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 , V_o > V₁ . This network acts as Voltage Amplifier

R₃ = R₁ , V_o = V₁ . This network acts as Voltage Follower

R₃ < R₁ , V_o < 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