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]

Dif. Pos.-fbk. op.-Amp.

see also : Pos. fbk. Op. Amp. - (spice) noise

 

\[V_a=V_S+\left({V_O-V_S}\right)\frac{R_0}{R_0+R_2}\]

\[V_b=V_R+\left({V_O-V_R}\right)\frac{R_1}{R_1+R_3}\]

\[\left({V_O-V_R}\right)\frac{R_1}{R_1+R_3}=
\left({V_S-V_R}\right)+\left({V_O-V_S}\right)\frac{R_0}{R_0+R_2}\]

\[\frac{V_O-V_R}{V_S-V_R}=
\frac{R_1+R_3}{R_1}\left({1+\frac{V_O-V_S}{V_S-V_R}·\frac{R_0}{R_0+R_2}}\right)\]

\[A_V·\frac{R_1}{R_1+R_3}=1+\frac{V_O-V_R+V_R-V_S}{V_S-V_R}·\frac{R_0}{R_0+R_2}=1+\left({A_V-1}\right)\frac{R_0}{R_0+R_2}\]

\[A_V\left({\frac{R_1}{R_1+R_3}-\frac{R_0}{R_0+R_2}}\right)=1-\frac{R_0}{R_0+R_2}\]

\[A_V=\frac{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)}{\left({R_0+R_2}\right)\left[{R_1\left({R_0+R_2}\right)-R_0\left({R_1+R_3}\right)}\right]}=\]

\[=\frac{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)}{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)\left[{\frac{R_1}{R_2}·\frac{R_0+R_2}{R_1+R_3}-\frac{R_0}{R_2}}\right]}=\]

\[=\left[{Def.:\ R\ ,\ \ \frac1{R_0}+\frac1{R_2}=\frac1{R_1}+\frac1{R_3}=\frac1{R}}\right]=\]

\[=\frac1{\frac{R_1R_0R_2}{R_2R_1R_3}-\frac{R_0}{R_2}}=\frac1{R_0\left({\frac1{R_3}-\frac1{R_2}}\right)}=\frac1{R_0\left({\frac1{R_0}-\frac1{R_1}}\right)}=\frac1{1-\frac{R_0}{R_1}}={A_V}^+\]

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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 V_S and V_R 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
\(R_2\)	\(R_0·\left({A_V-1}\right)\)	\(-A_V·R_0\)
\(\frac1R\)	\(\displaystyle{\frac1{R_0}+\frac1{R_2}=\frac1{R_1}+\frac1{R_3}}\)
\(R_1\)	\(\displaystyle{\frac{R_0}{1-\frac1{A_V}}}\)	\(R_0\left({1-\frac1{A_V}}\right)\)
\(R_3\)	\(\displaystyle{\frac1{\frac1R-\frac1{R_1}}}\)
\(R_3\)	\(\displaystyle{R_1\left({\frac1{1-{\left({1-\frac1{A_V}}\right)}^2}-1}\right)}\)	\(\displaystyle{\frac{R_1}{{\left({1-\frac1{A_V}}\right)}^2-1}}\)

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\[V_b=V_S+\left({V_O-V_S}\right)\frac{R_1}{R_1+R_3}\]

\[V_a=V_R+\left({V_O-V_R}\right)\frac{R_0}{R_0+R_2}\]

\[\left({V_O-V_R}\right)\frac{R_0}{R_0+R_2}=
\left({V_S-V_R}\right)+\left({V_O-V_S}\right)\frac{R_1}{R_1+R_3}\]

\[\frac{V_O-V_R}{V_S-V_R}·\frac{R_0}{R_0+R_2}=
1+\frac{V_O-V_R+V_R-V_S}{V_S-V_R}·\frac{R_1}{R_1+R_3}\]

\[A_V·\frac{R_0}{R_0+R_2}=1+\left({A_V-1}\right)·\frac{R_1}{R_1+R_3}\]

\[A_V·\left({\frac{R_0}{R_0+R_2}-\frac{R_1}{R_1+R_3}}\right)=1-\frac{R_1}{R_1+R_3}=\frac{R_3}{R_1+R_3}\]

\[A_V=\frac{R_3\left({R_1+R_3}\right)\left({R_0+R_2}\right)}{\left({R_1+R_3}\right)\left[{R_0\left({R_1+R_3}\right)-R_1\left({R_0+R_2}\right)}\right]}=\frac{\mathbb{Z}}{\mathbb{Z}\left[{\frac{R_0}{R_3}·\frac{R_1+R_3}{R_0+R_2}-\frac{R_1}{R_3}}\right]}=\]

\[=\frac1{\frac{R_0R_1R_3}{R_0R_2R_3}-\frac{R_1}{R_3}}=\frac1{R_1\left({\frac1{R_2}-\frac1{R_3}}\right)}=
\frac1{\frac{R_1}{R_1}-\frac{R_1}{R_0}}=\frac1{1-\frac{R_1}{R_0}}={A_V}^-\]

