More Op-Amp biasing schemes

\[\frac{V_X-V_R}{V_H-V_L}=\frac{R_F}{R_G}\]

\[IF\ \cases{V_X\ne V_R\\ R\ \rightarrow\ L\ ,\ U\ \rightarrow\ D\\ R_0=R_2\\ R_1=R_3}\]
\[V_L+\left({V_X-V_L}\right)\frac{R_G}{R_F+R_G}=V_H+\left({V_R-V_H}\right)\frac{R_G}{R_F+R_G}\]
\[\left({V_H-V_L}\right)\frac{R_F+R_G}{R_G}=\left({V_X-V_L}\right)-\left({V_R-V_H}\right)=\left({V_X-V_R}\right)+\left({V_H-V_L}\right)\]
\[\frac{V_X-V_R}{V_H-V_L}=\frac{R_F+R_G}{R_G}-1=\frac{R_F}{R_G}\]

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\[IF\ \cases{V_X=V_R\\ R\ \rightarrow\ L\ ,\ U\ \rightarrow\ D\\ R_0\ne R_2\\ R_1\ne R_3}\]
\[{}^{see\ the\ optimizations\ below\ :}\ \frac{V_X}{V_S}=\frac{R_BR_F}{R_D\left({R_B-R_F}\right)}\]

\[V_R+\left({V_X-V_R}\right)\frac{R_G}{R_F+R_G}=V_S+\left({V_X-V_S}\right)\frac{R_D}{R_B+R_D}\]
\[\left({V_S-V_R}\right)=\left({V_X-V_R}\right)\frac{R_G}{R_F+R_G}-\left({V_X-V_S}\right)\frac{R_D}{R_B+R_D}\]
\[\boxed{V_R=0\ :\ }\ V_S=V_X\frac{R_G}{R_F+R_G}-\left({V_X-V_S}\right)\frac{R_D}{R_B+R_D}\]
\[V_S\left({1-\frac{R_D}{R_B+R_D}}\right)=V_X\left({\frac{R_G}{R_F+R_G}-\frac{R_D}{R_B+R_D}}\right)\]
\[\frac{V_X}{V_S}=\frac{\frac{R_B}{R_B+R_D}}{\frac{R_G}{R_F+R_G}-\frac{R_D}{R_B+R_D}}=\frac{R_B}{\frac{R_B+R_D}{R_F+R_G}R_G-R_D}=\ ...\]
\[\left({preferably}\right)\ also\ :\ \frac1{R_F}+\frac1{R_G}=\frac1{R_B}+\frac1{R_D}\ ...\frac{R_B+R_D}{R_F+R_G}=\frac{R_BR_D}{R_FR_G}\]
\[...\ =\frac{R_B}{\frac{R_BR_D}{R_F\cancel{R_G}}\cancel{R_G}-R_D}=\frac{R_BR_F}{R_D\left({R_B-R_F}\right)}\]

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simulation example in Falstad
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[Eop]