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

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

$V=A{e}^{-}(\frac{1}{T})t$

$A=e^C$

T = RC

Circuit's Impedance in Polar coordinate

$Z=Z_R+Z_C$

$Z=R\mathrm{\angle }0+\frac{1}{\omega C}\mathrm{\angle }-90$

$\sqrt{R^2+(\frac{1}{\omega C}{)}^2}\mathrm{\angle }Ta{n}^{-}1\frac{1}{\omega RC}$

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

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

$A=e^c$

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

Circuit's Impedance In Polar Coordinate

$Z=Z_R+Z_L=R\mathrm{\angle }0+\omega L\mathrm{\angle }90$

$\sqrt{R^2+(\omega L{)}^2}\mathrm{\angle }Ta{n}^{-}1\omega \frac{L}{R}$

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

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

$A=e^c$

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

$Z=|Z|\mathrm{\angle }\theta$

$Z=|Z_L-Z_C|\mathrm{\angle }\pm 90$ . Z_L = Z_C

$Z=0\mathrm{\angle }0$ . Z_L = Z_C

Circuit's Natural Response at equilibrium

$L\frac{dI}{dt}+\frac{1}{C}\int Idt=0$

$\frac{d^{2}I}{d{t}^2}+\frac{1}{LC}=0$

$s^2+\frac{1}{LC}=0$

$s=\pm \sqrt{\frac{1}{LC}}t=\pm \omega t$

$I=e^{(}st)$

$I=e^j\omega t+e^{-}j\omega t$

$I=ASin\omega t$

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

$L\frac{dI}{dt}+IR+\frac{1}{C}\int Idt=0$

$\frac{dI}{dt}+I\frac{R}{L}+\frac{1}{LC}=0$

$\frac{d^{2}I}{d{t}^2}+\frac{R}{L}\frac{dI}{dt}+\frac{1}{LC}=0$

$s^2+\frac{R}{L}s+\frac{1}{LC}=0$

$s=(-\alpha \pm \lambda )t$

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 $\omega =oo{Z}_L=oo$ , 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

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

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

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