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 [Eop]

Re :: Pos. Feedback Op Amp

Definitions ::

\[\boxed{\frac1{R}=\frac1{R_1}+\frac1{R_2}=\frac1{R_3}+\frac1{R_4}\\ \frac1{R_{1_2}}{}^{↓}=^{↓}\frac1R-\frac1{R_{2_1}}\\ \frac1{R_{3_4}}{}^{↓}=^{↓}\frac1R-\frac1{R_{4_3}}}\]

\[{}^{↓↓}\ \frac{R_3}{R_1}+\frac{R_3}{R_2}-\frac{R_3}{R_3}=\frac{R_3}{R_4}=\\ =\mathbf{R_3·\frac{R_1+R_2}{R_1·R_2}-1}\]

\[V_B=V_X·\frac{R_1}{R_1+R_2}\\ V_A=V_S+\left({V_X-V_S}\right)·\frac{R_3}{R_3+R_4}\]

\[V_X·\frac{R_1}{R_1+R_2}=V_S+\left({V_X-V_S}\right)·\frac{R_3}{R_3+R_4}\\ V_X·\left({\frac{R_1}{R_1+R_2}-\frac{R_4}{R_3+R_4}}\right)=V_S·\left({1-\frac{R_4}{R_3+R_4}}\right)\]

\[A_V=N=\frac{V_X}{V_S}=\frac{\frac{\cancel{R_3}+R_4\cancel{-R_3}}{\bcancel{R_3+R_4}}}{\frac{\cancel{R_1·R_3}+R_1·R_4\cancel{-R_1·R_3}+R_2·R_3}{\left({R_1+R_2}\right)·\bcancel{\left({R_3+R_4}\right)}}}=\]

\[=\frac{R_1·R_4+R_2·R_4}{R_1·R_4-R_2·R_3}=\frac{1+\frac{R_2}{R_1}}{1-\frac{R_2}{R_1}·\frac{R_3}{R_4}}\]

\[N-N·\frac{R_2}{R_1}·\frac{R_3}{R_4}=1+\frac{R_2}{R_1}\]

\[\left({N-1}\right)·R_1·R_4=R_2·\left({N·R_3+R_4}\right)\]

---

\[\frac{R_2}{R_1}=\frac{\left({N-1}\right·R_4)}{N·R_3+R_4}=\frac{N-1}{N·\frac{R_3}{R_4}+1}\]

\[{}^{↑↑}\ \frac{R_2}{R_1}·\left({N·\mathbf{\left({R_3·\frac{R_1+R_2}{R_1·R_2}-1}\right)}+1}\right)=N-1\]

\[\frac{R_2}{R_1}·\left({N·R_3·\frac{R_1+R_2}{R_1·R_2}-\left({N-1}\right)}\right)=N-1\]

\[\frac{R_2}{R_1}=\frac1{\frac N{N-1}·R_3·\frac{R_1+R_2}{R_1·R_2}-1}\]

\[\frac N{N-1}·R_3·\frac{\frac{R_1}{R_2}+1}{R_1}-1=\frac{R_1}{R_2}\]

 \[\frac N{N-1}·R_3·\frac{1+\frac{R_2}{R_1}}{R_1}-\frac{R_2}{R_1}=1\]

 \[\frac N{N-1}·R_3·\frac{\bcancel{\frac{R_2}{R_1}+1}}{R_1}=\bcancel{\frac{R_2}{R_1}+1}\]

 \[\frac N{N-1}=\mathbf{\frac{R_1}{R_3}}\]

---

 \[\frac{R_2}{R_1}=\frac{N-1}{N·\frac{R_3}{R_4}+1}=\frac1{\frac N{N-1}·\frac{R_3}{R_4}+\frac1{N-1}}=\]

\[=\frac1{\mathbf{\frac{R_1}{\cancel{R_3}}}·\frac{\cancel{R_3}}{R_4}+\frac1{N-1}}=\frac1{\frac{R_1}{R_4}+\frac1{N-1}}\]

\[\frac{R_2}{R_1}·\left({\frac{R_1}{R_4}+\frac1{N-1}}\right)=1\\ \frac1{R_4}+\frac1{R_1}·\frac1{N-1}=\frac1{R_2}\\ \frac1{R_1}·\frac1{N-1}=\frac1{R_2}-\frac1{R_4}\]

\[N-1=\frac1{R_1·\left({\frac1{R_2}-\frac1{R_4}}\right)}\]

