Re deriving formulas for the quadratic eq.

${\Large x^2+ax+b=0\quad|\quad a=2s\\ \Large x^2+2sx+s^2=s^2-b\\ \Large {\left({x+s}\right)}^2=s^2-b}$

${\Large x+s=\sqrt{s^2-b}\\ \Large x=-s±\sqrt{s^2-b}\quad|\quad s=\frac a2\\ \Large x=-\frac a2±\sqrt{{\left({\frac a2}\right)}^2-b}}$

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[Eop]

re-worked the formulas for the OC(OD)/OE voltage comparator's resistive output delimitter

an arbitrary output range ver. of the below a related post + the formulas for the non supply median centered output limit :

$I_C=\frac{V_L}{R_3}\ \Rightarrow\ \boxed{R_3=\frac{V_L}{I_C}}\ ,\ def.\ :\ \boxed{{\large \chi}=R_3\left({\frac1{V_L}-\frac1{V_H}}\right)}\\ \cases{V_H=V_S\frac{R_1}{R_1+R_2}=V_S\frac1{1+\frac{R_2}{R_1}}\\ V_L=V_S\frac{R_1||R_3}{R_1||R_3+R_2}=V_S\frac1{1+\frac{R_2}{R_1||R_3}}=V_S\frac1{1+\frac{R_2}{R_1}\left({1+\frac{R_1}{R_3}}\right)}}$

$\begin{array}{l}{\cases{\frac{V_H}{V_S}\left({1+\frac{R_2}{R_1}}\right)=1\ \Rightarrow\ \frac{R_2}{R_1}=\frac{V_S}{V_H}-1\ ...\ =\frac{\frac{V_S}{V_L}-1}{\frac{R_1}{R_3}+1}\\ \frac{V_L}{V_S}\left({1+\frac{R_2}{R_1}\left({1+\frac{R_1}{R_3}}\right)}\right)=1\ \Rightarrow\ \frac{R_2}{R_1}\left({\frac{R_1}{R_3}+1}\right)=\frac{V_S}{V_L}-1}\ \Rightarrow\ \frac{R_1}{R_3}=\frac{\frac{V_S}{V_L}-1}{\frac{V_S}{V_H}-1}-1\ \Rightarrow}\end{array}$

$\Rightarrow\ \boxed{\boxed{R_1}=R_3\left({\frac{\frac{V_S}{V_L}-1}{\frac{V_S}{V_H}-1}-1}\right)=R_3\frac1{\frac1{V_H}-\frac1{V_S}}\left({\frac1{V_L}-\frac1{V_H}}\right)\boxed{={\large \chi}\cdot\frac1{\frac1{V_H}-\frac1{V_S}}}}$

$R_2=R_1\left({\frac{V_S}{V_L}-1}\right)=R_3\left({\frac{V_S}{V_L}-1}\right)\left({\frac{\frac{V_S}{V_L}-1}{\frac{V_S}{V_H}-1}-1}\right)=\\ =R_3\left({\frac{V_S}{V_L}-1-\frac{V_S}{V_H}+1}\right)=\boxed{R_3V_S\left({\frac1{V_L}-\frac1{V_H}}\right)=\boxed{R_2={\large \chi}\cdot V_S}}$

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---==≡==--- Do NOT mix the formulas from the above with the formulas from the below ---==≡==---

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• a Supply median centered output limit ver. -- Some definitions :

$\displaystyle{V_S=V_{CC} \quad I_C=I_{SNK}\\ V_H=\frac{R_1}{R_1+R_2}\\ V_L=\frac{R_1||R_3}{R_1||R_3+R_2}\\ I_C=\frac{V_L}{R_3}\Rightarrow \boxed{\mathbf{R_3}=\frac{V_L}{I_C}=\frac{V_S}{I_C}\cdot\frac{1-\frac1n}2}}$

$\displaystyle{\frac{R_2}{R_1}=\boxed{\mathbf{\Large \alpha}=\frac{V_L}{V_H}=\frac{n-1}{n+1}}=\frac{R_1||R_3}{R_2}=\frac{\left({\frac{R_1R_3}{R_1+R_3}}\right)}{R_2}=\frac1{\sqrt{1+\frac{R_1}{R_3}}}\ \Rightarrow\\ \Rightarrow\ \boxed{\mathbf{R_1}=R_3\left({\frac1{\alpha^2}-1}\right)=R_3\frac{4n}{{\left({n-1}\right)}^2}}\\ \Rightarrow\ \boxed{\mathbf{R_2}=\alpha\cdot R_1=R_3\frac{4n}{n^2-1}}}$

