Inductance depends on the number of turns in a solenoid. Is this the case with reluctance as well?

Question

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\$?

Answer 1

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.

Answer 2

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.

Answer 3

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.