OP-AMP Differential Amplifier circuit and derivation

Recently I have been studying the opamp circuits and their application. 

One of those circuits is a differential amplifier circuit. I wanted to make the article for the derivation of the differential amplifier circuit. Lets dive into the topic.

Differential amplifier is nothing but the combination of the inverting and non-inverting amplifier circuit.



The output equation of inverting amplifier is given by
`Vo1=-\frac{R2}{R1}\times V1`   --- (1)

The output equation of the non-inverting amplifier is given by
`Vo2=(1+\frac{R2}{R1})\times V2` ---(2)

Applying superposition theorem, the output of combining inverting and non-inverting amplifier is given by

`Vo=Vo1+Vo2` ---(3)
`Vo=-\frac{R2}{R1}\times V1 +(1+\frac{R2}{R1})\times V2` ---(4)

The equation of the differential amplifier is given by
`Vo=K (V2-V1)` ---(5)

From the equation (4), the amplication factor K can be derived if `-\frac{R2}{R1}=(1+\frac{R2}{R1})`

It can be done by adding the voltage divider circuit at the input of non-inverting terminal.
From the above image, the output equation can be written as
`Vo=-\frac{R2}{R1}\times V1 +(1+\frac{R2}{R1})\times V2`*  ---(6)

Here, V2*`=\frac{R2}{R1 +R2}\times V2`  

Applying V2* in the equation (6)

`Vo=-\frac{R2}{R1}\times V1 +\left[(1+\frac{R2}{R1})\times \frac{R2}{R1 +R2}\times V2\right]`

Solving the above equation, We can get differential circuit equation as
`Vo=\frac{R2}{R1}(V2-V1)`

Where the amplication factor K=`\frac{R2}{R1}`.

Example circuit simulation in LTSPICE:










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