---

\[N=1+\frac1{R_1·\left({\frac1{R_2}-\frac1{R_4}}\right)}{}^{↑}=^{↑}1+\frac1{R_1·\left({\frac1{R_2}+\frac1{R_3}-\frac1R}\right)}=1+\frac1{R_1·\left({\frac1{R_3}-\frac1{R_1}}\right)}=\]

\[=\boxed{A_V=1+\frac1{\frac{R_1}{R_3}-1}}\]

Reminder :: \(\frac1{R_4}=\frac1{R_2}+\frac1{R_1}-\frac1{R_3} \) or \(\cases{\frac1{R_2}=\frac1R-\frac1{R_1}\\ \frac1{R_4}=\frac1R-\frac1{R_3}}\)

\({Def\ ::\ d=\frac{R_1}{R_3}\\ then\ :\ N-1=\frac1{d-1}\ and\ d=1+\frac1{N-1}}\)

if R is also given the rest can be computed ... as \(R_1=d·R_3\)

in "general" situation the LM308 likes the biasing resistance of the double 7k5 Ω --e.g.-- the R = 3.75 kΩ (for the noisy input the 100 kΩ in parallel with the 5 pF or less may be more suitable/stable however - so ... )

Related Post : More Op-Amp biasing schemes

The LTSpice Example ::

PS! the negative gain can result by the reached formula for the voltage gain -- but in practise it won't work !!!

a slew-rate versus a common mode input impedance ::

Note :: the realistic/conventional Op.-Amp.-s have the impedance value a bit lower than shown in the simulation

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[Eop]

The variety of XOR-s

\[ A\oplus B=\left({A+B}\right)\overline{AB}=^1\]

\[ =^1\overline{\overline{A\overline{AB}}}+\overline{\overline{B\overline{AB}}}=^{21} \]

\[ =^{21}\overline{\overline{A\overline{AB}}\ ·\ \overline{B\overline{AB}}}=^{22} \]

\[ =^1=\left({A+B}\right)\overline{\overline{\overline{A}}\ ·\ \overline{\overline{B}}}=\left({A+B}\right)\overline{\overline{\overline{A}+\overline{B}}}=\left({A+B}\right)\left({\overline{A}+\overline{B}}\right)=^{31} \]

\[ =^{31}\overline{\overline{A+B}}\ ·\ \overline{\overline{\overline{A}+\overline{B}}}=\overline{\overline{A+B}\ +\ \overline{\overline{A}+\overline{B}}}=\overline{\overline{A+B}\ +\ AB}=^{32} \]

\[ =^{31}\cancel{A\overline{A}}+A\overline{B}+B\overline{A}+\cancel{B\overline{B}}=A\overline{B}+B\overline{A}=^{41} \]

\[  =^{41}\overline{\overline{A\overline{B}+B\overline{A}}}=\overline{\overline{A\overline{B}}\ ·\ \overline{B\overline{A}}}=^{42}\overline{\left({A+\overline{B}}\right)\left({\overline{A}+B}\right)}= \]

\[ =\overline{\cancel{A\overline{A}}+AB+\overline{A}\ \overline{B}+\cancel{B\overline{B}}}=\overline{AB+\overline{A}·\overline{B}}=^{32} \]

\[ =^{32}\overline{\overline{\overline{\overline{A+B}+A}}\ ·\ \overline{\overline{\overline{A+B}+B}}}=\overline{\overline{A+B}+A}+\overline{\overline{A+B}+B}=^{33} \]

---

note : since the Boolean Arithmetic has no priority in between the conjunction and the disjunction -- the following applies (the ⁽² might be used above ... somewhere) ::

\[ \left({A+B}\right)\left({C+D}\right)=AC+AD+BC+BD\qquad^{(1} \]

\[ A·B+C·D=\left({A+C}\right)·\left({A+D}\right)·\left({B+C}\right)·\left({B+D}\right)=_{"PROOF"}\qquad^{(2} \]

\[ =_{"PROOF"}\left({A+AD+CA+CD}\right)\left({B+BD+CB+CD}\right)= \]

\[ _{=AB+ABD_1+ACB_2+ACD_3+ADB_1+ADCB_4+ACD_3+CAB_2+CABD_4+CAB_2+CAD_3+CDB_5+CD=} \]

\[ =\mathbf{AB}_1+\mathbf{AB}D_2+\mathbf{AB}C_3+A\boxed{CD}+\mathbf{AB}\boxed{CD}_4+B\boxed{CD}+\boxed{CD}= \]

\[ =AB\left({1+D+C+CD}\right)+\left({A+AB+B+1}\right)CD=AB+CD\]

also

\[A+AX=A\left({X+\overline{X}}\right)+AX=AX+A\overline{X}=A=A\left({1+X}\right)=A\left({X+\overline{X}+X}\right) \]

etc. ...

