# 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

## Parallel RC

The total Impedance of the circuit

${\displaystyle Z=Z_{R}+Z_{C}=R+{\frac {1}{j\omega C}}={\frac {1+j\omega RC}{j\omega C}}}$
${\displaystyle Z={\frac {1}{j\omega C}}(1+j\omega T}$ )
T = RC

At Equilibrium sum of all voltages equal zero

${\displaystyle C{\frac {dV}{dt}}+{\frac {V}{R}}=0}$
${\displaystyle {\frac {dV}{dt}}=-{\frac {1}{RC}}V}$
${\displaystyle {\frac {1}{V}}dV=-{\frac {1}{RC}}dt}$
${\displaystyle \int {\frac {1}{V}}dV=-\int {\frac {1}{RC}}dt}$
ln V = ${\displaystyle -{\frac {1}{RC}}+C}$
${\displaystyle V=e^{-}({\frac {1}{RC}})t+C}$
${\displaystyle V=Ae^{-}({\frac {1}{T}})t}$
${\displaystyle A=e^{C}}$
T = RC

Circuit's Impedance in Polar coordinate

${\displaystyle Z=Z_{R}+Z_{C}}$
${\displaystyle Z=R\angle 0+{\frac {1}{\omega C}}\angle -90}$
${\displaystyle {\sqrt {R^{2}+({\frac {1}{\omega C}})^{2}}}\angle Tan^{-}1{\frac {1}{\omega RC}}}$

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

${\displaystyle Tan\theta ={\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}={\frac {1}{2\pi }}{\frac {t}{T}}}$

### Summary

RL series circuit has a first order differential equation of voltage

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

Which has one real root

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

The Natural Response of the circuit at equilibrium is a Exponential Decrease function

Phase Angle Difference Between Voltage and Current

${\displaystyle Tan\theta ={\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}={\frac {1}{2\pi }}{\frac {t}{T}}}$

## Parallel RL

The total Circuit's Impedance In Rectangular Coordinate

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

At Equilibrium sum of all voltages equal zero

${\displaystyle L{\frac {dI}{dt}}+IR=0}$
${\displaystyle {\frac {dI}{dt}}=-I{\frac {R}{L}}}$
${\displaystyle \int {\frac {1}{I}}dI=-\int {\frac {L}{R}}dt}$
ln I = ${\displaystyle (-{\frac {L}{R}}+c)}$
I = ${\displaystyle e^{(}-{\frac {L}{R}}+c)t}$
I = ${\displaystyle e^{c}e^{(}-{\frac {L}{R}}t)}$
I = ${\displaystyle Ae^{(}-{\frac {L}{R}}t)}$

Circuit's Impedance In Polar Coordinate

${\displaystyle Z=Z_{R}+Z_{L}=R\angle 0+\omega L\angle 90}$
${\displaystyle {\sqrt {R^{2}+(\omega L)^{2}}}\angle Tan^{-}1\omega {\frac {L}{R}}}$

Phase Angle of Difference Between Voltage and Current

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

### Summary

In summary RL series circuit has a first order differential equation of current

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

Which has one real root

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

The Natural Response of the circuit at equilibrium is a Exponential Decrease function

Phase Angle of Difference Between Voltage and Current

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

## Parallel LC

### Natural Response

The Total Circuit's Impedance in Rectangular Form

${\displaystyle Z=|Z|\angle \theta }$
${\displaystyle Z=|Z_{L}-Z_{C}|\angle \pm 90}$  . ZL = ZC
${\displaystyle Z=0\angle 0}$  . ZL = ZC

Circuit's Natural Response at equilibrium

${\displaystyle L{\frac {dI}{dt}}+{\frac {1}{C}}\int Idt=0}$
${\displaystyle {\frac {d^{2}I}{dt^{2}}}+{\frac {1}{LC}}=0}$
${\displaystyle s^{2}+{\frac {1}{LC}}=0}$
${\displaystyle s=\pm {\sqrt {\frac {1}{LC}}}t=\pm \omega t}$
${\displaystyle I=e^{(}st)}$
${\displaystyle I=e^{j}\omega t+e^{-}j\omega t}$
${\displaystyle I=ASin\omega t}$

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

${\displaystyle Z_{L}-Z_{C}=0}$
${\displaystyle \omega L={\frac {1}{\omega C}}}$
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$
${\displaystyle V_{L}+V_{C}=0}$
${\displaystyle V_{L}=-V_{C}}$

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

${\displaystyle L{\frac {dI}{dt}}+IR+{\frac {1}{C}}\int Idt=0}$
${\displaystyle {\frac {dI}{dt}}+I{\frac {R}{L}}+{\frac {1}{LC}}=0}$
${\displaystyle {\frac {d^{2}I}{dt^{2}}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0}$
${\displaystyle s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0}$
${\displaystyle s=(-\alpha \pm \lambda )t}$

Với

${\displaystyle \alpha ={\frac {R}{2L}}}$
${\displaystyle \beta ={\frac {1}{LC}}}$
${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$

Khi ${\displaystyle \alpha ^{2}=\beta ^{2}}$

${\displaystyle s=-\alpha t}$
${\displaystyle I=e^{(}-\alpha )t}$
The response of the circuit is an Exponential Deacy

Khi ${\displaystyle \alpha ^{2}>\beta ^{2}}$

${\displaystyle s=(-\alpha \pm \lambda )t}$
${\displaystyle I=e^{-}\alpha t\pm (e^{\lambda }t+e^{-}\lambda t)}$
The response of the circuit is an Exponential Deacy

Khi ${\displaystyle \alpha ^{2}<\beta ^{2}}$

${\displaystyle s=(-\alpha \pm \lambda )t}$
${\displaystyle I=e^{-}\alpha t\pm (e^{j}\lambda t+e^{-}j\lambda t)}$
The response of the circuit is an Exponential decay sinusoidal wave

Điện Kháng Tổng Mạch Điện

${\displaystyle Z=Z_{R}+Z_{L}+Z_{C}}$
${\displaystyle Z=R+j\omega L+{\frac {1}{j\omega C}}}$
${\displaystyle Z={\frac {1}{j\omega C}}(j\omega ^{2}+j\omega {\frac {R}{L}}+{\frac {1}{LC}})}$

### Resonance Response

The total impedance of the circuit

${\displaystyle Z=Z_{R}+Z_{L}+Z_{C}=R+0=R}$
${\displaystyle I={\frac {V}{R}}}$
${\displaystyle Z_{L}=Z_{C}}$
${\displaystyle j\omega L={\frac {1}{j\omega C}}}$
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$

At resonance frequency ${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$  the total impedance of the circuit is Z = R ; at its minimum value and current will be at its maximum value  : ${\displaystyle I={\frac {V}{R}}}$

Look at the circuit, at ${\displaystyle At\omega =0Z_{C}=oo}$  , Capacitor opens circuit . Therefore, current is equal to zero . At ${\displaystyle \omega =ooZ_{L}=oo}$  , 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

${\displaystyle {\frac {df(t)}{dt}}+\omega t=0}$

that has Decay exponential function as Natural Response

${\displaystyle f(x)=Ae^{(}-{\frac {t}{T}})}$
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

${\displaystyle {\frac {d^{2}f(t)}{dt}}+\omega t=0}$
${\displaystyle f(x)=e^{(}\pm \omega t)}$
${\displaystyle f(x)=e^{(}\omega t)+e^{(}-\omega t)=ASin\omega t}$

At equilibrium , the Natural Response of the circuit is Sinusoidal Wave

${\displaystyle f(x)=ASin\omega t}$

At Equilibrum , the Resonance Response is Standing Wave Reponse