Practical Electronics/π Network

Intro
A network consist of three resistors connected like a π

Analysis

 * $$I_1 = \frac{V_i}{R_1}$$
 * $$I_3 = \frac{V_o}{R_3}$$
 * $$I_2 = \frac{V_i - V_o}{R_2}$$
 * $$I_1 = I_2 + I_3$$
 * $$\frac{V_i}{R_1} = \frac{V_o}{R_3} + \frac{V_i - V_o}{R_2}$$
 * $$\frac{V_o}{V_i} = \frac{R_3 - R_2}{R_1 - R_2} $$

Formula

 * $$V_o = V_i \frac{R_3+R_2}{R_1+R_2}$$

Choosing Resistance's Value

 * R2 = 0
 * The fomula above now becomes
 * $$V_o = V_i \frac{R_3}{R_1}$$
 * R3 > R1, Vo > V1 . This network acts as Voltage Amplifier
 * R3 = R1, Vo = V1 . This network acts as Voltage Follower
 * R3 < R1, Vo < V1 . This network acts as Voltage Attenuator


 * R3 = 0
 * The fomula above now becomes
 * $$V_o = V_i \frac{R_2}{R_2+R_1}$$


 * This network acts as Voltage Attenuator


 * R1 = 0
 * The fomula above now becomes
 * $$V_o = V_i \frac{R_3 + R_2}{R_2}$$


 * This network acts as Voltage Amplifier 1 + $$\frac{R_3}{R_2}$$

Summary
This network can be used as Voltage Amplifier, Voltage Follower, Voltage Attenuator by choosing the right value for the resistors