---

the simulation ::

---

[EoP]

Swapping Display Memory on Desk Calculator

v.1
\[\begin{array}{llll}
math.&{key\\presses}&{DISP.\\B}&{MEM\\A}\\
M=M+D & [\ M+\ ] &B&A+B\\
D=-D & [\ ±\ ] &-B&A+B\\
D=D+M & [\ +\ ]\ [\ MR\ ]\ [\ =\ ] &-B+A+B&A+B\\
M=M-D & [\ M-\ ] &A&A+B-A\\
M ↔ D &done&A&B
\end{array}
\]

v.2
\[\begin{array}{llll}
math.&{key\\presses}&{DISP.\\B}&{MEM\\A}\\
D=-D & [\ ±\ ] &-B&A\\
D=D+M & [\ +\ ]\ [\ MR\ ]\ [\ =\ ] &-B+A&A\\
M=M-D & [\ M-\ ] &A-B&A-A+B\\
D="0,=" & [\ 0\ ]\ [\ =\ ] &0(+A)&B\\
M ↔ D &done&A&B
\end{array}
\]

v.3
\[\begin{array}{llll}
math.&{key\\presses}&{DISP.\\B}&{MEM\\A}\\
D=D-M & [\ -\ ]\ [\ MR\ ]\ [\ =\ ] &B-A&A\\
M=M+D & [\ M+\ ] &B-A&A+B-A\\
D=D-M & [\ -\ ]\ [\ MR\ ]\ [\ =\ ] &-A+B-B&B\\
D=-D & [\ ×\ ]\ [\ 1\ ]\ [\ -\ ]\ [\ =\ ]\ \left({\ _{[\ =\ ]}\ }\right)^{\ (1} &(-A)·1-((-A)·1)&B\\
M ↔ D &done&A&B
\end{array}\\
_{NOTE^{\ (1}\ :\ DEPENDING\ ON\ THE\ SPECIFIC\ CALCULATOR\ THE\ ALTERNATE\ SEQUENCE\\ FOR\ THE\ ARITHMETIC\ NEGATION\ [\ ±\ ]\ MAY\ REQUIRE\ DOUBLE\ [\ =\ ]\ PRESS} 
\]

---

[Eop]

Resistor Bridge

Simulation / Schematic :

\[U_R=\frac{U_LR_2R_4+U_UR_0R_4+U_DR_0R_2}{R_2R_4+R_0R_4+R_0R_2}=
\frac{U_L\left({R_1R_3+R_0R_3+R_0R_1}\right)-U_UR_0R_3-U_DR_0R_1}{R_1R_3}\]
\[U_L=\frac{U_RR_1R_3+U_UR_0R_3+U_DR_0R_1}{R_1R_3+R_0R_3+R_0R_1}=
\frac{U_R\left({R_2R_4+R_0R_4+R_0R_2}\right)-U_UR_0R_4-U_DR_0R_2}{R_2R_4}\]
\(U_RR_1R_3R_2R_4+U_UR_0R_3R_2R_4+U_DR_0R_1R_2R_4=U_R\sum_E^3\sum_O^3-U_UR_0R_4\sum_O^3-U_DR_0R_2\sum_O^3\)
\[U_R=R_0·\frac{\left({U_UR_3+U_DR_1}\right)R_2R_4+\left({U_UR_4+U_DR_2}\right)\sum_O^3}{\sum_E^3\sum_O^3-R_1R_2R_3R_4}\]
\(U_LR_1R_3R_2R_4+U_UR_0R_4R_1R_3+U_DR_0R_2R_1R_3=U_L\sum_O^3\sum_E^3-U_UR_0R_3\sum_E^3-U_DR_0R_1\sum_E^3\)
\[U_L=R_0·\frac{\left({U_UR_4+U_DR_2}\right)R_1R_3+\left({U_UR_3+U_DR_1}\right)\sum_E^3}{\sum_E^3\sum_O^3-R_1R_2R_3R_4}\]

---

[Eop]

