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Coil specification:
I need the number of turns to get an inductance of 150 mH.
While searching for calculating the number of turns, I found a lot of different formulas.
I know that we can calculate the inductance with the formula
$$L = \frac{\mu_0 N^2 A}{d}.$$
If I use it to get the number of turns, it gives 4 turns only.
I am also confused how this formula can calculate the inductance and not the turns.
Air core coil
Length 790mm
Diameter 290mm
Let me mention that air's magnetic permeability (\$\mu_0\$) is \$4\pi\times 10^{-7} \frac{H}{m}\$ and not unity.
Let's also start with the correct formula (and not what you wrote in the question): -
$$L = \dfrac{\mu_0\cdot N^2\cdot A}{\ell}$$
Where \$l\$ is the length of the solenoid. The enclosed area of a round solenoid is \$A = \pi d^2 / 4\$ for diameter \$d\$.
So, rearrange the formula to get: -
$$N = \sqrt{\dfrac{0.15\text{ H} \times 0.79\text{ m}}{4\pi\times 10^{-7} \frac{H}{m} \times 0.06605\text{ m}^2}} = 1195 \text{ turns}$$
Calculator double check: -
When I enter 1 for relative permeability the calculator inherently knows about \$4\pi\times 10^{-7} \frac{H}{m}\$.
I also confused how this formula can calculate the inductance and not the turns
It can calculate turns but you need to factor-in the correct value for the permeability of free-space. Good luck winding 1145 turns along a distance of 790 mm; I'd be considering a two-layer winding.
