Electronics Handbook/Circuits/RLC Series Analysis

Consider an RLC series circuit

1. If L = 0 then the cicuit is reduced to RC series
2. If C = 0 then the cicuit is reduced to RL series
3. If R = 0 then the cicuit is reduced to LC series
4. If R, L , C are not zero

RC Series

• Differential Equation

$C\frac{dV}{dt}+\frac{V}{R}=0$

$\frac{dV}{dt}=-\frac{1}{RC}V$

$\frac{1}{V}dV=-\frac{1}{RC}dt$

$\int \frac{1}{V}dV=-\int \frac{1}{RC}dt$

ln V = $-\frac{1}{RC}+C$

$V=e^{-}(\frac{1}{RC})t+C$

$V=A{e}^{-}(\frac{1}{RC})t$ với $A=e^C$

• Time Constant

t	V(t)	% Vo
0	A = eC = Vo	100%
$\frac{1}{RC}$	.63 Vo	60% Vo
$\frac{2}{RC}$	Vo
$\frac{3}{RC}$	Vo
$\frac{4}{RC}$	Vo
$\frac{5}{RC}$	.01 Vo	10% Vo

• Circuit Impedance

Z/_θ

$Z=Z_R+Z_C$

Z = R /_0 + ( 1 / ωC ) /_ - 90

Z = = |Z|/_θ = $\sqrt{R^2+(\frac{1}{\omega C}{)}^2}$ /_ Tan^-1 $\frac{1}{\omega RC}$

Z(jω)

$Z=Z_R+Z_C$

Z = $R+\frac{1}{j\omega C}=\frac{1+j\omega RC}{j\omega C}$

$Z=\frac{1}{X_C}(1+j\omega T$)

• Angle Difference Between Voltage and Current

There is a difference in angle Between Voltage and Current . Current leads Voltage by an angle θ

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

The difference in angle between Voltage and Current relates to the value of R , C and the Angular of Frequency ω which also relates to f and t . Therefore when change the value of R or C , the angle difference will be changed and so are ω , f , t

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

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

$t=2\pi RCTan\theta$

• First Order Equation of Circuit

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

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

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

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

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

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

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

• 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

• Circuit Impedance

$Z=Z_R+Z_L$ = R/_0 + ω L/_90

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

Z(jω)

$Z=Z_R+Z_L=R+j\omega L$

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

• 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

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

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

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

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

• In Summary

RL series circuit has a first order differential equation of current

Which has one real root of the form

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

$A=e^c$

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

Which has solution in the form

$f(t)=A{e}^{\frac{-t}{T}}$