3 - Resistor Voltage Divider

\[I_0+I_1+I_2=0\qquad,\qquad\text{! Notice the directions of the currents !}\]
\[{\large \cases{I_0=\frac{V_0-V_X}{R_0}\\ {\ }\\
I_1=\frac{V_1-V_X}{R_1}\\ {\ }\\
I_2=\frac{V_2-V_X}{R_2}}}\]
\[\frac{V_0-V_X}{R_0}+\frac{V_1-V_X}{R_1}+\frac{V_2-V_X}{R_2}=0\]
\[\frac{V_0R_1R_2-V_XR_1R_2+V_1R_0R_2-V_XR_0R_2+V_2R_0R_1-V_XR_0R_1}{R_0R_1R_2}=0\qquad|\qquad×R_0R_1R_2\]
\[\frac{V_0R_1R_2+V_1R_0R_2+V_2R_0R_1}{R_1R_2+R_0R_2+R_0R_1}=V_X\]
Note: the \(V_X\) is the unmarked voltage point on the schematic !

It is easy to see the calculation scheme easily adjusts for N voltages all connecting to the single \(V_X\)
\[V_{X_N}=\frac{\displaystyle{\sum_{i=0}^{N-1}V_i\frac{\displaystyle{\prod_{k=0}^{N-1}R_k}}{R_i}}}{\displaystyle{\sum_{i=0}^{N-1}\frac{\displaystyle{\prod_{k=0}^{N-1}R_k}}{R_i}}}\]
thus BWD chek ::
for N=2 :
\[V_X=\frac{V_1R_0+V_0R_1}{R_0+R_1}\]
incase of the \(V_0=0\) it reduces to a trivial 2 resistor divider formula :
\[V_X=V_1\frac{R_0}{R_0+R_1}\]

---

Simulation example/chk. for N=4 :

---

[Eop]

INA formula chk

\[E_1+\left({V_X-E_1}\right)·\frac{R_2}{R_6+R_2}=E_2+\left({V_R-E_2}\right)·\frac{R_5}{R_7+R_5}\]
\[E_1+\left({E_2-E_1}\right)·\frac{R_1}{R_1+R_4+R_3}=V_1\qquad\frac{E_1-V_1}{R_1}=\frac{E_1-E_2}{R_{\Sigma3}}\]
\[E_2+\left({E_1-E_2}\right)·\frac{R_4}{R_1+R_4+R_3}=V_2\qquad\frac{V_2-E_2}{R_4}=\frac{E_1-E_2}{R_{\Sigma3}}\]
\[E_1=\frac{R_1}{R_4}\left({V_2-E_2}\right)+V_1\qquad E_2=\frac{R_4}{R_1}\left({V_1-E_1}\right)+V_2\]
\[E_2-E_1+\frac{\left({E_1-E_2}\right)R_4+\left({E_1-E_2}\right)R_1}{R_{\Sigma3}}=V_2-V_1\]

\[1-\frac{R_1+R_4}{R_{\Sigma3}}=\frac{V_2-V_1}{E_2-E_1}=\frac{V_2-V_1}{E_2-\frac{R_1}{R_4}\left({V_2-E_2}\right)-V_1}=\frac{V_2-V_1}{\frac{R_4}{R_1}\left({V_1-E_1}\right)+V_2-E_1}\]
\[E_2\left({1+\frac{R_1}{R_4}}\right)-\left({V_2\frac{R_1}{R_4}+V_1}\right)=\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}\]
 \[E_1\left({1+\frac{R_4}{R_1}}\right)-\left({V_2+V_1\frac{R_4}{R_1}}\right)=-\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}\]
\[E_2=\frac{\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}+V_2\frac{R_1}{R_4}+V_1}{1+\frac{R_1}{R_4}}=\frac{\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}R_4+V_2R_1+V_1R_4}{R_1+R_4}\]
\[E_1=\frac{-\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}+V_2+V_1\frac{R_4}{R_1}}{1+\frac{R_4}{R_1}}=\frac{-\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}R_1+V_2R_1+V_1R_4}{R_1+R_4}\]
\[.\ .\ .\]
\[V_X\frac{R_2}{R_6+R_2}-V_R\frac{R_5}{R_7+R_5}=E_2\frac{R_7}{R_7+R_5}-E_1\frac{R_6}{R_6+R_2}\]

