# Electronics Handbook/Circuits/RLC Series Analysis

Consider an RLC series circuit
* If L = 0 then the cicuit is reduced to RC series
* If C = 0 then the cicuit is reduced to RL series
* If R = 0 then the cicuit is reduced to LC series
* 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 = $-{\frac {1}{RC}}+C$
    $V=e^{-}({\frac {1}{RC}})t+C$
    $V=Ae^{-}({\frac {1}{RC}})t$ với $A=e^{C}$
* Time Constant
| t | V(t) | % V$_o$ |
| 0 | A = e$^C$ = V$_o$ | 100% |
| ${\frac {1}{RC}}$ | .63 V$_o$ | 60% V$_o$ |
| ${\frac {2}{RC}}$ | V$_o$ | |
| ${\frac {3}{RC}}$ | V$_o$ | |
| ${\frac {4}{RC}}$ | V$_o$ | |
| ${\frac {5}{RC}}$ | .01 V$_o$ | 10% V$_o$ |
* 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 = $Ae^{-{\frac {L}{R}}t)$
* Time Constant RL
    τ = ${\frac {L}{R}}$
    I = A $e^{-}({\frac {L}{R}})t$
    tI(t) % I$_o$ 0A = e$^C$ = I$_o$100% ${\frac {1}{RC}}$.63 I$_o$63% I$_o$ ${\frac {2}{RC}}$I$_o$ ${\frac {3}{RC}}$I$_o$ ${\frac {4}{RC}}$I$_o$ ${\frac {5}{RC}}$.01 I$_o$10% I$_o$
* 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
    $I(t)=Ae^{\frac {t}{T}}$
    $A=e^{c}$
    ${\frac {d}{dt}}f(t)+{\frac {1}{T}}=0$
Which has solution in the form
    $f(t)=Ae^{\frac {-t}{T}}$