Electronics Handbook/Circuits/RLC Series Analysis

Consider an RLC series circuit
 * 1) If L = 0 then the cicuit is reduced to RC series
 * 2) If C = 0 then the cicuit is reduced to RL series
 * 3) If R = 0 then the cicuit is reduced to LC series
 * 4) If R, L, C are not zero

RC Series

 * [[Image:RC switch.svg|200px]]


 * Differential Equation
 * $$C \frac{dV}{dt} + \frac{V}{R} = 0$$
 * $$\frac{dV}{dt} = - \frac{1}{RC} V$$
 * $$\frac{1}{V} dV = - \frac{1}{RC} dt$$
 * $$\int \frac{1}{V} dV = - \int \frac{1}{RC} dt$$
 * ln V = $$ - \frac{1}{RC} + C$$
 * $$ V = e^- (\frac{1}{RC}) t + C$$
 * $$ V = A e^- (\frac{1}{RC}) t $$ với $$A = e^C$$


 * Time Constant

Z/_θ
 * Circuit Impedance
 * $$Z = Z_R + Z_C $$
 * Z = R /_0 + ( 1 / ωC ) /_ - 90
 * Z = = |Z|/_θ = $$\sqrt{R^2 + (\frac{1}{\omega C})^2}$$ /_ Tan-1 $$\frac{1}{\omega RC}$$

Z(jω)
 * $$Z = Z_R + Z_C $$
 * Z = $$ R + \frac{1}{j\omega C} = \frac{1 + j\omega RC}{j\omega C}$$
 * $$Z = \frac{1}{X_C} (1 + j\omega T$$)

''There is a difference in angle Between Voltage and Current. Current leads Voltage by an angle θ''
 * Angle Difference Between Voltage and Current
 * $$Tan\theta = \frac{1}{\omega RC} = \frac{1}{f} \frac{1}{2\pi RC} = t \frac{1}{2\pi RC}$$

''The difference in angle between Voltage and Current relates to the value of R, C and the Angular of Frequency ω which also relates to f and t. Therefore when change the value of R or C, the angle difference will be changed and so are ω , f , t ''
 * $$\omega= \frac{1}{RC} \frac{1}{Tan\theta}$$
 * $$f = \frac{1}{2\pi} \frac{1}{RC} \frac{1}{Tan\theta}$$
 * $$t = 2\pi RC Tan\theta$$




 * First Order Equation of Circuit
 * $$L\frac{dI}{dt} + IR = 0$$
 * $$\frac{dI}{dt} = - I \frac{R}{L}$$
 * $$\int \frac{1}{I} dI = - \int \frac{L}{R} dt$$
 * 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
 * }
 * }


 * Circuit Impedance


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

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

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
 * Angle of Difference Between Voltage and Current
 * $$Tan \theta = \omega \frac{L}{R} = 2 \pi f \frac{L}{R} = 2 \pi \frac{1}{t}  \frac{L}{R} $$
 * $$\omega = Tan \theta \frac{R}{L}$$
 * $$f = \frac{1}{2\pi} Tan \theta \frac{R}{L}$$
 * $$t = 2\pi \frac{1}{Tan \theta} \frac{L}{R}$$

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

Which has one real root of the form
 * $$I(t) = Ae^\frac{t}{T}$$
 * $$A = e^c$$


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

Which has solution in the form
 * $$f(t) = A e^\frac{-t}{T}$$