---

\[IF\ :\ \cases{R_1=R_4=R_A\\ R_3=R_D\\ R_6=R_7=R_F\\ R_2=R_5=R_G}\]
\[E_1=\frac{\cancel{R_A}\left({V_2+V_1}\right)-\frac{V_2-V_1}{1-\frac{2R_A}{2R_A+R_D}}\cancel{R_A}}{2\ \cancel{R_A}}=\frac{V_2+V_1}2-\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}\]
\[E_2=\frac{\cancel{R_A}\left({V_2+V_1}\right)+\frac{V_2-V_1}{1-\frac{2R_A}{2R_A+R_D}}\cancel{R_A}}{2\ \cancel{R_A}}=\frac{V_2+V_1}2+\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}\]
\[.\ .\ .\]

\[\left({V_X-V_R}\right)\frac{R_G}{\cancel{R_F+R_G}}=\left[{\left({\cancel{\frac{V_2+V_1}2}+\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}}\right)-\left({\cancel{\frac{V_2+V_1}2}-\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}}\right)}\right]\frac{R_F}{\cancel{R_F+R_G}}\]
\[\boxed{\frac{V_X-V_R}{V_2-V_1}=\frac{R_F}{R_G}·\frac{2R_A+R_D}{R_D}=\frac{R_6}{R_2}·\frac{2R_1+R_3}{R_3}}\]

---

dd

---

[Eop]

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]

Initialization fix for an old simple square root algorithm

. . . it's multiple proven that in general situation the iteration formula :
\[\sqrt{A}\leftarrow a_{n+1}=\frac{\frac{A}{a_n}+a_n}2\]
gives the fastest convergence , initialization :
\[a_0=A\rightarrow a_1=a_{1_1}=\frac{A+1}2\]

however i just explored a different approach with iterative formula :
\[\sqrt{A}\leftarrow a_{n+1}=\frac{2Aa_n}{A+a_n^2}=\frac2{\frac1{a_n}+\frac{a_n}A}=\frac{2A}{\frac A{a_n}+a_n}=\frac A{\frac{\frac{A}{a_n}+a_n}2}\]
which has an initialization :
\[a_0=1\rightarrow a_1=a_{1_2}=\frac{2A}{A+1}=1+\frac{A-1}{A+1}\]

the "Old" initialization gives quite large positive error ... while my "Latest" initialization gives quite large negative error . . . however the both averaged gives more reasonable positive error , where the :
\[a_1=a_{1_3}=\frac{a_{1_1}+a_{1_2}}2=\frac{A+1}4+\frac{A}{A+1}\]

while for each \(a_n\) where \(n>1\) the first iteration formula on this page applies

...

yet a lesser negative error is got by combining the two first initializations as :
\[a_1=a_{1_4}=\frac{2a_{1_1}a_{1_2}}{a_{1_1}+a_{1_2}}=\frac1{\frac{A+1}{4A}+\frac1{A+1}}=\frac{A}{a_{1_3}}\]

...

+ yet a significantly lesser positive error is got by combining the two last initializations as :
\[a_1=a_{1_5}=\frac{a_{1_3}+a_{1_4}}2=\frac{a_{1_3}+\frac{A}{a_{1_3}}}2\]

the last formula is actually the value of \(a_2\) for \(a_{1_3}\) . . . so , . . .

. . . anyway it give us quite accurate estimation formula for the near A=1 , as :

\[\sqrt{A}≈a=\frac{\frac{A+1}4+\frac A{A+1}+\frac A{\frac{A+1}4+\frac A{A+1}}}2\]

the error ≤ about 5% - from \(\frac1{20}...20\)

---

[Eop]

Right Triangle

\( α+ß=\frac{π}2=90°\ ,\ R=\frac{C}2 \)

\( r=\sqrt{{\left({\frac{C}2}\right)}^2+\frac{A·B}2}-\frac{C}2←\cases{A=r+x\\ B=r+y\\ C=x+y} \)

\( \frac{a}h=\frac{h}b=\frac{A}B=tan\ ß=\frac1{tan\ α}\)

\( a·b=h^2 \)

\( a+b=C \)

\( A^2+B^2=C^2={\left({a+b}\right)}^2=a^2+2·a·b+b^2 \)

\( A^2=a^2+h^2=a^2+a·b=a·\left({a+b}\right)=a·C \)

\( B^2=b^2+h^2=b^2+b·a=b·\left({b+a}\right)=b·C \)

\( a=\frac{A^2}C \)

\( b=\frac{B^2}C \)

\( h=\frac{A·B}C=\sqrt{a·b} \)

 \( S=\frac{A·B}2=\frac{\sqrt{a·C·b·C}}2=\frac{h·C}2=\frac{\frac{A·B}C·C}2 \)

 \( P=A+B+C=A+B+\sqrt{A^2+B^2}=\sqrt{a·C}+\sqrt{b·C}+\sqrt{C·C}=\sqrt{C}·\left({\sqrt{a}+\sqrt{b}+\sqrt{a+b}}\right) \)

