More Op-Amp biasing schemes

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

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

---

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

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

---

~~~~~~~~~~~~~~~
simulation example in Falstad
~~~~~~~~~~~~~~~

---

[Eop]

Another Diff. op.-Amp. circuit

$$\frac{V_X-V_R}{V_H-V_L}=\frac{R_AR_F}{R_AR_V+R_G\left({R_A+R_V}\right)}$$

$$V_D=\frac{V_CR_{11}R_3+V_XR_0R_3+V_LR_0R_{11}}{R_{11}R_3+R_0R_3+R_0R_{11}}$$
$$V_U=\frac{V_CR_{22}R_5+V_RR_4R_5+V_HR_4R_{22}}{R_{22}R_5+R_4R_5+R_4R_{22}}$$
$$FROM\ :\ V_X-V_R=\left({V_U-V_D}\right)\frac{R_F}{R_G}$$
$$IF\ :\ \cases{R_1=R_2=R_F\\ R_{10}=R_{20}=R_G\\ R_0=R_4=R_A\\ R_3=R_5=R_V}$$
$$\boxed{!\ note\ that\ the\ above\ condition\ makes\ the\ biasing\ invariant\ of\ the\ \mathbf{V_C}}$$
  $$\left({V_X-V_R}\right)\frac{R_G}{R_F}= \frac{\underline{V_C\left({R_F+R_G}\right)R_V}+\boxed{V_RR_AR_V}+V_HR_A\left({R_F+R_G}\right)}{\left({R_F+R_G}\right)R_V+R_AR_V+R_A\left({R_F+R_G}\right)}-\frac{\underline{V_C\left({R_F+R_G}\right)R_V}+\boxed{V_XR_AR_V}+V_LR_A\left({R_F+R_G}\right)}{\left({R_F+R_G}\right)R_V+R_AR_V+R_A\left({R_F+R_G}\right)}$$
$$\left({V_X-V_R}\right)\left({\frac{R_G}{R_F}+\frac{R_AR_V}{R_{\Sigma3}}}\right)=\left({V_H-V_L}\right)\frac{R_A\left({R_F+R_G}\right)}{R_{\Sigma3}}$$
$$\frac{V_X-V_R}{V_H-V_L}=\frac{R_A\left({R_F+R_G}\right)}{\cancel{R_{\Sigma3}}}·\frac{R_F·\cancel{R_{\Sigma3}}}{R_GR_{\Sigma3}+R_AR_VR_F}=$$
$$=\frac{R_A\left({R_F+R_G}\right)\frac{R_F}{R_G}}{\left({R_F+R_G}\right)R_V+R_AR_V+R_A\left({R_F+R_G}\right)+R_AR_V\frac{R_F}{R_G}}=$$
$$=\frac{R_A\cancel{\left({R_F+R_G}\right)}\frac{R_F}{R_G}}{\cancel{\left({R_F+R_G}\right)}\left({R_A+R_V}\right)+R_AR_V\frac{\cancel{\left({R_F+R_G}\right)}}{R_G}}=$$

$$=\frac{R_AR_F}{R_AR_V+R_G\left({R_A+R_V}\right)}$$

---

a simulation example ::

---

[Eop]

Differential Op-Amp formulas check

the case for :

$$\cases{signal\ :\ U_S=U_1\\ reference\ :\ U_R=U_0\\ gain\ :\ R_G=R_0\\ feedback\ :\ R_F=R_1\\ output\ :\ U_O=U_X}$$

$$U_R=U_\overline{IN}=U_S+\left({U_O-U_S}\right)·\frac{R_G}{R_G+R_F}$$

$$U_S-U_R=-\left({U_O-U_R+U_R-U_S}\right)·\frac{R_G}{R_G+R_F}$$

$$\frac{U_O-U_R}{U_S-U_R}-1=A_V-1=-1-\frac{R_F}{R_G}$$

$$\boxed{A_V=-\frac{R_F}{R_G}}$$

$$\begin{align*}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad  \\ \hline \end{align*}$$

$$\cases{signal\ :\ U_S=U_0\\ reference\ :\ U_R=U_1}$$

$$U_S=U_\overline{IN}=U_R+\left({U_O-U_R}\right)·\frac{R_G}{R_G+R_F}$$

$$U_S-U_R=\left({U_O-U_R}\right)·\frac{R_G}{R_G+R_F}$$

$$\frac{U_O-U_R}{U_S-U_R}=\boxed{A_V=1+\frac{R_F}{R_G}}$$

---

$$\frac{U_Y}{U_1}=\frac{R_0+R_1}{R_0}\qquad \frac{U_X-U_0}{U_Y-U_0}=-\frac{R_3}{R_2}\\ {\ }$$
$$\boxed{U_X=}\ U_0\left({\frac{R_2+R_3}{R_2}}\right)-\frac{R_3}{R_2}U_1\left({\frac{R_0+R_1}{R_0}}\right)\ {^?= {}_?}\ \mathbf{...}\ =\left({U_0-U_1}\right)\frac{R_2+R_3}{R_2}\ \boxed{=\Delta U_{IN}\left({1+\mathbf{M}}\right)}\\ {\ }$$
$$?\qquad \frac{R_2+R_3}{R_2}=\frac{R_3R_0+R_3R_1}{R_0R_0}\ {^?= {}_?}\ ...\ =\frac{R_3\frac{R_0}{R_1}+R_3}{\frac{R_0}{R_1}R_2}=\frac{R_3\frac{R_3}{R_2}+R_3}{\frac{R_3}{R_2}R_2}=\frac{R_2+R_3}{R_2}\\ {\ }$$
$$R_0=\frac{R_3\left({R_0+R_1}\right)}{R_3+R_2}\\ {\ }$$
$$\cases{\underline{R_0R_3}+R_0R_2=\underline{R_0R_3}+R_1R_3\\ {\ }\\ \boxed{R_0R_2=R_1R_3}}\qquad \frac{R_3}{R_2}=\frac{sR_0}{sR_1}=\mathbf{M}\qquad \mathbf{...} \uparrow$$

---

some tests :

MAX input impedance test :

Max. frequency TEST :

---

[Eop]