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]

(scheduled) derivation of the solution of the quadratic equation

 about (scheduled) :: every now and then (not too often) i set myself to re-figure it out (the history has proven there exists a variance of what i come up with each time ...)

... so -- Def.-s , etc. ... ::

$$\begin{array}{lcl} \left({x-a}\right)\left({x-b}\right)=0 &\ &\\ x^2-\left({a-b}\right)x+ab=0 &\ &\\ \begin{array}{l} x^2+px+q=0\qquad \qquad \qquad \qquad \rightarrow\\ x^2+2px+p^2=px-q+p^2\ |×3 &\\ x^2-2px+p^2=-3px-q+p^2\ |+\ \uparrow &\\ \hline 4x^2+4px+4p^2=0-4q+4p^2\ |-3p^2\\ 4x^2+4px+p^2=p^2-4q\ |÷4\\ \mathbf{x^2+2\frac p2x+{\left({\frac p2}\right)}^2={\left({\frac p2}\right)}^2-q} \end{array} &\ & \begin{array}{l} \mathbf{x^2+2\frac p2x+{\left({\frac p2}\right)}^2={\left({\frac p2}\right)}^2-q}\\ {\left({x+\frac p2}\right)}^2={\left({\frac p2}\right)}^2-q\\ \boxed{x=-\frac p2±\sqrt{{\left({\frac p2}\right)}^2-q}}\\{}\\{} \end{array} \end{array}$$

... (it) came out double ((at) this time) -- the short and the long -- way to (the solution) F;T
// is likely ↑↑ why ↑↑ in many blogs folks do not get a thing what i say
// (as an old school programmer i always live-compact my code(read: text))

see also the inner properties of @ About the Quadratic Equation

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[Eop]

Dif. Hi-Gain Amp. (not very practical)

$$V_b=V_Z+\left({V_n-V_Z}\right)\frac{R_g}{R_g+R_S}$$

$$V_a=V_S+\left({V_p-V_S}\right)\frac{R_g}{R_g+R_S}$$

$$Def\ :\ \frac1{\mathbf{R_X}}=\frac1{R_g+R_S}+\frac1{R_P}=\\ =\frac{R_g+R_S+R_P}{\left({R_g+R_S}\right)R_P}$$

$$V_n=V_Z+\left({V_O-V_Z}\right)\frac{R_X}{R_X+R_f}$$

$$V_p=V_S+\left({V_R-V_S}\right)\frac{R_X}{R_X+R_f}$$

$$V_b=V_Z+\left[{\cancel{V_Z}+\left({V_O-V_Z}\right)\frac{R_X}{R_X+R_f}\cancel{-V_Z}}\right]\frac{R_g}{R_g+R_S}=V_Z+\left({V_O-V_Z}\right)\frac{R_X}{R_X+R_f}·\frac{R_g}{R_g+R_S}$$

$$V_a=V_S+\left[{\cancel{V_S}+\left({V_R-V_S}\right)\frac{R_X}{R_X+R_f}\cancel{-V_S}}\right]\frac{R_g}{R_g+R_S}=V_S+\left({V_R-V_S}\right)\frac{R_X}{R_X+R_f}·\frac{R_g}{R_g+R_S}$$

$$V_S-V_Z=\left({V_O-V_R+V_S-V_Z}\right)\frac{R_X}{R_X+R_f}·\frac{R_g}{R_g+R_S}$$

$$\boxed{\quad A_V\quad}=\frac{V_O-V_R}{V_S-V_Z}=\left({1+\frac{R_f}{R_X}}\right)\left({1+\frac{R_S}{R_g}}\right)-1=\cancel{1}+\frac{R_f}{R_X}+\frac{R_S}{R_g}+\frac{R_fR_S}{R_XR_g}\cancel{-1}=$$

$$=\frac{R_gR_f+R_XR_S}{R_XR_g}+\frac{R_fR_S}{R_XR_g}=\frac{R_fR_S}{R_XR_g}\left({\frac{R_g}{R_S}+\frac{R_X}{R_f}+1}\right)=$$

$$=\frac{R_fR_S}{R_g}·\frac{R_g+R_S+R_P}{\left({R_g+R_S}\right)R_P}\left({\frac{R_g}{R_S}+1+\frac1{R_f}·\frac{\left({R_g+R_S}\right)R_P}{R_g+R_S+R_P}}\right)=$$

$$=R_f·\frac{R_S}{R_g}\left({\frac1{R_P}+\frac1{R_g+R_S}}\right)\left({\frac{R_g+R_S}{R_S}+\frac1{R_f}·\frac1{\frac1{R_P}+\frac1{R_g+R_S}}}\right)=$$

$$=R_f\left({\frac1{R_P}+\frac1{R_g+R_S}}\right)\frac{R_g+R_S}{R_g}+\frac{R_S}{R_g}=\boxed{\quad\frac1{R_g}\left[{R_f\left({\frac{R_g+R_S}{R_P}+1}\right)+R_S}\right]\quad}=$$

$$=R_f\left({\frac{1+\frac{R_S}{R_g}}{R_P}+\frac1{R_g}}\right)+\frac{R_S}{R_g}$$

$$\frac{A_V-\frac{R_S}{R_g}}{R_f}-\frac1{R_g}=\frac{1+\frac{R_S}{R_g}}{R_P}$$

$$\boxed{\quad R_P\quad}=\frac{1+\frac{R_S}{R_g}}{\frac{A_V-\frac{R_S}{R_g}}{R_f}-\frac1{R_g}}=\boxed{\quad \frac{R_g+R_S}{\frac{A_VR_g-R_S}{R_f}-1}\quad}$$

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about :

Uses the LM324 transistor model (the simulation)

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[Eop]