\( b=h^2\ ,\ a=1\ →\ A^2=C=h^2+1\ ,\ B^2=b·A^2=b·C=h^2·\left({h^2+1}\right)\ ,\ R=\frac{C}2 \)

---

[Eop]

About the Quadratic Equation

Derivation of the solutions ::

\(x^2+px+q=0=\left({x-u}\right)\left({x-v}\right)=x^2-\left({u+v}\right)x+uv\)

\(p=2a\)

\(x^2+2ax+a^2-a^2+q=0\)

\(x+a=±\sqrt{a^2-q}\)

\(a=\frac p2\)

\(x=-\frac p2±\sqrt{{\left({\frac p2}\right)}^2-q}\)

\(x_1=-\frac p2-\sqrt{{\left({\frac p2}\right)}^2-q}\)

\(x_2=-\frac p2+\sqrt{{\left({\frac p2}\right)}^2-q}\)

\(x_2-x_1=2\sqrt{{\left({\frac p2}\right)}^2-q}\)

\(x_2+x_1=-p\)

\(x_2=-p-x_1\)

\(x_1=-p-x_2\)

\(x+p+\frac qx=0\)

\(\displaystyle{x=-p-\frac qx\quad \rightarrow \quad x_n=x_n+x_\overline{n}-\frac q{x_n}}\)

\(\displaystyle{x_n=\frac q{x_\overline{n}}\ \equiv\ x_nx_\overline{n}=q\ |\ about:\ q=uv}\)

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Example ::

\(\mathbf{p}\)	\(\mathbf{q}\)	\(\mathbf{x_n}\)	\(\mathbf{x_1}\\ -p-x_2\\ \displaystyle{\frac q{x_\overline{n}}}\)	\(\mathbf{x_2}\\ -p-x_1\\ \displaystyle{\frac q{x_\overline{n}}}\)
\(-1\)	\(-1\)	\(-\frac{-1}2±\sqrt{{\left({\frac{-1}2}\right)}^2-\left({-1}\right)}=\) \(\frac{1±\sqrt{5}}2=\) \(=\cases{-\frac{\sqrt{5}-1}2\\ \frac{\sqrt{5}+1}2}\) \(=\cases{-\frac{5-1}{2\left({\sqrt{5}+1}\right)}\\ \frac{5-1}{2\left({\sqrt{5}-1}\right)}}\) \(=\cases{\frac{-2}{\sqrt{5}+1}\\ \frac2{\sqrt{5}-1}}\)	\(\frac{-2}{\sqrt{5}+1}\mathbf{=}\) \(\mathbf{=}-\left({-1}\right)-\frac{2}{\sqrt{5}-1}=\) \(=1-\frac{\sqrt{5}+1}2=\) \(=-\frac{-2+\sqrt{5}+1}2=\) \(=-\frac{\sqrt{5}-1}2=\) \(=-\frac2{\sqrt{5}+1}\) \(\mathbf{=}\frac{-1}{\left({\frac2{\sqrt{5}-1}}\right)}=\) \(=-\frac{\sqrt{5}-1}2=\) \(=\frac{-2}{\sqrt{5}+1}\)	\(\frac2{\sqrt{5}-1}\mathbf{=}\) \(\mathbf{=}-\left({-1}\right)-\frac{-2}{\sqrt{5}+1}=\) \(=1+\frac{\sqrt{5}-1}2=\) \(=\frac{2+\sqrt{5}-1}2=\) \(=\frac{\sqrt{5}+1}2=\) \(=\frac2{\sqrt{5}-1}\) \(\mathbf{=}\frac{-1}{\left({\frac{-2}{\sqrt{5}+1}}\right)}=\) \(=\frac{\sqrt{5}+1}2=\) \(=\frac2{\sqrt{5}-1}\)

Dynamic Formulas for the Capacitor and for the Inductor

Charging the capacitor from the CV(=Vs) src. ::

The voltage drop on the series resistor \[V_R=V_S-V_C\]

\[t_{0_{↑}^{↑}→V_C}=-R·C·ln\left({1-\frac{V_C}{V_S}}\right)=R·C·ln\frac{V_S}{V_R}\]

\[t_{V_{1_C}{}_{↑}^{↑}→V_{2_C}}=R·C·ln\frac{V_S-V_{1_C}}{V_S-V_{2_C}}=R·C·ln\frac{V_{1_R}}{V_{2_R}}\]

