# Questions about the formula used to determine inductive reactance and $Z_t$

I am currently on the inductors unit in my Navy schooling and I have two questions about these formulas that I learned about. As I'm aware, the ability of an inductor to concentrate a magnetic field is called an inductance. Inductive reactance measures the amount of opposition to the current flow in the circuit and is expressed as :

$X_L = 2\pi fL$ where $f$ is the frequency and $L$ represents the ohmic value from the inductor. I am not sure why we are multiplying by $2\pi$ here. The formula makes sense since frequency and current affects the inductor.

My other question has to do with this strange formula for $Z_t$ the total ohmic value of the resistors and inductors in a series parallel circuit:

$$Z_t = \frac{R \times X_L}{\sqrt{R^2 + X_{L}^2}}$$

Here $R$ represents the total resistance value and $X_L$ represents the total inductive value. I'm puzzled as to where this formula comes from.

In electric linear circuits theory, use is made of phasors to represent sinusoidal inputs (most common basic inputs to linear circuits).

For linear circuits the total output of a given input is just the sum of each output component associated with each input component (provided there is a way to decompose any given input to compoents) (this is the definition of a linear system)

i.e by definition a linear system $L$ has this property:

$$L \left( av_1 + bv_2\right) = aL(v_1)+bL(v_2)$$

(the output of the sum of inputs equals the sum of the outputs of each input)

The way to decompose the input is by using fourier series (effectively representing each input as a sum of sinusoidal signals)

So one sees now how the phasors make the analysis of any circuit simpler.

So, in this format, the resistance $R$ is just $R$. If one wants to combine inductors and capacitos in the phasor representation then the combined resistance is refered as impedance and allows to treat both inductors and capacitors as complex resistances (under a given frequency $f$)

Then the usual Ohm's laws and series/parallel formulas for resistors can be applied using instead the complex impendances stemming from analysing the input in phasors.

UPDATE:

The $X_L = 2\pi fL$ is actually $X_L = \omega L$ but $\omega = 2 \pi f$