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

1. RC Parallel
2. RL Parallel
3. LC Parallel
4. 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