Electronics Handbook/Circuits/RL Series

Series RL Circuit

Circuit Analysis

Circuit Impedance

In Polar Coordinate

$Z=Z_{R}+Z_{L}$ = R/_0 + ω L/_90

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

In Rectangular Coordinate

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

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

Phase Angle of Difference Between Voltage and Current

In RL series circuit, only L is the component that depends on frequency . There is no difference between voltage and current on R . There is an angle difference between voltage and current by 90 degree . When connect R and L in series , there is a difference in angle between voltage and current from 0 to 90 degree which can be expressed as a mathematic formula below

$Tan\theta =\omega {\frac {L}{R}}=2\pi f{\frac {L}{R}}=2\pi {\frac {1}{t}}{\frac {L}{R}}$

$\omega =Tan\theta {\frac {R}{L}}$

$f={\frac {1}{2\pi }}Tan\theta {\frac {R}{L}}$

$t=2\pi {\frac {1}{Tan\theta }}{\frac {L}{R}}$

Natural Response

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

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

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

ln I = $(-{\frac {L}{R}}+c)$

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

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

I = $Ae^{(}-{\frac {L}{R}}t)$

Time Constant RL

τ = ${\frac {L}{R}}$

I = A $e^{-}({\frac {L}{R}})t$

t	I(t)	% Io
0	A = eC = Io	100%
${\frac {1}{RC}}$	.63 Io	63% Io
${\frac {2}{RC}}$	Io
${\frac {3}{RC}}$	Io
${\frac {4}{RC}}$	Io
${\frac {5}{RC}}$	.01 Io	10% Io

Current of the circuit is decreased exponentially with time . At time t = 0 I = I_o . At time t = R/L I = 63% I_o . At time t = 5 time time constant I = 10% I_o

I = A $e^{-}({\frac {L}{R}})t$

Summary

In summary RL series circuit has a first order differential equation of current

${\frac {d}{dt}}f(t)+{\frac {1}{T}}=0$

Which has one real root

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

$A=e^{c}$