For a series RL of one resistor connected with one inductor in a closed loop

Circuit Impedance

In Polar Form Z/_θ

${\displaystyle Z=Z_{R}+Z_{L}}$ = R/_0 + ω L/_90

Z = |Z|/_θ = ${\displaystyle {\sqrt {R^{2}+(\omega L)^{2}}}}$/_Tan^-1${\displaystyle \omega {\frac {L}{R}}}$

In Complex Form Z(jω)

${\displaystyle Z=Z_{R}+Z_{L}=R+j\omega L}$

${\displaystyle Z=R+j\omega L=R(1+j\omega T)}$

${\displaystyle T={\frac {L}{R}}}$

Differential Equation of circuit at equilibrium

${\displaystyle L{\frac {dI}{dt}}+IR=0}$

${\displaystyle {\frac {dI}{dt}}=-I{\frac {R}{L}}}$

${\displaystyle \int {\frac {1}{I}}dI=-\int {\frac {L}{R}}dt}$

${\displaystyle lnI=(-{\frac {L}{R}}+c)}$

${\displaystyle I=e^{(}-{\frac {L}{R}}t+c)=e^{c}+e^{(}-{\frac {L}{R}}t}$

${\displaystyle I=Ae^{-}({\frac {t}{T}})}$

Time Constant

${\displaystyle T={\frac {L}{R}}}$

t	I(t)	% Io
0	A = eC = Io	100%
R/L	.63 Io	60% Io
2 R/L	Io
3 R/L	Io
4 R/L	Io
5 R/L	.01 Io	10% Io

Angle Difference between Voltage and Current

Voltage leads Current at an angle ? When a determining process is necessary many problems arise in a diagram. We need to expend on one process for the determing factor in this type of formulae

Tan? = ${\displaystyle {\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}=t{\frac {1}{2\pi RC}}}$

Change the value of R and L will change the value Angle Difference, Angular Frquency, Frequency, Time

${\displaystyle \omega ={\frac {1}{Tan\theta RC}}}$

${\displaystyle f={\frac {1}{2\pi Tan\theta RC}}}$

${\displaystyle t=2\pi Tan\theta RC}$