Dis-charging the capacitor through a fixed value resistor ::

\[t_{V_{2_C}{}_{↓}^{↓}→V_{1_C}}=-R·C·ln\frac{V_{1_C}}{V_{2_C}}=R·C·ln\frac{V_{2_C}}{V_{1_C}}\]

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"Charging" the inductor from the CV(=Vs) src. ::

The peak current \[I_{MAX}=\frac{V_S}R\]

\[t_{0_{↑}^{↑}→I_L}=-\frac LR·ln\left({1-\frac{I_L}{I_{MAX}}}\right)\]

\[t_{I_{1_L}{}_{↑}^{↑}→I_{2_L}}=\frac LR·ln\frac{I_{MAX}-I_{1_L}}{I_{MAX}-I_{2_L}}\]

"Dis-charging" the inductor through a fixed value resistor ::

\[t_{I_{2_L}{}_{↓}^{↓}→I_{1_L}}=\frac LR·ln\frac{I_{2_L}}{I_{1_L}}\]

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[Eop]

Diluting the solution

If there is solution with the concentration \(x=\frac b{b+a}\) and we need to get the solution with concentration \(y=\frac b{b+c}\) then from \(x\left({b+a}\right)=b=y\left({b+c}\right)\) we get the ratio of masses \(\displaystyle{\frac xy=\frac{b+c}{b+a}=\frac{m_{FINAL}}{m_{INITIAL}}}\) . . . so - if the \(\displaystyle{m_{FINAL}}\) is given we need the \(\displaystyle{m_{INITIAL}=\frac yx·m_{FINAL}}\) of the initial solution , the amount of water needed to be added is \(c-a=\mathbf{d}\) , from \(\displaystyle{\frac xy=\frac{m_F}{m_I}=1+\frac d{m_I}=\frac1{1-\frac d{m_F}}}\) we see that \(\displaystyle{d=\left({\frac xy-1}\right)·m_I=\left({1-\frac yx}\right)·m_F}\) . . . checking
\(\displaystyle{d=\left({\frac xy-1}\right)·m_I=\left({\frac xy-1}\right)·\frac yx·m_F}\) and
\(\displaystyle{d=\left({1-\frac yx}\right)·m_F=\left({1-\frac yx}\right)·\frac xy·m_I}\)

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PS! -- the below DEMO works and "in reverse" for the initial percentage being less than the final ... the result is given as a negative amount of the water to be "added" --e.g.-- describing the amount of water to be (evaporated/)extracted from the solution ... Now updated! to match the possible concentrating option ...

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. . . a Q.C. (in MSO Excel) for some non-intuitive cases ::

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Physics

Linear	Circular
Time \(t\ \left({s}\right)\)	Time \(t\ \left({s}\right)\)
Mass \(m\ \left({kg}\right)\)	Moment of inertia \(I=\sum{m_i·r_i}\ \left({kg·m^2}\right)\)
Distance \(s\ \left({m}\right)\)	Angle \(\overrightarrow{φ}\ \left({rad=1}\right)\)
Velocity \(v\ \left({\frac ms}\right)\)	Angular velocity \(\overrightarrow{ω}=\frac{∂φ}{∂t}=\frac vR\ \left({\frac {rad}s=\frac1s}\right)\)
Acceleration \(a\ \left({\frac m{s^2}}\right)\)	Angular_acceleration \(\overrightarrow{ɛ}=\frac{∂ω}{∂t}=\frac{a_{\tau}}R\ \left({\frac{rad}{s^2}=\frac1{s^2}}\right)\)
Impulse \(p=m·v=F·dt\ \left({N·s}\right)\)	Angular momentum \(N=I·\overrightarrow{ω}=\overrightarrow{R}×\overrightarrow{p}\ \left({\frac{kg·m^2}s}\right)\)
Force \(F=\frac{∂p}{∂t}=m·a\ \left({N}\right)\)	Torque \(M=I·\overrightarrow{ɛ}=\overrightarrow{R}×\overrightarrow{F}\ \left({N·m}\right)\)
Energy \(E=\frac{m·v^2}2=m·a·s\ \left({J}\right)\)	Energy \(W=\frac{I·ω^2}2=I·ɛ·φ\ \left({J}\right)\)
Work \(A=F·∂s·Cos\ φ\ \left({J}\right)\)	Work \(A=M·∂φ\ \left({J}\right)\)
Power \(P=\frac{∂A}{∂t}\ \left({W}\right)\)	Power \(P=\frac{∂W}{∂t}\ \left({W}\right)\)
\(d\left({m·\overrightarrow{v}}\right)=\overrightarrow{F}·dt\)	\(d\left({I·\overrightarrow{ω}}\right)=\overrightarrow{M}·dt\)