Electronics Handbook/Circuits/RL Series

Circuit Impedance
In Polar Coordinate
 * $$Z = Z_R + Z_L $$ = R/_0 + ω L/_90
 * Z = |Z|/_θ = $$\sqrt{R^2 + (\omega L)^2}$$/_Tan-1$$\omega\frac{L}{R}$$

In Rectangular Coordinate
 * $$Z = Z_R + Z_L = R + j \omega L$$
 * $$Z = R + j \omega L = R ( 1 + j \omega \frac{L}{R} )$$

Phase Angle of Difference Between Voltage and Current
In RL series circuit, only L is the component that depends on frequency. There is no difference between voltage and current on R. There is an angle difference between voltage and current by 90 degree. When connect R and L in series, there is a difference in angle between voltage and current from 0 to 90 degree which can be expressed as a mathematic formula below
 * $$Tan \theta = \omega \frac{L}{R} = 2 \pi f \frac{L}{R} = 2 \pi \frac{1}{t}  \frac{L}{R} $$
 * $$\omega = Tan \theta \frac{R}{L}$$
 * $$f = \frac{1}{2\pi} Tan \theta \frac{R}{L}$$
 * $$t = 2\pi \frac{1}{Tan \theta} \frac{L}{R}$$

Natural Response

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

Time Constant RL

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


 * {| class="wikitable" width=50%

! t !! I(t) !! % Io
 * 0 || A = eC = Io || 100%
 * $$\frac{1}{RC}$$ || .63 Io || 63% Io
 * $$\frac{2}{RC}$$ || Io ||
 * $$\frac{3}{RC}$$ || Io  ||
 * $$\frac{4}{RC}$$ || Io  ||
 * $$\frac{5}{RC}$$ || .01 Io || 10% Io
 * }
 * $$\frac{3}{RC}$$ || Io  ||
 * $$\frac{4}{RC}$$ || Io  ||
 * $$\frac{5}{RC}$$ || .01 Io || 10% Io
 * }
 * $$\frac{5}{RC}$$ || .01 Io || 10% Io
 * }
 * }

''Current of the circuit is decreased exponentially with time. At time t = 0 I = Io. At time t = R/L I = 63% Io. At time t = 5 time time constant I = 10% Io''
 * I = A $$e^-(\frac{L}{R}) t$$

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