# Electronics Handbook/Circuits/Parallel Circuit
## Series Circuit

Electronic components R,L,C can be connected in parallel to form RL, RC, LC, RLC series circuit
* RC Parallel
* RL Parallel
* LC Parallel
* RLC Parallel

## Parallel RC

The total Impedance of the circuit
    $Z=Z_{R}+Z_{C}=R+{\frac {1}{j\omega C}}={\frac {1+j\omega RC}{j\omega C}}$
    $Z={\frac {1}{j\omega C}}(1+j\omega T$)
    T = RC
At Equilibrium sum of all voltages equal zero
    $C{\frac {dV}{dt}}+{\frac {V}{R}}=0$
    ${\frac {dV}{dt}}=-{\frac {1}{RC}}V$
    ${\frac {1}{V}}dV=-{\frac {1}{RC}}dt$
    $\int {\frac {1}{V}}dV=-\int {\frac {1}{RC}}dt$
    ln V = $-{\frac {1}{RC}}+C$
    $V=e^{-}({\frac {1}{RC}})t+C$
    $V=Ae^{-}({\frac {1}{T}})t$
    $A=e^{C}$
    T = RC
Circuit's Impedance in Polar coordinate
    $Z=Z_{R}+Z_{C}$
    $Z=R\angle 0+{\frac {1}{\omega C}}\angle -90$
    ${\sqrt {R^{2}+({\frac {1}{\omega C}})^{2}}}\angle Tan^{-}1{\frac {1}{\omega RC}}$
Phase Angle Difference Between Voltage and Current There is a difference in angle Between Voltage and Current . Current leads Voltage by an angle θ
    $Tan\theta ={\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}={\frac {1}{2\pi }}{\frac {t}{T}}$

### Summary

RL series circuit has a first order differential equation of voltage
    ${\frac {d}{dt}}f(t)+{\frac {t}{T}}=0$
Which has one real root
    $V(t)=Ae^{\frac {-t}{T}}$
    $A=e^{c}$
The Natural Response of the circuit at equilibrium is a Exponential Decrease function
Phase Angle Difference Between Voltage and Current
    $Tan\theta ={\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}={\frac {1}{2\pi }}{\frac {t}{T}}$

## Parallel RL

The total Circuit's Impedance In Rectangular Coordinate
    $Z=Z_{R}+Z_{L}=R+j\omega L$
    $Z={\frac {1}{R}}(1+j\omega T)$
    $T={\frac {L}{R}}$
At Equilibrium sum of all voltages equal zero
    $L{\frac {dI}{dt}}+IR=0$
    ${\frac {dI}{dt}}=-I{\frac {R}{L}}$
    $\int {\frac {1}{I}}dI=-\int {\frac {L}{R}}dt$
    ln I = $(-{\frac {L}{R}}+c)$
    I = $e^{-{\frac {L}{R}}+c)t$
    I = $e^{c}e^{-{\frac {L}{R}}t)$
    I = $Ae^{-{\frac {L}{R}}t)$
Circuit's Impedance In Polar Coordinate
    $Z=Z_{R}+Z_{L}=R\angle 0+\omega L\angle 90$
    ${\sqrt {R^{2}+(\omega L)^{2}}}\angle Tan^{-}1\omega {\frac {L}{R}}$
Phase Angle of Difference Between Voltage and Current
    $Tan\theta =\omega {\frac {L}{R}}=2\pi f{\frac {L}{R}}=2\pi {\frac {T}{t}}$

### 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
    $I(t)=Ae^{\frac {t}{T}}$
    $A=e^{c}$
The Natural Response of the circuit at equilibrium is a Exponential Decrease function
Phase Angle of Difference Between Voltage and Current
    $Tan\theta =\omega {\frac {L}{R}}=2\pi f{\frac {L}{R}}=2\pi {\frac {T}{t}}$