• (↑ You only need to provide $\displaystyle{V_S\ ,\ n\ ,\ I_C}$ ↑ ,  anyway - ) Some more definitions :

$\displaystyle{V_L=V_S-V_H}$ , $\frac1n$ - is the dynamic output-range around the supply median --e.g.-- the 1 n-th of the total supply

$\cases{\displaystyle{\boxed{\frac{V_H-V_L}{V_S}=\mathbf{\frac1n}}=\frac{R_1}{R_1+R_2}-\frac{R_1||R_3}{R_1||R_3+R_2}=\frac1{1+\frac{R_2}{R_1}}-\frac1{1+\frac{R_2}{R_1||R_3}}}\\ \displaystyle{V_H+V_L=V_S\ \qquad\Rightarrow\qquad\Rightarrow\qquad\Rightarrow\qquad 1=\frac1{1+\frac{R_2}{R_1}}+\frac1{1+\frac{R_2}{R_1||R_3}}}}$

$\cases{\displaystyle{\frac{1+\frac1n}2=\frac1{1+\frac{R_2}{R_1}}=A\qquad \Rightarrow\quad \frac{R_1}A=R_1+R_2}\\ \displaystyle{\frac{1-\frac1n}2=\frac1{1+\frac{R_2}{R_1||R_3}}=B\Rightarrow\frac{R_1||R_3}B=R_1||R_3+R_2}}\displaystyle{\quad R_1||R_3=\frac1{\frac1{R_1}+\frac1{R_3}}=\frac{R_1\cdot R_3}{R_1+R_3}}$

$\begin{array}{l}{\displaystyle{R_2=R_1\left({\frac1A-1}\right)=R_1\frac{n-1}{n+1}=\\ =R_1||R_3\left({\frac1B-1}\right)=\frac{R_1R_3}{R_1+R_3}\cdot\frac{n+1}{n-1}}}\end{array}\displaystyle{\quad\Rightarrow\quad R_2=\sqrt{\frac{{R_1}^2R_3}{R_1+R_3}}=\frac{R_1}{\sqrt{1+\frac{R_1}{R_3}}}}$

• More backwards extending definitions :

$\begin{array}{l}{\cases{\displaystyle{\frac1A-1=\frac{2n}{n+1}-1=\frac{n-1}{n+1}=\alpha}\\ \displaystyle{\frac1B-1=\frac{2n}{n-1}-1=\frac{n+1}{n-1}=\frac1\alpha}}}\end{array}\qquad\begin{array}{l}{\cases{\displaystyle{V_H-V_L=\frac{V_S}n\\ V_H+V_L=V_S}}\quad\Rightarrow\quad\cases{V_H=V_S\frac{1+\frac1n}2\\V_L=V_S\frac{1-\frac1n}2}}\end{array}$

$\begin{array}{l}{\cases{\displaystyle{V_L=V_S\frac{R_1||R_3}{R_1||R_3+R_2}=\frac{V_S}{1+\frac{R_2}{R_1||R_3}}=\frac{V_S}{1+\frac1\alpha}=\frac{\alpha V_S}{1+\alpha}\\ V_H=V_S\frac{R_1}{R_1+R_2}=\frac{V_S}{1+\frac{R_2}{R_1}}=\frac{V_S}{1+\alpha}}}}\end{array}\quad\Rightarrow\quad\displaystyle{\boxed{\frac{V_L}{V_H}={\large \alpha}}}$

• About $\alpha_{{}_\text{Fn.of}}\left({n}\right)$ :

$\cases{\displaystyle{V_H-V_L=\frac{V_S}n\\ V_H+V_L=V_S}}\quad\Rightarrow\quad\cases{\displaystyle{1-\alpha=\frac1n\cdot\frac{V_S}{V_H}\\ 1+\alpha=\frac{V_S}{V_H}}}\quad\Rightarrow\quad\begin{array}{l}{\displaystyle{n\left({1-\alpha}\right)=1+\alpha\\ \boxed{\mathbf{n}=\frac{1+\alpha}{1-\alpha}}}}\end{array}$

$\displaystyle{n-n\alpha=1+\alpha\\ n-1=\alpha\left({n+1}\right)\\ \alpha=\frac{n-1}{n+1}}$

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[Eop]