫ f dΦ, where f is the partial current that causes the differential flux dΦ divided by the total current. I think, but am not positive, this equation is correct.

I had never seen this before so I was a little confused.

Thanks again!