# Practical Electronics/π Network

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## Intro

A network consist of three resistors connected like a π

## Analysis

${\displaystyle I_{1}={\frac {V_{i}}{R_{1}}}}$
${\displaystyle I_{3}={\frac {V_{o}}{R_{3}}}}$
${\displaystyle I_{2}={\frac {V_{i}-V_{o}}{R_{2}}}}$
${\displaystyle I_{1}=I_{2}+I_{3}}$
${\displaystyle {\frac {V_{i}}{R_{1}}}={\frac {V_{o}}{R_{3}}}+{\frac {V_{i}-V_{o}}{R_{2}}}}$
${\displaystyle {\frac {V_{o}}{V_{i}}}={\frac {R_{3}-R_{2}}{R_{1}-R_{2}}}}$

## Formula

${\displaystyle V_{o}=V_{i}{\frac {R_{3}+R_{2}}{R_{1}+R_{2}}}}$

## Choosing Resistance's Value

• R2 = 0
The fomula above now becomes
${\displaystyle V_{o}=V_{i}{\frac {R_{3}}{R_{1}}}}$
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
${\displaystyle V_{o}=V_{i}{\frac {R_{2}}{R_{2}+R_{1}}}}$
This network acts as Voltage Attenuator
• R1 = 0
The fomula above now becomes
${\displaystyle V_{o}=V_{i}{\frac {R_{3}+R_{2}}{R_{2}}}}$
This network acts as Voltage Amplifier 1 + ${\displaystyle {\frac {R_{3}}{R_{2}}}}$

## Summary

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