## Parallel LC
### Natural Response

The Total Circuit's Impedance in Rectangular Form
    $Z=|Z|\angle \theta$
       $Z=|Z_{L}-Z_{C}|\angle \pm 90$ . Z$_L$ = Z$_C$
       $Z=0\angle 0$ . Z$_L$ = Z$_C$
Circuit's Natural Response at equilibrium
    $L{\frac {dI}{dt}}+{\frac {1}{C}}\int Idt=0$
    ${\frac {d^{2}I}{dt^{2}}}+{\frac {1}{LC}}=0$
    $s^{2}+{\frac {1}{LC}}=0$
    $s=\pm {\sqrt {\frac {1}{LC}}}t=\pm \omega t$
    $I=e^{st)$
    $I=e^{j}\omega t+e^{-}j\omega t$
    $I=ASin\omega t$
The Natural Response at equilibrium of the circuit is a Sinusoidal Wave

### Resonance Response

At Resonance, The total Circuit's impedance is zero and the total volages are zero
    $Z_{L}-Z_{C}=0$
       $\omega L={\frac {1}{\omega C}}$
       $\omega ={\sqrt {\frac {1}{LC}}}$
    $V_{L}+V_{C}=0$
       $V_{L}=-V_{C}$
The Resonance Reponse of the circuit at resonance is a Standing (Sinusoidal) Wave

## Parallel RLC
### Natural Response

At Equilibrium, the sum of all voltages equal to zero
    $L{\frac {dI}{dt}}+IR+{\frac {1}{C}}\int Idt=0$
    ${\frac {dI}{dt}}+I{\frac {R}{L}}+{\frac {1}{LC}}=0$
    ${\frac {d^{2}I}{dt^{2}}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0$
    $s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0$
    $s=(-\alpha \pm \lambda )t$
Với
    $\alpha ={\frac {R}{2L}}$ và
    $\beta ={\frac {1}{LC}}$
    $\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}$
Khi $\alpha ^{2}=\beta ^{2}$
    $s=-\alpha t$
    $I=e^{-\alpha )t$
    The response of the circuit is an Exponential Deacy
Khi $\alpha ^{2}>\beta ^{2}$
    $s=(-\alpha \pm \lambda )t$
    $I=e^{-}\alpha t\pm (e^{\lambda }t+e^{-}\lambda t)$
    The response of the circuit is an Exponential Deacy
Khi $\alpha ^{2}<\beta ^{2}$
    $s=(-\alpha \pm \lambda )t$
    $I=e^{-}\alpha t\pm (e^{j}\lambda t+e^{-}j\lambda t)$
    The response of the circuit is an Exponential decay sinusoidal wave
Điện Kháng Tổng Mạch Điện
    $Z=Z_{R}+Z_{L}+Z_{C}$
    $Z=R+j\omega L+{\frac {1}{j\omega C}}$
    $Z={\frac {1}{j\omega C}}(j\omega ^{2}+j\omega {\frac {R}{L}}+{\frac {1}{LC}})$

### Resonance Response

The total impedance of the circuit
    $Z=Z_{R}+Z_{L}+Z_{C}=R+0=R$
    $I={\frac {V}{R}}$
    $Z_{L}=Z_{C}$
    $j\omega L={\frac {1}{j\omega C}}$
    $\omega ={\sqrt {\frac {1}{LC}}}$
At resonance frequency $\omega ={\sqrt {\frac {1}{LC}}}$ the total impedance of the circuit is Z = R ; at its minimum value and current will be at its maximum value : $I={\frac {V}{R}}$
Look at the circuit, at $At\omega =0Z_{C}=oo$ , Capacitor opens circuit . Therefore, current is equal to zero . At $\omega =ooZ_{L}=oo$ , Inductor opens circuit . Therefore, current is equal to zero

## Summary
### Series RL, RC

Series RC and RL has a Character first order differential equation of the form
    ${\frac {df(t)}{dt}}+\omega t=0$
that has Decay exponential function as Natural Response
    $f(x)=Ae^{-{\frac {t}{T}})$
    f(t) = i(t) for series RL
    f(t) = v(t) for series RC

### Series LC, RLC

Series LC and RLC has a Characteristic Second order differential equation of the form
    ${\frac {d^{2}f(t)}{dt}}+\omega t=0$
    $f(x)=e^{\pm \omega t)$
    $f(x)=e^{\omega t)+e^{-\omega t)=ASin\omega t$
At equilibrium , the Natural Response of the circuit is Sinusoidal Wave
    $f(x)=ASin\omega t$
At Equilibrum , the Resonance Response is Standing Wave Reponse