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The total flux (\$\Phi\$) through an solenoidal inductor of length \$l\$ and \$N\$ turns is proportional to the current through the inductor and the inductance \$L\$ of the inductor according to

$$\Phi =L \cdot I $$ $$\Rightarrow \Phi =\frac{\pi r^2\mu_0 N^2 }{l}\cdot I \tag{1}$$

Clearly, in this case, if we double the amount of turns (whilst holding the length and current constant) we increase the total flux by a factor of 4. This is because the inductance of a solenoid depends quadratically on \$N\$.

Now I have just learnt about the concept of reluctance (denoted \$R\$) and magnetomotive force (denoted \$m.m.f\$) where \$m.m.f\$ is defined to be \$m.m.f \equiv N\cdot I\$. These two quantities are related to each other by the total flux via the formula

$$m.m.f = R\cdot \Phi \tag{2}$$

Now applying the above formula (eq 2) to the case of the solenoidal inductor, if we hold the current constant but double the number of turns (N) of the inductor, the \$m.m.f\$ only doubles (since \$m.m.f \equiv N\cdot I\$). According to equation (1), the total flux must quadruple since \$\Phi \propto N^2\$. Thus in order for equation (2) to remain valid, the reluctance must necessarily half. That is, in order for equations 1 and 2 to be simultaneously valid, the reluctance must be inversely proportional to the number of turns so that \$R\propto \frac{1}{N}\$. However, everywhere I look, I find that reluctance is always given by equations that are independent of \$N\$. For instance, wikipedia states that \$R=\frac{l}{\mu A}\$.

How can this be? How can equations 1 and 2 be simultaneously true whilst reluctance is independent of \$N\$?

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The problem is your first formula \$\phi=L.I\$, it's wrong (according to wikipedia). That causes your equation 1 to be wrong. The correct version would be: $$L=\frac{N.\phi}{I}$$ Then, the correct equation 1 would be : $$\phi=\frac{\pi r^2.\mu_0.N}{l}I$$ That results in: $$\phi=\frac{L.I}{N}$$ Then... if you use your equation 2 to make \$\phi=\frac{m.m.f}{\Re}\$ and equates the flux term to that of your equation 1 and since \$m.m.f=N.I\$: $$\frac{N.I}{\Re}=\frac{\pi r^2.\mu_0.N}{l}I$$ You can replace the \$\Re\$ for the wikipedia's formula for reluctance and you will find the equality.

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There are two places to measure flux: in the circuit (at the winding terminals), and in the fields (integrated over some given cross-section; usually a magnetic core).

If you like, the fields flux has units of flux-per-turn, so that on multiplying it by number of turns, you get the circuit flux. (This follows with some other changes, like \$\mu_0\$ having units of H/(m.t), or other equivalent options.)

Magnetic path terms (MMF, R, etc.) are all in terms of fields -- hence the confusion.

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  • \$\begingroup\$ Thanks for the comment. Okay so am I correct in saying that we are actually dealing with two different fluxs in each equation? For \$\Phi =L I\$, the \$ \Phi \$ in this case is the total flux penetrating a surface/soap film covering the whole circuit. For clarity we could write \$\Phi_{tot} =L I\$. For the equation \$m.m.f = R \Phi \$ however, the flux here is actually not the total flux but rather the flux that penetrates a surface covering only one single turn. That is, it is the flux per turn. For clarity we could \$m.m.f = R \Phi_{perTurn}\$ So that \$\Phi_{tot}=N \Phi_{perTurn}\$ ? \$\endgroup\$ May 25 at 7:45
  • \$\begingroup\$ @SalahTheGoat Circuit flux is more abstract: 0-dimensional, measured at nodes in the circuit. Well, maybe that's not even too helpful of a description, since the purpose of magnetic circuit analysis is to reduce a 3D fields problem to a circuit representation as well. But the flux in that case is directly assumed from the 3D fields, while circuit flux is still the sum over turns. For the simple case of one core, fully enclosed by one winding, certainly, \$\Phi_{circuit} = N \Phi_{core}\$. \$\endgroup\$ May 25 at 18:58
  • \$\begingroup\$ The interesting stuff happens when more complicated networks of cores, flux paths and windings are assembled. In that case, you need to add up each case: how many turns, around each core, with what fluxes. But for each individual winding on a given piece of core, yeah, this. Luiz Oliveira's answer also explains the error succinctly: using the "fields" flux only, which is probably less confusing. \$\endgroup\$ May 25 at 19:03
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Reluctance/Permeance is a property of the magnetic circuit. The magnetic circuit is the circuit taken by the magnetic field loop, i.e. through the magnetic core and possible air gaps.

For a given coil orientation (that defines where the magnetic field loop actually is), the number of turns don't matter, because they don't change the magnetic circuit.

However, reluctance does depend on things like core material, core diameter or coil diameter as this changes where the magnetic circuit is.

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