# Electronics Handbook/Circuits/Series Circuit

## Series Circuit

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

## Series 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

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

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

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

### 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

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

## Series 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

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

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

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

## Series LC

### Natural Response

The Total Circuit's Impedance in Rectangular Form

$Z = |Z| \angle \theta$
$Z = |Z_L - Z_C| \angle \pm 90$ . ZL = ZC
$Z = 0 \angle 0$ . ZL = ZC

Circuit's Natural Response at equilibrium

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

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

## Series RLC

### Natural Response

At Equilibrium, the sum of all volatages equal to zero

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

Với

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

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 $At \omega = 0 Z_C = oo$ , 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

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

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

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

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

$f(x) = A Sin \omega t$

At Equilibrum , the Resonance Response is Standing Wave Reponse