\[V_a=V_S+\left({V_O-V_S}\right)\frac{R_0}{R_0+R_2}\]
\[V_b=V_R+\left({V_O-V_R}\right)\frac{R_1}{R_1+R_3}\]
\[\left({V_O-V_R}\right)\frac{R_1}{R_1+R_3}=
\left({V_S-V_R}\right)+\left({V_O-V_S}\right)\frac{R_0}{R_0+R_2}\]
\[\frac{V_O-V_R}{V_S-V_R}=
\frac{R_1+R_3}{R_1}\left({1+\frac{V_O-V_S}{V_S-V_R}·\frac{R_0}{R_0+R_2}}\right)\]
\[A_V·\frac{R_1}{R_1+R_3}=1+\frac{V_O-V_R+V_R-V_S}{V_S-V_R}·\frac{R_0}{R_0+R_2}=1+\left({A_V-1}\right)\frac{R_0}{R_0+R_2}\]
\[A_V\left({\frac{R_1}{R_1+R_3}-\frac{R_0}{R_0+R_2}}\right)=1-\frac{R_0}{R_0+R_2}\]
\[A_V=\frac{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)}{\left({R_0+R_2}\right)\left[{R_1\left({R_0+R_2}\right)-R_0\left({R_1+R_3}\right)}\right]}=\]
\[=\frac{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)}{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)\left[{\frac{R_1}{R_2}·\frac{R_0+R_2}{R_1+R_3}-\frac{R_0}{R_2}}\right]}=\]
\[=\left[{Def.:\ R\ ,\ \ \frac1{R_0}+\frac1{R_2}=\frac1{R_1}+\frac1{R_3}=\frac1{R}}\right]=\]
\[=\frac1{\frac{R_1R_0R_2}{R_2R_1R_3}-\frac{R_0}{R_2}}=\frac1{R_0\left({\frac1{R_3}-\frac1{R_2}}\right)}=\frac1{R_0\left({\frac1{R_0}-\frac1{R_1}}\right)}=\frac1{1-\frac{R_0}{R_1}}={A_V}^+\]
What the above means - is that in case of the shown configuration - the non-common mode signal is extracted from common mode one , amplified ... and added back to the common mode one . . . shortly put :
\[V_O=V_R+A_V·\left({V_S-V_R}\right)\]
PS! : It is also possible to show - as for the above positive voltage gain derivation - that when we swap VS and VR then the negative/inverting voltage gain becomes \(A_V=\frac1{1-\frac{R_1}{R_0}}\) (see below) . . .
assuming the relation \(R_2=R_0·\left({A_V-1}\right)\) for the positive gain and the relation \(R_2=-A_V·R_0\) for the inverting gain -- the following applies :
the neg. gain case :
\[{A_V}^+=1-{A_V}^-=1-\frac1{1-\frac{R_1}{R_0}}=\frac{1-\frac{R_1}{R_0}-1}{1-\frac{R_1}{R_0}}=\frac{-\frac{R_1}{R_0}}{1-\frac{R_1}{R_0}}=\frac1{1-\frac{R_0}{R_1}}\]
e.g. \(\quad\left|{{A_V}^-}\right|={A_V}^+-1\quad\) ← that
for the same resistor values or for the same R0 : R1 ratio
about formulas :
| parameter | non-inverting | inverting |
| \(A_V\quad\) | user set \(\displaystyle{\frac1{1-\frac{R_0}{R_1}}}\) | user set \(\displaystyle{\frac1{1-\frac{R_1}{R_0}}}\) |
| \(R_0\) | user set | user set |
| \(R_2\) | \(R_0·\left({A_V-1}\right)\) | \(-A_V·R_0\) |
| \(\frac1R\) | \(\displaystyle{\frac1{R_0}+\frac1{R_2}=\frac1{R_1}+\frac1{R_3}}\) | |
| \(R_1\) | \(\displaystyle{\frac{R_0}{1-\frac1{A_V}}}\) | \(R_0\left({1-\frac1{A_V}}\right)\) |
| \(R_3\) | \(\displaystyle{\frac1{\frac1R-\frac1{R_1}}}\) | |
| \(R_3\) | \(\displaystyle{R_1\left({\frac1{1-{\left({1-\frac1{A_V}}\right)}^2}-1}\right)}\) | \(\displaystyle{\frac{R_1}{{\left({1-\frac1{A_V}}\right)}^2-1}}\) |
\[V_b=V_S+\left({V_O-V_S}\right)\frac{R_1}{R_1+R_3}\]
\[V_a=V_R+\left({V_O-V_R}\right)\frac{R_0}{R_0+R_2}\]
\[\left({V_O-V_R}\right)\frac{R_0}{R_0+R_2}=
\left({V_S-V_R}\right)+\left({V_O-V_S}\right)\frac{R_1}{R_1+R_3}\]
\[\frac{V_O-V_R}{V_S-V_R}·\frac{R_0}{R_0+R_2}=
1+\frac{V_O-V_R+V_R-V_S}{V_S-V_R}·\frac{R_1}{R_1+R_3}\]
\[A_V·\frac{R_0}{R_0+R_2}=1+\left({A_V-1}\right)·\frac{R_1}{R_1+R_3}\]
\[A_V·\left({\frac{R_0}{R_0+R_2}-\frac{R_1}{R_1+R_3}}\right)=1-\frac{R_1}{R_1+R_3}=\frac{R_3}{R_1+R_3}\]
\[A_V=\frac{R_3\left({R_1+R_3}\right)\left({R_0+R_2}\right)}{\left({R_1+R_3}\right)\left[{R_0\left({R_1+R_3}\right)-R_1\left({R_0+R_2}\right)}\right]}=\frac{\mathbb{Z}}{\mathbb{Z}\left[{\frac{R_0}{R_3}·\frac{R_1+R_3}{R_0+R_2}-\frac{R_1}{R_3}}\right]}=\]
\[=\frac1{\frac{R_0R_1R_3}{R_0R_2R_3}-\frac{R_1}{R_3}}=\frac1{R_1\left({\frac1{R_2}-\frac1{R_3}}\right)}=
\frac1{\frac{R_1}{R_1}-\frac{R_1}{R_0}}=\frac1{1-\frac{R_1}{R_0}}={A_V}^-\]
[Eop]
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