Practical Electronics/Low Pass Filter

LR Network

 * $$\frac{V_o}{V_i} = \frac{Z_R}{Z_R + Z_L} = \frac{R}{R + j\omega L} = \frac{1}{1 + j\omega T}$$
 * $$T = \frac{L}{R}$$
 * $$\omega_o = \frac{1}{T}=\frac{R}{L}$$


 * $$\omega = 0 V_o = V_i$$
 * $$\omega_o = \sqrt{\frac{1}{LC}} V_o = \frac{V_i}{2}$$
 * $$\omega = 00 V_o = 0$$


 * Plot three points above we have a graph $$Vo - \omega$$ . From graph, we see voltage does not change with frequency on Low Frequency therefore LR network can be used as Low Pass Filter

RC Network

 * $$\frac{V_o}{V_i} = \frac{Z_C}{Z_R + Z_C} = \frac{\frac{1}{j\omega C}}{R + \frac{1}{j\omega C}} = \frac{1}{1 + j\omega T}$$
 * $$T = RC$$
 * $$\omega_o=\frac{1}{T} = \frac{1}{RC}$$


 * $$\omega = 0 V_o = V_i$$
 * $$\omega_o = \sqrt{\frac{1}{LC}} V_o = \frac{V_i}{2}$$
 * $$\omega = 00 V_o = 0$$


 * Plot three points above we have a graph $$Vo - \omega$$ . From graph, we see voltage does not change with frequency on Low Frequency therefore LR network can be used as Low Pass Filter

Summary
In general
 * 1) Low Pass Filter can be constructed from the two networks LR or RC.
 * 2) Low Pass Filter has stable voltage does not change with frequency on Low Frequency
 * 3) Low pass filter can be expressed in a mathematical form of
 * $$\frac{V_o}{V_i} = \frac{1}{1 + j\omega T}$$
 * T = RC for RC network
 * $$T = \frac{L}{R}$$ for RL network