Practical Electronics/Series RL

< Practical Electronics


Series RCEdit

Electric circuit of two components R and L connected in series

Circuit analysisEdit

Circuit's ImpedanceEdit

In Rectangular coordinate

  • Z = Z_R + Z_L
Z = R + \omega L = \frac{1}{R} (1 + j\omega T)
T = \frac{L}{R}

In Polar coordinate

  • Z = Z_R + Z_L
Z = R \angle 0 + \omega L \angle 90 = |Z| \angle \theta = \sqrt{R^2 + (\omega L)^2} \angle Tan^-1 \omega \frac{L}{R}
Tan \theta = \omega \frac{L}{R} = 2 \pi f \frac{L}{R} = \frac{t}{2 \pi  \frac{L}{R}}

The value of \theta , \omega , f depend on the value of R and L . Therefore when the value of R or L changed the value of Phase angle difference between Current and Voltage , Frequency , and Angular of Frequency also change

\omega =  \frac{R}{L} Tan \theta
f =  \frac{1}{2 \pi} \frac{R}{L} Tan \theta
t =  2 \pi \frac{L}{R} \frac{1}{Tan \theta}

Circuit's ResponseEdit

Natural Response of the cicuit can be obtained by setting the differential equation of the circuit to zero

C \frac{dV}{dt} + \frac{V}{R} = 0
\frac{dV}{dt} = -\frac{1}{RC} V
\int \frac{dV}{V} = -\frac{1}{RC} \int dt
Ln V = -(\frac{1}{RC})t + e^c
V = A e^ -(\frac{1}{RC})t
V = A e^ -(\frac{t}{T})
V = e^c = IR
T = \frac{L}{R}

The natural reponse of the circuit is an exponential decrease


Circuit Series RL
RLC series circuit.png
Impedance Z = Z_R + Z_L= R + j\omega L = \frac {1}{R} (1 + j\omega T)
T = \frac{L}{R}
Diferenial Equation C\frac{dV}{dt} + \frac{V}{R} = 0
Root of the equation V = A e^(-\frac{t}{T})
Z\angle\theta \sqrt{R^2 + (\omega L)^2} \angle Tan ^-1 \omega \frac{L}{C}
Phase Angle Difference between Voltage and Current Tan \theta = \omega \frac{L}{R}
\omega \omega = \frac{1}{Tan\theta} \frac{L}{R}
f \omega = \frac{Tan\theta}{2 \pi} \frac{L}{R}
t t = \frac{2 \pi}{Tan\theta} \frac{R}{L}